%------------------------------------------------------------------------------
% File     : ITP230_3 : TPTP v8.2.0. Released v8.0.0.
% Domain   : Interactive Theorem Proving
% Problem  : Sledgehammer problem VEBT_Insert 00815_052383
% Version  : [Des22] axioms.
% English  :

% Refs     : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
%          : [Des22] Desharnais (2022), Email to Geoff Sutcliffe
% Source   : [Des22]
% Names    : 0066_VEBT_Insert_00815_052383 [Des22]

% Status   : Theorem
% Rating   : 1.00 v8.1.0
% Syntax   : Number of formulae    : 12662 (4685 unt;1920 typ;   0 def)
%            Number of atoms       : 23680 (9226 equ)
%            Maximal formula atoms :   73 (   2 avg)
%            Number of connectives : 14405 (2219   ~; 547   |;1098   &)
%                                         (2779 <=>;7762  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   35 (   4 avg)
%            Maximal term depth    :   23 (   2 avg)
%            Number of FOOLs       : 1100 ( 512 fml; 588 var)
%            Number of X terms     :  671 (   0  []; 576 ite;  95 let)
%            Number of types       :  333 ( 332 usr)
%            Number of type conns  : 1756 (1180   >; 576   *;   0   +;   0  <<)
%            Number of predicates  :  105 ( 102 usr;   2 prp; 0-4 aty)
%            Number of functors    : 1500 (1500 usr; 422 con; 0-8 aty)
%            Number of variables   : 24267 (23698   !; 569   ?;24267   :)
% 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 20:21:43.928
%------------------------------------------------------------------------------
% Could-be-implicit typings (332)
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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: set_complex > fun_fu1358370060594899587plex_o ).

tff(sy_c_ATP_058Lamm__au____,type,
    aTP_Lamm_au: set_real > fun_fu9050450510664235267real_o ).

tff(sy_c_ATP_058Lamm__av____,type,
    aTP_Lamm_av: set_list_nat > fun_fu1994797156054922755_nat_o ).

tff(sy_c_ATP_058Lamm__aw____,type,
    aTP_Lamm_aw: set_set_nat > fun_fu2408719596384163971_nat_o ).

tff(sy_c_ATP_058Lamm__ax____,type,
    aTP_Lamm_ax: set_int > fun_fu3883539188510166275_int_o ).

tff(sy_c_ATP_058Lamm__ay____,type,
    aTP_Lamm_ay: set_nat > fun_fu4001317426178549763_nat_o ).

tff(sy_c_ATP_058Lamm__az____,type,
    aTP_Lamm_az: real > fun_real_real ).

tff(sy_c_ATP_058Lamm__ba____,type,
    aTP_Lamm_ba: rat > fun_rat_rat ).

tff(sy_c_ATP_058Lamm__bb____,type,
    aTP_Lamm_bb: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__bc____,type,
    aTP_Lamm_bc: fun_int_fun_int_int ).

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: num > fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__bf____,type,
    aTP_Lamm_bf: num > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__bg____,type,
    aTP_Lamm_bg: num > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__bh____,type,
    aTP_Lamm_bh: fun_o_o > fun_fu138107234227827011_int_o ).

tff(sy_c_ATP_058Lamm__bi____,type,
    aTP_Lamm_bi: fun_o_int > fun_fu6453395667394816515nt_int ).

tff(sy_c_ATP_058Lamm__bj____,type,
    aTP_Lamm_bj: fun_int_o > fun_fu1481857817119020493_int_o ).

tff(sy_c_ATP_058Lamm__bk____,type,
    aTP_Lamm_bk: fun_int_int > fun_fu6092417073193843705nt_int ).

tff(sy_c_ATP_058Lamm__bl____,type,
    aTP_Lamm_bl: fun_Pr3238374608701176162_int_o > fun_fu4592178885959468770_int_o ).

tff(sy_c_ATP_058Lamm__bm____,type,
    aTP_Lamm_bm: fun_Pr982841511892225912nt_int > fun_fu717549361983437348nt_int ).

tff(sy_c_ATP_058Lamm__bn____,type,
    aTP_Lamm_bn: fun_o_8988054265482332260nt_int > fun_fu8427683764667544036nt_int ).

tff(sy_c_ATP_058Lamm__bo____,type,
    aTP_Lamm_bo: fun_in4211310926729683942nt_int > fun_fu918106826131887214nt_int ).

tff(sy_c_ATP_058Lamm__bp____,type,
    aTP_Lamm_bp: fun_Pr2478283820680522243nt_int > fun_fu3898734532474705027nt_int ).

tff(sy_c_ATP_058Lamm__bq____,type,
    aTP_Lamm_bq: fun_fu3275390434952296369_nat_o > fun_fu6384561554008368101_nat_o ).

tff(sy_c_ATP_058Lamm__br____,type,
    aTP_Lamm_br: fun_Pr2477239776880934807at_nat > fun_na1668100129766719005at_nat ).

tff(sy_c_ATP_058Lamm__bs____,type,
    aTP_Lamm_bs: fun_Pr2028404973800837934_nat_o > fun_na5914473681949881652_nat_o ).

tff(sy_c_ATP_058Lamm__bt____,type,
    aTP_Lamm_bt: fun_Pr2478283820680522243nt_int > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__bu____,type,
    aTP_Lamm_bu: fun_Pr3238374608701176162_int_o > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__bv____,type,
    aTP_Lamm_bv: fun_Pr982841511892225912nt_int > fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__bw____,type,
    aTP_Lamm_bw: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__bx____,type,
    aTP_Lamm_bx: fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__by____,type,
    aTP_Lamm_by: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__bz____,type,
    aTP_Lamm_bz: fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__ca____,type,
    aTP_Lamm_ca: fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__cb____,type,
    aTP_Lamm_cb: fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__cc____,type,
    aTP_Lamm_cc: $o > fun_fu138107234227827011_int_o ).

tff(sy_c_ATP_058Lamm__cd____,type,
    aTP_Lamm_cd: fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__ce____,type,
    aTP_Lamm_ce: set_Pr448751882837621926eger_o > fun_Co480733329897759773un_o_o ).

tff(sy_c_ATP_058Lamm__cf____,type,
    aTP_Lamm_cf: set_Pr8218934625190621173um_num > fun_num_fun_num_o ).

tff(sy_c_ATP_058Lamm__cg____,type,
    aTP_Lamm_cg: set_Pr6200539531224447659at_num > fun_nat_fun_num_o ).

tff(sy_c_ATP_058Lamm__ch____,type,
    aTP_Lamm_ch: set_Pr1261947904930325089at_nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__ci____,type,
    aTP_Lamm_ci: set_Pr958786334691620121nt_int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__cj____,type,
    aTP_Lamm_cj: int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__ck____,type,
    aTP_Lamm_ck: int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__cl____,type,
    aTP_Lamm_cl: fun_complex_int > fun_complex_complex ).

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

tff(sy_c_ATP_058Lamm__cn____,type,
    aTP_Lamm_cn: fun_int_int > fun_int_real ).

tff(sy_c_ATP_058Lamm__co____,type,
    aTP_Lamm_co: fun_int_int > fun_int_rat ).

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

tff(sy_c_ATP_058Lamm__cq____,type,
    aTP_Lamm_cq: fun_int_int > fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__cr____,type,
    aTP_Lamm_cr: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__cs____,type,
    aTP_Lamm_cs: fun_int_int > fun_int_int ).

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_co8153928105989009288nteger > fun_fu1400560744564786701nteger ).

tff(sy_c_ATP_058Lamm__cv____,type,
    aTP_Lamm_cv: fun_re6978948838062957830nteger > fun_fu8643405727515436425nteger ).

tff(sy_c_ATP_058Lamm__cw____,type,
    aTP_Lamm_cw: fun_nat_Code_integer > fun_fu2499394402704830801nteger ).

tff(sy_c_ATP_058Lamm__cx____,type,
    aTP_Lamm_cx: fun_int_Code_integer > fun_fu3628599891403802633nteger ).

tff(sy_c_ATP_058Lamm__cy____,type,
    aTP_Lamm_cy: fun_complex_real > fun_fu7714627026455246123x_real ).

tff(sy_c_ATP_058Lamm__cz____,type,
    aTP_Lamm_cz: fun_real_real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__da____,type,
    aTP_Lamm_da: fun_int_real > fun_fu121957152452960295t_real ).

tff(sy_c_ATP_058Lamm__db____,type,
    aTP_Lamm_db: fun_complex_rat > fun_fu2990765671291504771ex_rat ).

tff(sy_c_ATP_058Lamm__dc____,type,
    aTP_Lamm_dc: fun_real_rat > fun_fu2598743018846156927al_rat ).

tff(sy_c_ATP_058Lamm__dd____,type,
    aTP_Lamm_dd: fun_nat_rat > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__de____,type,
    aTP_Lamm_de: fun_nat_real ).

tff(sy_c_ATP_058Lamm__df____,type,
    aTP_Lamm_df: fun_complex_nat > fun_fu705315015666092147ex_nat ).

tff(sy_c_ATP_058Lamm__dg____,type,
    aTP_Lamm_dg: fun_real_nat > fun_fu3214020956574444655al_nat ).

tff(sy_c_ATP_058Lamm__dh____,type,
    aTP_Lamm_dh: fun_set_nat_nat > fun_fu1590586450949517859at_nat ).

tff(sy_c_ATP_058Lamm__di____,type,
    aTP_Lamm_di: fun_int_nat > fun_fu8905809029841901679nt_nat ).

tff(sy_c_ATP_058Lamm__dj____,type,
    aTP_Lamm_dj: fun_nat_nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__dk____,type,
    aTP_Lamm_dk: nat > fun_co1148497455491383662plex_o ).

tff(sy_c_ATP_058Lamm__dl____,type,
    aTP_Lamm_dl: nat > fun_complex_o ).

tff(sy_c_ATP_058Lamm__dm____,type,
    aTP_Lamm_dm: fun_nat_nat > fun_nat_nat ).

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

tff(sy_c_ATP_058Lamm__do____,type,
    aTP_Lamm_do: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__dp____,type,
    aTP_Lamm_dp: fun_nat_real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__dq____,type,
    aTP_Lamm_dq: complex > fun_na1142814060743658952omplex ).

tff(sy_c_ATP_058Lamm__dr____,type,
    aTP_Lamm_dr: rat > fun_nat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__ds____,type,
    aTP_Lamm_ds: int > fun_nat_fun_nat_int ).

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

tff(sy_c_ATP_058Lamm__du____,type,
    aTP_Lamm_du: fun_nat_nat > fun_na3244735409550470327at_nat ).

tff(sy_c_ATP_058Lamm__dv____,type,
    aTP_Lamm_dv: fun_nat_real > fun_na7718645245721260691t_real ).

tff(sy_c_ATP_058Lamm__dw____,type,
    aTP_Lamm_dw: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__dx____,type,
    aTP_Lamm_dx: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__dy____,type,
    aTP_Lamm_dy: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__dz____,type,
    aTP_Lamm_dz: fun_nat_int > fun_nat_int ).

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

tff(sy_c_ATP_058Lamm__eb____,type,
    aTP_Lamm_eb: fun_nat_complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__ec____,type,
    aTP_Lamm_ec: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__ed____,type,
    aTP_Lamm_ed: fun_nat_int > fun_nat_int ).

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

tff(sy_c_ATP_058Lamm__ef____,type,
    aTP_Lamm_ef: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__eg____,type,
    aTP_Lamm_eg: fun_nat_int > fun_nat_int ).

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

tff(sy_c_ATP_058Lamm__ei____,type,
    aTP_Lamm_ei: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__ej____,type,
    aTP_Lamm_ej: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__ek____,type,
    aTP_Lamm_ek: fun_nat_int > fun_nat_int ).

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

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

tff(sy_c_ATP_058Lamm__en____,type,
    aTP_Lamm_en: int > fun_fu1143798476471108391nt_int ).

tff(sy_c_ATP_058Lamm__eo____,type,
    aTP_Lamm_eo: complex > fun_fu5956788528830221615omplex ).

tff(sy_c_ATP_058Lamm__ep____,type,
    aTP_Lamm_ep: nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__eq____,type,
    aTP_Lamm_eq: real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__er____,type,
    aTP_Lamm_er: fun_int_int > fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__es____,type,
    aTP_Lamm_es: fun_complex_complex > fun_co7673397065115134084omplex ).

tff(sy_c_ATP_058Lamm__et____,type,
    aTP_Lamm_et: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__eu____,type,
    aTP_Lamm_eu: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ev____,type,
    aTP_Lamm_ev: fun_int_int > fun_fu923337204662071120nt_int ).

tff(sy_c_ATP_058Lamm__ew____,type,
    aTP_Lamm_ew: fun_int_int > fun_fu3739269683040691718nt_int ).

tff(sy_c_ATP_058Lamm__ex____,type,
    aTP_Lamm_ex: fun_complex_complex > fun_fu8998703528969179354omplex ).

tff(sy_c_ATP_058Lamm__ey____,type,
    aTP_Lamm_ey: fun_complex_complex > fun_fu8084223252616112272omplex ).

tff(sy_c_ATP_058Lamm__ez____,type,
    aTP_Lamm_ez: fun_nat_nat > fun_fu4413992040155564420at_nat ).

tff(sy_c_ATP_058Lamm__fa____,type,
    aTP_Lamm_fa: fun_nat_nat > fun_fu2454420574033104442at_nat ).

tff(sy_c_ATP_058Lamm__fb____,type,
    aTP_Lamm_fb: fun_nat_real > fun_fu163975356400620604t_real ).

tff(sy_c_ATP_058Lamm__fc____,type,
    aTP_Lamm_fc: fun_nat_real > fun_fu3306995184869731826t_real ).

tff(sy_c_ATP_058Lamm__fd____,type,
    aTP_Lamm_fd: fun_int_int > fun_fu1143798476471108391nt_int ).

tff(sy_c_ATP_058Lamm__fe____,type,
    aTP_Lamm_fe: fun_complex_complex > fun_fu5956788528830221615omplex ).

tff(sy_c_ATP_058Lamm__ff____,type,
    aTP_Lamm_ff: fun_nat_nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__fg____,type,
    aTP_Lamm_fg: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__fh____,type,
    aTP_Lamm_fh: fun_complex_complex > fun_co7673397065115134084omplex ).

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

tff(sy_c_ATP_058Lamm__fj____,type,
    aTP_Lamm_fj: nat > fun_nat_fun_nat_nat ).

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

tff(sy_c_ATP_058Lamm__fl____,type,
    aTP_Lamm_fl: fun_complex_nat > fun_complex_complex ).

tff(sy_c_ATP_058Lamm__fm____,type,
    aTP_Lamm_fm: fun_nat_nat > fun_nat_int ).

tff(sy_c_ATP_058Lamm__fn____,type,
    aTP_Lamm_fn: fun_nat_nat > fun_nat_rat ).

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

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

tff(sy_c_ATP_058Lamm__fq____,type,
    aTP_Lamm_fq: complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__fr____,type,
    aTP_Lamm_fr: int > fun_int_fun_nat_int ).

tff(sy_c_ATP_058Lamm__fs____,type,
    aTP_Lamm_fs: rat > fun_rat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__ft____,type,
    aTP_Lamm_ft: nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__fu____,type,
    aTP_Lamm_fu: real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__fv____,type,
    aTP_Lamm_fv: fun_na1142814060743658952omplex ).

tff(sy_c_ATP_058Lamm__fw____,type,
    aTP_Lamm_fw: fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__fx____,type,
    aTP_Lamm_fx: fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__fy____,type,
    aTP_Lamm_fy: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__fz____,type,
    aTP_Lamm_fz: fun_nat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__ga____,type,
    aTP_Lamm_ga: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__gb____,type,
    aTP_Lamm_gb: fun_re4238438383274954274t_real ).

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

tff(sy_c_ATP_058Lamm__gd____,type,
    aTP_Lamm_gd: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__ge____,type,
    aTP_Lamm_ge: fun_nat_real ).

tff(sy_c_ATP_058Lamm__gf____,type,
    aTP_Lamm_gf: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__gg____,type,
    aTP_Lamm_gg: fun_nat_complex ).

tff(sy_c_ATP_058Lamm__gh____,type,
    aTP_Lamm_gh: fun_nat_int ).

tff(sy_c_ATP_058Lamm__gi____,type,
    aTP_Lamm_gi: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__gj____,type,
    aTP_Lamm_gj: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__gk____,type,
    aTP_Lamm_gk: rat > fun_ra5513060666784058567at_rat ).

tff(sy_c_ATP_058Lamm__gl____,type,
    aTP_Lamm_gl: rat > fun_ra5513060666784058567at_rat ).

tff(sy_c_ATP_058Lamm__gm____,type,
    aTP_Lamm_gm: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__gn____,type,
    aTP_Lamm_gn: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__go____,type,
    aTP_Lamm_go: set_nat > fun_set_nat_o ).

tff(sy_c_ATP_058Lamm__gp____,type,
    aTP_Lamm_gp: fun_rat_fun_rat_o ).

tff(sy_c_ATP_058Lamm__gq____,type,
    aTP_Lamm_gq: fun_num_fun_num_o ).

tff(sy_c_ATP_058Lamm__gr____,type,
    aTP_Lamm_gr: fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__gs____,type,
    aTP_Lamm_gs: fun_real_fun_real_o ).

tff(sy_c_ATP_058Lamm__gt____,type,
    aTP_Lamm_gt: fun_nat_nat > fun_nat_fun_nat_nat ).

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

tff(sy_c_ATP_058Lamm__gv____,type,
    aTP_Lamm_gv: fun_nat_int > fun_int_fun_nat_int ).

tff(sy_c_ATP_058Lamm__gw____,type,
    aTP_Lamm_gw: fun_nat_rat > fun_rat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__gx____,type,
    aTP_Lamm_gx: fun_nat_real > fun_re4238438383274954274t_real ).

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

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

tff(sy_c_ATP_058Lamm__ha____,type,
    aTP_Lamm_ha: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__hb____,type,
    aTP_Lamm_hb: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__hc____,type,
    aTP_Lamm_hc: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__hd____,type,
    aTP_Lamm_hd: rat > fun_ra5513060666784058567at_rat ).

tff(sy_c_ATP_058Lamm__he____,type,
    aTP_Lamm_he: rat > fun_ra5513060666784058567at_rat ).

tff(sy_c_ATP_058Lamm__hf____,type,
    aTP_Lamm_hf: int > fun_in6793278964266839919at_int ).

tff(sy_c_ATP_058Lamm__hg____,type,
    aTP_Lamm_hg: int > fun_in6793278964266839919at_int ).

tff(sy_c_ATP_058Lamm__hh____,type,
    aTP_Lamm_hh: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__hi____,type,
    aTP_Lamm_hi: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__hj____,type,
    aTP_Lamm_hj: complex > fun_na6632006986671646835omplex ).

tff(sy_c_ATP_058Lamm__hk____,type,
    aTP_Lamm_hk: rat > fun_na1148497095348402375at_rat ).

tff(sy_c_ATP_058Lamm__hl____,type,
    aTP_Lamm_hl: int > fun_na2382631619178260847at_int ).

tff(sy_c_ATP_058Lamm__hm____,type,
    aTP_Lamm_hm: real > fun_na5225841814499866095t_real ).

tff(sy_c_ATP_058Lamm__hn____,type,
    aTP_Lamm_hn: complex > fun_na6632006986671646835omplex ).

tff(sy_c_ATP_058Lamm__ho____,type,
    aTP_Lamm_ho: rat > fun_na1148497095348402375at_rat ).

tff(sy_c_ATP_058Lamm__hp____,type,
    aTP_Lamm_hp: int > fun_na2382631619178260847at_int ).

tff(sy_c_ATP_058Lamm__hq____,type,
    aTP_Lamm_hq: real > fun_na5225841814499866095t_real ).

tff(sy_c_ATP_058Lamm__hr____,type,
    aTP_Lamm_hr: complex > fun_na1142814060743658952omplex ).

tff(sy_c_ATP_058Lamm__hs____,type,
    aTP_Lamm_hs: rat > fun_nat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__ht____,type,
    aTP_Lamm_ht: int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__hu____,type,
    aTP_Lamm_hu: real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__hv____,type,
    aTP_Lamm_hv: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__hw____,type,
    aTP_Lamm_hw: fun_nat_real > fun_nat_real ).

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

tff(sy_c_ATP_058Lamm__hy____,type,
    aTP_Lamm_hy: fun_nat_real ).

tff(sy_c_ATP_058Lamm__hz____,type,
    aTP_Lamm_hz: nat > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__ia____,type,
    aTP_Lamm_ia: nat > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__ib____,type,
    aTP_Lamm_ib: nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__ic____,type,
    aTP_Lamm_ic: nat > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__id____,type,
    aTP_Lamm_id: complex > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__ie____,type,
    aTP_Lamm_ie: fun_nat_complex > fun_co3258106519541737638omplex ).

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

tff(sy_c_ATP_058Lamm__ig____,type,
    aTP_Lamm_ig: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ih____,type,
    aTP_Lamm_ih: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ii____,type,
    aTP_Lamm_ii: fun_nat_int > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__ij____,type,
    aTP_Lamm_ij: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ik____,type,
    aTP_Lamm_ik: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__il____,type,
    aTP_Lamm_il: fun_nat_complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__im____,type,
    aTP_Lamm_im: fun_nat_complex > fun_co8378349014452122532t_real ).

tff(sy_c_ATP_058Lamm__in____,type,
    aTP_Lamm_in: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__io____,type,
    aTP_Lamm_io: fun_nat_complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__ip____,type,
    aTP_Lamm_ip: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__iq____,type,
    aTP_Lamm_iq: fun_nat_complex > fun_na6632006986671646835omplex ).

tff(sy_c_ATP_058Lamm__ir____,type,
    aTP_Lamm_ir: fun_nat_real > fun_na5225841814499866095t_real ).

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

tff(sy_c_ATP_058Lamm__it____,type,
    aTP_Lamm_it: fun_nat_complex > fun_na1142814060743658952omplex ).

tff(sy_c_ATP_058Lamm__iu____,type,
    aTP_Lamm_iu: real > fun_fu7631928937237476593t_real ).

tff(sy_c_ATP_058Lamm__iv____,type,
    aTP_Lamm_iv: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__iw____,type,
    aTP_Lamm_iw: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ix____,type,
    aTP_Lamm_ix: real > fun_fu3477889739401271834t_real ).

tff(sy_c_ATP_058Lamm__iy____,type,
    aTP_Lamm_iy: real > fun_fu5589980185390252312t_real ).

tff(sy_c_ATP_058Lamm__iz____,type,
    aTP_Lamm_iz: real > fun_fu4806068833880117316t_real ).

tff(sy_c_ATP_058Lamm__ja____,type,
    aTP_Lamm_ja: real > fun_fu2115079063411676732t_real ).

tff(sy_c_ATP_058Lamm__jb____,type,
    aTP_Lamm_jb: real > fun_fu4540635375566031512t_real ).

tff(sy_c_ATP_058Lamm__jc____,type,
    aTP_Lamm_jc: real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__jd____,type,
    aTP_Lamm_jd: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__je____,type,
    aTP_Lamm_je: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__jf____,type,
    aTP_Lamm_jf: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__jg____,type,
    aTP_Lamm_jg: complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__jh____,type,
    aTP_Lamm_jh: complex > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__ji____,type,
    aTP_Lamm_ji: real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__jj____,type,
    aTP_Lamm_jj: fun_nat_complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__jk____,type,
    aTP_Lamm_jk: nat > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__jl____,type,
    aTP_Lamm_jl: nat > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__jm____,type,
    aTP_Lamm_jm: nat > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__jn____,type,
    aTP_Lamm_jn: nat > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__jo____,type,
    aTP_Lamm_jo: nat > fun_int_fun_nat_int ).

tff(sy_c_ATP_058Lamm__jp____,type,
    aTP_Lamm_jp: fun_nat_real ).

tff(sy_c_ATP_058Lamm__jq____,type,
    aTP_Lamm_jq: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__jr____,type,
    aTP_Lamm_jr: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__js____,type,
    aTP_Lamm_js: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__jt____,type,
    aTP_Lamm_jt: fun_nat_complex > fun_co3258106519541737638omplex ).

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

tff(sy_c_ATP_058Lamm__jv____,type,
    aTP_Lamm_jv: fun_nat_real > fun_re4238438383274954274t_real ).

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

tff(sy_c_ATP_058Lamm__jx____,type,
    aTP_Lamm_jx: real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__jy____,type,
    aTP_Lamm_jy: complex > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__jz____,type,
    aTP_Lamm_jz: complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__ka____,type,
    aTP_Lamm_ka: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__kb____,type,
    aTP_Lamm_kb: rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__kc____,type,
    aTP_Lamm_kc: complex > fun_na1591161823130732966omplex ).

tff(sy_c_ATP_058Lamm__kd____,type,
    aTP_Lamm_kd: int > fun_nat_fun_int_int ).

tff(sy_c_ATP_058Lamm__ke____,type,
    aTP_Lamm_ke: real > fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__kf____,type,
    aTP_Lamm_kf: rat > fun_nat_fun_rat_rat ).

tff(sy_c_ATP_058Lamm__kg____,type,
    aTP_Lamm_kg: nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__kh____,type,
    aTP_Lamm_kh: fun_complex_complex ).

tff(sy_c_ATP_058Lamm__ki____,type,
    aTP_Lamm_ki: fun_int_int ).

tff(sy_c_ATP_058Lamm__kj____,type,
    aTP_Lamm_kj: fun_real_real ).

tff(sy_c_ATP_058Lamm__kk____,type,
    aTP_Lamm_kk: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__kl____,type,
    aTP_Lamm_kl: fun_rat_rat ).

tff(sy_c_ATP_058Lamm__km____,type,
    aTP_Lamm_km: complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__kn____,type,
    aTP_Lamm_kn: rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__ko____,type,
    aTP_Lamm_ko: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__kp____,type,
    aTP_Lamm_kp: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__kq____,type,
    aTP_Lamm_kq: fun_nat_int ).

tff(sy_c_ATP_058Lamm__kr____,type,
    aTP_Lamm_kr: fun_int_int ).

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

tff(sy_c_ATP_058Lamm__kt____,type,
    aTP_Lamm_kt: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__ku____,type,
    aTP_Lamm_ku: fun_nat_nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__kv____,type,
    aTP_Lamm_kv: fun_nat_int > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__kw____,type,
    aTP_Lamm_kw: fun_int_int > fun_fu1143798476471108391nt_int ).

tff(sy_c_ATP_058Lamm__kx____,type,
    aTP_Lamm_kx: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ky____,type,
    aTP_Lamm_ky: fun_nat_int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__kz____,type,
    aTP_Lamm_kz: fun_int_int > fun_nat_fun_int_int ).

tff(sy_c_ATP_058Lamm__la____,type,
    aTP_Lamm_la: fun_nat_int > fun_int_fun_nat_int ).

tff(sy_c_ATP_058Lamm__lb____,type,
    aTP_Lamm_lb: fun_nat_complex > fun_na1591161823130732966omplex ).

tff(sy_c_ATP_058Lamm__lc____,type,
    aTP_Lamm_lc: fun_nat_real > fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__ld____,type,
    aTP_Lamm_ld: fun_nat_rat > fun_nat_fun_rat_rat ).

tff(sy_c_ATP_058Lamm__le____,type,
    aTP_Lamm_le: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__lf____,type,
    aTP_Lamm_lf: fun_nat_int > fun_nat_fun_int_int ).

tff(sy_c_ATP_058Lamm__lg____,type,
    aTP_Lamm_lg: real > fun_fu3106432010075488275t_real ).

tff(sy_c_ATP_058Lamm__lh____,type,
    aTP_Lamm_lh: complex > fun_fu4781654129426620053omplex ).

tff(sy_c_ATP_058Lamm__li____,type,
    aTP_Lamm_li: nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__lj____,type,
    aTP_Lamm_lj: int > fun_fu4512855384587262995at_int ).

tff(sy_c_ATP_058Lamm__lk____,type,
    aTP_Lamm_lk: int > fun_fu6415725884910174923nt_int ).

tff(sy_c_ATP_058Lamm__ll____,type,
    aTP_Lamm_ll: fun_nat_int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__lm____,type,
    aTP_Lamm_lm: fun_nat_int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__ln____,type,
    aTP_Lamm_ln: fun_nat_int > fun_na7654559027617327379at_int ).

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

tff(sy_c_ATP_058Lamm__lp____,type,
    aTP_Lamm_lp: rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__lq____,type,
    aTP_Lamm_lq: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__lr____,type,
    aTP_Lamm_lr: fun_nat_fun_nat_nat ).

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

tff(sy_c_ATP_058Lamm__lt____,type,
    aTP_Lamm_lt: fun_complex_real > fun_fu7714627026455246123x_real ).

tff(sy_c_ATP_058Lamm__lu____,type,
    aTP_Lamm_lu: fun_real_real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__lv____,type,
    aTP_Lamm_lv: fun_set_nat_real > fun_fu5354524256550068955t_real ).

tff(sy_c_ATP_058Lamm__lw____,type,
    aTP_Lamm_lw: fun_int_real > fun_fu121957152452960295t_real ).

tff(sy_c_ATP_058Lamm__lx____,type,
    aTP_Lamm_lx: fun_complex_complex > fun_fu8968174646369908141x_real ).

tff(sy_c_ATP_058Lamm__ly____,type,
    aTP_Lamm_ly: fun_real_complex > fun_fu6344110373156359337l_real ).

tff(sy_c_ATP_058Lamm__lz____,type,
    aTP_Lamm_lz: fun_set_nat_complex > fun_fu1617006900395023197t_real ).

tff(sy_c_ATP_058Lamm__ma____,type,
    aTP_Lamm_ma: fun_int_complex > fun_fu3594690983434520489t_real ).

tff(sy_c_ATP_058Lamm__mb____,type,
    aTP_Lamm_mb: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__mc____,type,
    aTP_Lamm_mc: fun_nat_complex > fun_fu7631928937237476593t_real ).

tff(sy_c_ATP_058Lamm__md____,type,
    aTP_Lamm_md: complex > fun_na1591161823130732966omplex ).

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

tff(sy_c_ATP_058Lamm__mf____,type,
    aTP_Lamm_mf: real > fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__mg____,type,
    aTP_Lamm_mg: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__mh____,type,
    aTP_Lamm_mh: rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__mi____,type,
    aTP_Lamm_mi: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__mj____,type,
    aTP_Lamm_mj: int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__mk____,type,
    aTP_Lamm_mk: real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__ml____,type,
    aTP_Lamm_ml: rat > fun_nat_fun_nat_rat ).

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

tff(sy_c_ATP_058Lamm__mn____,type,
    aTP_Lamm_mn: int > fun_nat_fun_nat_int ).

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

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

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

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

tff(sy_c_ATP_058Lamm__ms____,type,
    aTP_Lamm_ms: fun_nat_complex > fun_na1591161823130732966omplex ).

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

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

tff(sy_c_ATP_058Lamm__mv____,type,
    aTP_Lamm_mv: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__mw____,type,
    aTP_Lamm_mw: fun_nat_real > fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__mx____,type,
    aTP_Lamm_mx: nat > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__my____,type,
    aTP_Lamm_my: nat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__mz____,type,
    aTP_Lamm_mz: fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__na____,type,
    aTP_Lamm_na: nat > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__nb____,type,
    aTP_Lamm_nb: fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__nc____,type,
    aTP_Lamm_nc: nat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__nd____,type,
    aTP_Lamm_nd: fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__ne____,type,
    aTP_Lamm_ne: nat > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__nf____,type,
    aTP_Lamm_nf: fun_nat_fun_nat_int ).

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

tff(sy_c_ATP_058Lamm__nh____,type,
    aTP_Lamm_nh: fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__ni____,type,
    aTP_Lamm_ni: fun_real_fun_real_o ).

tff(sy_c_ATP_058Lamm__nj____,type,
    aTP_Lamm_nj: set_nat > fun_set_nat_o ).

tff(sy_c_ATP_058Lamm__nk____,type,
    aTP_Lamm_nk: fun_rat_fun_rat_o ).

tff(sy_c_ATP_058Lamm__nl____,type,
    aTP_Lamm_nl: fun_num_fun_num_o ).

tff(sy_c_ATP_058Lamm__nm____,type,
    aTP_Lamm_nm: fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__nn____,type,
    aTP_Lamm_nn: fun_nat_fun_nat_o ).

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

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

tff(sy_c_ATP_058Lamm__nq____,type,
    aTP_Lamm_nq: fun_nat_fun_nat_nat > fun_nat_nat ).

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

tff(sy_c_ATP_058Lamm__ns____,type,
    aTP_Lamm_ns: fun_nat_fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__nt____,type,
    aTP_Lamm_nt: nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__nu____,type,
    aTP_Lamm_nu: fun_nat_fun_nat_nat ).

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

tff(sy_c_ATP_058Lamm__nw____,type,
    aTP_Lamm_nw: fun_re4238438383274954274t_real ).

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

tff(sy_c_ATP_058Lamm__ny____,type,
    aTP_Lamm_ny: fun_nat_fun_nat_nat > fun_nat_fun_nat_nat ).

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

tff(sy_c_ATP_058Lamm__oa____,type,
    aTP_Lamm_oa: fun_nat_fun_nat_real > fun_nat_real ).

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

tff(sy_c_ATP_058Lamm__oc____,type,
    aTP_Lamm_oc: fun_nat_fun_nat_real > fun_nat_fun_nat_real ).

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

tff(sy_c_ATP_058Lamm__oe____,type,
    aTP_Lamm_oe: fun_nat_fun_nat_nat > fun_nat_fun_nat_nat ).

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

tff(sy_c_ATP_058Lamm__og____,type,
    aTP_Lamm_og: fun_nat_fun_nat_int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__oh____,type,
    aTP_Lamm_oh: fun_nat_fun_nat_int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__oi____,type,
    aTP_Lamm_oi: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__oj____,type,
    aTP_Lamm_oj: fun_nat_fun_nat_nat ).

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

tff(sy_c_ATP_058Lamm__ol____,type,
    aTP_Lamm_ol: rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__om____,type,
    aTP_Lamm_om: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__on____,type,
    aTP_Lamm_on: fun_nat_fun_nat_nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__oo____,type,
    aTP_Lamm_oo: fun_nat_fun_nat_real > fun_nat_fun_nat_real ).

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

tff(sy_c_ATP_058Lamm__oq____,type,
    aTP_Lamm_oq: nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__or____,type,
    aTP_Lamm_or: nat > fun_na3244735409550470327at_nat ).

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

tff(sy_c_ATP_058Lamm__ot____,type,
    aTP_Lamm_ot: fun_nat_int > fun_int_fun_nat_int ).

tff(sy_c_ATP_058Lamm__ou____,type,
    aTP_Lamm_ou: fun_nat_complex > fun_fu4528473944208963392omplex ).

tff(sy_c_ATP_058Lamm__ov____,type,
    aTP_Lamm_ov: fun_nat_complex > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__ow____,type,
    aTP_Lamm_ow: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__ox____,type,
    aTP_Lamm_ox: fun_nat_real > fun_fu163975356400620604t_real ).

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

tff(sy_c_ATP_058Lamm__oz____,type,
    aTP_Lamm_oz: nat > fun_na3244735409550470327at_nat ).

tff(sy_c_ATP_058Lamm__pa____,type,
    aTP_Lamm_pa: fun_nat_complex > fun_fu6652553491520710942omplex ).

tff(sy_c_ATP_058Lamm__pb____,type,
    aTP_Lamm_pb: fun_nat_rat > fun_fu6228432884689642900at_rat ).

tff(sy_c_ATP_058Lamm__pc____,type,
    aTP_Lamm_pc: fun_nat_rat > fun_fu4710970720072501148at_rat ).

tff(sy_c_ATP_058Lamm__pd____,type,
    aTP_Lamm_pd: fun_nat_int > fun_fu1567874110255095356at_int ).

tff(sy_c_ATP_058Lamm__pe____,type,
    aTP_Lamm_pe: fun_nat_int > fun_fu5519318738670804632at_int ).

tff(sy_c_ATP_058Lamm__pf____,type,
    aTP_Lamm_pf: fun_nat_real > fun_fu8064956510386623256t_real ).

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

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

tff(sy_c_ATP_058Lamm__pi____,type,
    aTP_Lamm_pi: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__pj____,type,
    aTP_Lamm_pj: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__pk____,type,
    aTP_Lamm_pk: int > fun_in6793278964266839919at_int ).

tff(sy_c_ATP_058Lamm__pl____,type,
    aTP_Lamm_pl: rat > fun_ra5513060666784058567at_rat ).

tff(sy_c_ATP_058Lamm__pm____,type,
    aTP_Lamm_pm: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__pn____,type,
    aTP_Lamm_pn: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__po____,type,
    aTP_Lamm_po: fun_nat_nat > fun_fu4413992040155564420at_nat ).

tff(sy_c_ATP_058Lamm__pp____,type,
    aTP_Lamm_pp: fun_nat_nat > fun_fu4413992040155564420at_nat ).

tff(sy_c_ATP_058Lamm__pq____,type,
    aTP_Lamm_pq: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__pr____,type,
    aTP_Lamm_pr: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ps____,type,
    aTP_Lamm_ps: nat > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__pt____,type,
    aTP_Lamm_pt: fun_real_fun_real_o ).

tff(sy_c_ATP_058Lamm__pu____,type,
    aTP_Lamm_pu: fun_real_fun_real_o ).

tff(sy_c_ATP_058Lamm__pv____,type,
    aTP_Lamm_pv: fun_real_o ).

tff(sy_c_ATP_058Lamm__pw____,type,
    aTP_Lamm_pw: real > fun_int_o ).

tff(sy_c_ATP_058Lamm__px____,type,
    aTP_Lamm_px: rat > fun_int_o ).

tff(sy_c_ATP_058Lamm__py____,type,
    aTP_Lamm_py: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__pz____,type,
    aTP_Lamm_pz: fun_nat_o > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__qa____,type,
    aTP_Lamm_qa: fun_nat_nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__qb____,type,
    aTP_Lamm_qb: int > fun_in2204242111582549184nt_int ).

tff(sy_c_ATP_058Lamm__qc____,type,
    aTP_Lamm_qc: rat > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__qd____,type,
    aTP_Lamm_qd: int > fun_in2204242111582549184nt_int ).

tff(sy_c_ATP_058Lamm__qe____,type,
    aTP_Lamm_qe: rat > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__qf____,type,
    aTP_Lamm_qf: int > fun_in2204242111582549184nt_int ).

tff(sy_c_ATP_058Lamm__qg____,type,
    aTP_Lamm_qg: rat > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__qh____,type,
    aTP_Lamm_qh: int > fun_in2204242111582549184nt_int ).

tff(sy_c_ATP_058Lamm__qi____,type,
    aTP_Lamm_qi: rat > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__qj____,type,
    aTP_Lamm_qj: int > fun_in5570076352459579423_int_o ).

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

tff(sy_c_ATP_058Lamm__ql____,type,
    aTP_Lamm_ql: int > fun_in5570076352459579423_int_o ).

tff(sy_c_ATP_058Lamm__qm____,type,
    aTP_Lamm_qm: rat > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__qn____,type,
    aTP_Lamm_qn: int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__qo____,type,
    aTP_Lamm_qo: int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__qp____,type,
    aTP_Lamm_qp: int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__qq____,type,
    aTP_Lamm_qq: int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__qr____,type,
    aTP_Lamm_qr: complex > fun_na1554462459066888140plex_o ).

tff(sy_c_ATP_058Lamm__qs____,type,
    aTP_Lamm_qs: fun_Co1092983284983208416er_int ).

tff(sy_c_ATP_058Lamm__qt____,type,
    aTP_Lamm_qt: fun_Co8521658056209057742er_num ).

tff(sy_c_ATP_058Lamm__qu____,type,
    aTP_Lamm_qu: fun_Co3583066429914935172er_nat ).

tff(sy_c_ATP_058Lamm__qv____,type,
    aTP_Lamm_qv: code_integer > fun_Co3544761417380347684eger_o ).

tff(sy_c_ATP_058Lamm__qw____,type,
    aTP_Lamm_qw: set_nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__qx____,type,
    aTP_Lamm_qx: set_nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__qy____,type,
    aTP_Lamm_qy: code_integer > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__qz____,type,
    aTP_Lamm_qz: code_integer > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__ra____,type,
    aTP_Lamm_ra: fun_nat_o ).

tff(sy_c_ATP_058Lamm__rb____,type,
    aTP_Lamm_rb: fun_nat_o ).

tff(sy_c_ATP_058Lamm__rc____,type,
    aTP_Lamm_rc: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__rd____,type,
    aTP_Lamm_rd: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__re____,type,
    aTP_Lamm_re: nat > fun_na8563630686737515764_nat_o ).

tff(sy_c_ATP_058Lamm__rf____,type,
    aTP_Lamm_rf: code_integer > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__rg____,type,
    aTP_Lamm_rg: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__rh____,type,
    aTP_Lamm_rh: nat > fun_na4143055713577223196at_nat ).

tff(sy_c_ATP_058Lamm__ri____,type,
    aTP_Lamm_ri: fun_na1668100129766719005at_nat ).

tff(sy_c_ATP_058Lamm__rj____,type,
    aTP_Lamm_rj: fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__rk____,type,
    aTP_Lamm_rk: nat > fun_na7928631014926962227_nat_o ).

tff(sy_c_ATP_058Lamm__rl____,type,
    aTP_Lamm_rl: fun_na5914473681949881652_nat_o ).

tff(sy_c_ATP_058Lamm__rm____,type,
    aTP_Lamm_rm: nat > fun_na7928631014926962227_nat_o ).

tff(sy_c_ATP_058Lamm__rn____,type,
    aTP_Lamm_rn: fun_na5914473681949881652_nat_o ).

tff(sy_c_ATP_058Lamm__ro____,type,
    aTP_Lamm_ro: nat > fun_na4143055713577223196at_nat ).

tff(sy_c_ATP_058Lamm__rp____,type,
    aTP_Lamm_rp: fun_na1668100129766719005at_nat ).

tff(sy_c_ATP_058Lamm__rq____,type,
    aTP_Lamm_rq: nat > fun_na4143055713577223196at_nat ).

tff(sy_c_ATP_058Lamm__rr____,type,
    aTP_Lamm_rr: fun_na1668100129766719005at_nat ).

tff(sy_c_ATP_058Lamm__rs____,type,
    aTP_Lamm_rs: fun_nat_fun_nat_nat ).

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_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__rv____,type,
    aTP_Lamm_rv: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__rw____,type,
    aTP_Lamm_rw: fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__rx____,type,
    aTP_Lamm_rx: fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__ry____,type,
    aTP_Lamm_ry: fun_real_real > fun_na6724076748692969250l_real ).

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

tff(sy_c_ATP_058Lamm__sa____,type,
    aTP_Lamm_sa: fun_real_real > fun_re7681423839302404990l_real ).

tff(sy_c_ATP_058Lamm__sb____,type,
    aTP_Lamm_sb: fun_real_real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__sc____,type,
    aTP_Lamm_sc: fun_re4238438383274954274t_real > fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__sd____,type,
    aTP_Lamm_sd: fun_re4238438383274954274t_real > fun_real_real ).

tff(sy_c_ATP_058Lamm__se____,type,
    aTP_Lamm_se: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__sf____,type,
    aTP_Lamm_sf: fun_nat_real > fun_re4238438383274954274t_real ).

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

tff(sy_c_ATP_058Lamm__sh____,type,
    aTP_Lamm_sh: fun_na6724076748692969250l_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__si____,type,
    aTP_Lamm_si: fun_na6724076748692969250l_real > fun_re8542384036533109323t_real ).

tff(sy_c_ATP_058Lamm__sj____,type,
    aTP_Lamm_sj: fun_na6724076748692969250l_real > fun_re8542384036533109323t_real ).

tff(sy_c_ATP_058Lamm__sk____,type,
    aTP_Lamm_sk: fun_na6724076748692969250l_real > fun_na5225841814499866095t_real ).

tff(sy_c_ATP_058Lamm__sl____,type,
    aTP_Lamm_sl: nat > fun_fu3612569900928539882l_real ).

tff(sy_c_ATP_058Lamm__sm____,type,
    aTP_Lamm_sm: fun_na6724076748692969250l_real > fun_na5225841814499866095t_real ).

tff(sy_c_ATP_058Lamm__sn____,type,
    aTP_Lamm_sn: fun_real_real ).

tff(sy_c_ATP_058Lamm__so____,type,
    aTP_Lamm_so: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__sp____,type,
    aTP_Lamm_sp: num > fun_num_int ).

tff(sy_c_ATP_058Lamm__sq____,type,
    aTP_Lamm_sq: fun_num_option_num ).

tff(sy_c_ATP_058Lamm__sr____,type,
    aTP_Lamm_sr: num > fun_nat_option_num ).

tff(sy_c_ATP_058Lamm__ss____,type,
    aTP_Lamm_ss: fun_nat_option_num ).

tff(sy_c_ATP_058Lamm__st____,type,
    aTP_Lamm_st: num > fun_nat_option_num ).

tff(sy_c_ATP_058Lamm__su____,type,
    aTP_Lamm_su: fun_num_option_num ).

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

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

tff(sy_c_ATP_058Lamm__sx____,type,
    aTP_Lamm_sx: num > fun_nat_option_num ).

tff(sy_c_ATP_058Lamm__sy____,type,
    aTP_Lamm_sy: fun_na7595719129879163854on_num ).

tff(sy_c_ATP_058Lamm__sz____,type,
    aTP_Lamm_sz: fun_real_real ).

tff(sy_c_ATP_058Lamm__ta____,type,
    aTP_Lamm_ta: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__tb____,type,
    aTP_Lamm_tb: fun_nat_real ).

tff(sy_c_ATP_058Lamm__tc____,type,
    aTP_Lamm_tc: fun_nat_real > fun_fu6899955116716589935t_real ).

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

tff(sy_c_ATP_058Lamm__te____,type,
    aTP_Lamm_te: fun_nat_real ).

tff(sy_c_ATP_058Lamm__tf____,type,
    aTP_Lamm_tf: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__tg____,type,
    aTP_Lamm_tg: fun_nat_real ).

tff(sy_c_ATP_058Lamm__th____,type,
    aTP_Lamm_th: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ti____,type,
    aTP_Lamm_ti: real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__tj____,type,
    aTP_Lamm_tj: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__tk____,type,
    aTP_Lamm_tk: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__tl____,type,
    aTP_Lamm_tl: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__tm____,type,
    aTP_Lamm_tm: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__tn____,type,
    aTP_Lamm_tn: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__to____,type,
    aTP_Lamm_to: fun_nat_real > fun_fu6119738925743952495t_real ).

tff(sy_c_ATP_058Lamm__tp____,type,
    aTP_Lamm_tp: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__tq____,type,
    aTP_Lamm_tq: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__tr____,type,
    aTP_Lamm_tr: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__ts____,type,
    aTP_Lamm_ts: real > fun_real_real ).

tff(sy_c_ATP_058Lamm__tt____,type,
    aTP_Lamm_tt: nat > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__tu____,type,
    aTP_Lamm_tu: real > fun_real_real ).

tff(sy_c_ATP_058Lamm__tv____,type,
    aTP_Lamm_tv: fun_nat_o > fun_nat_o ).

tff(sy_c_ATP_058Lamm__tw____,type,
    aTP_Lamm_tw: fun_nat_o > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__tx____,type,
    aTP_Lamm_tx: fun_real_o > fun_real_fun_real_o ).

tff(sy_c_ATP_058Lamm__ty____,type,
    aTP_Lamm_ty: nat > fun_li6914387844523535256_nat_o ).

tff(sy_c_ATP_058Lamm__tz____,type,
    aTP_Lamm_tz: set_int > fun_int_o ).

tff(sy_c_ATP_058Lamm__ua____,type,
    aTP_Lamm_ua: fun_real_real > fun_real_real ).

tff(sy_c_ATP_058Lamm__ub____,type,
    aTP_Lamm_ub: fun_real_real > fun_real_real ).

tff(sy_c_ATP_058Lamm__uc____,type,
    aTP_Lamm_uc: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ud____,type,
    aTP_Lamm_ud: fun_real_o ).

tff(sy_c_ATP_058Lamm__ue____,type,
    aTP_Lamm_ue: fun_real_o ).

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

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_nat_nat ).

tff(sy_c_ATP_058Lamm__ui____,type,
    aTP_Lamm_ui: nat > fun_na7234455742697417626_nat_o ).

tff(sy_c_ATP_058Lamm__uj____,type,
    aTP_Lamm_uj: nat > fun_na7234455742697417626_nat_o ).

tff(sy_c_ATP_058Lamm__uk____,type,
    aTP_Lamm_uk: nat > fun_na7234455742697417626_nat_o ).

tff(sy_c_Archimedean__Field_Oceiling_001t__Real__Oreal,type,
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tff(sy_c_Binomial_Ogbinomial_001t__Rat__Orat,type,
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tff(sy_c_Binomial_Ogbinomial_001t__Real__Oreal,type,
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tff(sy_c_Code__Numeral_Ointeger_Ointeger__of__int,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_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_Divides_Odivmod__nat,type,
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tff(sy_c_Filter_Oat__bot_001t__Real__Oreal,type,
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    inj_on_nat_char: fun_nat_char > fun_set_nat_o ).

tff(sy_c_Fun_Oinj__on_001t__Real__Oreal_001t__Real__Oreal,type,
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tff(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Set__Oset_It__Nat__Onat_J_001t__Real__Oreal,type,
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tff(sy_c_Groups__List_Ocomm__semiring__0__class_Ohorner__sum_001_Eo_001t__Int__Oint,type,
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tff(sy_c_Groups__List_Omonoid__add__class_Osum__list_001t__Nat__Onat,type,
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tff(sy_c_HOL_OThe_001t__Int__Oint,type,
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tff(sy_c_HOL_OThe_001t__Real__Oreal,type,
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tff(sy_c_Int_OAbs__Integ,type,
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tff(sy_c_Int_ORep__Integ,type,
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tff(sy_c_Int_Oint__ge__less__than,type,
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tff(sy_c_Int_Oint__ge__less__than2,type,
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tff(sy_c_Int_Onat,type,
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tff(sy_c_Int_Opower__int_001t__Real__Oreal,type,
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tff(sy_c_Int_Oring__1__class_OInts_001t__Real__Oreal,type,
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tff(sy_c_Int_Oring__1__class_Oof__int_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Int_Oring__1__class_Oof__int_001t__Complex__Ocomplex,type,
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tff(sy_c_Int_Oring__1__class_Oof__int_001t__Int__Oint,type,
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tff(sy_c_Int_Oring__1__class_Oof__int_001t__Rat__Orat,type,
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tff(sy_c_Int_Oring__1__class_Oof__int_001t__Real__Oreal,type,
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tff(sy_c_Lattices_Osemilattice__neutr__order_001t__Nat__Onat,type,
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tff(sy_c_Lattices_Osup__class_Osup_001t__Int__Oint,type,
    sup_sup_int: fun_int_fun_int_int ).

tff(sy_c_Lattices_Osup__class_Osup_001t__Nat__Onat,type,
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tff(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Nat__Onat_J,type,
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tff(sy_c_Limits_OBfun_001t__Nat__Onat_001t__Real__Oreal,type,
    bfun_nat_real: ( fun_nat_real * filter_nat ) > $o ).

tff(sy_c_List_Oappend_001t__Int__Oint,type,
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tff(sy_c_List_Oappend_001t__Nat__Onat,type,
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tff(sy_c_List_Odistinct_001t__Int__Oint,type,
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tff(sy_c_List_Odistinct_001t__Nat__Onat,type,
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tff(sy_c_List_Olinorder__class_Osorted__list__of__set_001t__Nat__Onat,type,
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tff(sy_c_List_Olist_OCons_001t__Int__Oint,type,
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tff(sy_c_List_Olist_OCons_001t__Nat__Onat,type,
    cons_nat: ( nat * list_nat ) > list_nat ).

tff(sy_c_List_Olist_ONil_001t__Int__Oint,type,
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tff(sy_c_List_Olist_ONil_001t__Nat__Onat,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_List_Olist_Oset_001_Eo,type,
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tff(sy_c_List_Olist_Oset_001t__Complex__Ocomplex,type,
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tff(sy_c_List_Olist_Oset_001t__Int__Oint,type,
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tff(sy_c_List_Olist_Oset_001t__Nat__Onat,type,
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tff(sy_c_List_Olist_Oset_001t__Real__Oreal,type,
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tff(sy_c_List_Olist_Oset_001t__Set__Oset_It__Nat__Onat_J,type,
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tff(sy_c_List_Olist_Oset_001t__VEBT____Definitions__OVEBT,type,
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tff(sy_c_List_Olist_Osize__list_001t__VEBT____Definitions__OVEBT,type,
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tff(sy_c_List_Olist_Otl_001t__Nat__Onat,type,
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tff(sy_c_List_Olist__update_001_Eo,type,
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tff(sy_c_List_Olist__update_001t__Complex__Ocomplex,type,
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tff(sy_c_List_Olist__update_001t__Int__Oint,type,
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tff(sy_c_List_Olist__update_001t__Nat__Onat,type,
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tff(sy_c_List_Olist__update_001t__Real__Oreal,type,
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tff(sy_c_List_Olist__update_001t__Set__Oset_It__Nat__Onat_J,type,
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tff(sy_c_List_Olist__update_001t__VEBT____Definitions__OVEBT,type,
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tff(sy_c_List_Onth_001_Eo,type,
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tff(sy_c_List_Onth_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_List_Onth_001t__Int__Oint,type,
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tff(sy_c_List_Onth_001t__Nat__Onat,type,
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tff(sy_c_List_Onth_001t__Num__Onum,type,
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tff(sy_c_List_Onth_001t__Set__Oset_It__Nat__Onat_J,type,
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tff(sy_c_List_Onth_001t__VEBT____Definitions__OVEBT,type,
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tff(sy_c_List_Oproduct_001_Eo_001_Eo,type,
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tff(sy_c_List_Oproduct_001_Eo_001t__Int__Oint,type,
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tff(sy_c_List_Oproduct_001_Eo_001t__Nat__Onat,type,
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tff(sy_c_List_Oproduct_001t__Code____Numeral__Ointeger_001_Eo,type,
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tff(sy_c_List_Oproduct_001t__Nat__Onat_001_Eo,type,
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tff(sy_c_List_Oproduct_001t__Num__Onum_001t__Num__Onum,type,
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tff(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001_Eo,type,
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tff(sy_c_List_Oremdups_001t__Nat__Onat,type,
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tff(sy_c_List_Oreplicate_001_Eo,type,
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tff(sy_c_List_Oreplicate_001t__Complex__Ocomplex,type,
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tff(sy_c_List_Oreplicate_001t__Int__Oint,type,
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tff(sy_c_List_Oreplicate_001t__Nat__Onat,type,
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tff(sy_c_List_Oreplicate_001t__Real__Oreal,type,
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tff(sy_c_List_Oreplicate_001t__Set__Oset_It__Nat__Onat_J,type,
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tff(sy_c_List_Oreplicate_001t__VEBT____Definitions__OVEBT,type,
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tff(sy_c_List_Osorted__wrt_001t__Int__Oint,type,
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tff(sy_c_List_Osorted__wrt_001t__Nat__Onat,type,
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tff(sy_c_List_Oupt,type,
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tff(sy_c_List_Oupto,type,
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tff(sy_c_List_Oupto__aux,type,
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tff(sy_c_List_Oupto__rel,type,
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tff(sy_c_Nat_OSuc,type,
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tff(sy_c_Nat_Ocompow_001t__fun_It__Nat__Onat_Mt__Nat__Onat_J,type,
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tff(sy_c_Nat_Onat_Ocase__nat_001_Eo,type,
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tff(sy_c_Nat_Onat_Ocase__nat_001t__Nat__Onat,type,
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tff(sy_c_Nat_Onat_Ocase__nat_001t__Option__Ooption_It__Num__Onum_J,type,
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tff(sy_c_Nat_Onat_Opred,type,
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tff(sy_c_Nat_Osemiring__1__class_ONats_001t__Int__Oint,type,
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tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Complex__Ocomplex,type,
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tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
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tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
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tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Rat__Orat,type,
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tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Real__Oreal,type,
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tff(sy_c_Nat__Bijection_Oset__encode,type,
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tff(sy_c_Nat__Bijection_Otriangle,type,
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tff(sy_c_NthRoot_Oroot,type,
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tff(sy_c_NthRoot_Osqrt,type,
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tff(sy_c_Num_OBitM,type,
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tff(sy_c_Num_Opow,type,
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tff(sy_c_Orderings_Oord__class_Omax_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Orderings_Oord__class_Omax_001t__Int__Oint,type,
    ord_max_int: fun_int_fun_int_int ).

tff(sy_c_Orderings_Oord__class_Omax_001t__Nat__Onat,type,
    ord_max_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Orderings_Oord__class_Omax_001t__Num__Onum,type,
    ord_max_num: num > fun_num_num ).

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

tff(sy_c_Orderings_Oord__class_Omax_001t__Real__Oreal,type,
    ord_max_real: real > fun_real_real ).

tff(sy_c_Orderings_Oord__class_Omax_001t__Set__Oset_It__Int__Oint_J,type,
    ord_max_set_int: ( set_int * set_int ) > set_int ).

tff(sy_c_Orderings_Oord__class_Omax_001t__Set__Oset_It__Nat__Onat_J,type,
    ord_max_set_nat: ( set_nat * set_nat ) > set_nat ).

tff(sy_c_Orderings_Oord__class_Omax_001t__Set__Oset_It__Real__Oreal_J,type,
    ord_max_set_real: ( set_real * set_real ) > set_real ).

tff(sy_c_Orderings_Oord__class_Omin_001t__Nat__Onat,type,
    ord_min_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Orderings_Oorder__class_OGreatest_001t__Nat__Onat,type,
    order_Greatest_nat: fun_nat_o > nat ).

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_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,
    top_top_set_int: set_int ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J,type,
    top_top_set_nat: set_nat ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Ounit_J,type,
    top_to1996260823553986621t_unit: set_Product_unit ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Real__Oreal_J,type,
    top_top_set_real: set_real ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__String__Ochar_J,type,
    top_top_set_char: set_char ).

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

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

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

tff(sy_c_Power_Opower__class_Opower_001t__Nat__Onat,type,
    power_power_nat: fun_nat_fun_nat_nat ).

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

tff(sy_c_Power_Opower__class_Opower_001t__Real__Oreal,type,
    power_power_real: fun_re4238438383274954274t_real ).

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

tff(sy_c_Product__Type_OPair_001_Eo_001t__Int__Oint,type,
    product_Pair_o_int: ( $o * int ) > product_prod_o_int ).

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

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

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

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

tff(sy_c_Product__Type_OPair_001t__Int__Oint_001t__Int__Oint,type,
    product_Pair_int_int: fun_in1696471567231710611nt_int ).

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

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

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

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

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

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

tff(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__Int__Oint,type,
    produc736041933913180425BT_int: ( vEBT_VEBT * int ) > produc4894624898956917775BT_int ).

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

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

tff(sy_c_Product__Type_OPair_001t__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    produc2818902965508115497at_nat: ( fun_nat_fun_nat_nat * produc7248412053542808358at_nat ) > produc9217583074051984497at_nat ).

tff(sy_c_Product__Type_OPair_001t__fun_It__Nat__Onat_Mt__fun_It__Num__Onum_Mt__Num__Onum_J_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J_J,type,
    produc460779904311030791at_num: ( fun_nat_fun_num_num * produc2963631642982155120at_num ) > produc8114805097830739791at_num ).

tff(sy_c_Product__Type_OSigma_001t__Nat__Onat_001t__Nat__Onat,type,
    produc457027306803732586at_nat: ( set_nat * fun_nat_set_nat ) > set_Pr1261947904930325089at_nat ).

tff(sy_c_Product__Type_Oapsnd_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
    produc6499014454317279255nteger: fun_fu4800041728187187127nteger ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001_Eo_001_Eo,type,
    produc7828578312038201481er_o_o: fun_Co480733329897759773un_o_o > fun_Pr3290071492984375169er_o_o ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001_Eo_001t__Set__Oset_It__Complex__Ocomplex_J,type,
    produc1043322548047392435omplex: ( fun_Co5015990272074005067omplex * produc6271795597528267376eger_o ) > set_complex ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001_Eo_001t__Set__Oset_It__Int__Oint_J,type,
    produc1253318751659547953et_int: ( fun_Co2587678820519358153et_int * produc6271795597528267376eger_o ) > set_int ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001_Eo_001t__Set__Oset_It__Nat__Onat_J,type,
    produc5431169771168744661et_nat: ( fun_Co7401227239307276909et_nat * produc6271795597528267376eger_o ) > set_nat ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001_Eo_001t__Set__Oset_It__Real__Oreal_J,type,
    produc242741666403216561t_real: ( fun_Co5368275265270938953t_real * produc6271795597528267376eger_o ) > set_real ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Int__Oint,type,
    produc1553301316500091796er_int: ( fun_Co1092983284983208416er_int * produc8923325533196201883nteger ) > int ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Nat__Onat,type,
    produc1555791787009142072er_nat: ( fun_Co3583066429914935172er_nat * produc8923325533196201883nteger ) > nat ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Num__Onum,type,
    produc7336495610019696514er_num: ( fun_Co8521658056209057742er_num * produc8923325533196201883nteger ) > num ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J,type,
    produc9125791028180074456eger_o: ( fun_Co3544761417380347684eger_o * produc8923325533196201883nteger ) > produc6271795597528267376eger_o ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    produc6916734918728496179nteger: ( fun_Co2470855966376689427nteger * produc8923325533196201883nteger ) > produc8923325533196201883nteger ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001_Eo,type,
    produc4947309494688390418_int_o: fun_int_fun_int_o > fun_Pr3238374608701176162_int_o ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001t__Int__Oint,type,
    produc8211389475949308722nt_int: fun_int_fun_int_int > fun_Pr982841511892225912nt_int ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    produc4245557441103728435nt_int: fun_in1696471567231710611nt_int > fun_Pr2478283820680522243nt_int ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001_Eo,type,
    produc6081775807080527818_nat_o: fun_nat_fun_nat_o > fun_Pr4796943543331071514_nat_o ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Complex__Ocomplex,type,
    produc1917071388513777916omplex: ( fun_na1142814060743658952omplex * product_prod_nat_nat ) > complex ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Int__Oint,type,
    produc6840382203811409530at_int: fun_nat_fun_nat_int > fun_Pr7647638413817135296at_int ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
    produc6842872674320459806at_nat: fun_nat_fun_nat_nat > fun_Pr2602117396471556196at_nat ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    produc2626176000494625587at_nat: fun_na4138050610385705875at_nat > fun_Pr2118491707810114563at_nat ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Rat__Orat,type,
    produc6207742614233964070at_rat: ( fun_nat_fun_nat_rat * product_prod_nat_nat ) > rat ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Real__Oreal,type,
    produc1703576794950452218t_real: fun_nat_fun_nat_real > fun_Pr672294278902479040t_real ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
    produc6395135316500328296_nat_o: fun_na5914473681949881652_nat_o > fun_Pr2028404973800837934_nat_o ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_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,type,
    produc4528391901023682385at_nat: fun_na1668100129766719005at_nat > fun_Pr2477239776880934807at_nat ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Num__Onum_001_Eo,type,
    produc4927758841916487424_num_o: fun_nat_fun_num_o > fun_Pr5555124106300582480_num_o ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Num__Onum_001t__Option__Ooption_It__Num__Onum_J,type,
    produc478579273971653890on_num: ( fun_na7595719129879163854on_num * product_prod_nat_num ) > option_num ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Num__Onum_001t__Set__Oset_It__Complex__Ocomplex_J,type,
    produc6231982587499038204omplex: ( fun_na3925014923085203400omplex * product_prod_nat_num ) > set_complex ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Num__Onum_001t__Set__Oset_It__Real__Oreal_J,type,
    produc1435849484188172666t_real: ( fun_na3666575854997362246t_real * product_prod_nat_num ) > set_real ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Num__Onum_001t__Num__Onum_001_Eo,type,
    produc5703948589228662326_num_o: fun_num_fun_num_o > fun_Pr6331313853612757382_num_o ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Num__Onum_001t__Num__Onum_001t__Set__Oset_It__Complex__Ocomplex_J,type,
    produc2866383454006189126omplex: ( fun_nu9041852840018610706omplex * product_prod_num_num ) > set_complex ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Num__Onum_001t__Num__Onum_001t__Set__Oset_It__Int__Oint_J,type,
    produc6406642877701697732et_int: ( fun_nu4940571614878387344et_int * product_prod_num_num ) > set_int ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Num__Onum_001t__Num__Onum_001t__Set__Oset_It__Nat__Onat_J,type,
    produc1361121860356118632et_nat: ( fun_nu530747996811530292et_nat * product_prod_num_num ) > set_nat ).

tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Num__Onum_001t__Num__Onum_001t__Set__Oset_It__Real__Oreal_J,type,
    produc8296048397933160132t_real: ( fun_nu4698968662094489488t_real * product_prod_num_num ) > set_real ).

tff(sy_c_Product__Type_Oprod_Ofst_001t__Int__Oint_001t__Int__Oint,type,
    product_fst_int_int: fun_Pr982841511892225912nt_int ).

tff(sy_c_Product__Type_Oprod_Ofst_001t__Nat__Onat_001t__Nat__Onat,type,
    product_fst_nat_nat: fun_Pr2602117396471556196at_nat ).

tff(sy_c_Product__Type_Oprod_Osnd_001t__Int__Oint_001t__Int__Oint,type,
    product_snd_int_int: fun_Pr982841511892225912nt_int ).

tff(sy_c_Product__Type_Oprod_Osnd_001t__Nat__Onat_001t__Nat__Onat,type,
    product_snd_nat_nat: fun_Pr2602117396471556196at_nat ).

tff(sy_c_Rat_OFract,type,
    fract: int > fun_int_rat ).

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

tff(sy_c_Rat_Ofield__char__0__class_ORats_001t__Real__Oreal,type,
    field_5140801741446780682s_real: set_real ).

tff(sy_c_Rat_Onormalize,type,
    normalize: fun_Pr2478283820680522243nt_int ).

tff(sy_c_Rat_Oof__int,type,
    of_int: fun_int_rat ).

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

tff(sy_c_Real__Vector__Spaces_OReals_001t__Complex__Ocomplex,type,
    real_V2521375963428798218omplex: set_complex ).

tff(sy_c_Real__Vector__Spaces_Obounded__linear_001t__Real__Oreal_001t__Real__Oreal,type,
    real_V5970128139526366754l_real: fun_real_real > $o ).

tff(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Complex__Ocomplex,type,
    real_V1022390504157884413omplex: fun_complex_real ).

tff(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Real__Oreal,type,
    real_V7735802525324610683m_real: fun_real_real ).

tff(sy_c_Real__Vector__Spaces_Oof__real_001t__Complex__Ocomplex,type,
    real_V4546457046886955230omplex: fun_real_complex ).

tff(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR_001t__Complex__Ocomplex,type,
    real_V2046097035970521341omplex: real > fun_complex_complex ).

tff(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR_001t__Real__Oreal,type,
    real_V1485227260804924795R_real: real > fun_real_real ).

tff(sy_c_Rings_Odivide__class_Odivide_001t__Code____Numeral__Ointeger,type,
    divide6298287555418463151nteger: ( code_integer * code_integer ) > code_integer ).

tff(sy_c_Rings_Odivide__class_Odivide_001t__Complex__Ocomplex,type,
    divide1717551699836669952omplex: complex > fun_complex_complex ).

tff(sy_c_Rings_Odivide__class_Odivide_001t__Int__Oint,type,
    divide_divide_int: fun_int_fun_int_int ).

tff(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
    divide_divide_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Rings_Odivide__class_Odivide_001t__Rat__Orat,type,
    divide_divide_rat: rat > fun_rat_rat ).

tff(sy_c_Rings_Odivide__class_Odivide_001t__Real__Oreal,type,
    divide_divide_real: real > fun_real_real ).

tff(sy_c_Rings_Odvd__class_Odvd_001t__Code____Numeral__Ointeger,type,
    dvd_dvd_Code_integer: fun_Co2081214437782444048eger_o ).

tff(sy_c_Rings_Odvd__class_Odvd_001t__Complex__Ocomplex,type,
    dvd_dvd_complex: complex > fun_complex_o ).

tff(sy_c_Rings_Odvd__class_Odvd_001t__Int__Oint,type,
    dvd_dvd_int: fun_int_fun_int_o ).

tff(sy_c_Rings_Odvd__class_Odvd_001t__Nat__Onat,type,
    dvd_dvd_nat: fun_nat_fun_nat_o ).

tff(sy_c_Rings_Odvd__class_Odvd_001t__Rat__Orat,type,
    dvd_dvd_rat: fun_rat_fun_rat_o ).

tff(sy_c_Rings_Odvd__class_Odvd_001t__Real__Oreal,type,
    dvd_dvd_real: fun_real_fun_real_o ).

tff(sy_c_Rings_Omodulo__class_Omodulo_001t__Code____Numeral__Ointeger,type,
    modulo364778990260209775nteger: ( code_integer * code_integer ) > code_integer ).

tff(sy_c_Rings_Omodulo__class_Omodulo_001t__Int__Oint,type,
    modulo_modulo_int: fun_int_fun_int_int ).

tff(sy_c_Rings_Omodulo__class_Omodulo_001t__Nat__Onat,type,
    modulo_modulo_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Code____Numeral__Ointeger,type,
    zero_n356916108424825756nteger: $o > code_integer ).

tff(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Complex__Ocomplex,type,
    zero_n1201886186963655149omplex: $o > complex ).

tff(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Int__Oint,type,
    zero_n2684676970156552555ol_int: fun_o_int ).

tff(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Nat__Onat,type,
    zero_n2687167440665602831ol_nat: $o > nat ).

tff(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Rat__Orat,type,
    zero_n2052037380579107095ol_rat: $o > rat ).

tff(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Real__Oreal,type,
    zero_n3304061248610475627l_real: $o > real ).

tff(sy_c_Series_Osuminf_001t__Complex__Ocomplex,type,
    suminf_complex: fun_nat_complex > complex ).

tff(sy_c_Series_Osuminf_001t__Int__Oint,type,
    suminf_int: fun_nat_int > int ).

tff(sy_c_Series_Osuminf_001t__Nat__Onat,type,
    suminf_nat: fun_nat_nat > nat ).

tff(sy_c_Series_Osuminf_001t__Real__Oreal,type,
    suminf_real: fun_nat_real > real ).

tff(sy_c_Series_Osummable_001t__Complex__Ocomplex,type,
    summable_complex: fun_nat_complex > $o ).

tff(sy_c_Series_Osummable_001t__Int__Oint,type,
    summable_int: fun_nat_int > $o ).

tff(sy_c_Series_Osummable_001t__Nat__Onat,type,
    summable_nat: fun_nat_nat > $o ).

tff(sy_c_Series_Osummable_001t__Real__Oreal,type,
    summable_real: fun_nat_real > $o ).

tff(sy_c_Series_Osums_001t__Complex__Ocomplex,type,
    sums_complex: fun_nat_complex > fun_complex_o ).

tff(sy_c_Series_Osums_001t__Int__Oint,type,
    sums_int: fun_nat_int > fun_int_o ).

tff(sy_c_Series_Osums_001t__Nat__Onat,type,
    sums_nat: fun_nat_nat > fun_nat_o ).

tff(sy_c_Series_Osums_001t__Real__Oreal,type,
    sums_real: fun_nat_real > fun_real_o ).

tff(sy_c_Set_OCollect_001t__Code____Numeral__Ointeger,type,
    collect_Code_integer: fun_Code_integer_o > set_Code_integer ).

tff(sy_c_Set_OCollect_001t__Complex__Ocomplex,type,
    collect_complex: fun_complex_o > set_complex ).

tff(sy_c_Set_OCollect_001t__Int__Oint,type,
    collect_int: fun_int_o > set_int ).

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,
    collect_nat: fun_nat_o > set_nat ).

tff(sy_c_Set_OCollect_001t__Num__Onum,type,
    collect_num: fun_num_o > set_num ).

tff(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    collec213857154873943460nt_int: fun_Pr3238374608701176162_int_o > set_Pr958786334691620121nt_int ).

tff(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    collec3392354462482085612at_nat: fun_Pr4796943543331071514_nat_o > set_Pr1261947904930325089at_nat ).

tff(sy_c_Set_OCollect_001t__Rat__Orat,type,
    collect_rat: fun_rat_o > set_rat ).

tff(sy_c_Set_OCollect_001t__Real__Oreal,type,
    collect_real: fun_real_o > set_real ).

tff(sy_c_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
    collect_set_nat: fun_set_nat_o > set_set_nat ).

tff(sy_c_Set_OPow_001t__Nat__Onat,type,
    pow_nat: set_nat > set_set_nat ).

tff(sy_c_Set_Oimage_001t__Int__Oint_001t__Int__Oint,type,
    image_int_int: ( fun_int_int * set_int ) > set_int ).

tff(sy_c_Set_Oimage_001t__Nat__Onat_001t__Int__Oint,type,
    image_nat_int: ( fun_nat_int * set_nat ) > set_int ).

tff(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
    image_nat_nat: ( fun_nat_nat * set_nat ) > set_nat ).

tff(sy_c_Set_Oimage_001t__Nat__Onat_001t__String__Ochar,type,
    image_nat_char: ( fun_nat_char * set_nat ) > set_char ).

tff(sy_c_Set_Oimage_001t__Real__Oreal_001t__Real__Oreal,type,
    image_real_real: ( fun_real_real * set_real ) > set_real ).

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__Complex__Ocomplex,type,
    insert_complex: ( complex * set_complex ) > set_complex ).

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_Oinsert_001t__Real__Oreal,type,
    insert_real: ( real * set_real ) > set_real ).

tff(sy_c_Set_Oinsert_001t__Set__Oset_It__Nat__Onat_J,type,
    insert_set_nat: ( set_nat * set_set_nat ) > set_set_nat ).

tff(sy_c_Set_Oinsert_001t__VEBT____Definitions__OVEBT,type,
    insert_VEBT_VEBT: ( vEBT_VEBT * set_VEBT_VEBT ) > set_VEBT_VEBT ).

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 ) > fun_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__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: real > fun_real_real ).

tff(sy_c_Transcendental_Opi,type,
    pi: real ).

tff(sy_c_Transcendental_Opowr_001t__Real__Oreal,type,
    powr_real: real > fun_real_real ).

tff(sy_c_Transcendental_Osin_001t__Complex__Ocomplex,type,
    sin_complex: fun_complex_complex ).

tff(sy_c_Transcendental_Osin_001t__Real__Oreal,type,
    sin_real: fun_real_real ).

tff(sy_c_Transcendental_Osin__coeff,type,
    sin_coeff: fun_nat_real ).

tff(sy_c_Transcendental_Osinh_001t__Real__Oreal,type,
    sinh_real: fun_real_real ).

tff(sy_c_Transcendental_Otan_001t__Complex__Ocomplex,type,
    tan_complex: fun_complex_complex ).

tff(sy_c_Transcendental_Otan_001t__Real__Oreal,type,
    tan_real: fun_real_real ).

tff(sy_c_Transcendental_Otanh_001t__Complex__Ocomplex,type,
    tanh_complex: fun_complex_complex ).

tff(sy_c_Transcendental_Otanh_001t__Real__Oreal,type,
    tanh_real: fun_real_real ).

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

tff(sy_c_VEBT__Insert_Ovebt__insert__rel,type,
    vEBT_vebt_insert_rel: fun_Pr8877776288780692068_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_OminNull,type,
    vEBT_VEBT_minNull: fun_VEBT_VEBT_o ).

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 > fun_Pr3238374608701176162_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 > fun_Pr4796943543331071514_nat_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_aa_001_Eo_001_Eo,type,
    aa_o_o: ( fun_o_o * $o ) > $o ).

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__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    aa_o_P3134438092330336605nt_int: ( fun_o_8988054265482332260nt_int * $o ) > product_prod_int_int ).

tff(sy_c_aa_001_Eo_001t__Set__Oset_It__Complex__Ocomplex_J,type,
    aa_o_set_complex: ( fun_o_set_complex * $o ) > set_complex ).

tff(sy_c_aa_001_Eo_001t__Set__Oset_It__Int__Oint_J,type,
    aa_o_set_int: ( fun_o_set_int * $o ) > set_int ).

tff(sy_c_aa_001_Eo_001t__Set__Oset_It__Nat__Onat_J,type,
    aa_o_set_nat: ( fun_o_set_nat * $o ) > set_nat ).

tff(sy_c_aa_001_Eo_001t__Set__Oset_It__Real__Oreal_J,type,
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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_m____,type,
    m: nat ).

tff(sy_v_ma____,type,
    ma: nat ).

tff(sy_v_mi____,type,
    mi: nat ).

tff(sy_v_na____,type,
    na: nat ).

tff(sy_v_summary____,type,
    summary: vEBT_VEBT ).

tff(sy_v_treeList____,type,
    treeList: list_VEBT_VEBT ).

tff(sy_v_xa____,type,
    xa: nat ).

tff(sy_v_ya____,type,
    ya: nat ).

% Relevant facts (10741)
tff(fact_0__C5_Ohyps_C_I3_J,axiom,
    m = aa_nat_nat(suc,na) ).

% "5.hyps"(3)
tff(fact_1_both__member__options__def,axiom,
    ! [Ta: vEBT_VEBT,Xb: nat] :
      ( aa_nat_o(vEBT_V8194947554948674370ptions(Ta),Xb)
    <=> ( aa_nat_o(vEBT_V5719532721284313246member(Ta),Xb)
        | aa_nat_o(vEBT_VEBT_membermima(Ta),Xb) ) ) ).

% both_member_options_def
tff(fact_2_valid__0__not,axiom,
    ! [Ta: vEBT_VEBT] : ~ aa_nat_o(vEBT_invar_vebt(Ta),zero_zero_nat) ).

% valid_0_not
tff(fact_3_valid__tree__deg__neq__0,axiom,
    ! [Ta: vEBT_VEBT] : ~ aa_nat_o(vEBT_invar_vebt(Ta),zero_zero_nat) ).

% valid_tree_deg_neq_0
tff(fact_4_even__odd__cases,axiom,
    ! [Xb: nat] :
      ( ! [N: nat] : Xb != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),N)
     => ~ ! [N: nat] : Xb != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_nat_nat(suc,N)) ) ).

% even_odd_cases
tff(fact_5__C5_Ohyps_C_I1_J,axiom,
    aa_nat_o(vEBT_invar_vebt(summary),m) ).

% "5.hyps"(1)
tff(fact_6_deg__deg__n,axiom,
    ! [Info: option4927543243414619207at_nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(vEBT_Node(Info,Dega,TreeLista,Summarya)),Nb)
     => ( Dega = Nb ) ) ).

% deg_deg_n
tff(fact_7__C5_Ohyps_C_I7_J,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,mi),ma) ).

% "5.hyps"(7)
tff(fact_8__C5_Ohyps_C_I4_J,axiom,
    deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,na),m) ).

% "5.hyps"(4)
tff(fact_9_False,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,mi),na) != aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,ya),na) ).

% False
tff(fact_10__C5_Ohyps_C_I6_J,axiom,
    ( ( mi = ma )
   => ! [X: vEBT_VEBT] :
        ( member_VEBT_VEBT(X,set_VEBT_VEBT2(treeList))
       => ~ ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X),X_1) ) ) ).

% "5.hyps"(6)
tff(fact_11_both__member__options__equiv__member,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( aa_nat_o(vEBT_V8194947554948674370ptions(Ta),Xb)
      <=> aa_nat_o(vEBT_vebt_member(Ta),Xb) ) ) ).

% both_member_options_equiv_member
tff(fact_12_valid__member__both__member__options,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( aa_nat_o(vEBT_V8194947554948674370ptions(Ta),Xb)
       => aa_nat_o(vEBT_vebt_member(Ta),Xb) ) ) ).

% valid_member_both_member_options
tff(fact_13__C000_C,axiom,
    aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,ya),na))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,ya),na)) ).

% "000"
tff(fact_14_deg__SUcn__Node,axiom,
    ! [Tree: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Tree),aa_nat_nat(suc,aa_nat_nat(suc,Nb)))
     => ? [Info2: option4927543243414619207at_nat,TreeList: list_VEBT_VEBT,S: vEBT_VEBT] : Tree = vEBT_Node(Info2,aa_nat_nat(suc,aa_nat_nat(suc,Nb)),TreeList,S) ) ).

% deg_SUcn_Node
tff(fact_15__092_060open_062x_A_060_Ami_092_060close_062,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,xa),mi) ).

% \<open>x < mi\<close>
tff(fact_16__C001_C,axiom,
    ( aa_nat_o(vEBT_invar_vebt(nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,ya),na))),na)
    & member_VEBT_VEBT(nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,ya),na)),set_VEBT_VEBT2(treeList)) ) ).

% "001"
tff(fact_17_mi__eq__ma__no__ch,axiom,
    ! [Mia: nat,Maa: nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT] :
      ( aa_nat_o(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya)),Dega)
     => ( ( Mia = Maa )
       => ( ! [X: vEBT_VEBT] :
              ( member_VEBT_VEBT(X,set_VEBT_VEBT2(TreeLista))
             => ~ ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X),X_1) )
          & ~ ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summarya),X_1) ) ) ) ).

% mi_eq_ma_no_ch
tff(fact_18__C5_Oprems_C_I3_J,axiom,
    aa_nat_o(vEBT_vebt_member(vEBT_vebt_insert(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(mi),ma)),deg,treeList,summary),xa)),ya) ).

% "5.prems"(3)
tff(fact_19__C5_Ohyps_C_I8_J,axiom,
    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))),deg)) ).

% "5.hyps"(8)
tff(fact_20__C5_Oprems_C_I2_J,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,ya),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),deg)) ).

% "5.prems"(2)
tff(fact_21__C004_C,axiom,
    ( aa_nat_o(vEBT_V5719532721284313246member(nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,ya),na))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,ya),na))
   => aa_nat_o(vEBT_V5719532721284313246member(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(mi),ma)),deg,treeList,summary)),ya) ) ).

% "004"
tff(fact_22_not__min__Null__member,axiom,
    ! [Ta: vEBT_VEBT] :
      ( ~ aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Ta)
     => ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Ta),X_12) ) ).

% not_min_Null_member
tff(fact_23__C5_Ohyps_C_I5_J,axiom,
    ! [I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m))
     => ( ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(treeList,I)),X_13)
      <=> aa_nat_o(vEBT_V8194947554948674370ptions(summary),I) ) ) ).

% "5.hyps"(5)
tff(fact_24__C5_Oprems_C_I1_J,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),deg)) ).

% "5.prems"(1)
tff(fact_25_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_26_option_Oinject,axiom,
    ! [X2: product_prod_nat_nat,Y2: product_prod_nat_nat] :
      ( ( some_P7363390416028606310at_nat(X2) = some_P7363390416028606310at_nat(Y2) )
    <=> ( X2 = Y2 ) ) ).

% option.inject
tff(fact_27_option_Oinject,axiom,
    ! [X2: num,Y2: num] :
      ( ( some_num(X2) = some_num(Y2) )
    <=> ( X2 = Y2 ) ) ).

% option.inject
tff(fact_28_prod_Oinject,axiom,
    ! [X1: code_integer,X2: $o,Y1: code_integer,Y2: $o] :
      ( ( produc6677183202524767010eger_o(X1,(X2)) = produc6677183202524767010eger_o(Y1,(Y2)) )
    <=> ( ( X1 = Y1 )
        & ( (X2)
        <=> (Y2) ) ) ) ).

% prod.inject
tff(fact_29_prod_Oinject,axiom,
    ! [X1: num,X2: num,Y1: num,Y2: num] :
      ( ( product_Pair_num_num(X1,X2) = product_Pair_num_num(Y1,Y2) )
    <=> ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
tff(fact_30_prod_Oinject,axiom,
    ! [X1: nat,X2: num,Y1: nat,Y2: num] :
      ( ( product_Pair_nat_num(X1,X2) = product_Pair_nat_num(Y1,Y2) )
    <=> ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
tff(fact_31_prod_Oinject,axiom,
    ! [X1: nat,X2: nat,Y1: nat,Y2: nat] :
      ( ( aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X1),X2) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Y1),Y2) )
    <=> ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
tff(fact_32_prod_Oinject,axiom,
    ! [X1: int,X2: int,Y1: int,Y2: int] :
      ( ( aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X1),X2) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Y1),Y2) )
    <=> ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
tff(fact_33_old_Oprod_Oinject,axiom,
    ! [A: code_integer,B: $o,A2: code_integer,B2: $o] :
      ( ( produc6677183202524767010eger_o(A,(B)) = produc6677183202524767010eger_o(A2,(B2)) )
    <=> ( ( A = A2 )
        & ( (B)
        <=> (B2) ) ) ) ).

% old.prod.inject
tff(fact_34_old_Oprod_Oinject,axiom,
    ! [A: num,B: num,A2: num,B2: num] :
      ( ( product_Pair_num_num(A,B) = product_Pair_num_num(A2,B2) )
    <=> ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
tff(fact_35_old_Oprod_Oinject,axiom,
    ! [A: nat,B: num,A2: nat,B2: num] :
      ( ( product_Pair_nat_num(A,B) = product_Pair_nat_num(A2,B2) )
    <=> ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
tff(fact_36_old_Oprod_Oinject,axiom,
    ! [A: nat,B: nat,A2: nat,B2: nat] :
      ( ( aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B2) )
    <=> ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
tff(fact_37_old_Oprod_Oinject,axiom,
    ! [A: int,B: int,A2: int,B2: int] :
      ( ( aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A2),B2) )
    <=> ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
tff(fact_38__C5_Ohyps_C_I2_J,axiom,
    size_s6755466524823107622T_VEBT(treeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m) ).

% "5.hyps"(2)
tff(fact_39_member__valid__both__member__options,axiom,
    ! [Tree: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Tree),Nb)
     => ( aa_nat_o(vEBT_vebt_member(Tree),Xb)
       => ( aa_nat_o(vEBT_V5719532721284313246member(Tree),Xb)
          | aa_nat_o(vEBT_VEBT_membermima(Tree),Xb) ) ) ) ).

% member_valid_both_member_options
tff(fact_40_min__Null__member,axiom,
    ! [Ta: vEBT_VEBT,Xb: nat] :
      ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Ta)
     => ~ aa_nat_o(vEBT_vebt_member(Ta),Xb) ) ).

% min_Null_member
tff(fact_41_deg__not__0,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).

% deg_not_0
tff(fact_42_inthall,axiom,
    ! [Xs: list_complex,P: fun_complex_o,Nb: nat] :
      ( ! [X3: complex] :
          ( member_complex(X3,set_complex2(Xs))
         => aa_complex_o(P,X3) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s3451745648224563538omplex(Xs))
       => aa_complex_o(P,aa_nat_complex(nth_complex(Xs),Nb)) ) ) ).

% inthall
tff(fact_43_inthall,axiom,
    ! [Xs: list_real,P: fun_real_o,Nb: nat] :
      ( ! [X3: real] :
          ( member_real(X3,set_real2(Xs))
         => aa_real_o(P,X3) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_real(Xs))
       => aa_real_o(P,aa_nat_real(nth_real(Xs),Nb)) ) ) ).

% inthall
tff(fact_44_inthall,axiom,
    ! [Xs: list_set_nat,P: fun_set_nat_o,Nb: nat] :
      ( ! [X3: set_nat] :
          ( member_set_nat(X3,set_set_nat2(Xs))
         => aa_set_nat_o(P,X3) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s3254054031482475050et_nat(Xs))
       => aa_set_nat_o(P,aa_nat_set_nat(nth_set_nat(Xs),Nb)) ) ) ).

% inthall
tff(fact_45_inthall,axiom,
    ! [Xs: list_VEBT_VEBT,P: fun_VEBT_VEBT_o,Nb: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(Xs))
         => aa_VEBT_VEBT_o(P,X3) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s6755466524823107622T_VEBT(Xs))
       => aa_VEBT_VEBT_o(P,nth_VEBT_VEBT(Xs,Nb)) ) ) ).

% inthall
tff(fact_46_inthall,axiom,
    ! [Xs: list_o,P: fun_o_o,Nb: nat] :
      ( ! [X3: $o] :
          ( member_o((X3),set_o2(Xs))
         => aa_o_o(P,(X3)) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_o(Xs))
       => aa_o_o(P,aa_nat_o(nth_o(Xs),Nb)) ) ) ).

% inthall
tff(fact_47_inthall,axiom,
    ! [Xs: list_nat,P: fun_nat_o,Nb: nat] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),set_nat2(Xs))
         => aa_nat_o(P,X3) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_nat(Xs))
       => aa_nat_o(P,aa_nat_nat(nth_nat(Xs),Nb)) ) ) ).

% inthall
tff(fact_48_inthall,axiom,
    ! [Xs: list_int,P: fun_int_o,Nb: nat] :
      ( ! [X3: int] :
          ( member_int(X3,set_int2(Xs))
         => aa_int_o(P,X3) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_int(Xs))
       => aa_int_o(P,aa_nat_int(nth_int(Xs),Nb)) ) ) ).

% inthall
tff(fact_49_bit__split__inv,axiom,
    ! [Xb: 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,Xb),D)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),D)),D) = Xb ).

% bit_split_inv
tff(fact_50_xyprop,axiom,
    ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),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_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,ya),na)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m)) ) ).

% xyprop
tff(fact_51_member__correct,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( aa_nat_o(vEBT_vebt_member(Ta),Xb)
      <=> aa_set_nat_o(member_nat(Xb),vEBT_set_vebt(Ta)) ) ) ).

% member_correct
tff(fact_52__C5_OIH_C_I1_J,axiom,
    ! [X: vEBT_VEBT] :
      ( member_VEBT_VEBT(X,set_VEBT_VEBT2(treeList))
     => ( aa_nat_o(vEBT_invar_vebt(X),na)
        & ! [Xa: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))
           => ! [Xb2: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb2),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(vEBT_vebt_member(vEBT_vebt_insert(X,Xa)),Xb2)
                 => ( aa_nat_o(vEBT_vebt_member(X),Xb2)
                    | ( Xa = Xb2 ) ) ) ) ) ) ) ).

% "5.IH"(1)
tff(fact_53__C00_C,axiom,
    ( ( deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,na),m) )
    & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),na)
    & ( aa_nat_nat(suc,na) = m )
    & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),deg)
    & ( 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) )
    & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),na) ) ).

% "00"
tff(fact_54_high__bound__aux,axiom,
    ! [Maa: nat,Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Maa),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,Nb),Mb)))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Maa),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)) ) ).

% high_bound_aux
tff(fact_55_member__bound,axiom,
    ! [Tree: vEBT_VEBT,Xb: nat,Nb: nat] :
      ( aa_nat_o(vEBT_vebt_member(Tree),Xb)
     => ( aa_nat_o(vEBT_invar_vebt(Tree),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ) ) ).

% member_bound
tff(fact_56_insert__simp__mima,axiom,
    ! [Xb: nat,Mia: nat,Maa: nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT] :
      ( ( ( Xb = Mia )
        | ( Xb = Maa ) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)
       => ( vEBT_vebt_insert(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya),Xb) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya) ) ) ) ).

% insert_simp_mima
tff(fact_57_valid__insert__both__member__options__pres,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat,Yb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Yb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))
         => ( aa_nat_o(vEBT_V8194947554948674370ptions(Ta),Xb)
           => aa_nat_o(vEBT_V8194947554948674370ptions(vEBT_vebt_insert(Ta,Yb)),Xb) ) ) ) ) ).

% valid_insert_both_member_options_pres
tff(fact_58_valid__insert__both__member__options__add,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))
       => aa_nat_o(vEBT_V8194947554948674370ptions(vEBT_vebt_insert(Ta,Xb)),Xb) ) ) ).

% valid_insert_both_member_options_add
tff(fact_59__C5_Ohyps_C_I9_J,axiom,
    ( ( mi != ma )
   => ! [I: nat] :
        ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),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) = I )
           => aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(treeList,I)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,ma),na)) )
          & ! [X: nat] :
              ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),na) = I )
                & aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(treeList,I)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),na)) )
             => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,mi),X)
                & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),ma) ) ) ) ) ) ).

% "5.hyps"(9)
tff(fact_60_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_61_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_62_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_63_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_64_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_65_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_66_Collect__mem__eq,axiom,
    ! [A3: set_complex] : collect_complex(aTP_Lamm_a(A3)) = A3 ).

% Collect_mem_eq
tff(fact_67_Collect__mem__eq,axiom,
    ! [A3: set_real] : collect_real(aTP_Lamm_aa(A3)) = A3 ).

% Collect_mem_eq
tff(fact_68_Collect__mem__eq,axiom,
    ! [A3: set_list_nat] : collect_list_nat(aTP_Lamm_ab(A3)) = A3 ).

% Collect_mem_eq
tff(fact_69_Collect__mem__eq,axiom,
    ! [A3: set_set_nat] : collect_set_nat(aTP_Lamm_ac(A3)) = A3 ).

% Collect_mem_eq
tff(fact_70_Collect__mem__eq,axiom,
    ! [A3: set_nat] : collect_nat(aTP_Lamm_ad(A3)) = A3 ).

% Collect_mem_eq
tff(fact_71_Collect__mem__eq,axiom,
    ! [A3: set_int] : collect_int(aTP_Lamm_ae(A3)) = A3 ).

% Collect_mem_eq
tff(fact_72_Collect__cong,axiom,
    ! [P: fun_real_o,Q: fun_real_o] :
      ( ! [X3: real] :
          ( aa_real_o(P,X3)
        <=> aa_real_o(Q,X3) )
     => ( collect_real(P) = collect_real(Q) ) ) ).

% Collect_cong
tff(fact_73_Collect__cong,axiom,
    ! [P: fun_list_nat_o,Q: fun_list_nat_o] :
      ( ! [X3: list_nat] :
          ( aa_list_nat_o(P,X3)
        <=> aa_list_nat_o(Q,X3) )
     => ( collect_list_nat(P) = collect_list_nat(Q) ) ) ).

% Collect_cong
tff(fact_74_Collect__cong,axiom,
    ! [P: fun_set_nat_o,Q: fun_set_nat_o] :
      ( ! [X3: set_nat] :
          ( aa_set_nat_o(P,X3)
        <=> aa_set_nat_o(Q,X3) )
     => ( collect_set_nat(P) = collect_set_nat(Q) ) ) ).

% Collect_cong
tff(fact_75_Collect__cong,axiom,
    ! [P: fun_nat_o,Q: fun_nat_o] :
      ( ! [X3: nat] :
          ( aa_nat_o(P,X3)
        <=> aa_nat_o(Q,X3) )
     => ( collect_nat(P) = collect_nat(Q) ) ) ).

% Collect_cong
tff(fact_76_Collect__cong,axiom,
    ! [P: fun_int_o,Q: fun_int_o] :
      ( ! [X3: int] :
          ( aa_int_o(P,X3)
        <=> aa_int_o(Q,X3) )
     => ( collect_int(P) = collect_int(Q) ) ) ).

% Collect_cong
tff(fact_77_mi__ma__2__deg,axiom,
    ! [Mia: nat,Maa: nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya)),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mia),Maa)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Maa),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_78__C5_OIH_C_I2_J,axiom,
    ! [Xb: nat,Yb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),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_nat,Yb),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(vEBT_vebt_member(vEBT_vebt_insert(summary,Xb)),Yb)
         => ( aa_nat_o(vEBT_vebt_member(summary),Yb)
            | ( Xb = Yb ) ) ) ) ) ).

% "5.IH"(2)
tff(fact_79__092_060open_062low_Ax_An_A_060_A2_A_094_An_A_092_060and_062_Alow_Ay_An_A_060_A2_A_094_An_092_060close_062,axiom,
    ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xa),na)),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,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,ya),na)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na)) ) ).

% \<open>low x n < 2 ^ n \<and> low y n < 2 ^ n\<close>
tff(fact_80_set__n__deg__not__0,axiom,
    ! [TreeLista: list_VEBT_VEBT,Nb: nat,Mb: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeLista))
         => aa_nat_o(vEBT_invar_vebt(X3),Nb) )
     => ( ( 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))),Mb) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Nb) ) ) ).

% set_n_deg_not_0
tff(fact_81_mimaxyprop,axiom,
    ( ~ ( ( xa = mi )
        | ( xa = ma ) )
    & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),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_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,mi),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_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xa),na)),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,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,mi),na)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na))
    & ( 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) ) ) ).

% mimaxyprop
tff(fact_82_VEBT__internal_Oexp__split__high__low_I2_J,axiom,
    ! [Xb: nat,Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),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,Nb),Mb)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ) ) ) ).

% VEBT_internal.exp_split_high_low(2)
tff(fact_83_VEBT__internal_Oexp__split__high__low_I1_J,axiom,
    ! [Xb: nat,Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),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,Nb),Mb)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)) ) ) ) ).

% VEBT_internal.exp_split_high_low(1)
tff(fact_84_vebt__buildup_Ocases,axiom,
    ! [Xb: nat] :
      ( ( Xb != zero_zero_nat )
     => ( ( Xb != aa_nat_nat(suc,zero_zero_nat) )
       => ~ ! [Va: nat] : Xb != aa_nat_nat(suc,aa_nat_nat(suc,Va)) ) ) ).

% vebt_buildup.cases
tff(fact_85_vebt__insert_Osimps_I3_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S2: vEBT_VEBT,Xb: nat] : vEBT_vebt_insert(vEBT_Node(Info,aa_nat_nat(suc,zero_zero_nat),Ts,S2),Xb) = vEBT_Node(Info,aa_nat_nat(suc,zero_zero_nat),Ts,S2) ).

% vebt_insert.simps(3)
tff(fact_86_invar__vebt_Ointros_I5_J,axiom,
    ! [TreeLista: list_VEBT_VEBT,Nb: nat,Summarya: vEBT_VEBT,Mb: nat,Dega: nat,Mia: nat,Maa: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeLista))
         => aa_nat_o(vEBT_invar_vebt(X3),Nb) )
     => ( aa_nat_o(vEBT_invar_vebt(Summarya),Mb)
       => ( ( 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))),Mb) )
         => ( ( Mb = aa_nat_nat(suc,Nb) )
           => ( ( Dega = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb) )
             => ( ! [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))),Mb))
                   => ( ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeLista,I2)),X_13)
                    <=> aa_nat_o(vEBT_V8194947554948674370ptions(Summarya),I2) ) )
               => ( ( ( Mia = Maa )
                   => ! [X3: vEBT_VEBT] :
                        ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeLista))
                       => ~ ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X3),X_12) ) )
                 => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mia),Maa)
                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Maa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega))
                     => ( ( ( Mia != Maa )
                         => ! [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))),Mb))
                             => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Maa),Nb) = I2 )
                                 => aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeLista,I2)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Maa),Nb)) )
                                & ! [X3: nat] :
                                    ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),Nb) = I2 )
                                      & aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeLista,I2)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),Nb)) )
                                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mia),X3)
                                      & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Maa) ) ) ) ) )
                       => aa_nat_o(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya)),Dega) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(5)
tff(fact_87_invar__vebt_Ointros_I4_J,axiom,
    ! [TreeLista: list_VEBT_VEBT,Nb: nat,Summarya: vEBT_VEBT,Mb: nat,Dega: nat,Mia: nat,Maa: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeLista))
         => aa_nat_o(vEBT_invar_vebt(X3),Nb) )
     => ( aa_nat_o(vEBT_invar_vebt(Summarya),Mb)
       => ( ( 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))),Mb) )
         => ( ( Mb = Nb )
           => ( ( Dega = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb) )
             => ( ! [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))),Mb))
                   => ( ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeLista,I2)),X_13)
                    <=> aa_nat_o(vEBT_V8194947554948674370ptions(Summarya),I2) ) )
               => ( ( ( Mia = Maa )
                   => ! [X3: vEBT_VEBT] :
                        ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeLista))
                       => ~ ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X3),X_12) ) )
                 => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mia),Maa)
                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Maa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega))
                     => ( ( ( Mia != Maa )
                         => ! [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))),Mb))
                             => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Maa),Nb) = I2 )
                                 => aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeLista,I2)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Maa),Nb)) )
                                & ! [X3: nat] :
                                    ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),Nb) = I2 )
                                      & aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeLista,I2)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),Nb)) )
                                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mia),X3)
                                      & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Maa) ) ) ) ) )
                       => aa_nat_o(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya)),Dega) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(4)
tff(fact_88_vebt__insert_Osimps_I2_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S2: vEBT_VEBT,Xb: nat] : vEBT_vebt_insert(vEBT_Node(Info,zero_zero_nat,Ts,S2),Xb) = vEBT_Node(Info,zero_zero_nat,Ts,S2) ).

% vebt_insert.simps(2)
tff(fact_89_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_90_Pair__inject,axiom,
    ! [A: code_integer,B: $o,A2: code_integer,B2: $o] :
      ( ( produc6677183202524767010eger_o(A,(B)) = produc6677183202524767010eger_o(A2,(B2)) )
     => ~ ( ( A = A2 )
         => ( (B)
          <=> ~ (B2) ) ) ) ).

% Pair_inject
tff(fact_91_Pair__inject,axiom,
    ! [A: num,B: num,A2: num,B2: num] :
      ( ( product_Pair_num_num(A,B) = product_Pair_num_num(A2,B2) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
tff(fact_92_Pair__inject,axiom,
    ! [A: nat,B: num,A2: nat,B2: num] :
      ( ( product_Pair_nat_num(A,B) = product_Pair_nat_num(A2,B2) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
tff(fact_93_Pair__inject,axiom,
    ! [A: nat,B: nat,A2: nat,B2: nat] :
      ( ( aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B2) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
tff(fact_94_Pair__inject,axiom,
    ! [A: int,B: int,A2: int,B2: int] :
      ( ( aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A2),B2) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
tff(fact_95_prod__cases,axiom,
    ! [P: fun_Pr3290071492984375169er_o_o,P2: produc6271795597528267376eger_o] :
      ( ! [A4: code_integer,B3: $o] : aa_Pro5286387174248033258er_o_o(P,produc6677183202524767010eger_o(A4,(B3)))
     => aa_Pro5286387174248033258er_o_o(P,P2) ) ).

% prod_cases
tff(fact_96_prod__cases,axiom,
    ! [P: fun_Pr6331313853612757382_num_o,P2: product_prod_num_num] :
      ( ! [A4: num,B3: num] : aa_Pro7685007914727650815_num_o(P,product_Pair_num_num(A4,B3))
     => aa_Pro7685007914727650815_num_o(P,P2) ) ).

% prod_cases
tff(fact_97_prod__cases,axiom,
    ! [P: fun_Pr5555124106300582480_num_o,P2: product_prod_nat_num] :
      ( ! [A4: nat,B3: num] : aa_Pro6908818167415475913_num_o(P,product_Pair_nat_num(A4,B3))
     => aa_Pro6908818167415475913_num_o(P,P2) ) ).

% prod_cases
tff(fact_98_prod__cases,axiom,
    ! [P: fun_Pr4796943543331071514_nat_o,P2: product_prod_nat_nat] :
      ( ! [A4: nat,B3: nat] : aa_Pro6150637604445964947_nat_o(P,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A4),B3))
     => aa_Pro6150637604445964947_nat_o(P,P2) ) ).

% prod_cases
tff(fact_99_prod__cases,axiom,
    ! [P: fun_Pr3238374608701176162_int_o,P2: product_prod_int_int] :
      ( ! [A4: int,B3: int] : aa_Pro4592068669816069595_int_o(P,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A4),B3))
     => aa_Pro4592068669816069595_int_o(P,P2) ) ).

% prod_cases
tff(fact_100_surj__pair,axiom,
    ! [P2: produc6271795597528267376eger_o] :
    ? [X3: code_integer,Y: $o] : P2 = produc6677183202524767010eger_o(X3,(Y)) ).

% surj_pair
tff(fact_101_surj__pair,axiom,
    ! [P2: product_prod_num_num] :
    ? [X3: num,Y: num] : P2 = product_Pair_num_num(X3,Y) ).

% surj_pair
tff(fact_102_surj__pair,axiom,
    ! [P2: product_prod_nat_num] :
    ? [X3: nat,Y: num] : P2 = product_Pair_nat_num(X3,Y) ).

% surj_pair
tff(fact_103_surj__pair,axiom,
    ! [P2: product_prod_nat_nat] :
    ? [X3: nat,Y: nat] : P2 = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),Y) ).

% surj_pair
tff(fact_104_surj__pair,axiom,
    ! [P2: product_prod_int_int] :
    ? [X3: int,Y: int] : P2 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X3),Y) ).

% surj_pair
tff(fact_105_old_Oprod_Oexhaust,axiom,
    ! [Yb: produc6271795597528267376eger_o] :
      ~ ! [A4: code_integer,B3: $o] : Yb != produc6677183202524767010eger_o(A4,(B3)) ).

% old.prod.exhaust
tff(fact_106_old_Oprod_Oexhaust,axiom,
    ! [Yb: product_prod_num_num] :
      ~ ! [A4: num,B3: num] : Yb != product_Pair_num_num(A4,B3) ).

% old.prod.exhaust
tff(fact_107_old_Oprod_Oexhaust,axiom,
    ! [Yb: product_prod_nat_num] :
      ~ ! [A4: nat,B3: num] : Yb != product_Pair_nat_num(A4,B3) ).

% old.prod.exhaust
tff(fact_108_old_Oprod_Oexhaust,axiom,
    ! [Yb: product_prod_nat_nat] :
      ~ ! [A4: nat,B3: nat] : Yb != aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A4),B3) ).

% old.prod.exhaust
tff(fact_109_old_Oprod_Oexhaust,axiom,
    ! [Yb: product_prod_int_int] :
      ~ ! [A4: int,B3: int] : Yb != aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A4),B3) ).

% old.prod.exhaust
tff(fact_110_VEBT__internal_Omembermima_Osimps_I3_J,axiom,
    ! [Mia: nat,Maa: nat,Va2: list_VEBT_VEBT,Vb: vEBT_VEBT,Xb: nat] :
      ( aa_nat_o(vEBT_VEBT_membermima(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),zero_zero_nat,Va2,Vb)),Xb)
    <=> ( ( Xb = Mia )
        | ( Xb = Maa ) ) ) ).

% VEBT_internal.membermima.simps(3)
tff(fact_111_in__children__def,axiom,
    ! [Nb: nat,TreeLista: list_VEBT_VEBT,Xb: nat] :
      ( aa_nat_o(vEBT_V5917875025757280293ildren(Nb,TreeLista),Xb)
    <=> aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),Nb))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),Nb)) ) ).

% in_children_def
tff(fact_112_sum__power2__eq__zero__iff,axiom,
    ! [Xb: rat,Yb: rat] :
      ( ( aa_rat_rat(plus_plus_rat(aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = zero_zero_rat )
    <=> ( ( Xb = zero_zero_rat )
        & ( Yb = zero_zero_rat ) ) ) ).

% sum_power2_eq_zero_iff
tff(fact_113_sum__power2__eq__zero__iff,axiom,
    ! [Xb: int,Yb: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(power_power_int(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = zero_zero_int )
    <=> ( ( Xb = zero_zero_int )
        & ( Yb = zero_zero_int ) ) ) ).

% sum_power2_eq_zero_iff
tff(fact_114_sum__power2__eq__zero__iff,axiom,
    ! [Xb: real,Yb: real] :
      ( ( aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = zero_zero_real )
    <=> ( ( Xb = zero_zero_real )
        & ( Yb = zero_zero_real ) ) ) ).

% sum_power2_eq_zero_iff
tff(fact_115_zero__less__power2,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
    <=> ( A != zero_zero_real ) ) ).

% zero_less_power2
tff(fact_116_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_117_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_118_power2__less__eq__zero__iff,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_real)
    <=> ( A = zero_zero_real ) ) ).

% power2_less_eq_zero_iff
tff(fact_119_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_120_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_121_power2__eq__iff__nonneg,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Yb)
       => ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
        <=> ( Xb = Yb ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_122_power2__eq__iff__nonneg,axiom,
    ! [Xb: rat,Yb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Yb)
       => ( ( aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_rat(power_power_rat(Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
        <=> ( Xb = Yb ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_123_power2__eq__iff__nonneg,axiom,
    ! [Xb: nat,Yb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Xb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Yb)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
        <=> ( Xb = Yb ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_124_power2__eq__iff__nonneg,axiom,
    ! [Xb: int,Yb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Yb)
       => ( ( aa_nat_int(power_power_int(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_int(power_power_int(Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
        <=> ( Xb = Yb ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_125_add__2__eq__Suc,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) = aa_nat_nat(suc,aa_nat_nat(suc,Nb)) ).

% add_2_eq_Suc
tff(fact_126_add__2__eq__Suc_H,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(suc,aa_nat_nat(suc,Nb)) ).

% add_2_eq_Suc'
tff(fact_127_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_128_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_129_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_130_zero__eq__power2,axiom,
    ! [A: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_real )
    <=> ( A = zero_zero_real ) ) ).

% zero_eq_power2
tff(fact_131_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_132_power__mono__iff,axiom,
    ! [A: real,B: real,Nb: 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),Nb)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb))
          <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ) ) ).

% power_mono_iff
tff(fact_133_power__mono__iff,axiom,
    ! [A: rat,B: rat,Nb: 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),Nb)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),Nb)),aa_nat_rat(power_power_rat(B),Nb))
          <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ) ) ).

% power_mono_iff
tff(fact_134_power__mono__iff,axiom,
    ! [A: nat,B: nat,Nb: 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),Nb)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb))
          <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ) ) ) ).

% power_mono_iff
tff(fact_135_power__mono__iff,axiom,
    ! [A: int,B: int,Nb: 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),Nb)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),Nb)),aa_nat_int(power_power_int(B),Nb))
          <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ) ) ) ).

% power_mono_iff
tff(fact_136_member__inv,axiom,
    ! [Mia: nat,Maa: nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xb: nat] :
      ( aa_nat_o(vEBT_vebt_member(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya)),Xb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)
        & ( ( Xb = Mia )
          | ( Xb = Maa )
          | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Maa)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mia),Xb)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),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,Xb),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,Xb),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_137_power__eq__0__iff,axiom,
    ! [A: rat,Nb: nat] :
      ( ( aa_nat_rat(power_power_rat(A),Nb) = zero_zero_rat )
    <=> ( ( A = zero_zero_rat )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% power_eq_0_iff
tff(fact_138_power__eq__0__iff,axiom,
    ! [A: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb) = zero_zero_nat )
    <=> ( ( A = zero_zero_nat )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% power_eq_0_iff
tff(fact_139_power__eq__0__iff,axiom,
    ! [A: int,Nb: nat] :
      ( ( aa_nat_int(power_power_int(A),Nb) = zero_zero_int )
    <=> ( ( A = zero_zero_int )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% power_eq_0_iff
tff(fact_140_power__eq__0__iff,axiom,
    ! [A: real,Nb: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb) = zero_zero_real )
    <=> ( ( A = zero_zero_real )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% power_eq_0_iff
tff(fact_141_power__eq__0__iff,axiom,
    ! [A: complex,Nb: nat] :
      ( ( aa_nat_complex(power_power_complex(A),Nb) = zero_zero_complex )
    <=> ( ( A = zero_zero_complex )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% power_eq_0_iff
tff(fact_142__092_060open_062vebt__member_A_ItreeList_091high_Ami_An_A_058_061_Avebt__insert_A_ItreeList_A_B_Ahigh_Ami_An_J_A_Ilow_Ami_An_J_093_A_B_Ahigh_Ay_An_J_A_Ilow_Ay_An_J_092_060close_062,axiom,
    aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,mi),na),vEBT_vebt_insert(nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,mi),na)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,mi),na))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,ya),na))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,ya),na)) ).

% \<open>vebt_member (treeList[high mi n := vebt_insert (treeList ! high mi n) (low mi n)] ! high y n) (low y n)\<close>
tff(fact_143_numeral__eq__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( ( numera6690914467698888265omplex(Mb) = numera6690914467698888265omplex(Nb) )
    <=> ( Mb = Nb ) ) ).

% numeral_eq_iff
tff(fact_144_numeral__eq__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( ( aa_num_real(numeral_numeral_real,Mb) = aa_num_real(numeral_numeral_real,Nb) )
    <=> ( Mb = Nb ) ) ).

% numeral_eq_iff
tff(fact_145_numeral__eq__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( ( aa_num_rat(numeral_numeral_rat,Mb) = aa_num_rat(numeral_numeral_rat,Nb) )
    <=> ( Mb = Nb ) ) ).

% numeral_eq_iff
tff(fact_146_numeral__eq__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( ( aa_num_nat(numeral_numeral_nat,Mb) = aa_num_nat(numeral_numeral_nat,Nb) )
    <=> ( Mb = Nb ) ) ).

% numeral_eq_iff
tff(fact_147_numeral__eq__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( ( aa_num_int(numeral_numeral_int,Mb) = aa_num_int(numeral_numeral_int,Nb) )
    <=> ( Mb = Nb ) ) ).

% numeral_eq_iff
tff(fact_148_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_149_high__def,axiom,
    ! [Xb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% high_def
tff(fact_150_numeral__le__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,Mb)),aa_num_real(numeral_numeral_real,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Mb),Nb) ) ).

% numeral_le_iff
tff(fact_151_numeral__le__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,Mb)),aa_num_rat(numeral_numeral_rat,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Mb),Nb) ) ).

% numeral_le_iff
tff(fact_152_numeral__le__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Mb),Nb) ) ).

% numeral_le_iff
tff(fact_153_numeral__le__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Mb),Nb) ) ).

% numeral_le_iff
tff(fact_154_numeral__less__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,Mb)),aa_num_real(numeral_numeral_real,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Nb) ) ).

% numeral_less_iff
tff(fact_155_numeral__less__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,Mb)),aa_num_rat(numeral_numeral_rat,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Nb) ) ).

% numeral_less_iff
tff(fact_156_numeral__less__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Nb) ) ).

% numeral_less_iff
tff(fact_157_numeral__less__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Nb) ) ).

% numeral_less_iff
tff(fact_158_add__numeral__left,axiom,
    ! [V: num,W: num,Z: complex] : aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(V)),aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(W)),Z)) = aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(aa_num_num(plus_plus_num(V),W))),Z) ).

% add_numeral_left
tff(fact_159_add__numeral__left,axiom,
    ! [V: num,W: num,Z: real] : aa_real_real(plus_plus_real(aa_num_real(numeral_numeral_real,V)),aa_real_real(plus_plus_real(aa_num_real(numeral_numeral_real,W)),Z)) = aa_real_real(plus_plus_real(aa_num_real(numeral_numeral_real,aa_num_num(plus_plus_num(V),W))),Z) ).

% add_numeral_left
tff(fact_160_add__numeral__left,axiom,
    ! [V: num,W: num,Z: rat] : aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,V)),aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,W)),Z)) = aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(plus_plus_num(V),W))),Z) ).

% add_numeral_left
tff(fact_161_add__numeral__left,axiom,
    ! [V: num,W: num,Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,V)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,W)),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(V),W))),Z) ).

% add_numeral_left
tff(fact_162_add__numeral__left,axiom,
    ! [V: num,W: num,Z: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,V)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,W)),Z)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,aa_num_num(plus_plus_num(V),W))),Z) ).

% add_numeral_left
tff(fact_163_numeral__plus__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(Mb)),numera6690914467698888265omplex(Nb)) = numera6690914467698888265omplex(aa_num_num(plus_plus_num(Mb),Nb)) ).

% numeral_plus_numeral
tff(fact_164_numeral__plus__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_real_real(plus_plus_real(aa_num_real(numeral_numeral_real,Mb)),aa_num_real(numeral_numeral_real,Nb)) = aa_num_real(numeral_numeral_real,aa_num_num(plus_plus_num(Mb),Nb)) ).

% numeral_plus_numeral
tff(fact_165_numeral__plus__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,Mb)),aa_num_rat(numeral_numeral_rat,Nb)) = aa_num_rat(numeral_numeral_rat,aa_num_num(plus_plus_num(Mb),Nb)) ).

% numeral_plus_numeral
tff(fact_166_numeral__plus__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Nb)) = aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(Mb),Nb)) ).

% numeral_plus_numeral
tff(fact_167_numeral__plus__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,Nb)) = aa_num_int(numeral_numeral_int,aa_num_num(plus_plus_num(Mb),Nb)) ).

% numeral_plus_numeral
tff(fact_168_power__one,axiom,
    ! [Nb: nat] : aa_nat_rat(power_power_rat(one_one_rat),Nb) = one_one_rat ).

% power_one
tff(fact_169_power__one,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,one_one_nat),Nb) = one_one_nat ).

% power_one
tff(fact_170_power__one,axiom,
    ! [Nb: nat] : aa_nat_int(power_power_int(one_one_int),Nb) = one_one_int ).

% power_one
tff(fact_171_power__one,axiom,
    ! [Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,one_one_real),Nb) = one_one_real ).

% power_one
tff(fact_172_power__one,axiom,
    ! [Nb: nat] : aa_nat_complex(power_power_complex(one_one_complex),Nb) = one_one_complex ).

% power_one
tff(fact_173_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_174_power__one__right,axiom,
    ! [A: int] : aa_nat_int(power_power_int(A),one_one_nat) = A ).

% power_one_right
tff(fact_175_power__one__right,axiom,
    ! [A: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),one_one_nat) = A ).

% power_one_right
tff(fact_176_power__one__right,axiom,
    ! [A: complex] : aa_nat_complex(power_power_complex(A),one_one_nat) = A ).

% power_one_right
tff(fact_177_set__vebt__set__vebt_H__valid,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( vEBT_set_vebt(Ta) = vEBT_VEBT_set_vebt(Ta) ) ) ).

% set_vebt_set_vebt'_valid
tff(fact_178_one__eq__numeral__iff,axiom,
    ! [Nb: num] :
      ( ( one_one_complex = numera6690914467698888265omplex(Nb) )
    <=> ( one = Nb ) ) ).

% one_eq_numeral_iff
tff(fact_179_one__eq__numeral__iff,axiom,
    ! [Nb: num] :
      ( ( one_one_real = aa_num_real(numeral_numeral_real,Nb) )
    <=> ( one = Nb ) ) ).

% one_eq_numeral_iff
tff(fact_180_one__eq__numeral__iff,axiom,
    ! [Nb: num] :
      ( ( one_one_rat = aa_num_rat(numeral_numeral_rat,Nb) )
    <=> ( one = Nb ) ) ).

% one_eq_numeral_iff
tff(fact_181_one__eq__numeral__iff,axiom,
    ! [Nb: num] :
      ( ( one_one_nat = aa_num_nat(numeral_numeral_nat,Nb) )
    <=> ( one = Nb ) ) ).

% one_eq_numeral_iff
tff(fact_182_one__eq__numeral__iff,axiom,
    ! [Nb: num] :
      ( ( one_one_int = aa_num_int(numeral_numeral_int,Nb) )
    <=> ( one = Nb ) ) ).

% one_eq_numeral_iff
tff(fact_183_numeral__eq__one__iff,axiom,
    ! [Nb: num] :
      ( ( numera6690914467698888265omplex(Nb) = one_one_complex )
    <=> ( Nb = one ) ) ).

% numeral_eq_one_iff
tff(fact_184_numeral__eq__one__iff,axiom,
    ! [Nb: num] :
      ( ( aa_num_real(numeral_numeral_real,Nb) = one_one_real )
    <=> ( Nb = one ) ) ).

% numeral_eq_one_iff
tff(fact_185_numeral__eq__one__iff,axiom,
    ! [Nb: num] :
      ( ( aa_num_rat(numeral_numeral_rat,Nb) = one_one_rat )
    <=> ( Nb = one ) ) ).

% numeral_eq_one_iff
tff(fact_186_numeral__eq__one__iff,axiom,
    ! [Nb: num] :
      ( ( aa_num_nat(numeral_numeral_nat,Nb) = one_one_nat )
    <=> ( Nb = one ) ) ).

% numeral_eq_one_iff
tff(fact_187_numeral__eq__one__iff,axiom,
    ! [Nb: num] :
      ( ( aa_num_int(numeral_numeral_int,Nb) = one_one_int )
    <=> ( Nb = one ) ) ).

% numeral_eq_one_iff
tff(fact_188_power__inject__exp,axiom,
    ! [A: real,Mb: nat,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Mb) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb) )
      <=> ( Mb = Nb ) ) ) ).

% power_inject_exp
tff(fact_189_power__inject__exp,axiom,
    ! [A: rat,Mb: nat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A)
     => ( ( aa_nat_rat(power_power_rat(A),Mb) = aa_nat_rat(power_power_rat(A),Nb) )
      <=> ( Mb = Nb ) ) ) ).

% power_inject_exp
tff(fact_190_power__inject__exp,axiom,
    ! [A: nat,Mb: nat,Nb: 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),Mb) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb) )
      <=> ( Mb = Nb ) ) ) ).

% power_inject_exp
tff(fact_191_power__inject__exp,axiom,
    ! [A: int,Mb: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
     => ( ( aa_nat_int(power_power_int(A),Mb) = aa_nat_int(power_power_int(A),Nb) )
      <=> ( Mb = Nb ) ) ) ).

% power_inject_exp
tff(fact_192_power__0__Suc,axiom,
    ! [Nb: nat] : aa_nat_rat(power_power_rat(zero_zero_rat),aa_nat_nat(suc,Nb)) = zero_zero_rat ).

% power_0_Suc
tff(fact_193_power__0__Suc,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),aa_nat_nat(suc,Nb)) = zero_zero_nat ).

% power_0_Suc
tff(fact_194_power__0__Suc,axiom,
    ! [Nb: nat] : aa_nat_int(power_power_int(zero_zero_int),aa_nat_nat(suc,Nb)) = zero_zero_int ).

% power_0_Suc
tff(fact_195_power__0__Suc,axiom,
    ! [Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),aa_nat_nat(suc,Nb)) = zero_zero_real ).

% power_0_Suc
tff(fact_196_power__0__Suc,axiom,
    ! [Nb: nat] : aa_nat_complex(power_power_complex(zero_zero_complex),aa_nat_nat(suc,Nb)) = zero_zero_complex ).

% power_0_Suc
tff(fact_197_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_198_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_199_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_200_power__zero__numeral,axiom,
    ! [K: num] : aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),aa_num_nat(numeral_numeral_nat,K)) = zero_zero_real ).

% power_zero_numeral
tff(fact_201_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_202_Suc__numeral,axiom,
    ! [Nb: num] : aa_nat_nat(suc,aa_num_nat(numeral_numeral_nat,Nb)) = aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(Nb),one)) ).

% Suc_numeral
tff(fact_203_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_204_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_205_power__Suc0__right,axiom,
    ! [A: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,zero_zero_nat)) = A ).

% power_Suc0_right
tff(fact_206_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_207_nat__power__eq__Suc__0__iff,axiom,
    ! [Xb: nat,Mb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),Mb) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ( ( Mb = zero_zero_nat )
        | ( Xb = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% nat_power_eq_Suc_0_iff
tff(fact_208_power__Suc__0,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(suc,zero_zero_nat)),Nb) = aa_nat_nat(suc,zero_zero_nat) ).

% power_Suc_0
tff(fact_209_nat__zero__less__power__iff,axiom,
    ! [Xb: nat,Nb: 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,Xb),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Xb)
        | ( Nb = zero_zero_nat ) ) ) ).

% nat_zero_less_power_iff
tff(fact_210_both__member__options__ding,axiom,
    ! [Info: option4927543243414619207at_nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(vEBT_Node(Info,Dega,TreeLista,Summarya)),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),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,Xb),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,Xb),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(Info,Dega,TreeLista,Summarya)),Xb) ) ) ) ).

% both_member_options_ding
tff(fact_211_both__member__options__from__complete__tree__to__child,axiom,
    ! [Dega: nat,Mia: nat,Maa: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xb: 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(Mia),Maa)),Dega,TreeLista,Summarya)),Xb)
       => ( aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),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,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
          | ( Xb = Mia )
          | ( Xb = Maa ) ) ) ) ).

% both_member_options_from_complete_tree_to_child
tff(fact_212_numeral__le__one__iff,axiom,
    ! [Nb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,Nb)),one_one_real)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Nb),one) ) ).

% numeral_le_one_iff
tff(fact_213_numeral__le__one__iff,axiom,
    ! [Nb: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,Nb)),one_one_rat)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Nb),one) ) ).

% numeral_le_one_iff
tff(fact_214_numeral__le__one__iff,axiom,
    ! [Nb: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,Nb)),one_one_nat)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Nb),one) ) ).

% numeral_le_one_iff
tff(fact_215_numeral__le__one__iff,axiom,
    ! [Nb: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Nb)),one_one_int)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Nb),one) ) ).

% numeral_le_one_iff
tff(fact_216_one__less__numeral__iff,axiom,
    ! [Nb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_num_real(numeral_numeral_real,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,one),Nb) ) ).

% one_less_numeral_iff
tff(fact_217_one__less__numeral__iff,axiom,
    ! [Nb: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_num_rat(numeral_numeral_rat,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,one),Nb) ) ).

% one_less_numeral_iff
tff(fact_218_one__less__numeral__iff,axiom,
    ! [Nb: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,one),Nb) ) ).

% one_less_numeral_iff
tff(fact_219_one__less__numeral__iff,axiom,
    ! [Nb: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_num_int(numeral_numeral_int,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,one),Nb) ) ).

% one_less_numeral_iff
tff(fact_220_one__plus__numeral,axiom,
    ! [Nb: num] : aa_complex_complex(plus_plus_complex(one_one_complex),numera6690914467698888265omplex(Nb)) = numera6690914467698888265omplex(aa_num_num(plus_plus_num(one),Nb)) ).

% one_plus_numeral
tff(fact_221_one__plus__numeral,axiom,
    ! [Nb: num] : aa_real_real(plus_plus_real(one_one_real),aa_num_real(numeral_numeral_real,Nb)) = aa_num_real(numeral_numeral_real,aa_num_num(plus_plus_num(one),Nb)) ).

% one_plus_numeral
tff(fact_222_one__plus__numeral,axiom,
    ! [Nb: num] : aa_rat_rat(plus_plus_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,Nb)) = aa_num_rat(numeral_numeral_rat,aa_num_num(plus_plus_num(one),Nb)) ).

% one_plus_numeral
tff(fact_223_one__plus__numeral,axiom,
    ! [Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,Nb)) = aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(one),Nb)) ).

% one_plus_numeral
tff(fact_224_one__plus__numeral,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_num_int(numeral_numeral_int,Nb)) = aa_num_int(numeral_numeral_int,aa_num_num(plus_plus_num(one),Nb)) ).

% one_plus_numeral
tff(fact_225_numeral__plus__one,axiom,
    ! [Nb: num] : aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(Nb)),one_one_complex) = numera6690914467698888265omplex(aa_num_num(plus_plus_num(Nb),one)) ).

% numeral_plus_one
tff(fact_226_numeral__plus__one,axiom,
    ! [Nb: num] : aa_real_real(plus_plus_real(aa_num_real(numeral_numeral_real,Nb)),one_one_real) = aa_num_real(numeral_numeral_real,aa_num_num(plus_plus_num(Nb),one)) ).

% numeral_plus_one
tff(fact_227_numeral__plus__one,axiom,
    ! [Nb: num] : aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,Nb)),one_one_rat) = aa_num_rat(numeral_numeral_rat,aa_num_num(plus_plus_num(Nb),one)) ).

% numeral_plus_one
tff(fact_228_numeral__plus__one,axiom,
    ! [Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,Nb)),one_one_nat) = aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(Nb),one)) ).

% numeral_plus_one
tff(fact_229_numeral__plus__one,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,Nb)),one_one_int) = aa_num_int(numeral_numeral_int,aa_num_num(plus_plus_num(Nb),one)) ).

% numeral_plus_one
tff(fact_230_power__strict__increasing__iff,axiom,
    ! [B: real,Xb: nat,Yb: 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(aa_real_fun_nat_real(power_power_real,B),Xb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Yb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Yb) ) ) ).

% power_strict_increasing_iff
tff(fact_231_power__strict__increasing__iff,axiom,
    ! [B: rat,Xb: nat,Yb: 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),Xb)),aa_nat_rat(power_power_rat(B),Yb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Yb) ) ) ).

% power_strict_increasing_iff
tff(fact_232_power__strict__increasing__iff,axiom,
    ! [B: nat,Xb: nat,Yb: 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),Xb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Yb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Yb) ) ) ).

% power_strict_increasing_iff
tff(fact_233_power__strict__increasing__iff,axiom,
    ! [B: int,Xb: nat,Yb: 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),Xb)),aa_nat_int(power_power_int(B),Yb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Yb) ) ) ).

% power_strict_increasing_iff
tff(fact_234_both__member__options__from__chilf__to__complete__tree,axiom,
    ! [Xb: nat,Dega: nat,TreeLista: list_VEBT_VEBT,Mia: nat,Maa: 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,Xb),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,Xb),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,Xb),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(Mia),Maa)),Dega,TreeLista,Summarya)),Xb) ) ) ) ).

% both_member_options_from_chilf_to_complete_tree
tff(fact_235_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_236_one__add__one,axiom,
    aa_real_real(plus_plus_real(one_one_real),one_one_real) = aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)) ).

% one_add_one
tff(fact_237_one__add__one,axiom,
    aa_rat_rat(plus_plus_rat(one_one_rat),one_one_rat) = aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)) ).

% one_add_one
tff(fact_238_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_239_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_240_power__strict__decreasing__iff,axiom,
    ! [B: real,Mb: nat,Nb: 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(aa_real_fun_nat_real(power_power_real,B),Mb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_241_power__strict__decreasing__iff,axiom,
    ! [B: rat,Mb: nat,Nb: 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),Mb)),aa_nat_rat(power_power_rat(B),Nb))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_242_power__strict__decreasing__iff,axiom,
    ! [B: nat,Mb: nat,Nb: 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),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_243_power__strict__decreasing__iff,axiom,
    ! [B: int,Mb: nat,Nb: 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),Mb)),aa_nat_int(power_power_int(B),Nb))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_244_power__increasing__iff,axiom,
    ! [B: real,Xb: nat,Yb: 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(aa_real_fun_nat_real(power_power_real,B),Xb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Yb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Yb) ) ) ).

% power_increasing_iff
tff(fact_245_power__increasing__iff,axiom,
    ! [B: rat,Xb: nat,Yb: 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),Xb)),aa_nat_rat(power_power_rat(B),Yb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Yb) ) ) ).

% power_increasing_iff
tff(fact_246_power__increasing__iff,axiom,
    ! [B: nat,Xb: nat,Yb: 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),Xb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Yb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Yb) ) ) ).

% power_increasing_iff
tff(fact_247_power__increasing__iff,axiom,
    ! [B: int,Xb: nat,Yb: 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),Xb)),aa_nat_int(power_power_int(B),Yb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Yb) ) ) ).

% power_increasing_iff
tff(fact_248_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_249_power__decreasing__iff,axiom,
    ! [B: real,Mb: nat,Nb: 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(aa_real_fun_nat_real(power_power_real,B),Mb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb) ) ) ) ).

% power_decreasing_iff
tff(fact_250_power__decreasing__iff,axiom,
    ! [B: rat,Mb: nat,Nb: 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),Mb)),aa_nat_rat(power_power_rat(B),Nb))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb) ) ) ) ).

% power_decreasing_iff
tff(fact_251_power__decreasing__iff,axiom,
    ! [B: nat,Mb: nat,Nb: 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),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb) ) ) ) ).

% power_decreasing_iff
tff(fact_252_power__decreasing__iff,axiom,
    ! [B: int,Mb: nat,Nb: 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),Mb)),aa_nat_int(power_power_int(B),Nb))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb) ) ) ) ).

% power_decreasing_iff
tff(fact_253_power__one__over,axiom,
    ! [A: complex,Nb: nat] : aa_nat_complex(power_power_complex(aa_complex_complex(divide1717551699836669952omplex(one_one_complex),A)),Nb) = aa_complex_complex(divide1717551699836669952omplex(one_one_complex),aa_nat_complex(power_power_complex(A),Nb)) ).

% power_one_over
tff(fact_254_power__one__over,axiom,
    ! [A: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(divide_divide_real(one_one_real),A)),Nb) = aa_real_real(divide_divide_real(one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ).

% power_one_over
tff(fact_255_power__one__over,axiom,
    ! [A: rat,Nb: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(divide_divide_rat(one_one_rat),A)),Nb) = aa_rat_rat(divide_divide_rat(one_one_rat),aa_nat_rat(power_power_rat(A),Nb)) ).

% power_one_over
tff(fact_256_add__One__commute,axiom,
    ! [Nb: num] : aa_num_num(plus_plus_num(one),Nb) = aa_num_num(plus_plus_num(Nb),one) ).

% add_One_commute
tff(fact_257_power__divide,axiom,
    ! [A: complex,B: complex,Nb: nat] : aa_nat_complex(power_power_complex(aa_complex_complex(divide1717551699836669952omplex(A),B)),Nb) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(power_power_complex(A),Nb)),aa_nat_complex(power_power_complex(B),Nb)) ).

% power_divide
tff(fact_258_power__divide,axiom,
    ! [A: real,B: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(divide_divide_real(A),B)),Nb) = aa_real_real(divide_divide_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb)) ).

% power_divide
tff(fact_259_power__divide,axiom,
    ! [A: rat,B: rat,Nb: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(divide_divide_rat(A),B)),Nb) = aa_rat_rat(divide_divide_rat(aa_nat_rat(power_power_rat(A),Nb)),aa_nat_rat(power_power_rat(B),Nb)) ).

% power_divide
tff(fact_260_le__num__One__iff,axiom,
    ! [Xb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),one)
    <=> ( Xb = one ) ) ).

% le_num_One_iff
tff(fact_261_divide__numeral__1,axiom,
    ! [A: complex] : aa_complex_complex(divide1717551699836669952omplex(A),numera6690914467698888265omplex(one)) = A ).

% divide_numeral_1
tff(fact_262_divide__numeral__1,axiom,
    ! [A: real] : aa_real_real(divide_divide_real(A),aa_num_real(numeral_numeral_real,one)) = A ).

% divide_numeral_1
tff(fact_263_divide__numeral__1,axiom,
    ! [A: rat] : aa_rat_rat(divide_divide_rat(A),aa_num_rat(numeral_numeral_rat,one)) = A ).

% divide_numeral_1
tff(fact_264_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_265_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_266_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_267_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_268_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_269_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_270_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_271_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_272_Suc__nat__number__of__add,axiom,
    ! [V: num,Nb: nat] : aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,V)),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(V),one))),Nb) ).

% Suc_nat_number_of_add
tff(fact_273_is__num__normalize_I1_J,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(plus_plus_real(aa_real_real(plus_plus_real(A),B)),C) = aa_real_real(plus_plus_real(A),aa_real_real(plus_plus_real(B),C)) ).

% is_num_normalize(1)
tff(fact_274_is__num__normalize_I1_J,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(A),B)),C) = aa_rat_rat(plus_plus_rat(A),aa_rat_rat(plus_plus_rat(B),C)) ).

% is_num_normalize(1)
tff(fact_275_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_276_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_277_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_278_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_279_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_280_one__le__numeral,axiom,
    ! [Nb: num] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_num_real(numeral_numeral_real,Nb)) ).

% one_le_numeral
tff(fact_281_one__le__numeral,axiom,
    ! [Nb: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_num_rat(numeral_numeral_rat,Nb)) ).

% one_le_numeral
tff(fact_282_one__le__numeral,axiom,
    ! [Nb: num] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,Nb)) ).

% one_le_numeral
tff(fact_283_one__le__numeral,axiom,
    ! [Nb: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_num_int(numeral_numeral_int,Nb)) ).

% one_le_numeral
tff(fact_284_not__numeral__less__one,axiom,
    ! [Nb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,Nb)),one_one_real) ).

% not_numeral_less_one
tff(fact_285_not__numeral__less__one,axiom,
    ! [Nb: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,Nb)),one_one_rat) ).

% not_numeral_less_one
tff(fact_286_not__numeral__less__one,axiom,
    ! [Nb: num] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,Nb)),one_one_nat) ).

% not_numeral_less_one
tff(fact_287_not__numeral__less__one,axiom,
    ! [Nb: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Nb)),one_one_int) ).

% not_numeral_less_one
tff(fact_288_half__gt__zero__iff,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(divide_divide_real(A),aa_num_real(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_289_half__gt__zero__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(A),aa_num_rat(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_290_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(divide_divide_real(A),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ).

% half_gt_zero
tff(fact_291_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),aa_rat_rat(divide_divide_rat(A),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))) ) ).

% half_gt_zero
tff(fact_292_one__plus__numeral__commute,axiom,
    ! [Xb: num] : aa_complex_complex(plus_plus_complex(one_one_complex),numera6690914467698888265omplex(Xb)) = aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(Xb)),one_one_complex) ).

% one_plus_numeral_commute
tff(fact_293_one__plus__numeral__commute,axiom,
    ! [Xb: num] : aa_real_real(plus_plus_real(one_one_real),aa_num_real(numeral_numeral_real,Xb)) = aa_real_real(plus_plus_real(aa_num_real(numeral_numeral_real,Xb)),one_one_real) ).

% one_plus_numeral_commute
tff(fact_294_one__plus__numeral__commute,axiom,
    ! [Xb: num] : aa_rat_rat(plus_plus_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,Xb)) = aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,Xb)),one_one_rat) ).

% one_plus_numeral_commute
tff(fact_295_one__plus__numeral__commute,axiom,
    ! [Xb: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,Xb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,Xb)),one_one_nat) ).

% one_plus_numeral_commute
tff(fact_296_one__plus__numeral__commute,axiom,
    ! [Xb: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_num_int(numeral_numeral_int,Xb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,Xb)),one_one_int) ).

% one_plus_numeral_commute
tff(fact_297_numeral__One,axiom,
    numera6690914467698888265omplex(one) = one_one_complex ).

% numeral_One
tff(fact_298_numeral__One,axiom,
    aa_num_real(numeral_numeral_real,one) = one_one_real ).

% numeral_One
tff(fact_299_numeral__One,axiom,
    aa_num_rat(numeral_numeral_rat,one) = one_one_rat ).

% numeral_One
tff(fact_300_numeral__One,axiom,
    aa_num_nat(numeral_numeral_nat,one) = one_one_nat ).

% numeral_One
tff(fact_301_numeral__One,axiom,
    aa_num_int(numeral_numeral_int,one) = one_one_int ).

% numeral_One
tff(fact_302_one__le__power,axiom,
    ! [A: real,Nb: 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(aa_real_fun_nat_real(power_power_real,A),Nb)) ) ).

% one_le_power
tff(fact_303_one__le__power,axiom,
    ! [A: rat,Nb: 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),Nb)) ) ).

% one_le_power
tff(fact_304_one__le__power,axiom,
    ! [A: nat,Nb: 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),Nb)) ) ).

% one_le_power
tff(fact_305_one__le__power,axiom,
    ! [A: int,Nb: 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),Nb)) ) ).

% one_le_power
tff(fact_306_numerals_I1_J,axiom,
    aa_num_nat(numeral_numeral_nat,one) = one_one_nat ).

% numerals(1)
tff(fact_307_power__0,axiom,
    ! [A: rat] : aa_nat_rat(power_power_rat(A),zero_zero_nat) = one_one_rat ).

% power_0
tff(fact_308_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_309_power__0,axiom,
    ! [A: int] : aa_nat_int(power_power_int(A),zero_zero_nat) = one_one_int ).

% power_0
tff(fact_310_power__0,axiom,
    ! [A: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),zero_zero_nat) = one_one_real ).

% power_0
tff(fact_311_power__0,axiom,
    ! [A: complex] : aa_nat_complex(power_power_complex(A),zero_zero_nat) = one_one_complex ).

% power_0
tff(fact_312_power__le__one,axiom,
    ! [A: real,Nb: 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(aa_real_fun_nat_real(power_power_real,A),Nb)),one_one_real) ) ) ).

% power_le_one
tff(fact_313_power__le__one,axiom,
    ! [A: rat,Nb: 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),Nb)),one_one_rat) ) ) ).

% power_le_one
tff(fact_314_power__le__one,axiom,
    ! [A: nat,Nb: 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),Nb)),one_one_nat) ) ) ).

% power_le_one
tff(fact_315_power__le__one,axiom,
    ! [A: int,Nb: 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),Nb)),one_one_int) ) ) ).

% power_le_one
tff(fact_316_power__0__left,axiom,
    ! [Nb: nat] :
      aa_nat_rat(power_power_rat(zero_zero_rat),Nb) = $ite(Nb = zero_zero_nat,one_one_rat,zero_zero_rat) ).

% power_0_left
tff(fact_317_power__0__left,axiom,
    ! [Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),Nb) = $ite(Nb = zero_zero_nat,one_one_nat,zero_zero_nat) ).

% power_0_left
tff(fact_318_power__0__left,axiom,
    ! [Nb: nat] :
      aa_nat_int(power_power_int(zero_zero_int),Nb) = $ite(Nb = zero_zero_nat,one_one_int,zero_zero_int) ).

% power_0_left
tff(fact_319_power__0__left,axiom,
    ! [Nb: nat] :
      aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),Nb) = $ite(Nb = zero_zero_nat,one_one_real,zero_zero_real) ).

% power_0_left
tff(fact_320_power__0__left,axiom,
    ! [Nb: nat] :
      aa_nat_complex(power_power_complex(zero_zero_complex),Nb) = $ite(Nb = zero_zero_nat,one_one_complex,zero_zero_complex) ).

% power_0_left
tff(fact_321_power__gt1,axiom,
    ! [A: real,Nb: 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(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,Nb))) ) ).

% power_gt1
tff(fact_322_power__gt1,axiom,
    ! [A: rat,Nb: 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,Nb))) ) ).

% power_gt1
tff(fact_323_power__gt1,axiom,
    ! [A: nat,Nb: 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,Nb))) ) ).

% power_gt1
tff(fact_324_power__gt1,axiom,
    ! [A: int,Nb: 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,Nb))) ) ).

% power_gt1
tff(fact_325_power__strict__increasing,axiom,
    ! [Nb: nat,N2: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),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(aa_real_fun_nat_real(power_power_real,A),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N2)) ) ) ).

% power_strict_increasing
tff(fact_326_power__strict__increasing,axiom,
    ! [Nb: nat,N2: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),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),Nb)),aa_nat_rat(power_power_rat(A),N2)) ) ) ).

% power_strict_increasing
tff(fact_327_power__strict__increasing,axiom,
    ! [Nb: nat,N2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),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),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N2)) ) ) ).

% power_strict_increasing
tff(fact_328_power__strict__increasing,axiom,
    ! [Nb: nat,N2: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),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),Nb)),aa_nat_int(power_power_int(A),N2)) ) ) ).

% power_strict_increasing
tff(fact_329_power__less__imp__less__exp,axiom,
    ! [A: real,Mb: nat,Nb: 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(aa_real_fun_nat_real(power_power_real,A),Mb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ) ).

% power_less_imp_less_exp
tff(fact_330_power__less__imp__less__exp,axiom,
    ! [A: rat,Mb: nat,Nb: 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),Mb)),aa_nat_rat(power_power_rat(A),Nb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ) ).

% power_less_imp_less_exp
tff(fact_331_power__less__imp__less__exp,axiom,
    ! [A: nat,Mb: nat,Nb: 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),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ) ).

% power_less_imp_less_exp
tff(fact_332_power__less__imp__less__exp,axiom,
    ! [A: int,Mb: nat,Nb: 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),Mb)),aa_nat_int(power_power_int(A),Nb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ) ).

% power_less_imp_less_exp
tff(fact_333_power__increasing,axiom,
    ! [Nb: nat,N2: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),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(aa_real_fun_nat_real(power_power_real,A),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),N2)) ) ) ).

% power_increasing
tff(fact_334_power__increasing,axiom,
    ! [Nb: nat,N2: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),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),Nb)),aa_nat_rat(power_power_rat(A),N2)) ) ) ).

% power_increasing
tff(fact_335_power__increasing,axiom,
    ! [Nb: nat,N2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),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),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N2)) ) ) ).

% power_increasing
tff(fact_336_power__increasing,axiom,
    ! [Nb: nat,N2: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),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),Nb)),aa_nat_int(power_power_int(A),N2)) ) ) ).

% power_increasing
tff(fact_337_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_338_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_339_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_340_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_341_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_342_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_343_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_344_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_345_zero__neq__numeral,axiom,
    ! [Nb: num] : zero_zero_complex != numera6690914467698888265omplex(Nb) ).

% zero_neq_numeral
tff(fact_346_zero__neq__numeral,axiom,
    ! [Nb: num] : zero_zero_real != aa_num_real(numeral_numeral_real,Nb) ).

% zero_neq_numeral
tff(fact_347_zero__neq__numeral,axiom,
    ! [Nb: num] : zero_zero_rat != aa_num_rat(numeral_numeral_rat,Nb) ).

% zero_neq_numeral
tff(fact_348_zero__neq__numeral,axiom,
    ! [Nb: num] : zero_zero_nat != aa_num_nat(numeral_numeral_nat,Nb) ).

% zero_neq_numeral
tff(fact_349_zero__neq__numeral,axiom,
    ! [Nb: num] : zero_zero_int != aa_num_int(numeral_numeral_int,Nb) ).

% zero_neq_numeral
tff(fact_350_power__not__zero,axiom,
    ! [A: rat,Nb: nat] :
      ( ( A != zero_zero_rat )
     => ( aa_nat_rat(power_power_rat(A),Nb) != zero_zero_rat ) ) ).

% power_not_zero
tff(fact_351_power__not__zero,axiom,
    ! [A: nat,Nb: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb) != zero_zero_nat ) ) ).

% power_not_zero
tff(fact_352_power__not__zero,axiom,
    ! [A: int,Nb: nat] :
      ( ( A != zero_zero_int )
     => ( aa_nat_int(power_power_int(A),Nb) != zero_zero_int ) ) ).

% power_not_zero
tff(fact_353_power__not__zero,axiom,
    ! [A: real,Nb: nat] :
      ( ( A != zero_zero_real )
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb) != zero_zero_real ) ) ).

% power_not_zero
tff(fact_354_power__not__zero,axiom,
    ! [A: complex,Nb: nat] :
      ( ( A != zero_zero_complex )
     => ( aa_nat_complex(power_power_complex(A),Nb) != zero_zero_complex ) ) ).

% power_not_zero
tff(fact_355_power__Suc__le__self,axiom,
    ! [A: real,Nb: 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(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,Nb))),A) ) ) ).

% power_Suc_le_self
tff(fact_356_power__Suc__le__self,axiom,
    ! [A: rat,Nb: 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,Nb))),A) ) ) ).

% power_Suc_le_self
tff(fact_357_power__Suc__le__self,axiom,
    ! [A: nat,Nb: 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,Nb))),A) ) ) ).

% power_Suc_le_self
tff(fact_358_power__Suc__le__self,axiom,
    ! [A: int,Nb: 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,Nb))),A) ) ) ).

% power_Suc_le_self
tff(fact_359_power__Suc__less__one,axiom,
    ! [A: real,Nb: 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(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,Nb))),one_one_real) ) ) ).

% power_Suc_less_one
tff(fact_360_power__Suc__less__one,axiom,
    ! [A: rat,Nb: 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,Nb))),one_one_rat) ) ) ).

% power_Suc_less_one
tff(fact_361_power__Suc__less__one,axiom,
    ! [A: nat,Nb: 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,Nb))),one_one_nat) ) ) ).

% power_Suc_less_one
tff(fact_362_power__Suc__less__one,axiom,
    ! [A: int,Nb: 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,Nb))),one_one_int) ) ) ).

% power_Suc_less_one
tff(fact_363_power__strict__decreasing,axiom,
    ! [Nb: nat,N2: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),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(aa_real_fun_nat_real(power_power_real,A),N2)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ) ) ) ).

% power_strict_decreasing
tff(fact_364_power__strict__decreasing,axiom,
    ! [Nb: nat,N2: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),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),Nb)) ) ) ) ).

% power_strict_decreasing
tff(fact_365_power__strict__decreasing,axiom,
    ! [Nb: nat,N2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),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),Nb)) ) ) ) ).

% power_strict_decreasing
tff(fact_366_power__strict__decreasing,axiom,
    ! [Nb: nat,N2: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),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),Nb)) ) ) ) ).

% power_strict_decreasing
tff(fact_367_power__decreasing,axiom,
    ! [Nb: nat,N2: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),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(aa_real_fun_nat_real(power_power_real,A),N2)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ) ) ) ).

% power_decreasing
tff(fact_368_power__decreasing,axiom,
    ! [Nb: nat,N2: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),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),Nb)) ) ) ) ).

% power_decreasing
tff(fact_369_power__decreasing,axiom,
    ! [Nb: nat,N2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),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),Nb)) ) ) ) ).

% power_decreasing
tff(fact_370_power__decreasing,axiom,
    ! [Nb: nat,N2: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),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),Nb)) ) ) ) ).

% power_decreasing
tff(fact_371_power__le__imp__le__exp,axiom,
    ! [A: real,Mb: nat,Nb: 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(aa_real_fun_nat_real(power_power_real,A),Mb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ).

% power_le_imp_le_exp
tff(fact_372_power__le__imp__le__exp,axiom,
    ! [A: rat,Mb: nat,Nb: 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),Mb)),aa_nat_rat(power_power_rat(A),Nb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ).

% power_le_imp_le_exp
tff(fact_373_power__le__imp__le__exp,axiom,
    ! [A: nat,Mb: nat,Nb: 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),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ).

% power_le_imp_le_exp
tff(fact_374_power__le__imp__le__exp,axiom,
    ! [A: int,Mb: nat,Nb: 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),Mb)),aa_nat_int(power_power_int(A),Nb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ).

% power_le_imp_le_exp
tff(fact_375_self__le__power,axiom,
    ! [A: real,Nb: 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),Nb)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ) ) ).

% self_le_power
tff(fact_376_self__le__power,axiom,
    ! [A: rat,Nb: 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),Nb)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_nat_rat(power_power_rat(A),Nb)) ) ) ).

% self_le_power
tff(fact_377_self__le__power,axiom,
    ! [A: nat,Nb: 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),Nb)
       => 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),Nb)) ) ) ).

% self_le_power
tff(fact_378_self__le__power,axiom,
    ! [A: int,Nb: 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),Nb)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(power_power_int(A),Nb)) ) ) ).

% self_le_power
tff(fact_379_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_380_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_381_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_382_one__power2,axiom,
    aa_nat_real(aa_real_fun_nat_real(power_power_real,one_one_real),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_real ).

% one_power2
tff(fact_383_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_384_one__less__power,axiom,
    ! [A: real,Nb: 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),Nb)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ) ) ).

% one_less_power
tff(fact_385_one__less__power,axiom,
    ! [A: rat,Nb: 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),Nb)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_nat_rat(power_power_rat(A),Nb)) ) ) ).

% one_less_power
tff(fact_386_one__less__power,axiom,
    ! [A: nat,Nb: 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),Nb)
       => 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),Nb)) ) ) ).

% one_less_power
tff(fact_387_one__less__power,axiom,
    ! [A: int,Nb: 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),Nb)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_nat_int(power_power_int(A),Nb)) ) ) ).

% one_less_power
tff(fact_388_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_389_not__numeral__le__zero,axiom,
    ! [Nb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,Nb)),zero_zero_real) ).

% not_numeral_le_zero
tff(fact_390_not__numeral__le__zero,axiom,
    ! [Nb: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,Nb)),zero_zero_rat) ).

% not_numeral_le_zero
tff(fact_391_not__numeral__le__zero,axiom,
    ! [Nb: num] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,Nb)),zero_zero_nat) ).

% not_numeral_le_zero
tff(fact_392_not__numeral__le__zero,axiom,
    ! [Nb: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Nb)),zero_zero_int) ).

% not_numeral_le_zero
tff(fact_393_zero__le__numeral,axiom,
    ! [Nb: num] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_num_real(numeral_numeral_real,Nb)) ).

% zero_le_numeral
tff(fact_394_zero__le__numeral,axiom,
    ! [Nb: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_num_rat(numeral_numeral_rat,Nb)) ).

% zero_le_numeral
tff(fact_395_zero__le__numeral,axiom,
    ! [Nb: num] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,Nb)) ).

% zero_le_numeral
tff(fact_396_zero__le__numeral,axiom,
    ! [Nb: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_num_int(numeral_numeral_int,Nb)) ).

% zero_le_numeral
tff(fact_397_not__numeral__less__zero,axiom,
    ! [Nb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,Nb)),zero_zero_real) ).

% not_numeral_less_zero
tff(fact_398_not__numeral__less__zero,axiom,
    ! [Nb: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,Nb)),zero_zero_rat) ).

% not_numeral_less_zero
tff(fact_399_not__numeral__less__zero,axiom,
    ! [Nb: num] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,Nb)),zero_zero_nat) ).

% not_numeral_less_zero
tff(fact_400_not__numeral__less__zero,axiom,
    ! [Nb: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Nb)),zero_zero_int) ).

% not_numeral_less_zero
tff(fact_401_zero__less__numeral,axiom,
    ! [Nb: num] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_num_real(numeral_numeral_real,Nb)) ).

% zero_less_numeral
tff(fact_402_zero__less__numeral,axiom,
    ! [Nb: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_num_rat(numeral_numeral_rat,Nb)) ).

% zero_less_numeral
tff(fact_403_zero__less__numeral,axiom,
    ! [Nb: num] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,Nb)) ).

% zero_less_numeral
tff(fact_404_zero__less__numeral,axiom,
    ! [Nb: num] : aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_num_int(numeral_numeral_int,Nb)) ).

% zero_less_numeral
tff(fact_405_zero__le__power,axiom,
    ! [A: real,Nb: 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(aa_real_fun_nat_real(power_power_real,A),Nb)) ) ).

% zero_le_power
tff(fact_406_zero__le__power,axiom,
    ! [A: rat,Nb: 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),Nb)) ) ).

% zero_le_power
tff(fact_407_zero__le__power,axiom,
    ! [A: nat,Nb: 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),Nb)) ) ).

% zero_le_power
tff(fact_408_zero__le__power,axiom,
    ! [A: int,Nb: 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),Nb)) ) ).

% zero_le_power
tff(fact_409_power__mono,axiom,
    ! [A: real,B: real,Nb: 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(aa_real_fun_nat_real(power_power_real,A),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb)) ) ) ).

% power_mono
tff(fact_410_power__mono,axiom,
    ! [A: rat,B: rat,Nb: 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),Nb)),aa_nat_rat(power_power_rat(B),Nb)) ) ) ).

% power_mono
tff(fact_411_power__mono,axiom,
    ! [A: nat,B: nat,Nb: 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),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb)) ) ) ).

% power_mono
tff(fact_412_power__mono,axiom,
    ! [A: int,B: int,Nb: 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),Nb)),aa_nat_int(power_power_int(B),Nb)) ) ) ).

% power_mono
tff(fact_413_zero__less__power,axiom,
    ! [A: real,Nb: 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(aa_real_fun_nat_real(power_power_real,A),Nb)) ) ).

% zero_less_power
tff(fact_414_zero__less__power,axiom,
    ! [A: rat,Nb: 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),Nb)) ) ).

% zero_less_power
tff(fact_415_zero__less__power,axiom,
    ! [A: nat,Nb: 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),Nb)) ) ).

% zero_less_power
tff(fact_416_zero__less__power,axiom,
    ! [A: int,Nb: 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),Nb)) ) ).

% zero_less_power
tff(fact_417_numeral__Bit0,axiom,
    ! [Nb: num] : numera6690914467698888265omplex(aa_num_num(bit0,Nb)) = aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(Nb)),numera6690914467698888265omplex(Nb)) ).

% numeral_Bit0
tff(fact_418_numeral__Bit0,axiom,
    ! [Nb: num] : aa_num_real(numeral_numeral_real,aa_num_num(bit0,Nb)) = aa_real_real(plus_plus_real(aa_num_real(numeral_numeral_real,Nb)),aa_num_real(numeral_numeral_real,Nb)) ).

% numeral_Bit0
tff(fact_419_numeral__Bit0,axiom,
    ! [Nb: num] : aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,Nb)) = aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,Nb)),aa_num_rat(numeral_numeral_rat,Nb)) ).

% numeral_Bit0
tff(fact_420_numeral__Bit0,axiom,
    ! [Nb: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,Nb)),aa_num_nat(numeral_numeral_nat,Nb)) ).

% numeral_Bit0
tff(fact_421_numeral__Bit0,axiom,
    ! [Nb: num] : aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,Nb)),aa_num_int(numeral_numeral_int,Nb)) ).

% numeral_Bit0
tff(fact_422_nat__power__less__imp__less,axiom,
    ! [I3: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),I3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I3),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I3),Nb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ) ).

% nat_power_less_imp_less
tff(fact_423_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_424_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_425_power__less__imp__less__base,axiom,
    ! [A: real,Nb: nat,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb))
     => ( 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_426_power__less__imp__less__base,axiom,
    ! [A: rat,Nb: nat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),Nb)),aa_nat_rat(power_power_rat(B),Nb))
     => ( 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_427_power__less__imp__less__base,axiom,
    ! [A: nat,Nb: 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),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb))
     => ( 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_428_power__less__imp__less__base,axiom,
    ! [A: int,Nb: nat,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),Nb)),aa_nat_int(power_power_int(B),Nb))
     => ( 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_429_power__le__imp__le__base,axiom,
    ! [A: real,Nb: nat,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),aa_nat_nat(suc,Nb)))
     => ( 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_430_power__le__imp__le__base,axiom,
    ! [A: rat,Nb: 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,Nb))),aa_nat_rat(power_power_rat(B),aa_nat_nat(suc,Nb)))
     => ( 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_431_power__le__imp__le__base,axiom,
    ! [A: nat,Nb: 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,Nb))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(suc,Nb)))
     => ( 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_432_power__le__imp__le__base,axiom,
    ! [A: int,Nb: 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,Nb))),aa_nat_int(power_power_int(B),aa_nat_nat(suc,Nb)))
     => ( 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_433_power__inject__base,axiom,
    ! [A: real,Nb: nat,B: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,B),aa_nat_nat(suc,Nb)) )
     => ( 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_434_power__inject__base,axiom,
    ! [A: rat,Nb: nat,B: rat] :
      ( ( aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,Nb)) = aa_nat_rat(power_power_rat(B),aa_nat_nat(suc,Nb)) )
     => ( 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_435_power__inject__base,axiom,
    ! [A: nat,Nb: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),aa_nat_nat(suc,Nb)) )
     => ( 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_436_power__inject__base,axiom,
    ! [A: int,Nb: nat,B: int] :
      ( ( aa_nat_int(power_power_int(A),aa_nat_nat(suc,Nb)) = aa_nat_int(power_power_int(B),aa_nat_nat(suc,Nb)) )
     => ( 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_437_zero__power,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_rat(power_power_rat(zero_zero_rat),Nb) = zero_zero_rat ) ) ).

% zero_power
tff(fact_438_zero__power,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),Nb) = zero_zero_nat ) ) ).

% zero_power
tff(fact_439_zero__power,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_int(power_power_int(zero_zero_int),Nb) = zero_zero_int ) ) ).

% zero_power
tff(fact_440_zero__power,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),Nb) = zero_zero_real ) ) ).

% zero_power
tff(fact_441_zero__power,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_complex(power_power_complex(zero_zero_complex),Nb) = zero_zero_complex ) ) ).

% zero_power
tff(fact_442_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_443_power__gt__expt,axiom,
    ! [Nb: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Nb),K)) ) ).

% power_gt_expt
tff(fact_444_nat__one__le__power,axiom,
    ! [I3: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),I3)
     => 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,I3),Nb)) ) ).

% nat_one_le_power
tff(fact_445_power__eq__imp__eq__base,axiom,
    ! [A: real,Nb: nat,B: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb) = aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb) )
     => ( 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),Nb)
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_446_power__eq__imp__eq__base,axiom,
    ! [A: rat,Nb: nat,B: rat] :
      ( ( aa_nat_rat(power_power_rat(A),Nb) = aa_nat_rat(power_power_rat(B),Nb) )
     => ( 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),Nb)
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_447_power__eq__imp__eq__base,axiom,
    ! [A: nat,Nb: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb) )
     => ( 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),Nb)
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_448_power__eq__imp__eq__base,axiom,
    ! [A: int,Nb: nat,B: int] :
      ( ( aa_nat_int(power_power_int(A),Nb) = aa_nat_int(power_power_int(B),Nb) )
     => ( 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),Nb)
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_449_power__eq__iff__eq__base,axiom,
    ! [Nb: nat,A: real,B: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( 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(aa_real_fun_nat_real(power_power_real,A),Nb) = aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb) )
          <=> ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_450_power__eq__iff__eq__base,axiom,
    ! [Nb: nat,A: rat,B: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( 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),Nb) = aa_nat_rat(power_power_rat(B),Nb) )
          <=> ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_451_power__eq__iff__eq__base,axiom,
    ! [Nb: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( 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),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb) )
          <=> ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_452_power__eq__iff__eq__base,axiom,
    ! [Nb: nat,A: int,B: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( 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),Nb) = aa_nat_int(power_power_int(B),Nb) )
          <=> ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_453_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_454_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_455_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_456_zero__power2,axiom,
    aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_real ).

% zero_power2
tff(fact_457_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_458_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_459_less__exp,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% less_exp
tff(fact_460_power2__nat__le__imp__le,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% power2_nat_le_imp_le
tff(fact_461_power2__nat__le__eq__le,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% power2_nat_le_eq_le
tff(fact_462_self__le__ge2__pow,axiom,
    ! [K: nat,Mb: 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,Mb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K),Mb)) ) ).

% self_le_ge2_pow
tff(fact_463_power2__le__imp__le,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Yb),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),Yb)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Yb) ) ) ).

% power2_le_imp_le
tff(fact_464_power2__le__imp__le,axiom,
    ! [Xb: rat,Yb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Yb),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),Yb)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Yb) ) ) ).

% power2_le_imp_le
tff(fact_465_power2__le__imp__le,axiom,
    ! [Xb: nat,Yb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Yb),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),Yb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Yb) ) ) ).

% power2_le_imp_le
tff(fact_466_power2__le__imp__le,axiom,
    ! [Xb: int,Yb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Yb),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),Yb)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Yb) ) ) ).

% power2_le_imp_le
tff(fact_467_power2__eq__imp__eq,axiom,
    ! [Xb: real,Yb: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Yb),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),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Yb)
         => ( Xb = Yb ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_468_power2__eq__imp__eq,axiom,
    ! [Xb: rat,Yb: rat] :
      ( ( aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_rat(power_power_rat(Yb),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),Xb)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Yb)
         => ( Xb = Yb ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_469_power2__eq__imp__eq,axiom,
    ! [Xb: nat,Yb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Yb),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),Xb)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Yb)
         => ( Xb = Yb ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_470_power2__eq__imp__eq,axiom,
    ! [Xb: int,Yb: int] :
      ( ( aa_nat_int(power_power_int(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_int(power_power_int(Yb),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),Xb)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Yb)
         => ( Xb = Yb ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_471_zero__le__power2,axiom,
    ! [A: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% zero_le_power2
tff(fact_472_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_473_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_474_power__strict__mono,axiom,
    ! [A: real,B: real,Nb: 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),Nb)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb)) ) ) ) ).

% power_strict_mono
tff(fact_475_power__strict__mono,axiom,
    ! [A: rat,B: rat,Nb: 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),Nb)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),Nb)),aa_nat_rat(power_power_rat(B),Nb)) ) ) ) ).

% power_strict_mono
tff(fact_476_power__strict__mono,axiom,
    ! [A: nat,B: nat,Nb: 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),Nb)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb)) ) ) ) ).

% power_strict_mono
tff(fact_477_power__strict__mono,axiom,
    ! [A: int,B: int,Nb: 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),Nb)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),Nb)),aa_nat_int(power_power_int(B),Nb)) ) ) ) ).

% power_strict_mono
tff(fact_478_power2__less__0,axiom,
    ! [A: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_real) ).

% power2_less_0
tff(fact_479_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_480_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_481_less__2__cases__iff,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))
    <=> ( ( Nb = zero_zero_nat )
        | ( Nb = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% less_2_cases_iff
tff(fact_482_less__2__cases,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))
     => ( ( Nb = zero_zero_nat )
        | ( Nb = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% less_2_cases
tff(fact_483_power2__less__imp__less,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Yb),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),Yb)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Yb) ) ) ).

% power2_less_imp_less
tff(fact_484_power2__less__imp__less,axiom,
    ! [Xb: rat,Yb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Yb),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),Yb)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Yb) ) ) ).

% power2_less_imp_less
tff(fact_485_power2__less__imp__less,axiom,
    ! [Xb: nat,Yb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Yb),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),Yb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Yb) ) ) ).

% power2_less_imp_less
tff(fact_486_power2__less__imp__less,axiom,
    ! [Xb: int,Yb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Yb),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),Yb)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Yb) ) ) ).

% power2_less_imp_less
tff(fact_487_sum__power2__le__zero__iff,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_real)
    <=> ( ( Xb = zero_zero_real )
        & ( Yb = zero_zero_real ) ) ) ).

% sum_power2_le_zero_iff
tff(fact_488_sum__power2__le__zero__iff,axiom,
    ! [Xb: rat,Yb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_rat)
    <=> ( ( Xb = zero_zero_rat )
        & ( Yb = zero_zero_rat ) ) ) ).

% sum_power2_le_zero_iff
tff(fact_489_sum__power2__le__zero__iff,axiom,
    ! [Xb: int,Yb: 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(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_int)
    <=> ( ( Xb = zero_zero_int )
        & ( Yb = zero_zero_int ) ) ) ).

% sum_power2_le_zero_iff
tff(fact_490_sum__power2__ge__zero,axiom,
    ! [Xb: real,Yb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% sum_power2_ge_zero
tff(fact_491_sum__power2__ge__zero,axiom,
    ! [Xb: rat,Yb: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% sum_power2_ge_zero
tff(fact_492_sum__power2__ge__zero,axiom,
    ! [Xb: int,Yb: 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(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% sum_power2_ge_zero
tff(fact_493_sum__power2__gt__zero__iff,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
    <=> ( ( Xb != zero_zero_real )
        | ( Yb != zero_zero_real ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_494_sum__power2__gt__zero__iff,axiom,
    ! [Xb: rat,Yb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
    <=> ( ( Xb != zero_zero_rat )
        | ( Yb != zero_zero_rat ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_495_sum__power2__gt__zero__iff,axiom,
    ! [Xb: int,Yb: 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(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
    <=> ( ( Xb != zero_zero_int )
        | ( Yb != zero_zero_int ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_496_not__sum__power2__lt__zero,axiom,
    ! [Xb: real,Yb: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_real) ).

% not_sum_power2_lt_zero
tff(fact_497_not__sum__power2__lt__zero,axiom,
    ! [Xb: rat,Yb: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_rat) ).

% not_sum_power2_lt_zero
tff(fact_498_not__sum__power2__lt__zero,axiom,
    ! [Xb: int,Yb: 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(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_int) ).

% not_sum_power2_lt_zero
tff(fact_499__092_060open_062vebt__insert_A_INode_A_ISome_A_Imi_M_Ama_J_J_Adeg_AtreeList_Asummary_J_Ax_A_061_ANode_A_ISome_A_Ix_M_Amax_Ami_Ama_J_J_Adeg_A_ItreeList_091high_Ami_An_A_058_061_Avebt__insert_A_ItreeList_A_B_Ahigh_Ami_An_J_A_Ilow_Ami_An_J_093_J_A_Iif_AminNull_A_ItreeList_A_B_Ahigh_Ami_An_J_Athen_Avebt__insert_Asummary_A_Ihigh_Ami_An_J_Aelse_Asummary_J_092_060close_062,axiom,
    vEBT_vebt_insert(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(mi),ma)),deg,treeList,summary),xa) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(xa),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,mi),ma))),deg,list_u1324408373059187874T_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,mi),na),vEBT_vebt_insert(nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,mi),na)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,mi),na))),
      $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,mi),na))),vEBT_vebt_insert(summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,mi),na)),summary)) ).

% \<open>vebt_insert (Node (Some (mi, ma)) deg treeList summary) x = Node (Some (x, max mi ma)) deg (treeList[high mi n := vebt_insert (treeList ! high mi n) (low mi n)]) (if minNull (treeList ! high mi n) then vebt_insert summary (high mi n) else summary)\<close>
tff(fact_500_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_501_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_502_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_503_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_504_set__swap,axiom,
    ! [I3: nat,Xs: list_VEBT_VEBT,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_s6755466524823107622T_VEBT(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),size_s6755466524823107622T_VEBT(Xs))
       => ( set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(list_u1324408373059187874T_VEBT(Xs,I3,nth_VEBT_VEBT(Xs,J)),J,nth_VEBT_VEBT(Xs,I3))) = set_VEBT_VEBT2(Xs) ) ) ) ).

% set_swap
tff(fact_505_set__swap,axiom,
    ! [I3: nat,Xs: list_o,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_o(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),size_size_list_o(Xs))
       => ( set_o2(list_update_o(list_update_o(Xs,I3,aa_nat_o(nth_o(Xs),J)),J,aa_nat_o(nth_o(Xs),I3))) = set_o2(Xs) ) ) ) ).

% set_swap
tff(fact_506_set__swap,axiom,
    ! [I3: nat,Xs: list_nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_nat(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),size_size_list_nat(Xs))
       => ( set_nat2(list_update_nat(list_update_nat(Xs,I3,aa_nat_nat(nth_nat(Xs),J)),J,aa_nat_nat(nth_nat(Xs),I3))) = set_nat2(Xs) ) ) ) ).

% set_swap
tff(fact_507_set__swap,axiom,
    ! [I3: nat,Xs: list_int,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_int(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),size_size_list_int(Xs))
       => ( set_int2(list_update_int(list_update_int(Xs,I3,aa_nat_int(nth_int(Xs),J)),J,aa_nat_int(nth_int(Xs),I3))) = set_int2(Xs) ) ) ) ).

% set_swap
tff(fact_508_add__self__div__2,axiom,
    ! [Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Mb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = Mb ).

% add_self_div_2
tff(fact_509_div2__Suc__Suc,axiom,
    ! [Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Mb))),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,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% div2_Suc_Suc
tff(fact_510_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(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_511_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),aa_rat_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_512_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(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_513_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),aa_rat_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_514_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(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_515_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,aa_rat_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_516_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(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_517_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,aa_rat_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_518_insert__simp__excp,axiom,
    ! [Mia: nat,Dega: nat,TreeLista: list_VEBT_VEBT,Xb: nat,Maa: 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,Mia),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_nat,Xb),Mia)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)
         => ( ( Xb != Maa )
           => ( vEBT_vebt_insert(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya),Xb) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xb),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Mia),Maa))),Dega,list_u1324408373059187874T_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mia),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),vEBT_vebt_insert(nth_VEBT_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mia),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,Mia),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),
                  $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mia),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_insert(Summarya,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mia),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),Summarya)) ) ) ) ) ) ).

% insert_simp_excp
tff(fact_519_insert__simp__norm,axiom,
    ! [Xb: nat,Dega: nat,TreeLista: list_VEBT_VEBT,Mia: nat,Maa: 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,Xb),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_nat,Mia),Xb)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)
         => ( ( Xb != Maa )
           => ( vEBT_vebt_insert(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya),Xb) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xb),Maa))),Dega,list_u1324408373059187874T_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),vEBT_vebt_insert(nth_VEBT_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),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,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),
                  $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_insert(Summarya,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),Summarya)) ) ) ) ) ) ).

% insert_simp_norm
tff(fact_520_list__update__overwrite,axiom,
    ! [Xs: list_VEBT_VEBT,I3: nat,Xb: vEBT_VEBT,Yb: vEBT_VEBT] : list_u1324408373059187874T_VEBT(list_u1324408373059187874T_VEBT(Xs,I3,Xb),I3,Yb) = list_u1324408373059187874T_VEBT(Xs,I3,Yb) ).

% list_update_overwrite
tff(fact_521_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_522_divide__eq__0__iff,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(divide_divide_real(A),B) = zero_zero_real )
    <=> ( ( A = zero_zero_real )
        | ( B = zero_zero_real ) ) ) ).

% divide_eq_0_iff
tff(fact_523_divide__eq__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(divide_divide_rat(A),B) = zero_zero_rat )
    <=> ( ( A = zero_zero_rat )
        | ( B = zero_zero_rat ) ) ) ).

% divide_eq_0_iff
tff(fact_524_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_525_divide__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( aa_real_real(divide_divide_real(C),A) = aa_real_real(divide_divide_real(C),B) )
    <=> ( ( C = zero_zero_real )
        | ( A = B ) ) ) ).

% divide_cancel_left
tff(fact_526_divide__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( aa_rat_rat(divide_divide_rat(C),A) = aa_rat_rat(divide_divide_rat(C),B) )
    <=> ( ( C = zero_zero_rat )
        | ( A = B ) ) ) ).

% divide_cancel_left
tff(fact_527_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_528_divide__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( ( aa_real_real(divide_divide_real(A),C) = aa_real_real(divide_divide_real(B),C) )
    <=> ( ( C = zero_zero_real )
        | ( A = B ) ) ) ).

% divide_cancel_right
tff(fact_529_divide__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( aa_rat_rat(divide_divide_rat(A),C) = aa_rat_rat(divide_divide_rat(B),C) )
    <=> ( ( C = zero_zero_rat )
        | ( A = B ) ) ) ).

% divide_cancel_right
tff(fact_530_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_531_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_532_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_533_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_534_division__ring__divide__zero,axiom,
    ! [A: complex] : aa_complex_complex(divide1717551699836669952omplex(A),zero_zero_complex) = zero_zero_complex ).

% division_ring_divide_zero
tff(fact_535_division__ring__divide__zero,axiom,
    ! [A: real] : aa_real_real(divide_divide_real(A),zero_zero_real) = zero_zero_real ).

% division_ring_divide_zero
tff(fact_536_division__ring__divide__zero,axiom,
    ! [A: rat] : aa_rat_rat(divide_divide_rat(A),zero_zero_rat) = zero_zero_rat ).

% division_ring_divide_zero
tff(fact_537_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_538_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_539_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_540_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_541_length__list__update,axiom,
    ! [Xs: list_VEBT_VEBT,I3: nat,Xb: vEBT_VEBT] : size_s6755466524823107622T_VEBT(list_u1324408373059187874T_VEBT(Xs,I3,Xb)) = size_s6755466524823107622T_VEBT(Xs) ).

% length_list_update
tff(fact_542_length__list__update,axiom,
    ! [Xs: list_o,I3: nat,Xb: $o] : size_size_list_o(list_update_o(Xs,I3,(Xb))) = size_size_list_o(Xs) ).

% length_list_update
tff(fact_543_length__list__update,axiom,
    ! [Xs: list_nat,I3: nat,Xb: nat] : size_size_list_nat(list_update_nat(Xs,I3,Xb)) = size_size_list_nat(Xs) ).

% length_list_update
tff(fact_544_length__list__update,axiom,
    ! [Xs: list_int,I3: nat,Xb: int] : size_size_list_int(list_update_int(Xs,I3,Xb)) = size_size_list_int(Xs) ).

% length_list_update
tff(fact_545_list__update__id,axiom,
    ! [Xs: list_nat,I3: nat] : list_update_nat(Xs,I3,aa_nat_nat(nth_nat(Xs),I3)) = Xs ).

% list_update_id
tff(fact_546_list__update__id,axiom,
    ! [Xs: list_int,I3: nat] : list_update_int(Xs,I3,aa_nat_int(nth_int(Xs),I3)) = Xs ).

% list_update_id
tff(fact_547_list__update__id,axiom,
    ! [Xs: list_VEBT_VEBT,I3: nat] : list_u1324408373059187874T_VEBT(Xs,I3,nth_VEBT_VEBT(Xs,I3)) = Xs ).

% list_update_id
tff(fact_548_nth__list__update__neq,axiom,
    ! [I3: nat,J: nat,Xs: list_nat,Xb: nat] :
      ( ( I3 != J )
     => ( aa_nat_nat(nth_nat(list_update_nat(Xs,I3,Xb)),J) = aa_nat_nat(nth_nat(Xs),J) ) ) ).

% nth_list_update_neq
tff(fact_549_nth__list__update__neq,axiom,
    ! [I3: nat,J: nat,Xs: list_int,Xb: int] :
      ( ( I3 != J )
     => ( aa_nat_int(nth_int(list_update_int(Xs,I3,Xb)),J) = aa_nat_int(nth_int(Xs),J) ) ) ).

% nth_list_update_neq
tff(fact_550_nth__list__update__neq,axiom,
    ! [I3: nat,J: nat,Xs: list_VEBT_VEBT,Xb: vEBT_VEBT] :
      ( ( I3 != J )
     => ( nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(Xs,I3,Xb),J) = nth_VEBT_VEBT(Xs,J) ) ) ).

% nth_list_update_neq
tff(fact_551_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_552_divide__eq__1__iff,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(divide_divide_real(A),B) = one_one_real )
    <=> ( ( B != zero_zero_real )
        & ( A = B ) ) ) ).

% divide_eq_1_iff
tff(fact_553_divide__eq__1__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(divide_divide_rat(A),B) = one_one_rat )
    <=> ( ( B != zero_zero_rat )
        & ( A = B ) ) ) ).

% divide_eq_1_iff
tff(fact_554_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_555_one__eq__divide__iff,axiom,
    ! [A: real,B: real] :
      ( ( one_one_real = aa_real_real(divide_divide_real(A),B) )
    <=> ( ( B != zero_zero_real )
        & ( A = B ) ) ) ).

% one_eq_divide_iff
tff(fact_556_one__eq__divide__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( one_one_rat = aa_rat_rat(divide_divide_rat(A),B) )
    <=> ( ( B != zero_zero_rat )
        & ( A = B ) ) ) ).

% one_eq_divide_iff
tff(fact_557_divide__self,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(A),A) = one_one_complex ) ) ).

% divide_self
tff(fact_558_divide__self,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(divide_divide_real(A),A) = one_one_real ) ) ).

% divide_self
tff(fact_559_divide__self,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(A),A) = one_one_rat ) ) ).

% divide_self
tff(fact_560_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_561_divide__self__if,axiom,
    ! [A: real] :
      aa_real_real(divide_divide_real(A),A) = $ite(A = zero_zero_real,zero_zero_real,one_one_real) ).

% divide_self_if
tff(fact_562_divide__self__if,axiom,
    ! [A: rat] :
      aa_rat_rat(divide_divide_rat(A),A) = $ite(A = zero_zero_rat,zero_zero_rat,one_one_rat) ).

% divide_self_if
tff(fact_563_divide__eq__eq__1,axiom,
    ! [B: real,A: real] :
      ( ( aa_real_real(divide_divide_real(B),A) = one_one_real )
    <=> ( ( A != zero_zero_real )
        & ( A = B ) ) ) ).

% divide_eq_eq_1
tff(fact_564_divide__eq__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( ( aa_rat_rat(divide_divide_rat(B),A) = one_one_rat )
    <=> ( ( A != zero_zero_rat )
        & ( A = B ) ) ) ).

% divide_eq_eq_1
tff(fact_565_eq__divide__eq__1,axiom,
    ! [B: real,A: real] :
      ( ( one_one_real = aa_real_real(divide_divide_real(B),A) )
    <=> ( ( A != zero_zero_real )
        & ( A = B ) ) ) ).

% eq_divide_eq_1
tff(fact_566_eq__divide__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( ( one_one_rat = aa_rat_rat(divide_divide_rat(B),A) )
    <=> ( ( A != zero_zero_rat )
        & ( A = B ) ) ) ).

% eq_divide_eq_1
tff(fact_567_one__divide__eq__0__iff,axiom,
    ! [A: real] :
      ( ( aa_real_real(divide_divide_real(one_one_real),A) = zero_zero_real )
    <=> ( A = zero_zero_real ) ) ).

% one_divide_eq_0_iff
tff(fact_568_one__divide__eq__0__iff,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(divide_divide_rat(one_one_rat),A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% one_divide_eq_0_iff
tff(fact_569_zero__eq__1__divide__iff,axiom,
    ! [A: real] :
      ( ( zero_zero_real = aa_real_real(divide_divide_real(one_one_real),A) )
    <=> ( A = zero_zero_real ) ) ).

% zero_eq_1_divide_iff
tff(fact_570_zero__eq__1__divide__iff,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat = aa_rat_rat(divide_divide_rat(one_one_rat),A) )
    <=> ( A = zero_zero_rat ) ) ).

% zero_eq_1_divide_iff
tff(fact_571_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      aa_real_real(ord_max_real(aa_num_real(numeral_numeral_real,U)),aa_num_real(numeral_numeral_real,V)) = $ite(aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,U)),aa_num_real(numeral_numeral_real,V)),aa_num_real(numeral_numeral_real,V),aa_num_real(numeral_numeral_real,U)) ).

% max_number_of(1)
tff(fact_572_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      aa_rat_rat(ord_max_rat(aa_num_rat(numeral_numeral_rat,U)),aa_num_rat(numeral_numeral_rat,V)) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,U)),aa_num_rat(numeral_numeral_rat,V)),aa_num_rat(numeral_numeral_rat,V),aa_num_rat(numeral_numeral_rat,U)) ).

% max_number_of(1)
tff(fact_573_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(numeral_numeral_nat,U)),aa_num_nat(numeral_numeral_nat,V)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,U)),aa_num_nat(numeral_numeral_nat,V)),aa_num_nat(numeral_numeral_nat,V),aa_num_nat(numeral_numeral_nat,U)) ).

% max_number_of(1)
tff(fact_574_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      aa_int_int(aa_int_fun_int_int(ord_max_int,aa_num_int(numeral_numeral_int,U)),aa_num_int(numeral_numeral_int,V)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,U)),aa_num_int(numeral_numeral_int,V)),aa_num_int(numeral_numeral_int,V),aa_num_int(numeral_numeral_int,U)) ).

% max_number_of(1)
tff(fact_575_max__0__1_I4_J,axiom,
    ! [Xb: num] : aa_real_real(ord_max_real(aa_num_real(numeral_numeral_real,Xb)),zero_zero_real) = aa_num_real(numeral_numeral_real,Xb) ).

% max_0_1(4)
tff(fact_576_max__0__1_I4_J,axiom,
    ! [Xb: num] : aa_rat_rat(ord_max_rat(aa_num_rat(numeral_numeral_rat,Xb)),zero_zero_rat) = aa_num_rat(numeral_numeral_rat,Xb) ).

% max_0_1(4)
tff(fact_577_max__0__1_I4_J,axiom,
    ! [Xb: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(numeral_numeral_nat,Xb)),zero_zero_nat) = aa_num_nat(numeral_numeral_nat,Xb) ).

% max_0_1(4)
tff(fact_578_max__0__1_I4_J,axiom,
    ! [Xb: num] : aa_int_int(aa_int_fun_int_int(ord_max_int,aa_num_int(numeral_numeral_int,Xb)),zero_zero_int) = aa_num_int(numeral_numeral_int,Xb) ).

% max_0_1(4)
tff(fact_579_max__0__1_I3_J,axiom,
    ! [Xb: num] : aa_real_real(ord_max_real(zero_zero_real),aa_num_real(numeral_numeral_real,Xb)) = aa_num_real(numeral_numeral_real,Xb) ).

% max_0_1(3)
tff(fact_580_max__0__1_I3_J,axiom,
    ! [Xb: num] : aa_rat_rat(ord_max_rat(zero_zero_rat),aa_num_rat(numeral_numeral_rat,Xb)) = aa_num_rat(numeral_numeral_rat,Xb) ).

% max_0_1(3)
tff(fact_581_max__0__1_I3_J,axiom,
    ! [Xb: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,Xb)) = aa_num_nat(numeral_numeral_nat,Xb) ).

% max_0_1(3)
tff(fact_582_max__0__1_I3_J,axiom,
    ! [Xb: num] : aa_int_int(aa_int_fun_int_int(ord_max_int,zero_zero_int),aa_num_int(numeral_numeral_int,Xb)) = aa_num_int(numeral_numeral_int,Xb) ).

% max_0_1(3)
tff(fact_583_max__0__1_I2_J,axiom,
    aa_real_real(ord_max_real(one_one_real),zero_zero_real) = one_one_real ).

% max_0_1(2)
tff(fact_584_max__0__1_I2_J,axiom,
    aa_rat_rat(ord_max_rat(one_one_rat),zero_zero_rat) = one_one_rat ).

% max_0_1(2)
tff(fact_585_max__0__1_I2_J,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,one_one_nat),zero_zero_nat) = one_one_nat ).

% max_0_1(2)
tff(fact_586_max__0__1_I2_J,axiom,
    aa_int_int(aa_int_fun_int_int(ord_max_int,one_one_int),zero_zero_int) = one_one_int ).

% max_0_1(2)
tff(fact_587_max__0__1_I1_J,axiom,
    aa_real_real(ord_max_real(zero_zero_real),one_one_real) = one_one_real ).

% max_0_1(1)
tff(fact_588_max__0__1_I1_J,axiom,
    aa_rat_rat(ord_max_rat(zero_zero_rat),one_one_rat) = one_one_rat ).

% max_0_1(1)
tff(fact_589_max__0__1_I1_J,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,zero_zero_nat),one_one_nat) = one_one_nat ).

% max_0_1(1)
tff(fact_590_max__0__1_I1_J,axiom,
    aa_int_int(aa_int_fun_int_int(ord_max_int,zero_zero_int),one_one_int) = one_one_int ).

% max_0_1(1)
tff(fact_591_max__0__1_I6_J,axiom,
    ! [Xb: num] : aa_real_real(ord_max_real(aa_num_real(numeral_numeral_real,Xb)),one_one_real) = aa_num_real(numeral_numeral_real,Xb) ).

% max_0_1(6)
tff(fact_592_max__0__1_I6_J,axiom,
    ! [Xb: num] : aa_rat_rat(ord_max_rat(aa_num_rat(numeral_numeral_rat,Xb)),one_one_rat) = aa_num_rat(numeral_numeral_rat,Xb) ).

% max_0_1(6)
tff(fact_593_max__0__1_I6_J,axiom,
    ! [Xb: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(numeral_numeral_nat,Xb)),one_one_nat) = aa_num_nat(numeral_numeral_nat,Xb) ).

% max_0_1(6)
tff(fact_594_max__0__1_I6_J,axiom,
    ! [Xb: num] : aa_int_int(aa_int_fun_int_int(ord_max_int,aa_num_int(numeral_numeral_int,Xb)),one_one_int) = aa_num_int(numeral_numeral_int,Xb) ).

% max_0_1(6)
tff(fact_595_max__0__1_I5_J,axiom,
    ! [Xb: num] : aa_real_real(ord_max_real(one_one_real),aa_num_real(numeral_numeral_real,Xb)) = aa_num_real(numeral_numeral_real,Xb) ).

% max_0_1(5)
tff(fact_596_max__0__1_I5_J,axiom,
    ! [Xb: num] : aa_rat_rat(ord_max_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,Xb)) = aa_num_rat(numeral_numeral_rat,Xb) ).

% max_0_1(5)
tff(fact_597_max__0__1_I5_J,axiom,
    ! [Xb: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,Xb)) = aa_num_nat(numeral_numeral_nat,Xb) ).

% max_0_1(5)
tff(fact_598_max__0__1_I5_J,axiom,
    ! [Xb: num] : aa_int_int(aa_int_fun_int_int(ord_max_int,one_one_int),aa_num_int(numeral_numeral_int,Xb)) = aa_num_int(numeral_numeral_int,Xb) ).

% max_0_1(5)
tff(fact_599_div__by__Suc__0,axiom,
    ! [Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),aa_nat_nat(suc,zero_zero_nat)) = Mb ).

% div_by_Suc_0
tff(fact_600_div__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb) = zero_zero_nat ) ) ).

% div_less
tff(fact_601_list__update__beyond,axiom,
    ! [Xs: list_VEBT_VEBT,I3: nat,Xb: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_s6755466524823107622T_VEBT(Xs)),I3)
     => ( list_u1324408373059187874T_VEBT(Xs,I3,Xb) = Xs ) ) ).

% list_update_beyond
tff(fact_602_list__update__beyond,axiom,
    ! [Xs: list_o,I3: nat,Xb: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_o(Xs)),I3)
     => ( list_update_o(Xs,I3,(Xb)) = Xs ) ) ).

% list_update_beyond
tff(fact_603_list__update__beyond,axiom,
    ! [Xs: list_nat,I3: nat,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_nat(Xs)),I3)
     => ( list_update_nat(Xs,I3,Xb) = Xs ) ) ).

% list_update_beyond
tff(fact_604_list__update__beyond,axiom,
    ! [Xs: list_int,I3: nat,Xb: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_int(Xs)),I3)
     => ( list_update_int(Xs,I3,Xb) = Xs ) ) ).

% list_update_beyond
tff(fact_605_divide__le__0__1__iff,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_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_606_divide__le__0__1__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_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_607_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(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_608_zero__le__divide__1__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_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_609_divide__less__0__1__iff,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_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_610_divide__less__0__1__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_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_611_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(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_612_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,aa_rat_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_613_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(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_614_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,aa_rat_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_615_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(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_616_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),aa_rat_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_617_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(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_618_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),aa_rat_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_619_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(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_620_zero__less__divide__1__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_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_621_nth__list__update__eq,axiom,
    ! [I3: nat,Xs: list_VEBT_VEBT,Xb: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_s6755466524823107622T_VEBT(Xs))
     => ( nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(Xs,I3,Xb),I3) = Xb ) ) ).

% nth_list_update_eq
tff(fact_622_nth__list__update__eq,axiom,
    ! [I3: nat,Xs: list_o,Xb: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_o(Xs))
     => ( aa_nat_o(nth_o(list_update_o(Xs,I3,(Xb))),I3)
      <=> (Xb) ) ) ).

% nth_list_update_eq
tff(fact_623_nth__list__update__eq,axiom,
    ! [I3: nat,Xs: list_nat,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_nat(Xs))
     => ( aa_nat_nat(nth_nat(list_update_nat(Xs,I3,Xb)),I3) = Xb ) ) ).

% nth_list_update_eq
tff(fact_624_nth__list__update__eq,axiom,
    ! [I3: nat,Xs: list_int,Xb: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_int(Xs))
     => ( aa_nat_int(nth_int(list_update_int(Xs,I3,Xb)),I3) = Xb ) ) ).

% nth_list_update_eq
tff(fact_625_not__exp__less__eq__0__int,axiom,
    ! [Nb: 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))),Nb)),zero_zero_int) ).

% not_exp_less_eq_0_int
tff(fact_626_linordered__field__no__lb,axiom,
    ! [X: real] :
    ? [Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,Y),X) ).

% linordered_field_no_lb
tff(fact_627_linordered__field__no__lb,axiom,
    ! [X: rat] :
    ? [Y: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),X) ).

% linordered_field_no_lb
tff(fact_628_linordered__field__no__ub,axiom,
    ! [X: real] :
    ? [X_12: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,X),X_12) ).

% linordered_field_no_ub
tff(fact_629_linordered__field__no__ub,axiom,
    ! [X: rat] :
    ? [X_12: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),X_12) ).

% linordered_field_no_ub
tff(fact_630_subset__code_I1_J,axiom,
    ! [Xs: list_complex,B4: set_complex] :
      ( ord_le211207098394363844omplex(set_complex2(Xs),B4)
    <=> ! [X4: complex] :
          ( member_complex(X4,set_complex2(Xs))
         => member_complex(X4,B4) ) ) ).

% subset_code(1)
tff(fact_631_subset__code_I1_J,axiom,
    ! [Xs: list_real,B4: set_real] :
      ( ord_less_eq_set_real(set_real2(Xs),B4)
    <=> ! [X4: real] :
          ( member_real(X4,set_real2(Xs))
         => member_real(X4,B4) ) ) ).

% subset_code(1)
tff(fact_632_subset__code_I1_J,axiom,
    ! [Xs: list_set_nat,B4: set_set_nat] :
      ( ord_le6893508408891458716et_nat(set_set_nat2(Xs),B4)
    <=> ! [X4: set_nat] :
          ( member_set_nat(X4,set_set_nat2(Xs))
         => member_set_nat(X4,B4) ) ) ).

% subset_code(1)
tff(fact_633_subset__code_I1_J,axiom,
    ! [Xs: list_VEBT_VEBT,B4: set_VEBT_VEBT] :
      ( ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(Xs),B4)
    <=> ! [X4: vEBT_VEBT] :
          ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(Xs))
         => member_VEBT_VEBT(X4,B4) ) ) ).

% subset_code(1)
tff(fact_634_subset__code_I1_J,axiom,
    ! [Xs: list_int,B4: set_int] :
      ( ord_less_eq_set_int(set_int2(Xs),B4)
    <=> ! [X4: int] :
          ( member_int(X4,set_int2(Xs))
         => member_int(X4,B4) ) ) ).

% subset_code(1)
tff(fact_635_subset__code_I1_J,axiom,
    ! [Xs: list_nat,B4: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(set_nat2(Xs)),B4)
    <=> ! [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),set_nat2(Xs))
         => aa_set_nat_o(member_nat(X4),B4) ) ) ).

% subset_code(1)
tff(fact_636_Ex__list__of__length,axiom,
    ! [Nb: nat] :
    ? [Xs2: list_VEBT_VEBT] : size_s6755466524823107622T_VEBT(Xs2) = Nb ).

% Ex_list_of_length
tff(fact_637_Ex__list__of__length,axiom,
    ! [Nb: nat] :
    ? [Xs2: list_o] : size_size_list_o(Xs2) = Nb ).

% Ex_list_of_length
tff(fact_638_Ex__list__of__length,axiom,
    ! [Nb: nat] :
    ? [Xs2: list_nat] : size_size_list_nat(Xs2) = Nb ).

% Ex_list_of_length
tff(fact_639_Ex__list__of__length,axiom,
    ! [Nb: nat] :
    ? [Xs2: list_int] : size_size_list_int(Xs2) = Nb ).

% Ex_list_of_length
tff(fact_640_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_641_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_642_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_643_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_644_list__update__swap,axiom,
    ! [I3: nat,I4: nat,Xs: list_VEBT_VEBT,Xb: vEBT_VEBT,X5: vEBT_VEBT] :
      ( ( I3 != I4 )
     => ( list_u1324408373059187874T_VEBT(list_u1324408373059187874T_VEBT(Xs,I3,Xb),I4,X5) = list_u1324408373059187874T_VEBT(list_u1324408373059187874T_VEBT(Xs,I4,X5),I3,Xb) ) ) ).

% list_update_swap
tff(fact_645_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_646_add__divide__distrib,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(A),B)),C) = aa_real_real(plus_plus_real(aa_real_real(divide_divide_real(A),C)),aa_real_real(divide_divide_real(B),C)) ).

% add_divide_distrib
tff(fact_647_add__divide__distrib,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(A),B)),C) = aa_rat_rat(plus_plus_rat(aa_rat_rat(divide_divide_rat(A),C)),aa_rat_rat(divide_divide_rat(B),C)) ).

% add_divide_distrib
tff(fact_648_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_649_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_650_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_651_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_652_div__le__mono,axiom,
    ! [Mb: nat,Nb: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),K)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),K)) ) ).

% div_le_mono
tff(fact_653_div__le__dividend,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)),Mb) ).

% div_le_dividend
tff(fact_654_set__update__subsetI,axiom,
    ! [Xs: list_complex,A3: set_complex,Xb: complex,I3: nat] :
      ( ord_le211207098394363844omplex(set_complex2(Xs),A3)
     => ( member_complex(Xb,A3)
       => ord_le211207098394363844omplex(set_complex2(list_update_complex(Xs,I3,Xb)),A3) ) ) ).

% set_update_subsetI
tff(fact_655_set__update__subsetI,axiom,
    ! [Xs: list_real,A3: set_real,Xb: real,I3: nat] :
      ( ord_less_eq_set_real(set_real2(Xs),A3)
     => ( member_real(Xb,A3)
       => ord_less_eq_set_real(set_real2(list_update_real(Xs,I3,Xb)),A3) ) ) ).

% set_update_subsetI
tff(fact_656_set__update__subsetI,axiom,
    ! [Xs: list_set_nat,A3: set_set_nat,Xb: set_nat,I3: nat] :
      ( ord_le6893508408891458716et_nat(set_set_nat2(Xs),A3)
     => ( member_set_nat(Xb,A3)
       => ord_le6893508408891458716et_nat(set_set_nat2(list_update_set_nat(Xs,I3,Xb)),A3) ) ) ).

% set_update_subsetI
tff(fact_657_set__update__subsetI,axiom,
    ! [Xs: list_int,A3: set_int,Xb: int,I3: nat] :
      ( ord_less_eq_set_int(set_int2(Xs),A3)
     => ( member_int(Xb,A3)
       => ord_less_eq_set_int(set_int2(list_update_int(Xs,I3,Xb)),A3) ) ) ).

% set_update_subsetI
tff(fact_658_set__update__subsetI,axiom,
    ! [Xs: list_VEBT_VEBT,A3: set_VEBT_VEBT,Xb: vEBT_VEBT,I3: nat] :
      ( ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(Xs),A3)
     => ( member_VEBT_VEBT(Xb,A3)
       => ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(Xs,I3,Xb)),A3) ) ) ).

% set_update_subsetI
tff(fact_659_set__update__subsetI,axiom,
    ! [Xs: list_nat,A3: set_nat,Xb: nat,I3: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(set_nat2(Xs)),A3)
     => ( aa_set_nat_o(member_nat(Xb),A3)
       => aa_set_nat_o(ord_less_eq_set_nat(set_nat2(list_update_nat(Xs,I3,Xb))),A3) ) ) ).

% set_update_subsetI
tff(fact_660_divide__le__0__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_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_661_divide__le__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_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_662_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(divide_divide_real(A),C)),aa_real_real(divide_divide_real(B),C)) ) ) ).

% divide_right_mono
tff(fact_663_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,aa_rat_rat(divide_divide_rat(A),C)),aa_rat_rat(divide_divide_rat(B),C)) ) ) ).

% divide_right_mono
tff(fact_664_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(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_665_zero__le__divide__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_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_666_divide__nonneg__nonneg,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Yb)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(divide_divide_real(Xb),Yb)) ) ) ).

% divide_nonneg_nonneg
tff(fact_667_divide__nonneg__nonneg,axiom,
    ! [Xb: rat,Yb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Yb)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(Xb),Yb)) ) ) ).

% divide_nonneg_nonneg
tff(fact_668_divide__nonneg__nonpos,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Yb),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(Xb),Yb)),zero_zero_real) ) ) ).

% divide_nonneg_nonpos
tff(fact_669_divide__nonneg__nonpos,axiom,
    ! [Xb: rat,Yb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Yb),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(Xb),Yb)),zero_zero_rat) ) ) ).

% divide_nonneg_nonpos
tff(fact_670_divide__nonpos__nonneg,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Yb)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(Xb),Yb)),zero_zero_real) ) ) ).

% divide_nonpos_nonneg
tff(fact_671_divide__nonpos__nonneg,axiom,
    ! [Xb: rat,Yb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Yb)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(Xb),Yb)),zero_zero_rat) ) ) ).

% divide_nonpos_nonneg
tff(fact_672_divide__nonpos__nonpos,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Yb),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(divide_divide_real(Xb),Yb)) ) ) ).

% divide_nonpos_nonpos
tff(fact_673_divide__nonpos__nonpos,axiom,
    ! [Xb: rat,Yb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Yb),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(Xb),Yb)) ) ) ).

% divide_nonpos_nonpos
tff(fact_674_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(divide_divide_real(B),C)),aa_real_real(divide_divide_real(A),C)) ) ) ).

% divide_right_mono_neg
tff(fact_675_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,aa_rat_rat(divide_divide_rat(B),C)),aa_rat_rat(divide_divide_rat(A),C)) ) ) ).

% divide_right_mono_neg
tff(fact_676_divide__neg__neg,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Yb),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(divide_divide_real(Xb),Yb)) ) ) ).

% divide_neg_neg
tff(fact_677_divide__neg__neg,axiom,
    ! [Xb: rat,Yb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Yb),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(Xb),Yb)) ) ) ).

% divide_neg_neg
tff(fact_678_divide__neg__pos,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Yb)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(Xb),Yb)),zero_zero_real) ) ) ).

% divide_neg_pos
tff(fact_679_divide__neg__pos,axiom,
    ! [Xb: rat,Yb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Yb)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(Xb),Yb)),zero_zero_rat) ) ) ).

% divide_neg_pos
tff(fact_680_divide__pos__neg,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Yb),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(Xb),Yb)),zero_zero_real) ) ) ).

% divide_pos_neg
tff(fact_681_divide__pos__neg,axiom,
    ! [Xb: rat,Yb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Yb),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(Xb),Yb)),zero_zero_rat) ) ) ).

% divide_pos_neg
tff(fact_682_divide__pos__pos,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Yb)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(divide_divide_real(Xb),Yb)) ) ) ).

% divide_pos_pos
tff(fact_683_divide__pos__pos,axiom,
    ! [Xb: rat,Yb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Yb)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(Xb),Yb)) ) ) ).

% divide_pos_pos
tff(fact_684_divide__less__0__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_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_685_divide__less__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_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_686_divide__less__cancel,axiom,
    ! [A: real,C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(A),C)),aa_real_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_687_divide__less__cancel,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(A),C)),aa_rat_rat(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_688_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(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_689_zero__less__divide__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_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_690_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(divide_divide_real(A),C)),aa_real_real(divide_divide_real(B),C)) ) ) ).

% divide_strict_right_mono
tff(fact_691_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,aa_rat_rat(divide_divide_rat(A),C)),aa_rat_rat(divide_divide_rat(B),C)) ) ) ).

% divide_strict_right_mono
tff(fact_692_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(divide_divide_real(A),C)),aa_real_real(divide_divide_real(B),C)) ) ) ).

% divide_strict_right_mono_neg
tff(fact_693_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,aa_rat_rat(divide_divide_rat(A),C)),aa_rat_rat(divide_divide_rat(B),C)) ) ) ).

% divide_strict_right_mono_neg
tff(fact_694_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_695_right__inverse__eq,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( ( aa_real_real(divide_divide_real(A),B) = one_one_real )
      <=> ( A = B ) ) ) ).

% right_inverse_eq
tff(fact_696_right__inverse__eq,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( ( aa_rat_rat(divide_divide_rat(A),B) = one_one_rat )
      <=> ( A = B ) ) ) ).

% right_inverse_eq
tff(fact_697_Euclidean__Division_Odiv__eq__0__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb) = zero_zero_nat )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
        | ( Nb = zero_zero_nat ) ) ) ).

% Euclidean_Division.div_eq_0_iff
tff(fact_698_Suc__div__le__mono,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,Mb)),Nb)) ).

% Suc_div_le_mono
tff(fact_699_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_700_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_701_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_702_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_703_Skolem__list__nth,axiom,
    ! [K: nat,P: fun_na6887940067898170868VEBT_o] :
      ( ! [I5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),K)
         => ? [X_13: vEBT_VEBT] : aa_VEBT_VEBT_o(aa_nat1925708948643195573VEBT_o(P,I5),X_13) )
    <=> ? [Xs3: list_VEBT_VEBT] :
          ( ( size_s6755466524823107622T_VEBT(Xs3) = K )
          & ! [I5: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),K)
             => aa_VEBT_VEBT_o(aa_nat1925708948643195573VEBT_o(P,I5),nth_VEBT_VEBT(Xs3,I5)) ) ) ) ).

% Skolem_list_nth
tff(fact_704_Skolem__list__nth,axiom,
    ! [K: nat,P: fun_nat_fun_o_o] :
      ( ! [I5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),K)
         => ? [X_13: $o] : aa_o_o(aa_nat_fun_o_o(P,I5),(X_13)) )
    <=> ? [Xs3: list_o] :
          ( ( size_size_list_o(Xs3) = K )
          & ! [I5: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),K)
             => aa_o_o(aa_nat_fun_o_o(P,I5),aa_nat_o(nth_o(Xs3),I5)) ) ) ) ).

% Skolem_list_nth
tff(fact_705_Skolem__list__nth,axiom,
    ! [K: nat,P: fun_nat_fun_nat_o] :
      ( ! [I5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),K)
         => ? [X_13: nat] : aa_nat_o(aa_nat_fun_nat_o(P,I5),X_13) )
    <=> ? [Xs3: list_nat] :
          ( ( size_size_list_nat(Xs3) = K )
          & ! [I5: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),K)
             => aa_nat_o(aa_nat_fun_nat_o(P,I5),aa_nat_nat(nth_nat(Xs3),I5)) ) ) ) ).

% Skolem_list_nth
tff(fact_706_Skolem__list__nth,axiom,
    ! [K: nat,P: fun_nat_fun_int_o] :
      ( ! [I5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),K)
         => ? [X_13: int] : aa_int_o(aa_nat_fun_int_o(P,I5),X_13) )
    <=> ? [Xs3: list_int] :
          ( ( size_size_list_int(Xs3) = K )
          & ! [I5: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),K)
             => aa_int_o(aa_nat_fun_int_o(P,I5),aa_nat_int(nth_int(Xs3),I5)) ) ) ) ).

% Skolem_list_nth
tff(fact_707_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( ( Xs = Ys )
    <=> ( ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Ys) )
        & ! [I5: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),size_s6755466524823107622T_VEBT(Xs))
           => ( nth_VEBT_VEBT(Xs,I5) = nth_VEBT_VEBT(Ys,I5) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_708_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_o,Ys: list_o] :
      ( ( Xs = Ys )
    <=> ( ( size_size_list_o(Xs) = size_size_list_o(Ys) )
        & ! [I5: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),size_size_list_o(Xs))
           => ( aa_nat_o(nth_o(Xs),I5)
            <=> aa_nat_o(nth_o(Ys),I5) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_709_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( Xs = Ys )
    <=> ( ( size_size_list_nat(Xs) = size_size_list_nat(Ys) )
        & ! [I5: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),size_size_list_nat(Xs))
           => ( aa_nat_nat(nth_nat(Xs),I5) = aa_nat_nat(nth_nat(Ys),I5) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_710_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( Xs = Ys )
    <=> ( ( size_size_list_int(Xs) = size_size_list_int(Ys) )
        & ! [I5: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),size_size_list_int(Xs))
           => ( aa_nat_int(nth_int(Xs),I5) = aa_nat_int(nth_int(Ys),I5) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_711_field__le__epsilon,axiom,
    ! [Xb: real,Yb: 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,Xb),aa_real_real(plus_plus_real(Yb),E)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Yb) ) ).

% field_le_epsilon
tff(fact_712_field__le__epsilon,axiom,
    ! [Xb: rat,Yb: 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,Xb),aa_rat_rat(plus_plus_rat(Yb),E)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Yb) ) ).

% field_le_epsilon
tff(fact_713_frac__le,axiom,
    ! [Yb: real,Xb: real,W: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Yb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Yb)
       => ( 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(divide_divide_real(Xb),Z)),aa_real_real(divide_divide_real(Yb),W)) ) ) ) ) ).

% frac_le
tff(fact_714_frac__le,axiom,
    ! [Yb: rat,Xb: rat,W: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Yb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Yb)
       => ( 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,aa_rat_rat(divide_divide_rat(Xb),Z)),aa_rat_rat(divide_divide_rat(Yb),W)) ) ) ) ) ).

% frac_le
tff(fact_715_frac__less,axiom,
    ! [Xb: real,Yb: real,W: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Yb)
       => ( 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(divide_divide_real(Xb),Z)),aa_real_real(divide_divide_real(Yb),W)) ) ) ) ) ).

% frac_less
tff(fact_716_frac__less,axiom,
    ! [Xb: rat,Yb: rat,W: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Yb)
       => ( 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,aa_rat_rat(divide_divide_rat(Xb),Z)),aa_rat_rat(divide_divide_rat(Yb),W)) ) ) ) ) ).

% frac_less
tff(fact_717_frac__less2,axiom,
    ! [Xb: real,Yb: real,W: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Yb)
       => ( 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(divide_divide_real(Xb),Z)),aa_real_real(divide_divide_real(Yb),W)) ) ) ) ) ).

% frac_less2
tff(fact_718_frac__less2,axiom,
    ! [Xb: rat,Yb: rat,W: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Yb)
       => ( 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,aa_rat_rat(divide_divide_rat(Xb),Z)),aa_rat_rat(divide_divide_rat(Yb),W)) ) ) ) ) ).

% frac_less2
tff(fact_719_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(divide_divide_real(A),C)),aa_real_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_720_divide__le__cancel,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(A),C)),aa_rat_rat(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_721_divide__nonneg__neg,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Yb),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(Xb),Yb)),zero_zero_real) ) ) ).

% divide_nonneg_neg
tff(fact_722_divide__nonneg__neg,axiom,
    ! [Xb: rat,Yb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Yb),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(Xb),Yb)),zero_zero_rat) ) ) ).

% divide_nonneg_neg
tff(fact_723_divide__nonneg__pos,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Yb)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(divide_divide_real(Xb),Yb)) ) ) ).

% divide_nonneg_pos
tff(fact_724_divide__nonneg__pos,axiom,
    ! [Xb: rat,Yb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Yb)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(Xb),Yb)) ) ) ).

% divide_nonneg_pos
tff(fact_725_divide__nonpos__neg,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Yb),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(divide_divide_real(Xb),Yb)) ) ) ).

% divide_nonpos_neg
tff(fact_726_divide__nonpos__neg,axiom,
    ! [Xb: rat,Yb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Yb),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(Xb),Yb)) ) ) ).

% divide_nonpos_neg
tff(fact_727_divide__nonpos__pos,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Yb)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(Xb),Yb)),zero_zero_real) ) ) ).

% divide_nonpos_pos
tff(fact_728_divide__nonpos__pos,axiom,
    ! [Xb: rat,Yb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Yb)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(Xb),Yb)),zero_zero_rat) ) ) ).

% divide_nonpos_pos
tff(fact_729_divide__less__eq__1,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_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_730_divide__less__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_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_731_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(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_732_less__divide__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_rat_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_733_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_734_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_735_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_736_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_737_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(divide_divide_real(aa_real_real(plus_plus_real(A),B)),aa_real_real(plus_plus_real(one_one_real),one_one_real))),B) ) ).

% gt_half_sum
tff(fact_738_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,aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(A),B)),aa_rat_rat(plus_plus_rat(one_one_rat),one_one_rat))),B) ) ).

% gt_half_sum
tff(fact_739_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(divide_divide_real(aa_real_real(plus_plus_real(A),B)),aa_real_real(plus_plus_real(one_one_real),one_one_real))) ) ).

% less_half_sum
tff(fact_740_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),aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(A),B)),aa_rat_rat(plus_plus_rat(one_one_rat),one_one_rat))) ) ).

% less_half_sum
tff(fact_741_numeral__Bit0__div__2,axiom,
    ! [Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Nb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_num_nat(numeral_numeral_nat,Nb) ).

% numeral_Bit0_div_2
tff(fact_742_numeral__Bit0__div__2,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_num_int(numeral_numeral_int,Nb) ).

% numeral_Bit0_div_2
tff(fact_743_length__pos__if__in__set,axiom,
    ! [Xb: complex,Xs: list_complex] :
      ( member_complex(Xb,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_744_length__pos__if__in__set,axiom,
    ! [Xb: real,Xs: list_real] :
      ( member_real(Xb,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_745_length__pos__if__in__set,axiom,
    ! [Xb: set_nat,Xs: list_set_nat] :
      ( member_set_nat(Xb,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_746_length__pos__if__in__set,axiom,
    ! [Xb: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( member_VEBT_VEBT(Xb,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_747_length__pos__if__in__set,axiom,
    ! [Xb: $o,Xs: list_o] :
      ( member_o((Xb),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_748_length__pos__if__in__set,axiom,
    ! [Xb: nat,Xs: list_nat] :
      ( aa_set_nat_o(member_nat(Xb),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_749_length__pos__if__in__set,axiom,
    ! [Xb: int,Xs: list_int] :
      ( member_int(Xb,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_750_div__le__mono2,axiom,
    ! [Mb: nat,Nb: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,K),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,K),Mb)) ) ) ).

% div_le_mono2
tff(fact_751_div__greater__zero__iff,axiom,
    ! [Mb: nat,Nb: 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,Mb),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% div_greater_zero_iff
tff(fact_752_nth__mem,axiom,
    ! [Nb: nat,Xs: list_complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s3451745648224563538omplex(Xs))
     => member_complex(aa_nat_complex(nth_complex(Xs),Nb),set_complex2(Xs)) ) ).

% nth_mem
tff(fact_753_nth__mem,axiom,
    ! [Nb: nat,Xs: list_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_real(Xs))
     => member_real(aa_nat_real(nth_real(Xs),Nb),set_real2(Xs)) ) ).

% nth_mem
tff(fact_754_nth__mem,axiom,
    ! [Nb: nat,Xs: list_set_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s3254054031482475050et_nat(Xs))
     => member_set_nat(aa_nat_set_nat(nth_set_nat(Xs),Nb),set_set_nat2(Xs)) ) ).

% nth_mem
tff(fact_755_nth__mem,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s6755466524823107622T_VEBT(Xs))
     => member_VEBT_VEBT(nth_VEBT_VEBT(Xs,Nb),set_VEBT_VEBT2(Xs)) ) ).

% nth_mem
tff(fact_756_nth__mem,axiom,
    ! [Nb: nat,Xs: list_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_o(Xs))
     => member_o(aa_nat_o(nth_o(Xs),Nb),set_o2(Xs)) ) ).

% nth_mem
tff(fact_757_nth__mem,axiom,
    ! [Nb: nat,Xs: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_nat(Xs))
     => aa_set_nat_o(member_nat(aa_nat_nat(nth_nat(Xs),Nb)),set_nat2(Xs)) ) ).

% nth_mem
tff(fact_758_nth__mem,axiom,
    ! [Nb: nat,Xs: list_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_int(Xs))
     => member_int(aa_nat_int(nth_int(Xs),Nb),set_int2(Xs)) ) ).

% nth_mem
tff(fact_759_list__ball__nth,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBT,P: fun_VEBT_VEBT_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s6755466524823107622T_VEBT(Xs))
     => ( ! [X3: vEBT_VEBT] :
            ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(Xs))
           => aa_VEBT_VEBT_o(P,X3) )
       => aa_VEBT_VEBT_o(P,nth_VEBT_VEBT(Xs,Nb)) ) ) ).

% list_ball_nth
tff(fact_760_list__ball__nth,axiom,
    ! [Nb: nat,Xs: list_o,P: fun_o_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_o(Xs))
     => ( ! [X3: $o] :
            ( member_o((X3),set_o2(Xs))
           => aa_o_o(P,(X3)) )
       => aa_o_o(P,aa_nat_o(nth_o(Xs),Nb)) ) ) ).

% list_ball_nth
tff(fact_761_list__ball__nth,axiom,
    ! [Nb: nat,Xs: list_nat,P: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_nat(Xs))
     => ( ! [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),set_nat2(Xs))
           => aa_nat_o(P,X3) )
       => aa_nat_o(P,aa_nat_nat(nth_nat(Xs),Nb)) ) ) ).

% list_ball_nth
tff(fact_762_list__ball__nth,axiom,
    ! [Nb: nat,Xs: list_int,P: fun_int_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_int(Xs))
     => ( ! [X3: int] :
            ( member_int(X3,set_int2(Xs))
           => aa_int_o(P,X3) )
       => aa_int_o(P,aa_nat_int(nth_int(Xs),Nb)) ) ) ).

% list_ball_nth
tff(fact_763_in__set__conv__nth,axiom,
    ! [Xb: complex,Xs: list_complex] :
      ( member_complex(Xb,set_complex2(Xs))
    <=> ? [I5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),size_s3451745648224563538omplex(Xs))
          & ( aa_nat_complex(nth_complex(Xs),I5) = Xb ) ) ) ).

% in_set_conv_nth
tff(fact_764_in__set__conv__nth,axiom,
    ! [Xb: real,Xs: list_real] :
      ( member_real(Xb,set_real2(Xs))
    <=> ? [I5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),size_size_list_real(Xs))
          & ( aa_nat_real(nth_real(Xs),I5) = Xb ) ) ) ).

% in_set_conv_nth
tff(fact_765_in__set__conv__nth,axiom,
    ! [Xb: set_nat,Xs: list_set_nat] :
      ( member_set_nat(Xb,set_set_nat2(Xs))
    <=> ? [I5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),size_s3254054031482475050et_nat(Xs))
          & ( aa_nat_set_nat(nth_set_nat(Xs),I5) = Xb ) ) ) ).

% in_set_conv_nth
tff(fact_766_in__set__conv__nth,axiom,
    ! [Xb: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( member_VEBT_VEBT(Xb,set_VEBT_VEBT2(Xs))
    <=> ? [I5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),size_s6755466524823107622T_VEBT(Xs))
          & ( nth_VEBT_VEBT(Xs,I5) = Xb ) ) ) ).

% in_set_conv_nth
tff(fact_767_in__set__conv__nth,axiom,
    ! [Xb: $o,Xs: list_o] :
      ( member_o((Xb),set_o2(Xs))
    <=> ? [I5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),size_size_list_o(Xs))
          & ( aa_nat_o(nth_o(Xs),I5)
          <=> (Xb) ) ) ) ).

% in_set_conv_nth
tff(fact_768_in__set__conv__nth,axiom,
    ! [Xb: nat,Xs: list_nat] :
      ( aa_set_nat_o(member_nat(Xb),set_nat2(Xs))
    <=> ? [I5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),size_size_list_nat(Xs))
          & ( aa_nat_nat(nth_nat(Xs),I5) = Xb ) ) ) ).

% in_set_conv_nth
tff(fact_769_in__set__conv__nth,axiom,
    ! [Xb: int,Xs: list_int] :
      ( member_int(Xb,set_int2(Xs))
    <=> ? [I5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),size_size_list_int(Xs))
          & ( aa_nat_int(nth_int(Xs),I5) = Xb ) ) ) ).

% in_set_conv_nth
tff(fact_770_all__nth__imp__all__set,axiom,
    ! [Xs: list_complex,P: fun_complex_o,Xb: 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(Xb,set_complex2(Xs))
       => aa_complex_o(P,Xb) ) ) ).

% all_nth_imp_all_set
tff(fact_771_all__nth__imp__all__set,axiom,
    ! [Xs: list_real,P: fun_real_o,Xb: 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(Xb,set_real2(Xs))
       => aa_real_o(P,Xb) ) ) ).

% all_nth_imp_all_set
tff(fact_772_all__nth__imp__all__set,axiom,
    ! [Xs: list_set_nat,P: fun_set_nat_o,Xb: 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(Xb,set_set_nat2(Xs))
       => aa_set_nat_o(P,Xb) ) ) ).

% all_nth_imp_all_set
tff(fact_773_all__nth__imp__all__set,axiom,
    ! [Xs: list_VEBT_VEBT,P: fun_VEBT_VEBT_o,Xb: 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(Xb,set_VEBT_VEBT2(Xs))
       => aa_VEBT_VEBT_o(P,Xb) ) ) ).

% all_nth_imp_all_set
tff(fact_774_all__nth__imp__all__set,axiom,
    ! [Xs: list_o,P: fun_o_o,Xb: $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((Xb),set_o2(Xs))
       => aa_o_o(P,(Xb)) ) ) ).

% all_nth_imp_all_set
tff(fact_775_all__nth__imp__all__set,axiom,
    ! [Xs: list_nat,P: fun_nat_o,Xb: 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(Xb),set_nat2(Xs))
       => aa_nat_o(P,Xb) ) ) ).

% all_nth_imp_all_set
tff(fact_776_all__nth__imp__all__set,axiom,
    ! [Xs: list_int,P: fun_int_o,Xb: 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(Xb,set_int2(Xs))
       => aa_int_o(P,Xb) ) ) ).

% all_nth_imp_all_set
tff(fact_777_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) )
    <=> ! [I5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),size_s6755466524823107622T_VEBT(Xs))
         => aa_VEBT_VEBT_o(P,nth_VEBT_VEBT(Xs,I5)) ) ) ).

% all_set_conv_all_nth
tff(fact_778_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)) )
    <=> ! [I5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),size_size_list_o(Xs))
         => aa_o_o(P,aa_nat_o(nth_o(Xs),I5)) ) ) ).

% all_set_conv_all_nth
tff(fact_779_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) )
    <=> ! [I5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),size_size_list_nat(Xs))
         => aa_nat_o(P,aa_nat_nat(nth_nat(Xs),I5)) ) ) ).

% all_set_conv_all_nth
tff(fact_780_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) )
    <=> ! [I5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),size_size_list_int(Xs))
         => aa_int_o(P,aa_nat_int(nth_int(Xs),I5)) ) ) ).

% all_set_conv_all_nth
tff(fact_781_div__less__dividend,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)),Mb) ) ) ).

% div_less_dividend
tff(fact_782_div__eq__dividend__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb) = Mb )
      <=> ( Nb = one_one_nat ) ) ) ).

% div_eq_dividend_iff
tff(fact_783_set__update__memI,axiom,
    ! [Nb: nat,Xs: list_complex,Xb: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s3451745648224563538omplex(Xs))
     => member_complex(Xb,set_complex2(list_update_complex(Xs,Nb,Xb))) ) ).

% set_update_memI
tff(fact_784_set__update__memI,axiom,
    ! [Nb: nat,Xs: list_real,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_real(Xs))
     => member_real(Xb,set_real2(list_update_real(Xs,Nb,Xb))) ) ).

% set_update_memI
tff(fact_785_set__update__memI,axiom,
    ! [Nb: nat,Xs: list_set_nat,Xb: set_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s3254054031482475050et_nat(Xs))
     => member_set_nat(Xb,set_set_nat2(list_update_set_nat(Xs,Nb,Xb))) ) ).

% set_update_memI
tff(fact_786_set__update__memI,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBT,Xb: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s6755466524823107622T_VEBT(Xs))
     => member_VEBT_VEBT(Xb,set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(Xs,Nb,Xb))) ) ).

% set_update_memI
tff(fact_787_set__update__memI,axiom,
    ! [Nb: nat,Xs: list_o,Xb: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_o(Xs))
     => member_o((Xb),set_o2(list_update_o(Xs,Nb,(Xb)))) ) ).

% set_update_memI
tff(fact_788_set__update__memI,axiom,
    ! [Nb: nat,Xs: list_nat,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_nat(Xs))
     => aa_set_nat_o(member_nat(Xb),set_nat2(list_update_nat(Xs,Nb,Xb))) ) ).

% set_update_memI
tff(fact_789_set__update__memI,axiom,
    ! [Nb: nat,Xs: list_int,Xb: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_int(Xs))
     => member_int(Xb,set_int2(list_update_int(Xs,Nb,Xb))) ) ).

% set_update_memI
tff(fact_790_nth__list__update,axiom,
    ! [I3: nat,Xs: list_VEBT_VEBT,Xb: vEBT_VEBT,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_s6755466524823107622T_VEBT(Xs))
     => ( nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(Xs,I3,Xb),J) = $ite(I3 = J,Xb,nth_VEBT_VEBT(Xs,J)) ) ) ).

% nth_list_update
tff(fact_791_nth__list__update,axiom,
    ! [I3: nat,Xs: list_o,Xb: $o,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_o(Xs))
     => ( aa_nat_o(nth_o(list_update_o(Xs,I3,(Xb))),J)
      <=> $ite(I3 = J,(Xb),aa_nat_o(nth_o(Xs),J)) ) ) ).

% nth_list_update
tff(fact_792_nth__list__update,axiom,
    ! [I3: nat,Xs: list_nat,Xb: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_nat(Xs))
     => ( aa_nat_nat(nth_nat(list_update_nat(Xs,I3,Xb)),J) = $ite(I3 = J,Xb,aa_nat_nat(nth_nat(Xs),J)) ) ) ).

% nth_list_update
tff(fact_793_nth__list__update,axiom,
    ! [I3: nat,Xs: list_int,Xb: int,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_int(Xs))
     => ( aa_nat_int(nth_int(list_update_int(Xs,I3,Xb)),J) = $ite(I3 = J,Xb,aa_nat_int(nth_int(Xs),J)) ) ) ).

% nth_list_update
tff(fact_794_list__update__same__conv,axiom,
    ! [I3: nat,Xs: list_VEBT_VEBT,Xb: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_s6755466524823107622T_VEBT(Xs))
     => ( ( list_u1324408373059187874T_VEBT(Xs,I3,Xb) = Xs )
      <=> ( nth_VEBT_VEBT(Xs,I3) = Xb ) ) ) ).

% list_update_same_conv
tff(fact_795_list__update__same__conv,axiom,
    ! [I3: nat,Xs: list_o,Xb: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_o(Xs))
     => ( ( list_update_o(Xs,I3,(Xb)) = Xs )
      <=> ( aa_nat_o(nth_o(Xs),I3)
        <=> (Xb) ) ) ) ).

% list_update_same_conv
tff(fact_796_list__update__same__conv,axiom,
    ! [I3: nat,Xs: list_nat,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_nat(Xs))
     => ( ( list_update_nat(Xs,I3,Xb) = Xs )
      <=> ( aa_nat_nat(nth_nat(Xs),I3) = Xb ) ) ) ).

% list_update_same_conv
tff(fact_797_list__update__same__conv,axiom,
    ! [I3: nat,Xs: list_int,Xb: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_int(Xs))
     => ( ( list_update_int(Xs,I3,Xb) = Xs )
      <=> ( aa_nat_int(nth_int(Xs),I3) = Xb ) ) ) ).

% list_update_same_conv
tff(fact_798_divide__le__eq__1,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_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_799_divide__le__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_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_800_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(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_801_le__divide__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_rat_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_802_exp__add__not__zero__imp__left,axiom,
    ! [Mb: nat,Nb: 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,Mb),Nb)) != 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))),Mb) != zero_zero_nat ) ) ).

% exp_add_not_zero_imp_left
tff(fact_803_exp__add__not__zero__imp__left,axiom,
    ! [Mb: nat,Nb: 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,Mb),Nb)) != zero_zero_int )
     => ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Mb) != zero_zero_int ) ) ).

% exp_add_not_zero_imp_left
tff(fact_804_exp__add__not__zero__imp__right,axiom,
    ! [Mb: nat,Nb: 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,Mb),Nb)) != 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))),Nb) != zero_zero_nat ) ) ).

% exp_add_not_zero_imp_right
tff(fact_805_exp__add__not__zero__imp__right,axiom,
    ! [Mb: nat,Nb: 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,Mb),Nb)) != zero_zero_int )
     => ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb) != zero_zero_int ) ) ).

% exp_add_not_zero_imp_right
tff(fact_806_div__exp__eq,axiom,
    ! [A: nat,Mb: nat,Nb: 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))),Mb))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = 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,Mb),Nb))) ).

% div_exp_eq
tff(fact_807_div__exp__eq,axiom,
    ! [A: int,Mb: nat,Nb: 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))),Mb))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) = 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,Mb),Nb))) ).

% div_exp_eq
tff(fact_808_div__2__gt__zero,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Nb)
     => 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,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% div_2_gt_zero
tff(fact_809_Suc__n__div__2__gt__zero,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => 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,Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% Suc_n_div_2_gt_zero
tff(fact_810_semiring__norm_I69_J,axiom,
    ! [Mb: num] : ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(bit0,Mb)),one) ).

% semiring_norm(69)
tff(fact_811_semiring__norm_I76_J,axiom,
    ! [Nb: num] : aa_num_o(aa_num_fun_num_o(ord_less_num,one),aa_num_num(bit0,Nb)) ).

% semiring_norm(76)
tff(fact_812_semiring__norm_I2_J,axiom,
    aa_num_num(plus_plus_num(one),one) = aa_num_num(bit0,one) ).

% semiring_norm(2)
tff(fact_813_less__one,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),one_one_nat)
    <=> ( Nb = zero_zero_nat ) ) ).

% less_one
tff(fact_814_add__gr__0,axiom,
    ! [Mb: nat,Nb: 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,Mb),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% add_gr_0
tff(fact_815_less__Suc0,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(suc,zero_zero_nat))
    <=> ( Nb = zero_zero_nat ) ) ).

% less_Suc0
tff(fact_816_zero__less__Suc,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(suc,Nb)) ).

% zero_less_Suc
tff(fact_817_div__self,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(A),A) = one_one_complex ) ) ).

% div_self
tff(fact_818_div__self,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(divide_divide_real(A),A) = one_one_real ) ) ).

% div_self
tff(fact_819_div__self,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(A),A) = one_one_rat ) ) ).

% div_self
tff(fact_820_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_821_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_822_buildup__gives__valid,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_nat_o(vEBT_invar_vebt(vEBT_vebt_buildup(Nb)),Nb) ) ).

% buildup_gives_valid
tff(fact_823_add__less__same__cancel1,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_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_824_add__less__same__cancel1,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_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_825_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_826_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_827_add__less__same__cancel2,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_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_828_add__less__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_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_829_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_830_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_831_buildup__nothing__in__min__max,axiom,
    ! [Nb: nat,Xb: nat] : ~ aa_nat_o(vEBT_VEBT_membermima(vEBT_vebt_buildup(Nb)),Xb) ).

% buildup_nothing_in_min_max
tff(fact_832_buildup__nothing__in__leaf,axiom,
    ! [Nb: nat,Xb: nat] : ~ aa_nat_o(vEBT_V5719532721284313246member(vEBT_vebt_buildup(Nb)),Xb) ).

% buildup_nothing_in_leaf
tff(fact_833_semiring__norm_I87_J,axiom,
    ! [Mb: num,Nb: num] :
      ( ( aa_num_num(bit0,Mb) = aa_num_num(bit0,Nb) )
    <=> ( Mb = Nb ) ) ).

% semiring_norm(87)
tff(fact_834_add__right__cancel,axiom,
    ! [B: real,A: real,C: real] :
      ( ( aa_real_real(plus_plus_real(B),A) = aa_real_real(plus_plus_real(C),A) )
    <=> ( B = C ) ) ).

% add_right_cancel
tff(fact_835_add__right__cancel,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( aa_rat_rat(plus_plus_rat(B),A) = aa_rat_rat(plus_plus_rat(C),A) )
    <=> ( B = C ) ) ).

% add_right_cancel
tff(fact_836_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_837_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_838_add__left__cancel,axiom,
    ! [A: real,B: real,C: real] :
      ( ( aa_real_real(plus_plus_real(A),B) = aa_real_real(plus_plus_real(A),C) )
    <=> ( B = C ) ) ).

% add_left_cancel
tff(fact_839_add__left__cancel,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( aa_rat_rat(plus_plus_rat(A),B) = aa_rat_rat(plus_plus_rat(A),C) )
    <=> ( B = C ) ) ).

% add_left_cancel
tff(fact_840_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_841_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_842_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_843_nat_Oinject,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( aa_nat_nat(suc,X2) = aa_nat_nat(suc,Y2) )
    <=> ( X2 = Y2 ) ) ).

% nat.inject
tff(fact_844_le__zero__eq,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),zero_zero_nat)
    <=> ( Nb = zero_zero_nat ) ) ).

% le_zero_eq
tff(fact_845_not__gr__zero,axiom,
    ! [Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
    <=> ( Nb = zero_zero_nat ) ) ).

% not_gr_zero
tff(fact_846_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(plus_plus_real(A),C)),aa_real_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_847_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(plus_plus_rat(A),C)),aa_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_848_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_849_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_850_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(plus_plus_real(C),A)),aa_real_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_851_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(plus_plus_rat(C),A)),aa_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_852_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_853_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_854_add__0,axiom,
    ! [A: complex] : aa_complex_complex(plus_plus_complex(zero_zero_complex),A) = A ).

% add_0
tff(fact_855_add__0,axiom,
    ! [A: real] : aa_real_real(plus_plus_real(zero_zero_real),A) = A ).

% add_0
tff(fact_856_add__0,axiom,
    ! [A: rat] : aa_rat_rat(plus_plus_rat(zero_zero_rat),A) = A ).

% add_0
tff(fact_857_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_858_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_859_zero__eq__add__iff__both__eq__0,axiom,
    ! [Xb: nat,Yb: nat] :
      ( ( zero_zero_nat = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xb),Yb) )
    <=> ( ( Xb = zero_zero_nat )
        & ( Yb = zero_zero_nat ) ) ) ).

% zero_eq_add_iff_both_eq_0
tff(fact_860_add__eq__0__iff__both__eq__0,axiom,
    ! [Xb: nat,Yb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xb),Yb) = zero_zero_nat )
    <=> ( ( Xb = zero_zero_nat )
        & ( Yb = zero_zero_nat ) ) ) ).

% add_eq_0_iff_both_eq_0
tff(fact_861_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_862_add__cancel__right__right,axiom,
    ! [A: real,B: real] :
      ( ( A = aa_real_real(plus_plus_real(A),B) )
    <=> ( B = zero_zero_real ) ) ).

% add_cancel_right_right
tff(fact_863_add__cancel__right__right,axiom,
    ! [A: rat,B: rat] :
      ( ( A = aa_rat_rat(plus_plus_rat(A),B) )
    <=> ( B = zero_zero_rat ) ) ).

% add_cancel_right_right
tff(fact_864_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_865_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_866_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_867_add__cancel__right__left,axiom,
    ! [A: real,B: real] :
      ( ( A = aa_real_real(plus_plus_real(B),A) )
    <=> ( B = zero_zero_real ) ) ).

% add_cancel_right_left
tff(fact_868_add__cancel__right__left,axiom,
    ! [A: rat,B: rat] :
      ( ( A = aa_rat_rat(plus_plus_rat(B),A) )
    <=> ( B = zero_zero_rat ) ) ).

% add_cancel_right_left
tff(fact_869_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_870_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_871_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_872_add__cancel__left__right,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(plus_plus_real(A),B) = A )
    <=> ( B = zero_zero_real ) ) ).

% add_cancel_left_right
tff(fact_873_add__cancel__left__right,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(plus_plus_rat(A),B) = A )
    <=> ( B = zero_zero_rat ) ) ).

% add_cancel_left_right
tff(fact_874_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_875_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_876_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_877_add__cancel__left__left,axiom,
    ! [B: real,A: real] :
      ( ( aa_real_real(plus_plus_real(B),A) = A )
    <=> ( B = zero_zero_real ) ) ).

% add_cancel_left_left
tff(fact_878_add__cancel__left__left,axiom,
    ! [B: rat,A: rat] :
      ( ( aa_rat_rat(plus_plus_rat(B),A) = A )
    <=> ( B = zero_zero_rat ) ) ).

% add_cancel_left_left
tff(fact_879_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_880_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_881_double__zero__sym,axiom,
    ! [A: real] :
      ( ( zero_zero_real = aa_real_real(plus_plus_real(A),A) )
    <=> ( A = zero_zero_real ) ) ).

% double_zero_sym
tff(fact_882_double__zero__sym,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat = aa_rat_rat(plus_plus_rat(A),A) )
    <=> ( A = zero_zero_rat ) ) ).

% double_zero_sym
tff(fact_883_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_884_add_Oright__neutral,axiom,
    ! [A: complex] : aa_complex_complex(plus_plus_complex(A),zero_zero_complex) = A ).

% add.right_neutral
tff(fact_885_add_Oright__neutral,axiom,
    ! [A: real] : aa_real_real(plus_plus_real(A),zero_zero_real) = A ).

% add.right_neutral
tff(fact_886_add_Oright__neutral,axiom,
    ! [A: rat] : aa_rat_rat(plus_plus_rat(A),zero_zero_rat) = A ).

% add.right_neutral
tff(fact_887_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_888_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_889_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(plus_plus_real(A),C)),aa_real_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_890_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(plus_plus_rat(A),C)),aa_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_891_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_892_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_893_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(plus_plus_real(C),A)),aa_real_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_894_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(plus_plus_rat(C),A)),aa_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_895_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_896_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_897_div__by__0,axiom,
    ! [A: complex] : aa_complex_complex(divide1717551699836669952omplex(A),zero_zero_complex) = zero_zero_complex ).

% div_by_0
tff(fact_898_div__by__0,axiom,
    ! [A: real] : aa_real_real(divide_divide_real(A),zero_zero_real) = zero_zero_real ).

% div_by_0
tff(fact_899_div__by__0,axiom,
    ! [A: rat] : aa_rat_rat(divide_divide_rat(A),zero_zero_rat) = zero_zero_rat ).

% div_by_0
tff(fact_900_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_901_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_902_div__0,axiom,
    ! [A: complex] : aa_complex_complex(divide1717551699836669952omplex(zero_zero_complex),A) = zero_zero_complex ).

% div_0
tff(fact_903_div__0,axiom,
    ! [A: real] : aa_real_real(divide_divide_real(zero_zero_real),A) = zero_zero_real ).

% div_0
tff(fact_904_div__0,axiom,
    ! [A: rat] : aa_rat_rat(divide_divide_rat(zero_zero_rat),A) = zero_zero_rat ).

% div_0
tff(fact_905_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_906_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_907_semiring__norm_I83_J,axiom,
    ! [Nb: num] : one != aa_num_num(bit0,Nb) ).

% semiring_norm(83)
tff(fact_908_semiring__norm_I85_J,axiom,
    ! [Mb: num] : aa_num_num(bit0,Mb) != one ).

% semiring_norm(85)
tff(fact_909_div__by__1,axiom,
    ! [A: complex] : aa_complex_complex(divide1717551699836669952omplex(A),one_one_complex) = A ).

% div_by_1
tff(fact_910_div__by__1,axiom,
    ! [A: real] : aa_real_real(divide_divide_real(A),one_one_real) = A ).

% div_by_1
tff(fact_911_div__by__1,axiom,
    ! [A: rat] : aa_rat_rat(divide_divide_rat(A),one_one_rat) = A ).

% div_by_1
tff(fact_912_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_913_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_914_Suc__less__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Mb)),aa_nat_nat(suc,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% Suc_less_eq
tff(fact_915_Suc__mono,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Mb)),aa_nat_nat(suc,Nb)) ) ).

% Suc_mono
tff(fact_916_lessI,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(suc,Nb)) ).

% lessI
tff(fact_917_less__nat__zero__code,axiom,
    ! [Nb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),zero_zero_nat) ).

% less_nat_zero_code
tff(fact_918_neq0__conv,axiom,
    ! [Nb: nat] :
      ( ( Nb != zero_zero_nat )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).

% neq0_conv
tff(fact_919_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_920_Suc__le__mono,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Nb)),aa_nat_nat(suc,Mb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb) ) ).

% Suc_le_mono
tff(fact_921_le0,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Nb) ).

% le0
tff(fact_922_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_923_add__Suc__right,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),aa_nat_nat(suc,Nb)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) ).

% add_Suc_right
tff(fact_924_Nat_Oadd__0__right,axiom,
    ! [Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),zero_zero_nat) = Mb ).

% Nat.add_0_right
tff(fact_925_add__is__0,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb) = zero_zero_nat )
    <=> ( ( Mb = zero_zero_nat )
        & ( Nb = zero_zero_nat ) ) ) ).

% add_is_0
tff(fact_926_nat__add__left__cancel__less,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% nat_add_left_cancel_less
tff(fact_927_nat__add__left__cancel__le,axiom,
    ! [K: nat,Mb: nat,Nb: 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),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% nat_add_left_cancel_le
tff(fact_928_semiring__norm_I6_J,axiom,
    ! [Mb: num,Nb: num] : aa_num_num(plus_plus_num(aa_num_num(bit0,Mb)),aa_num_num(bit0,Nb)) = aa_num_num(bit0,aa_num_num(plus_plus_num(Mb),Nb)) ).

% semiring_norm(6)
tff(fact_929_max__Suc__Suc,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(suc,Mb)),aa_nat_nat(suc,Nb)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Mb),Nb)) ).

% max_Suc_Suc
tff(fact_930_max__0R,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Nb),zero_zero_nat) = Nb ).

% max_0R
tff(fact_931_max__0L,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,zero_zero_nat),Nb) = Nb ).

% max_0L
tff(fact_932_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_933_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_934_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_935_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_936_semiring__norm_I78_J,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(bit0,Mb)),aa_num_num(bit0,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Nb) ) ).

% semiring_norm(78)
tff(fact_937_semiring__norm_I71_J,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(bit0,Mb)),aa_num_num(bit0,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Mb),Nb) ) ).

% semiring_norm(71)
tff(fact_938_semiring__norm_I75_J,axiom,
    ! [Mb: num] : ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),one) ).

% semiring_norm(75)
tff(fact_939_semiring__norm_I68_J,axiom,
    ! [Nb: num] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,one),Nb) ).

% semiring_norm(68)
tff(fact_940_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(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_941_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(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_942_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_943_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(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_944_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(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_945_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_946_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(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_947_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(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_948_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_949_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_950_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(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_951_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(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_952_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_953_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_954_add__le__same__cancel2,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_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_955_add__le__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_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_956_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_957_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_958_add__le__same__cancel1,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_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_959_add__le__same__cancel1,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_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_960_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_961_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_962_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(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_963_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(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_964_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_965_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(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_966_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(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_967_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_968_less__add__same__cancel2,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_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_969_less__add__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_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_970_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_971_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_972_less__add__same__cancel1,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_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_973_less__add__same__cancel1,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_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_974_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_975_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_976_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_977_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_978_zero__reorient,axiom,
    ! [Xb: complex] :
      ( ( zero_zero_complex = Xb )
    <=> ( Xb = zero_zero_complex ) ) ).

% zero_reorient
tff(fact_979_zero__reorient,axiom,
    ! [Xb: real] :
      ( ( zero_zero_real = Xb )
    <=> ( Xb = zero_zero_real ) ) ).

% zero_reorient
tff(fact_980_zero__reorient,axiom,
    ! [Xb: rat] :
      ( ( zero_zero_rat = Xb )
    <=> ( Xb = zero_zero_rat ) ) ).

% zero_reorient
tff(fact_981_zero__reorient,axiom,
    ! [Xb: nat] :
      ( ( zero_zero_nat = Xb )
    <=> ( Xb = zero_zero_nat ) ) ).

% zero_reorient
tff(fact_982_zero__reorient,axiom,
    ! [Xb: int] :
      ( ( zero_zero_int = Xb )
    <=> ( Xb = zero_zero_int ) ) ).

% zero_reorient
tff(fact_983_linorder__neqE__linordered__idom,axiom,
    ! [Xb: real,Yb: real] :
      ( ( Xb != Yb )
     => ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Yb)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Yb),Xb) ) ) ).

% linorder_neqE_linordered_idom
tff(fact_984_linorder__neqE__linordered__idom,axiom,
    ! [Xb: rat,Yb: rat] :
      ( ( Xb != Yb )
     => ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Yb)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Yb),Xb) ) ) ).

% linorder_neqE_linordered_idom
tff(fact_985_linorder__neqE__linordered__idom,axiom,
    ! [Xb: int,Yb: int] :
      ( ( Xb != Yb )
     => ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Yb)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,Yb),Xb) ) ) ).

% linorder_neqE_linordered_idom
tff(fact_986_one__reorient,axiom,
    ! [Xb: complex] :
      ( ( one_one_complex = Xb )
    <=> ( Xb = one_one_complex ) ) ).

% one_reorient
tff(fact_987_one__reorient,axiom,
    ! [Xb: real] :
      ( ( one_one_real = Xb )
    <=> ( Xb = one_one_real ) ) ).

% one_reorient
tff(fact_988_one__reorient,axiom,
    ! [Xb: rat] :
      ( ( one_one_rat = Xb )
    <=> ( Xb = one_one_rat ) ) ).

% one_reorient
tff(fact_989_one__reorient,axiom,
    ! [Xb: nat] :
      ( ( one_one_nat = Xb )
    <=> ( Xb = one_one_nat ) ) ).

% one_reorient
tff(fact_990_one__reorient,axiom,
    ! [Xb: int] :
      ( ( one_one_int = Xb )
    <=> ( Xb = one_one_int ) ) ).

% one_reorient
tff(fact_991_add__right__imp__eq,axiom,
    ! [B: real,A: real,C: real] :
      ( ( aa_real_real(plus_plus_real(B),A) = aa_real_real(plus_plus_real(C),A) )
     => ( B = C ) ) ).

% add_right_imp_eq
tff(fact_992_add__right__imp__eq,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( aa_rat_rat(plus_plus_rat(B),A) = aa_rat_rat(plus_plus_rat(C),A) )
     => ( B = C ) ) ).

% add_right_imp_eq
tff(fact_993_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_994_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_995_add__left__imp__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( aa_real_real(plus_plus_real(A),B) = aa_real_real(plus_plus_real(A),C) )
     => ( B = C ) ) ).

% add_left_imp_eq
tff(fact_996_add__left__imp__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( aa_rat_rat(plus_plus_rat(A),B) = aa_rat_rat(plus_plus_rat(A),C) )
     => ( B = C ) ) ).

% add_left_imp_eq
tff(fact_997_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_998_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_999_add_Oleft__commute,axiom,
    ! [B: real,A: real,C: real] : aa_real_real(plus_plus_real(B),aa_real_real(plus_plus_real(A),C)) = aa_real_real(plus_plus_real(A),aa_real_real(plus_plus_real(B),C)) ).

% add.left_commute
tff(fact_1000_add_Oleft__commute,axiom,
    ! [B: rat,A: rat,C: rat] : aa_rat_rat(plus_plus_rat(B),aa_rat_rat(plus_plus_rat(A),C)) = aa_rat_rat(plus_plus_rat(A),aa_rat_rat(plus_plus_rat(B),C)) ).

% add.left_commute
tff(fact_1001_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_1002_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_1003_add_Ocommute,axiom,
    ! [A: real,B: real] : aa_real_real(plus_plus_real(A),B) = aa_real_real(plus_plus_real(B),A) ).

% add.commute
tff(fact_1004_add_Ocommute,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(plus_plus_rat(A),B) = aa_rat_rat(plus_plus_rat(B),A) ).

% add.commute
tff(fact_1005_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_1006_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_1007_add_Oright__cancel,axiom,
    ! [B: real,A: real,C: real] :
      ( ( aa_real_real(plus_plus_real(B),A) = aa_real_real(plus_plus_real(C),A) )
    <=> ( B = C ) ) ).

% add.right_cancel
tff(fact_1008_add_Oright__cancel,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( aa_rat_rat(plus_plus_rat(B),A) = aa_rat_rat(plus_plus_rat(C),A) )
    <=> ( B = C ) ) ).

% add.right_cancel
tff(fact_1009_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_1010_add_Oleft__cancel,axiom,
    ! [A: real,B: real,C: real] :
      ( ( aa_real_real(plus_plus_real(A),B) = aa_real_real(plus_plus_real(A),C) )
    <=> ( B = C ) ) ).

% add.left_cancel
tff(fact_1011_add_Oleft__cancel,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( aa_rat_rat(plus_plus_rat(A),B) = aa_rat_rat(plus_plus_rat(A),C) )
    <=> ( B = C ) ) ).

% add.left_cancel
tff(fact_1012_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_1013_add_Oassoc,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(plus_plus_real(aa_real_real(plus_plus_real(A),B)),C) = aa_real_real(plus_plus_real(A),aa_real_real(plus_plus_real(B),C)) ).

% add.assoc
tff(fact_1014_add_Oassoc,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(A),B)),C) = aa_rat_rat(plus_plus_rat(A),aa_rat_rat(plus_plus_rat(B),C)) ).

% add.assoc
tff(fact_1015_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_1016_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_1017_group__cancel_Oadd2,axiom,
    ! [B4: real,K: real,B: real,A: real] :
      ( ( B4 = aa_real_real(plus_plus_real(K),B) )
     => ( aa_real_real(plus_plus_real(A),B4) = aa_real_real(plus_plus_real(K),aa_real_real(plus_plus_real(A),B)) ) ) ).

% group_cancel.add2
tff(fact_1018_group__cancel_Oadd2,axiom,
    ! [B4: rat,K: rat,B: rat,A: rat] :
      ( ( B4 = aa_rat_rat(plus_plus_rat(K),B) )
     => ( aa_rat_rat(plus_plus_rat(A),B4) = aa_rat_rat(plus_plus_rat(K),aa_rat_rat(plus_plus_rat(A),B)) ) ) ).

% group_cancel.add2
tff(fact_1019_group__cancel_Oadd2,axiom,
    ! [B4: nat,K: nat,B: nat,A: nat] :
      ( ( B4 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),B) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B4) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).

% group_cancel.add2
tff(fact_1020_group__cancel_Oadd2,axiom,
    ! [B4: int,K: int,B: int,A: int] :
      ( ( B4 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),B) )
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B4) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ) ).

% group_cancel.add2
tff(fact_1021_group__cancel_Oadd1,axiom,
    ! [A3: real,K: real,A: real,B: real] :
      ( ( A3 = aa_real_real(plus_plus_real(K),A) )
     => ( aa_real_real(plus_plus_real(A3),B) = aa_real_real(plus_plus_real(K),aa_real_real(plus_plus_real(A),B)) ) ) ).

% group_cancel.add1
tff(fact_1022_group__cancel_Oadd1,axiom,
    ! [A3: rat,K: rat,A: rat,B: rat] :
      ( ( A3 = aa_rat_rat(plus_plus_rat(K),A) )
     => ( aa_rat_rat(plus_plus_rat(A3),B) = aa_rat_rat(plus_plus_rat(K),aa_rat_rat(plus_plus_rat(A),B)) ) ) ).

% group_cancel.add1
tff(fact_1023_group__cancel_Oadd1,axiom,
    ! [A3: nat,K: nat,A: nat,B: nat] :
      ( ( A3 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),A) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A3),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).

% group_cancel.add1
tff(fact_1024_group__cancel_Oadd1,axiom,
    ! [A3: int,K: int,A: int,B: int] :
      ( ( A3 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),A) )
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A3),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ) ).

% group_cancel.add1
tff(fact_1025_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I3: real,J: real,K: real,L: real] :
      ( ( ( I3 = J )
        & ( K = L ) )
     => ( aa_real_real(plus_plus_real(I3),K) = aa_real_real(plus_plus_real(J),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_1026_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I3: rat,J: rat,K: rat,L: rat] :
      ( ( ( I3 = J )
        & ( K = L ) )
     => ( aa_rat_rat(plus_plus_rat(I3),K) = aa_rat_rat(plus_plus_rat(J),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_1027_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I3: nat,J: nat,K: nat,L: nat] :
      ( ( ( I3 = J )
        & ( K = L ) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I3),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_1028_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I3: int,J: int,K: int,L: int] :
      ( ( ( I3 = J )
        & ( K = L ) )
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,I3),K) = aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_1029_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(plus_plus_real(aa_real_real(plus_plus_real(A),B)),C) = aa_real_real(plus_plus_real(A),aa_real_real(plus_plus_real(B),C)) ).

% ab_semigroup_add_class.add_ac(1)
tff(fact_1030_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(A),B)),C) = aa_rat_rat(plus_plus_rat(A),aa_rat_rat(plus_plus_rat(B),C)) ).

% ab_semigroup_add_class.add_ac(1)
tff(fact_1031_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_1032_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_1033_n__not__Suc__n,axiom,
    ! [Nb: nat] : Nb != aa_nat_nat(suc,Nb) ).

% n_not_Suc_n
tff(fact_1034_Suc__inject,axiom,
    ! [Xb: nat,Yb: nat] :
      ( ( aa_nat_nat(suc,Xb) = aa_nat_nat(suc,Yb) )
     => ( Xb = Yb ) ) ).

% Suc_inject
tff(fact_1035_linorder__neqE__nat,axiom,
    ! [Xb: nat,Yb: nat] :
      ( ( Xb != Yb )
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Yb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Yb),Xb) ) ) ).

% linorder_neqE_nat
tff(fact_1036_infinite__descent,axiom,
    ! [P: fun_nat_o,Nb: nat] :
      ( ! [N: nat] :
          ( ~ aa_nat_o(P,N)
         => ? [M: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N)
              & ~ aa_nat_o(P,M) ) )
     => aa_nat_o(P,Nb) ) ).

% infinite_descent
tff(fact_1037_nat__less__induct,axiom,
    ! [P: fun_nat_o,Nb: nat] :
      ( ! [N: nat] :
          ( ! [M: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N)
             => aa_nat_o(P,M) )
         => aa_nat_o(P,N) )
     => aa_nat_o(P,Nb) ) ).

% nat_less_induct
tff(fact_1038_less__irrefl__nat,axiom,
    ! [Nb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Nb) ).

% less_irrefl_nat
tff(fact_1039_less__not__refl3,axiom,
    ! [S2: nat,Ta: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,S2),Ta)
     => ( S2 != Ta ) ) ).

% less_not_refl3
tff(fact_1040_less__not__refl2,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb)
     => ( Mb != Nb ) ) ).

% less_not_refl2
tff(fact_1041_less__not__refl,axiom,
    ! [Nb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Nb) ).

% less_not_refl
tff(fact_1042_nat__neq__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( Mb != Nb )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb) ) ) ).

% nat_neq_iff
tff(fact_1043_Nat_Oex__has__greatest__nat,axiom,
    ! [P: fun_nat_o,K: nat,B: nat] :
      ( aa_nat_o(P,K)
     => ( ! [Y: nat] :
            ( aa_nat_o(P,Y)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),B) )
       => ? [X3: nat] :
            ( aa_nat_o(P,X3)
            & ! [Y3: nat] :
                ( aa_nat_o(P,Y3)
               => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y3),X3) ) ) ) ) ).

% Nat.ex_has_greatest_nat
tff(fact_1044_nat__le__linear,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
      | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb) ) ).

% nat_le_linear
tff(fact_1045_le__antisym,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => ( Mb = Nb ) ) ) ).

% le_antisym
tff(fact_1046_eq__imp__le,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( Mb = Nb )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% eq_imp_le
tff(fact_1047_le__trans,axiom,
    ! [I3: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I3),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,I3),K) ) ) ).

% le_trans
tff(fact_1048_le__refl,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Nb) ).

% le_refl
tff(fact_1049_size__neq__size__imp__neq,axiom,
    ! [Xb: list_VEBT_VEBT,Yb: list_VEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT(Xb) != size_s6755466524823107622T_VEBT(Yb) )
     => ( Xb != Yb ) ) ).

% size_neq_size_imp_neq
tff(fact_1050_size__neq__size__imp__neq,axiom,
    ! [Xb: list_o,Yb: list_o] :
      ( ( size_size_list_o(Xb) != size_size_list_o(Yb) )
     => ( Xb != Yb ) ) ).

% size_neq_size_imp_neq
tff(fact_1051_size__neq__size__imp__neq,axiom,
    ! [Xb: list_nat,Yb: list_nat] :
      ( ( size_size_list_nat(Xb) != size_size_list_nat(Yb) )
     => ( Xb != Yb ) ) ).

% size_neq_size_imp_neq
tff(fact_1052_size__neq__size__imp__neq,axiom,
    ! [Xb: list_int,Yb: list_int] :
      ( ( size_size_list_int(Xb) != size_size_list_int(Yb) )
     => ( Xb != Yb ) ) ).

% size_neq_size_imp_neq
tff(fact_1053_size__neq__size__imp__neq,axiom,
    ! [Xb: num,Yb: num] :
      ( ( aa_num_nat(size_size_num,Xb) != aa_num_nat(size_size_num,Yb) )
     => ( Xb != Yb ) ) ).

% size_neq_size_imp_neq
tff(fact_1054_zero__le,axiom,
    ! [Xb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Xb) ).

% zero_le
tff(fact_1055_zero__less__iff__neq__zero,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
    <=> ( Nb != zero_zero_nat ) ) ).

% zero_less_iff_neq_zero
tff(fact_1056_gr__implies__not__zero,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => ( Nb != zero_zero_nat ) ) ).

% gr_implies_not_zero
tff(fact_1057_not__less__zero,axiom,
    ! [Nb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),zero_zero_nat) ).

% not_less_zero
tff(fact_1058_gr__zeroI,axiom,
    ! [Nb: nat] :
      ( ( Nb != zero_zero_nat )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).

% gr_zeroI
tff(fact_1059_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(plus_plus_real(A),C)),aa_real_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_1060_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(plus_plus_rat(A),C)),aa_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_1061_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_1062_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_1063_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(plus_plus_real(C),A)),aa_real_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_1064_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(plus_plus_rat(C),A)),aa_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_1065_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_1066_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_1067_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_1068_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(plus_plus_real(A),C)),aa_real_real(plus_plus_real(B),C)) ) ).

% add_right_mono
tff(fact_1069_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(plus_plus_rat(A),C)),aa_rat_rat(plus_plus_rat(B),C)) ) ).

% add_right_mono
tff(fact_1070_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_1071_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_1072_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_1073_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(plus_plus_real(C),A)),aa_real_real(plus_plus_real(C),B)) ) ).

% add_left_mono
tff(fact_1074_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(plus_plus_rat(C),A)),aa_rat_rat(plus_plus_rat(C),B)) ) ).

% add_left_mono
tff(fact_1075_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_1076_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_1077_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(plus_plus_real(A),C)),aa_real_real(plus_plus_real(B),D)) ) ) ).

% add_mono
tff(fact_1078_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(plus_plus_rat(A),C)),aa_rat_rat(plus_plus_rat(B),D)) ) ) ).

% add_mono
tff(fact_1079_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_1080_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_1081_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I3: real,J: real,K: real,L: real] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,I3),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(plus_plus_real(I3),K)),aa_real_real(plus_plus_real(J),L)) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_1082_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I3: rat,J: rat,K: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,I3),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(plus_plus_rat(I3),K)),aa_rat_rat(plus_plus_rat(J),L)) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_1083_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I3: nat,J: nat,K: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I3),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,I3),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_1084_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I3: int,J: int,K: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I3),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,I3),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_1085_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I3: real,J: real,K: real,L: real] :
      ( ( ( I3 = 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(plus_plus_real(I3),K)),aa_real_real(plus_plus_real(J),L)) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_1086_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I3: rat,J: rat,K: rat,L: rat] :
      ( ( ( I3 = 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(plus_plus_rat(I3),K)),aa_rat_rat(plus_plus_rat(J),L)) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_1087_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I3: nat,J: nat,K: nat,L: nat] :
      ( ( ( I3 = 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,I3),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_1088_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I3: int,J: int,K: int,L: int] :
      ( ( ( I3 = 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,I3),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_1089_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I3: real,J: real,K: real,L: real] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,I3),J)
        & ( K = L ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(I3),K)),aa_real_real(plus_plus_real(J),L)) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_1090_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I3: rat,J: rat,K: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,I3),J)
        & ( K = L ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(I3),K)),aa_rat_rat(plus_plus_rat(J),L)) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_1091_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I3: nat,J: nat,K: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I3),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,I3),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_1092_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I3: int,J: int,K: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I3),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,I3),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_1093_zero__neq__one,axiom,
    zero_zero_complex != one_one_complex ).

% zero_neq_one
tff(fact_1094_zero__neq__one,axiom,
    zero_zero_real != one_one_real ).

% zero_neq_one
tff(fact_1095_zero__neq__one,axiom,
    zero_zero_rat != one_one_rat ).

% zero_neq_one
tff(fact_1096_zero__neq__one,axiom,
    zero_zero_nat != one_one_nat ).

% zero_neq_one
tff(fact_1097_zero__neq__one,axiom,
    zero_zero_int != one_one_int ).

% zero_neq_one
tff(fact_1098_add_Ogroup__left__neutral,axiom,
    ! [A: complex] : aa_complex_complex(plus_plus_complex(zero_zero_complex),A) = A ).

% add.group_left_neutral
tff(fact_1099_add_Ogroup__left__neutral,axiom,
    ! [A: real] : aa_real_real(plus_plus_real(zero_zero_real),A) = A ).

% add.group_left_neutral
tff(fact_1100_add_Ogroup__left__neutral,axiom,
    ! [A: rat] : aa_rat_rat(plus_plus_rat(zero_zero_rat),A) = A ).

% add.group_left_neutral
tff(fact_1101_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_1102_add_Ocomm__neutral,axiom,
    ! [A: complex] : aa_complex_complex(plus_plus_complex(A),zero_zero_complex) = A ).

% add.comm_neutral
tff(fact_1103_add_Ocomm__neutral,axiom,
    ! [A: real] : aa_real_real(plus_plus_real(A),zero_zero_real) = A ).

% add.comm_neutral
tff(fact_1104_add_Ocomm__neutral,axiom,
    ! [A: rat] : aa_rat_rat(plus_plus_rat(A),zero_zero_rat) = A ).

% add.comm_neutral
tff(fact_1105_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_1106_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_1107_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_1108_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: real] : aa_real_real(plus_plus_real(zero_zero_real),A) = A ).

% comm_monoid_add_class.add_0
tff(fact_1109_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: rat] : aa_rat_rat(plus_plus_rat(zero_zero_rat),A) = A ).

% comm_monoid_add_class.add_0
tff(fact_1110_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_1111_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_1112_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(plus_plus_real(A),C)),aa_real_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_1113_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(plus_plus_rat(A),C)),aa_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_1114_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_1115_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_1116_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(plus_plus_real(C),A)),aa_real_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_1117_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(plus_plus_rat(C),A)),aa_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_1118_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_1119_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_1120_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(plus_plus_real(A),C)),aa_real_real(plus_plus_real(B),C)) ) ).

% add_strict_right_mono
tff(fact_1121_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(plus_plus_rat(A),C)),aa_rat_rat(plus_plus_rat(B),C)) ) ).

% add_strict_right_mono
tff(fact_1122_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_1123_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_1124_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(plus_plus_real(C),A)),aa_real_real(plus_plus_real(C),B)) ) ).

% add_strict_left_mono
tff(fact_1125_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(plus_plus_rat(C),A)),aa_rat_rat(plus_plus_rat(C),B)) ) ).

% add_strict_left_mono
tff(fact_1126_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_1127_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_1128_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(plus_plus_real(A),C)),aa_real_real(plus_plus_real(B),D)) ) ) ).

% add_strict_mono
tff(fact_1129_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(plus_plus_rat(A),C)),aa_rat_rat(plus_plus_rat(B),D)) ) ) ).

% add_strict_mono
tff(fact_1130_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_1131_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_1132_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I3: real,J: real,K: real,L: real] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,I3),J)
        & ( K = L ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(I3),K)),aa_real_real(plus_plus_real(J),L)) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_1133_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I3: rat,J: rat,K: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,I3),J)
        & ( K = L ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(I3),K)),aa_rat_rat(plus_plus_rat(J),L)) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_1134_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I3: nat,J: nat,K: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),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,I3),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_1135_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I3: int,J: int,K: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,I3),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,I3),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_1136_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I3: real,J: real,K: real,L: real] :
      ( ( ( I3 = 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(plus_plus_real(I3),K)),aa_real_real(plus_plus_real(J),L)) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_1137_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I3: rat,J: rat,K: rat,L: rat] :
      ( ( ( I3 = 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(plus_plus_rat(I3),K)),aa_rat_rat(plus_plus_rat(J),L)) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_1138_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I3: nat,J: nat,K: nat,L: nat] :
      ( ( ( I3 = 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,I3),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_1139_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I3: int,J: int,K: int,L: int] :
      ( ( ( I3 = 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,I3),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_1140_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I3: real,J: real,K: real,L: real] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,I3),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(plus_plus_real(I3),K)),aa_real_real(plus_plus_real(J),L)) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_1141_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I3: rat,J: rat,K: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,I3),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(plus_plus_rat(I3),K)),aa_rat_rat(plus_plus_rat(J),L)) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_1142_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I3: nat,J: nat,K: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),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,I3),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_1143_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I3: int,J: int,K: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,I3),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,I3),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_1144_not0__implies__Suc,axiom,
    ! [Nb: nat] :
      ( ( Nb != zero_zero_nat )
     => ? [M2: nat] : Nb = aa_nat_nat(suc,M2) ) ).

% not0_implies_Suc
tff(fact_1145_Zero__not__Suc,axiom,
    ! [Mb: nat] : zero_zero_nat != aa_nat_nat(suc,Mb) ).

% Zero_not_Suc
tff(fact_1146_Zero__neq__Suc,axiom,
    ! [Mb: nat] : zero_zero_nat != aa_nat_nat(suc,Mb) ).

% Zero_neq_Suc
tff(fact_1147_Suc__neq__Zero,axiom,
    ! [Mb: nat] : aa_nat_nat(suc,Mb) != zero_zero_nat ).

% Suc_neq_Zero
tff(fact_1148_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_1149_diff__induct,axiom,
    ! [P: fun_nat_fun_nat_o,Mb: nat,Nb: nat] :
      ( ! [X3: nat] : aa_nat_o(aa_nat_fun_nat_o(P,X3),zero_zero_nat)
     => ( ! [Y: nat] : aa_nat_o(aa_nat_fun_nat_o(P,zero_zero_nat),aa_nat_nat(suc,Y))
       => ( ! [X3: nat,Y: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(P,X3),Y)
             => aa_nat_o(aa_nat_fun_nat_o(P,aa_nat_nat(suc,X3)),aa_nat_nat(suc,Y)) )
         => aa_nat_o(aa_nat_fun_nat_o(P,Mb),Nb) ) ) ) ).

% diff_induct
tff(fact_1150_nat__induct,axiom,
    ! [P: fun_nat_o,Nb: 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,Nb) ) ) ).

% nat_induct
tff(fact_1151_old_Onat_Oexhaust,axiom,
    ! [Yb: nat] :
      ( ( Yb != zero_zero_nat )
     => ~ ! [Nat3: nat] : Yb != aa_nat_nat(suc,Nat3) ) ).

% old.nat.exhaust
tff(fact_1152_nat_OdiscI,axiom,
    ! [Nat: nat,X2: nat] :
      ( ( Nat = aa_nat_nat(suc,X2) )
     => ( Nat != zero_zero_nat ) ) ).

% nat.discI
tff(fact_1153_old_Onat_Odistinct_I1_J,axiom,
    ! [Nat2: nat] : zero_zero_nat != aa_nat_nat(suc,Nat2) ).

% old.nat.distinct(1)
tff(fact_1154_old_Onat_Odistinct_I2_J,axiom,
    ! [Nat2: nat] : aa_nat_nat(suc,Nat2) != zero_zero_nat ).

% old.nat.distinct(2)
tff(fact_1155_nat_Odistinct_I1_J,axiom,
    ! [X2: nat] : zero_zero_nat != aa_nat_nat(suc,X2) ).

% nat.distinct(1)
tff(fact_1156_not__less__less__Suc__eq,axiom,
    ! [Nb: nat,Mb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(suc,Mb))
      <=> ( Nb = Mb ) ) ) ).

% not_less_less_Suc_eq
tff(fact_1157_strict__inc__induct,axiom,
    ! [I3: nat,J: nat,P: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),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,I3) ) ) ) ).

% strict_inc_induct
tff(fact_1158_less__Suc__induct,axiom,
    ! [I3: nat,J: nat,P: fun_nat_fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),J)
     => ( ! [I2: nat] : aa_nat_o(aa_nat_fun_nat_o(P,I2),aa_nat_nat(suc,I2))
       => ( ! [I2: nat,J2: nat,K2: 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),K2)
               => ( aa_nat_o(aa_nat_fun_nat_o(P,I2),J2)
                 => ( aa_nat_o(aa_nat_fun_nat_o(P,J2),K2)
                   => aa_nat_o(aa_nat_fun_nat_o(P,I2),K2) ) ) ) )
         => aa_nat_o(aa_nat_fun_nat_o(P,I3),J) ) ) ) ).

% less_Suc_induct
tff(fact_1159_less__trans__Suc,axiom,
    ! [I3: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),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,I3)),K) ) ) ).

% less_trans_Suc
tff(fact_1160_Suc__less__SucD,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Mb)),aa_nat_nat(suc,Nb))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% Suc_less_SucD
tff(fact_1161_less__antisym,axiom,
    ! [Nb: nat,Mb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(suc,Mb))
       => ( Mb = Nb ) ) ) ).

% less_antisym
tff(fact_1162_Suc__less__eq2,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),Mb)
    <=> ? [M3: nat] :
          ( ( Mb = aa_nat_nat(suc,M3) )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),M3) ) ) ).

% Suc_less_eq2
tff(fact_1163_All__less__Suc,axiom,
    ! [Nb: nat,P: fun_nat_o] :
      ( ! [I5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),aa_nat_nat(suc,Nb))
         => aa_nat_o(P,I5) )
    <=> ( aa_nat_o(P,Nb)
        & ! [I5: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),Nb)
           => aa_nat_o(P,I5) ) ) ) ).

% All_less_Suc
tff(fact_1164_not__less__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(suc,Mb)) ) ).

% not_less_eq
tff(fact_1165_less__Suc__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(suc,Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
        | ( Mb = Nb ) ) ) ).

% less_Suc_eq
tff(fact_1166_Ex__less__Suc,axiom,
    ! [Nb: nat,P: fun_nat_o] :
      ( ? [I5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),aa_nat_nat(suc,Nb))
          & aa_nat_o(P,I5) )
    <=> ( aa_nat_o(P,Nb)
        | ? [I5: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),Nb)
            & aa_nat_o(P,I5) ) ) ) ).

% Ex_less_Suc
tff(fact_1167_less__SucI,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(suc,Nb)) ) ).

% less_SucI
tff(fact_1168_less__SucE,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(suc,Nb))
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
       => ( Mb = Nb ) ) ) ).

% less_SucE
tff(fact_1169_Suc__lessI,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => ( ( aa_nat_nat(suc,Mb) != Nb )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Mb)),Nb) ) ) ).

% Suc_lessI
tff(fact_1170_Suc__lessE,axiom,
    ! [I3: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,I3)),K)
     => ~ ! [J2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),J2)
           => ( K != aa_nat_nat(suc,J2) ) ) ) ).

% Suc_lessE
tff(fact_1171_Suc__lessD,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Mb)),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% Suc_lessD
tff(fact_1172_Nat_OlessE,axiom,
    ! [I3: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),K)
     => ( ( K != aa_nat_nat(suc,I3) )
       => ~ ! [J2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),J2)
             => ( K != aa_nat_nat(suc,J2) ) ) ) ) ).

% Nat.lessE
tff(fact_1173_infinite__descent0,axiom,
    ! [P: fun_nat_o,Nb: 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)
             => ? [M: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),N)
                  & ~ aa_nat_o(P,M) ) ) )
       => aa_nat_o(P,Nb) ) ) ).

% infinite_descent0
tff(fact_1174_gr__implies__not0,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => ( Nb != zero_zero_nat ) ) ).

% gr_implies_not0
tff(fact_1175_less__zeroE,axiom,
    ! [Nb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),zero_zero_nat) ).

% less_zeroE
tff(fact_1176_not__less0,axiom,
    ! [Nb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),zero_zero_nat) ).

% not_less0
tff(fact_1177_not__gr0,axiom,
    ! [Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
    <=> ( Nb = zero_zero_nat ) ) ).

% not_gr0
tff(fact_1178_gr0I,axiom,
    ! [Nb: nat] :
      ( ( Nb != zero_zero_nat )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).

% gr0I
tff(fact_1179_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_1180_transitive__stepwise__le,axiom,
    ! [Mb: nat,Nb: nat,R: fun_nat_fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( ! [X3: nat] : aa_nat_o(aa_nat_fun_nat_o(R,X3),X3)
       => ( ! [X3: nat,Y: nat,Z2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(R,X3),Y)
             => ( aa_nat_o(aa_nat_fun_nat_o(R,Y),Z2)
               => aa_nat_o(aa_nat_fun_nat_o(R,X3),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,Mb),Nb) ) ) ) ) ).

% transitive_stepwise_le
tff(fact_1181_nat__induct__at__least,axiom,
    ! [Mb: nat,Nb: nat,P: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_o(P,Mb)
       => ( ! [N: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),N)
             => ( aa_nat_o(P,N)
               => aa_nat_o(P,aa_nat_nat(suc,N)) ) )
         => aa_nat_o(P,Nb) ) ) ) ).

% nat_induct_at_least
tff(fact_1182_full__nat__induct,axiom,
    ! [P: fun_nat_o,Nb: nat] :
      ( ! [N: nat] :
          ( ! [M: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,M)),N)
             => aa_nat_o(P,M) )
         => aa_nat_o(P,N) )
     => aa_nat_o(P,Nb) ) ).

% full_nat_induct
tff(fact_1183_not__less__eq__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Nb)),Mb) ) ).

% not_less_eq_eq
tff(fact_1184_Suc__n__not__le__n,axiom,
    ! [Nb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Nb)),Nb) ).

% Suc_n_not_le_n
tff(fact_1185_le__Suc__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
        | ( Mb = aa_nat_nat(suc,Nb) ) ) ) ).

% le_Suc_eq
tff(fact_1186_Suc__le__D,axiom,
    ! [Nb: nat,M4: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Nb)),M4)
     => ? [M2: nat] : M4 = aa_nat_nat(suc,M2) ) ).

% Suc_le_D
tff(fact_1187_le__SucI,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb)) ) ).

% le_SucI
tff(fact_1188_le__SucE,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb))
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
       => ( Mb = aa_nat_nat(suc,Nb) ) ) ) ).

% le_SucE
tff(fact_1189_Suc__leD,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Mb)),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% Suc_leD
tff(fact_1190_le__0__eq,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),zero_zero_nat)
    <=> ( Nb = zero_zero_nat ) ) ).

% le_0_eq
tff(fact_1191_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_1192_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_1193_less__eq__nat_Osimps_I1_J,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Nb) ).

% less_eq_nat.simps(1)
tff(fact_1194_add__Suc__shift,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(suc,Mb)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),aa_nat_nat(suc,Nb)) ).

% add_Suc_shift
tff(fact_1195_add__Suc,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(suc,Mb)),Nb) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) ).

% add_Suc
tff(fact_1196_nat__arith_Osuc1,axiom,
    ! [A3: nat,K: nat,A: nat] :
      ( ( A3 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),A) )
     => ( aa_nat_nat(suc,A3) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_nat_nat(suc,A)) ) ) ).

% nat_arith.suc1
tff(fact_1197_less__mono__imp__le__mono,axiom,
    ! [F: fun_nat_nat,I3: 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,I3),J)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,I3)),aa_nat_nat(F,J)) ) ) ).

% less_mono_imp_le_mono
tff(fact_1198_le__neq__implies__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( ( Mb != Nb )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ) ).

% le_neq_implies_less
tff(fact_1199_less__or__eq__imp__le,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
        | ( Mb = Nb ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% less_or_eq_imp_le
tff(fact_1200_le__eq__less__or__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
        | ( Mb = Nb ) ) ) ).

% le_eq_less_or_eq
tff(fact_1201_less__imp__le__nat,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% less_imp_le_nat
tff(fact_1202_nat__less__le,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
        & ( Mb != Nb ) ) ) ).

% nat_less_le
tff(fact_1203_add__eq__self__zero,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb) = Mb )
     => ( Nb = zero_zero_nat ) ) ).

% add_eq_self_zero
tff(fact_1204_plus__nat_Oadd__0,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,zero_zero_nat),Nb) = Nb ).

% plus_nat.add_0
tff(fact_1205_less__add__eq__less,axiom,
    ! [K: nat,L: nat,Mb: nat,Nb: 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,Mb),L) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Nb) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ) ).

% less_add_eq_less
tff(fact_1206_trans__less__add2,axiom,
    ! [I3: nat,J: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),J)) ) ).

% trans_less_add2
tff(fact_1207_trans__less__add1,axiom,
    ! [I3: nat,J: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),Mb)) ) ).

% trans_less_add1
tff(fact_1208_add__less__mono1,axiom,
    ! [I3: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I3),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K)) ) ).

% add_less_mono1
tff(fact_1209_not__add__less2,axiom,
    ! [J: nat,I3: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),I3)),I3) ).

% not_add_less2
tff(fact_1210_not__add__less1,axiom,
    ! [I3: 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,I3),J)),I3) ).

% not_add_less1
tff(fact_1211_add__less__mono,axiom,
    ! [I3: nat,J: nat,K: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),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,I3),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ) ).

% add_less_mono
tff(fact_1212_add__lessD1,axiom,
    ! [I3: 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,I3),J)),K)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),K) ) ).

% add_lessD1
tff(fact_1213_nat__le__iff__add,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
    <=> ? [K3: nat] : Nb = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),K3) ) ).

% nat_le_iff_add
tff(fact_1214_trans__le__add2,axiom,
    ! [I3: nat,J: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I3),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I3),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),J)) ) ).

% trans_le_add2
tff(fact_1215_trans__le__add1,axiom,
    ! [I3: nat,J: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I3),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I3),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),Mb)) ) ).

% trans_le_add1
tff(fact_1216_add__le__mono1,axiom,
    ! [I3: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I3),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I3),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K)) ) ).

% add_le_mono1
tff(fact_1217_add__le__mono,axiom,
    ! [I3: nat,J: nat,K: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I3),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,I3),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ) ).

% add_le_mono
tff(fact_1218_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_1219_add__leD2,axiom,
    ! [Mb: nat,K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),K)),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb) ) ).

% add_leD2
tff(fact_1220_add__leD1,axiom,
    ! [Mb: nat,K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),K)),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% add_leD1
tff(fact_1221_le__add2,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) ).

% le_add2
tff(fact_1222_le__add1,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)) ).

% le_add1
tff(fact_1223_add__leE,axiom,
    ! [Mb: nat,K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),K)),Nb)
     => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
         => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb) ) ) ).

% add_leE
tff(fact_1224_max__add__distrib__right,axiom,
    ! [Xb: real,Yb: real,Z: real] : aa_real_real(plus_plus_real(Xb),aa_real_real(ord_max_real(Yb),Z)) = aa_real_real(ord_max_real(aa_real_real(plus_plus_real(Xb),Yb)),aa_real_real(plus_plus_real(Xb),Z)) ).

% max_add_distrib_right
tff(fact_1225_max__add__distrib__right,axiom,
    ! [Xb: rat,Yb: rat,Z: rat] : aa_rat_rat(plus_plus_rat(Xb),aa_rat_rat(ord_max_rat(Yb),Z)) = aa_rat_rat(ord_max_rat(aa_rat_rat(plus_plus_rat(Xb),Yb)),aa_rat_rat(plus_plus_rat(Xb),Z)) ).

% max_add_distrib_right
tff(fact_1226_max__add__distrib__right,axiom,
    ! [Xb: nat,Yb: nat,Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Yb),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xb),Yb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xb),Z)) ).

% max_add_distrib_right
tff(fact_1227_max__add__distrib__right,axiom,
    ! [Xb: int,Yb: int,Z: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),aa_int_int(aa_int_fun_int_int(ord_max_int,Yb),Z)) = aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Yb)),aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Z)) ).

% max_add_distrib_right
tff(fact_1228_max__add__distrib__left,axiom,
    ! [Xb: real,Yb: real,Z: real] : aa_real_real(plus_plus_real(aa_real_real(ord_max_real(Xb),Yb)),Z) = aa_real_real(ord_max_real(aa_real_real(plus_plus_real(Xb),Z)),aa_real_real(plus_plus_real(Yb),Z)) ).

% max_add_distrib_left
tff(fact_1229_max__add__distrib__left,axiom,
    ! [Xb: rat,Yb: rat,Z: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(ord_max_rat(Xb),Yb)),Z) = aa_rat_rat(ord_max_rat(aa_rat_rat(plus_plus_rat(Xb),Z)),aa_rat_rat(plus_plus_rat(Yb),Z)) ).

% max_add_distrib_left
tff(fact_1230_max__add__distrib__left,axiom,
    ! [Xb: nat,Yb: nat,Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xb),Yb)),Z) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xb),Z)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Yb),Z)) ).

% max_add_distrib_left
tff(fact_1231_max__add__distrib__left,axiom,
    ! [Xb: int,Yb: int,Z: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(ord_max_int,Xb),Yb)),Z) = aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Z)),aa_int_int(aa_int_fun_int_int(plus_plus_int,Yb),Z)) ).

% max_add_distrib_left
tff(fact_1232_nat__add__max__right,axiom,
    ! [Mb: nat,Nb: nat,Q2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Nb),Q2)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Q2)) ).

% nat_add_max_right
tff(fact_1233_nat__add__max__left,axiom,
    ! [Mb: nat,Nb: nat,Q2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Mb),Nb)),Q2) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Q2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Q2)) ).

% nat_add_max_left
tff(fact_1234_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_1235_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_1236_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_1237_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_1238_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_1239_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_1240_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_1241_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_1242_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_1243_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_1244_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_1245_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_1246_add__nonpos__eq__0__iff,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Yb),zero_zero_real)
       => ( ( aa_real_real(plus_plus_real(Xb),Yb) = zero_zero_real )
        <=> ( ( Xb = zero_zero_real )
            & ( Yb = zero_zero_real ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_1247_add__nonpos__eq__0__iff,axiom,
    ! [Xb: rat,Yb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Yb),zero_zero_rat)
       => ( ( aa_rat_rat(plus_plus_rat(Xb),Yb) = zero_zero_rat )
        <=> ( ( Xb = zero_zero_rat )
            & ( Yb = zero_zero_rat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_1248_add__nonpos__eq__0__iff,axiom,
    ! [Xb: nat,Yb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Yb),zero_zero_nat)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xb),Yb) = zero_zero_nat )
        <=> ( ( Xb = zero_zero_nat )
            & ( Yb = zero_zero_nat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_1249_add__nonpos__eq__0__iff,axiom,
    ! [Xb: int,Yb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Yb),zero_zero_int)
       => ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Yb) = zero_zero_int )
        <=> ( ( Xb = zero_zero_int )
            & ( Yb = zero_zero_int ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_1250_add__nonneg__eq__0__iff,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Yb)
       => ( ( aa_real_real(plus_plus_real(Xb),Yb) = zero_zero_real )
        <=> ( ( Xb = zero_zero_real )
            & ( Yb = zero_zero_real ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_1251_add__nonneg__eq__0__iff,axiom,
    ! [Xb: rat,Yb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Yb)
       => ( ( aa_rat_rat(plus_plus_rat(Xb),Yb) = zero_zero_rat )
        <=> ( ( Xb = zero_zero_rat )
            & ( Yb = zero_zero_rat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_1252_add__nonneg__eq__0__iff,axiom,
    ! [Xb: nat,Yb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Xb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Yb)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xb),Yb) = zero_zero_nat )
        <=> ( ( Xb = zero_zero_nat )
            & ( Yb = zero_zero_nat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_1253_add__nonneg__eq__0__iff,axiom,
    ! [Xb: int,Yb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Yb)
       => ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Yb) = zero_zero_int )
        <=> ( ( Xb = zero_zero_int )
            & ( Yb = zero_zero_int ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_1254_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(plus_plus_real(A),B)),zero_zero_real) ) ) ).

% add_nonpos_nonpos
tff(fact_1255_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(plus_plus_rat(A),B)),zero_zero_rat) ) ) ).

% add_nonpos_nonpos
tff(fact_1256_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_1257_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_1258_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(plus_plus_real(A),B)) ) ) ).

% add_nonneg_nonneg
tff(fact_1259_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(plus_plus_rat(A),B)) ) ) ).

% add_nonneg_nonneg
tff(fact_1260_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_1261_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_1262_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(plus_plus_real(A),C)) ) ) ).

% add_increasing2
tff(fact_1263_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(plus_plus_rat(A),C)) ) ) ).

% add_increasing2
tff(fact_1264_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_1265_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_1266_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(plus_plus_real(A),C)),B) ) ) ).

% add_decreasing2
tff(fact_1267_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(plus_plus_rat(A),C)),B) ) ) ).

% add_decreasing2
tff(fact_1268_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_1269_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_1270_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(plus_plus_real(A),C)) ) ) ).

% add_increasing
tff(fact_1271_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(plus_plus_rat(A),C)) ) ) ).

% add_increasing
tff(fact_1272_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_1273_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_1274_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(plus_plus_real(A),C)),B) ) ) ).

% add_decreasing
tff(fact_1275_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(plus_plus_rat(A),C)),B) ) ) ).

% add_decreasing
tff(fact_1276_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_1277_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_1278_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(plus_plus_real(A),C)),aa_real_real(plus_plus_real(B),D)) ) ) ).

% add_less_le_mono
tff(fact_1279_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(plus_plus_rat(A),C)),aa_rat_rat(plus_plus_rat(B),D)) ) ) ).

% add_less_le_mono
tff(fact_1280_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_1281_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_1282_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(plus_plus_real(A),C)),aa_real_real(plus_plus_real(B),D)) ) ) ).

% add_le_less_mono
tff(fact_1283_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(plus_plus_rat(A),C)),aa_rat_rat(plus_plus_rat(B),D)) ) ) ).

% add_le_less_mono
tff(fact_1284_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_1285_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_1286_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I3: real,J: real,K: real,L: real] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,I3),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(plus_plus_real(I3),K)),aa_real_real(plus_plus_real(J),L)) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_1287_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I3: rat,J: rat,K: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,I3),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(plus_plus_rat(I3),K)),aa_rat_rat(plus_plus_rat(J),L)) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_1288_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I3: nat,J: nat,K: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),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,I3),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_1289_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I3: int,J: int,K: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,I3),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,I3),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_1290_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I3: real,J: real,K: real,L: real] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,I3),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(plus_plus_real(I3),K)),aa_real_real(plus_plus_real(J),L)) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_1291_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I3: rat,J: rat,K: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,I3),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(plus_plus_rat(I3),K)),aa_rat_rat(plus_plus_rat(J),L)) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_1292_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I3: nat,J: nat,K: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I3),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,I3),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_1293_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I3: int,J: int,K: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I3),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,I3),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_1294_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_1295_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_1296_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_1297_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_1298_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_1299_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_1300_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_1301_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_1302_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(plus_plus_real(A),C)) ) ) ).

% pos_add_strict
tff(fact_1303_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(plus_plus_rat(A),C)) ) ) ).

% pos_add_strict
tff(fact_1304_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_1305_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_1306_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_1307_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(plus_plus_real(A),B)) ) ) ).

% add_pos_pos
tff(fact_1308_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(plus_plus_rat(A),B)) ) ) ).

% add_pos_pos
tff(fact_1309_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_1310_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_1311_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(plus_plus_real(A),B)),zero_zero_real) ) ) ).

% add_neg_neg
tff(fact_1312_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(plus_plus_rat(A),B)),zero_zero_rat) ) ) ).

% add_neg_neg
tff(fact_1313_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_1314_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_1315_add__less__zeroD,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(Xb),Yb)),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real)
        | aa_real_o(aa_real_fun_real_o(ord_less_real,Yb),zero_zero_real) ) ) ).

% add_less_zeroD
tff(fact_1316_add__less__zeroD,axiom,
    ! [Xb: rat,Yb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(Xb),Yb)),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),zero_zero_rat)
        | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Yb),zero_zero_rat) ) ) ).

% add_less_zeroD
tff(fact_1317_add__less__zeroD,axiom,
    ! [Xb: int,Yb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Yb)),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),zero_zero_int)
        | aa_int_o(aa_int_fun_int_o(ord_less_int,Yb),zero_zero_int) ) ) ).

% add_less_zeroD
tff(fact_1318_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(plus_plus_real(A),one_one_real)),aa_real_real(plus_plus_real(B),one_one_real)) ) ).

% add_mono1
tff(fact_1319_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(plus_plus_rat(A),one_one_rat)),aa_rat_rat(plus_plus_rat(B),one_one_rat)) ) ).

% add_mono1
tff(fact_1320_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_1321_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_1322_less__add__one,axiom,
    ! [A: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(plus_plus_real(A),one_one_real)) ).

% less_add_one
tff(fact_1323_less__add__one,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(plus_plus_rat(A),one_one_rat)) ).

% less_add_one
tff(fact_1324_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_1325_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_1326_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_real,Nb: nat,Mb: 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,Nb)),aa_nat_real(F,Mb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_1327_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_rat,Nb: nat,Mb: 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,Nb)),aa_nat_rat(F,Mb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_1328_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_num,Nb: nat,Mb: nat] :
      ( ! [N: nat] : aa_num_o(aa_num_fun_num_o(ord_less_num,aa_nat_num(F,N)),aa_nat_num(F,aa_nat_nat(suc,N)))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_nat_num(F,Nb)),aa_nat_num(F,Mb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_1329_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_nat,Nb: nat,Mb: 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,Nb)),aa_nat_nat(F,Mb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_1330_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_int,Nb: nat,Mb: 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,Nb)),aa_nat_int(F,Mb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_1331_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_real,Nb: 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,Nb),N3)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,Nb)),aa_nat_real(F,N3)) ) ) ).

% lift_Suc_mono_less
tff(fact_1332_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_rat,Nb: 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,Nb),N3)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,Nb)),aa_nat_rat(F,N3)) ) ) ).

% lift_Suc_mono_less
tff(fact_1333_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_num,Nb: nat,N3: nat] :
      ( ! [N: nat] : aa_num_o(aa_num_fun_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,Nb),N3)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_nat_num(F,Nb)),aa_nat_num(F,N3)) ) ) ).

% lift_Suc_mono_less
tff(fact_1334_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_nat,Nb: 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,Nb),N3)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,Nb)),aa_nat_nat(F,N3)) ) ) ).

% lift_Suc_mono_less
tff(fact_1335_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_int,Nb: 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,Nb),N3)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,Nb)),aa_nat_int(F,N3)) ) ) ).

% lift_Suc_mono_less
tff(fact_1336_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_set_nat,Nb: 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,Nb),N3)
       => aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(F,N3)),aa_nat_set_nat(F,Nb)) ) ) ).

% lift_Suc_antimono_le
tff(fact_1337_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_rat,Nb: 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,Nb),N3)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,N3)),aa_nat_rat(F,Nb)) ) ) ).

% lift_Suc_antimono_le
tff(fact_1338_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_num,Nb: nat,N3: nat] :
      ( ! [N: nat] : aa_num_o(aa_num_fun_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,Nb),N3)
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,N3)),aa_nat_num(F,Nb)) ) ) ).

% lift_Suc_antimono_le
tff(fact_1339_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_nat,Nb: 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,Nb),N3)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,N3)),aa_nat_nat(F,Nb)) ) ) ).

% lift_Suc_antimono_le
tff(fact_1340_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_int,Nb: 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,Nb),N3)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,N3)),aa_nat_int(F,Nb)) ) ) ).

% lift_Suc_antimono_le
tff(fact_1341_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_set_nat,Nb: 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,Nb),N3)
       => aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(F,Nb)),aa_nat_set_nat(F,N3)) ) ) ).

% lift_Suc_mono_le
tff(fact_1342_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_rat,Nb: 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,Nb),N3)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,Nb)),aa_nat_rat(F,N3)) ) ) ).

% lift_Suc_mono_le
tff(fact_1343_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_num,Nb: nat,N3: nat] :
      ( ! [N: nat] : aa_num_o(aa_num_fun_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,Nb),N3)
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,Nb)),aa_nat_num(F,N3)) ) ) ).

% lift_Suc_mono_le
tff(fact_1344_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_nat,Nb: 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,Nb),N3)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,Nb)),aa_nat_nat(F,N3)) ) ) ).

% lift_Suc_mono_le
tff(fact_1345_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_int,Nb: 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,Nb),N3)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,Nb)),aa_nat_int(F,N3)) ) ) ).

% lift_Suc_mono_le
tff(fact_1346_less__Suc__eq__0__disj,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(suc,Nb))
    <=> ( ( Mb = zero_zero_nat )
        | ? [J3: nat] :
            ( ( Mb = aa_nat_nat(suc,J3) )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),Nb) ) ) ) ).

% less_Suc_eq_0_disj
tff(fact_1347_gr0__implies__Suc,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ? [M2: nat] : Nb = aa_nat_nat(suc,M2) ) ).

% gr0_implies_Suc
tff(fact_1348_All__less__Suc2,axiom,
    ! [Nb: nat,P: fun_nat_o] :
      ( ! [I5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),aa_nat_nat(suc,Nb))
         => aa_nat_o(P,I5) )
    <=> ( aa_nat_o(P,zero_zero_nat)
        & ! [I5: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),Nb)
           => aa_nat_o(P,aa_nat_nat(suc,I5)) ) ) ) ).

% All_less_Suc2
tff(fact_1349_gr0__conv__Suc,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
    <=> ? [M5: nat] : Nb = aa_nat_nat(suc,M5) ) ).

% gr0_conv_Suc
tff(fact_1350_Ex__less__Suc2,axiom,
    ! [Nb: nat,P: fun_nat_o] :
      ( ? [I5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),aa_nat_nat(suc,Nb))
          & aa_nat_o(P,I5) )
    <=> ( aa_nat_o(P,zero_zero_nat)
        | ? [I5: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),Nb)
            & aa_nat_o(P,aa_nat_nat(suc,I5)) ) ) ) ).

% Ex_less_Suc2
tff(fact_1351_le__imp__less__Suc,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(suc,Nb)) ) ).

% le_imp_less_Suc
tff(fact_1352_less__eq__Suc__le,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Nb)),Mb) ) ).

% less_eq_Suc_le
tff(fact_1353_less__Suc__eq__le,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(suc,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% less_Suc_eq_le
tff(fact_1354_le__less__Suc__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(suc,Mb))
      <=> ( Nb = Mb ) ) ) ).

% le_less_Suc_eq
tff(fact_1355_Suc__le__lessD,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Mb)),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% Suc_le_lessD
tff(fact_1356_inc__induct,axiom,
    ! [I3: nat,J: nat,P: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I3),J)
     => ( aa_nat_o(P,J)
       => ( ! [N: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I3),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,I3) ) ) ) ).

% inc_induct
tff(fact_1357_dec__induct,axiom,
    ! [I3: nat,J: nat,P: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I3),J)
     => ( aa_nat_o(P,I3)
       => ( ! [N: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I3),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_1358_Suc__le__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Mb)),Nb)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% Suc_le_eq
tff(fact_1359_Suc__leI,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Mb)),Nb) ) ).

% Suc_leI
tff(fact_1360_one__is__add,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(suc,zero_zero_nat) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb) )
    <=> ( ( ( Mb = aa_nat_nat(suc,zero_zero_nat) )
          & ( Nb = zero_zero_nat ) )
        | ( ( Mb = zero_zero_nat )
          & ( Nb = aa_nat_nat(suc,zero_zero_nat) ) ) ) ) ).

% one_is_add
tff(fact_1361_add__is__1,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ( ( ( Mb = aa_nat_nat(suc,zero_zero_nat) )
          & ( Nb = zero_zero_nat ) )
        | ( ( Mb = zero_zero_nat )
          & ( Nb = aa_nat_nat(suc,zero_zero_nat) ) ) ) ) ).

% add_is_1
tff(fact_1362_ex__least__nat__le,axiom,
    ! [P: fun_nat_o,Nb: nat] :
      ( aa_nat_o(P,Nb)
     => ( ~ aa_nat_o(P,zero_zero_nat)
       => ? [K2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
            & ! [I: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),K2)
               => ~ aa_nat_o(P,I) )
            & aa_nat_o(P,K2) ) ) ) ).

% ex_least_nat_le
tff(fact_1363_less__imp__Suc__add,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => ? [K2: nat] : Nb = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),K2)) ) ).

% less_imp_Suc_add
tff(fact_1364_less__iff__Suc__add,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
    <=> ? [K3: nat] : Nb = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),K3)) ) ).

% less_iff_Suc_add
tff(fact_1365_less__add__Suc2,axiom,
    ! [I3: nat,Mb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),I3))) ).

% less_add_Suc2
tff(fact_1366_less__add__Suc1,axiom,
    ! [I3: nat,Mb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I3),Mb))) ).

% less_add_Suc1
tff(fact_1367_less__natE,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => ~ ! [Q3: nat] : Nb != aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Q3)) ) ).

% less_natE
tff(fact_1368_less__imp__add__positive,axiom,
    ! [I3: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),J)
     => ? [K2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K2)
          & ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I3),K2) = J ) ) ) ).

% less_imp_add_positive
tff(fact_1369_mono__nat__linear__lb,axiom,
    ! [F: fun_nat_nat,Mb: nat,K: nat] :
      ( ! [M2: nat,N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),N)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,M2)),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,Mb)),K)),aa_nat_nat(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),K))) ) ).

% mono_nat_linear_lb
tff(fact_1370_One__nat__def,axiom,
    one_one_nat = aa_nat_nat(suc,zero_zero_nat) ).

% One_nat_def
tff(fact_1371_Suc__eq__plus1__left,axiom,
    ! [Nb: nat] : aa_nat_nat(suc,Nb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),Nb) ).

% Suc_eq_plus1_left
tff(fact_1372_plus__1__eq__Suc,axiom,
    aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat) = suc ).

% plus_1_eq_Suc
tff(fact_1373_Suc__eq__plus1,axiom,
    ! [Nb: nat] : aa_nat_nat(suc,Nb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat) ).

% Suc_eq_plus1
tff(fact_1374_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(plus_plus_real(A),C)) ) ) ).

% add_strict_increasing2
tff(fact_1375_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(plus_plus_rat(A),C)) ) ) ).

% add_strict_increasing2
tff(fact_1376_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_1377_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_1378_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(plus_plus_real(A),C)) ) ) ).

% add_strict_increasing
tff(fact_1379_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(plus_plus_rat(A),C)) ) ) ).

% add_strict_increasing
tff(fact_1380_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_1381_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_1382_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(plus_plus_real(A),B)) ) ) ).

% add_pos_nonneg
tff(fact_1383_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(plus_plus_rat(A),B)) ) ) ).

% add_pos_nonneg
tff(fact_1384_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_1385_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_1386_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(plus_plus_real(A),B)),zero_zero_real) ) ) ).

% add_nonpos_neg
tff(fact_1387_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(plus_plus_rat(A),B)),zero_zero_rat) ) ) ).

% add_nonpos_neg
tff(fact_1388_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_1389_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_1390_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(plus_plus_real(A),B)) ) ) ).

% add_nonneg_pos
tff(fact_1391_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(plus_plus_rat(A),B)) ) ) ).

% add_nonneg_pos
tff(fact_1392_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_1393_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_1394_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(plus_plus_real(A),B)),zero_zero_real) ) ) ).

% add_neg_nonpos
tff(fact_1395_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(plus_plus_rat(A),B)),zero_zero_rat) ) ) ).

% add_neg_nonpos
tff(fact_1396_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_1397_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_1398_zero__less__two,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(plus_plus_real(one_one_real),one_one_real)) ).

% zero_less_two
tff(fact_1399_zero__less__two,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(one_one_rat),one_one_rat)) ).

% zero_less_two
tff(fact_1400_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_1401_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_1402_ex__least__nat__less,axiom,
    ! [P: fun_nat_o,Nb: nat] :
      ( aa_nat_o(P,Nb)
     => ( ~ aa_nat_o(P,zero_zero_nat)
       => ? [K2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K2),Nb)
            & ! [I: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),K2)
               => ~ aa_nat_o(P,I) )
            & aa_nat_o(P,aa_nat_nat(suc,K2)) ) ) ) ).

% ex_least_nat_less
tff(fact_1403_nat__induct__non__zero,axiom,
    ! [Nb: nat,P: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( 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,Nb) ) ) ) ).

% nat_induct_non_zero
tff(fact_1404_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_1405_nat__induct2,axiom,
    ! [P: fun_nat_o,Nb: 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,Nb) ) ) ) ).

% nat_induct2
tff(fact_1406_field__less__half__sum,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Yb)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(Xb),Yb)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ).

% field_less_half_sum
tff(fact_1407_field__less__half__sum,axiom,
    ! [Xb: rat,Yb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Yb)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(Xb),Yb)),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))) ) ).

% field_less_half_sum
tff(fact_1408_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_1409_low__inv,axiom,
    ! [Xb: nat,Nb: nat,Yb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))
     => ( 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,Yb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))),Xb)),Nb) = Xb ) ) ).

% low_inv
tff(fact_1410_high__inv,axiom,
    ! [Xb: nat,Nb: nat,Yb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))
     => ( 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,Yb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))),Xb)),Nb) = Yb ) ) ).

% high_inv
tff(fact_1411_vebt__member_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT,Xb: nat] : ~ aa_nat_o(vEBT_vebt_member(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vb,Vc)),Xb) ).

% vebt_member.simps(4)
tff(fact_1412_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_1413_max__less__iff__conj,axiom,
    ! [Xb: real,Yb: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(ord_max_real(Xb),Yb)),Z)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Z)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,Yb),Z) ) ) ).

% max_less_iff_conj
tff(fact_1414_max__less__iff__conj,axiom,
    ! [Xb: rat,Yb: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(ord_max_rat(Xb),Yb)),Z)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Z)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Yb),Z) ) ) ).

% max_less_iff_conj
tff(fact_1415_max__less__iff__conj,axiom,
    ! [Xb: num,Yb: num,Z: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(ord_max_num(Xb),Yb)),Z)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Z)
        & aa_num_o(aa_num_fun_num_o(ord_less_num,Yb),Z) ) ) ).

% max_less_iff_conj
tff(fact_1416_max__less__iff__conj,axiom,
    ! [Xb: nat,Yb: nat,Z: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xb),Yb)),Z)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Z)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Yb),Z) ) ) ).

% max_less_iff_conj
tff(fact_1417_max__less__iff__conj,axiom,
    ! [Xb: int,Yb: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(ord_max_int,Xb),Yb)),Z)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Z)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,Yb),Z) ) ) ).

% max_less_iff_conj
tff(fact_1418_max_Oabsorb4,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_real_real(ord_max_real(A),B) = B ) ) ).

% max.absorb4
tff(fact_1419_max_Oabsorb4,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_rat_rat(ord_max_rat(A),B) = B ) ) ).

% max.absorb4
tff(fact_1420_max_Oabsorb4,axiom,
    ! [A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
     => ( aa_num_num(ord_max_num(A),B) = B ) ) ).

% max.absorb4
tff(fact_1421_max_Oabsorb4,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(ord_max_nat,A),B) = B ) ) ).

% max.absorb4
tff(fact_1422_max_Oabsorb4,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(ord_max_int,A),B) = B ) ) ).

% max.absorb4
tff(fact_1423_max_Oabsorb3,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
     => ( aa_real_real(ord_max_real(A),B) = A ) ) ).

% max.absorb3
tff(fact_1424_max_Oabsorb3,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => ( aa_rat_rat(ord_max_rat(A),B) = A ) ) ).

% max.absorb3
tff(fact_1425_max_Oabsorb3,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
     => ( aa_num_num(ord_max_num(A),B) = A ) ) ).

% max.absorb3
tff(fact_1426_max_Oabsorb3,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = A ) ) ).

% max.absorb3
tff(fact_1427_max_Oabsorb3,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => ( aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) = A ) ) ).

% max.absorb3
tff(fact_1428_max_Oidem,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),A) = A ).

% max.idem
tff(fact_1429_max_Oidem,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(ord_max_int,A),A) = A ).

% max.idem
tff(fact_1430_max_Oleft__idem,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) ).

% max.left_idem
tff(fact_1431_max_Oleft__idem,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(ord_max_int,A),aa_int_int(aa_int_fun_int_int(ord_max_int,A),B)) = aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) ).

% max.left_idem
tff(fact_1432_max_Oright__idem,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B)),B) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) ).

% max.right_idem
tff(fact_1433_max_Oright__idem,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(aa_int_fun_int_int(ord_max_int,A),B)),B) = aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) ).

% max.right_idem
tff(fact_1434_mult__zero__left,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(zero_zero_complex),A) = zero_zero_complex ).

% mult_zero_left
tff(fact_1435_mult__zero__left,axiom,
    ! [A: real] : aa_real_real(times_times_real(zero_zero_real),A) = zero_zero_real ).

% mult_zero_left
tff(fact_1436_mult__zero__left,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(zero_zero_rat),A) = zero_zero_rat ).

% mult_zero_left
tff(fact_1437_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_1438_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_1439_mult__zero__right,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(A),zero_zero_complex) = zero_zero_complex ).

% mult_zero_right
tff(fact_1440_mult__zero__right,axiom,
    ! [A: real] : aa_real_real(times_times_real(A),zero_zero_real) = zero_zero_real ).

% mult_zero_right
tff(fact_1441_mult__zero__right,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(A),zero_zero_rat) = zero_zero_rat ).

% mult_zero_right
tff(fact_1442_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_1443_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_1444_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_1445_mult__eq__0__iff,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(times_times_real(A),B) = zero_zero_real )
    <=> ( ( A = zero_zero_real )
        | ( B = zero_zero_real ) ) ) ).

% mult_eq_0_iff
tff(fact_1446_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_1447_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_1448_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_1449_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_1450_mult__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( aa_real_real(times_times_real(C),A) = aa_real_real(times_times_real(C),B) )
    <=> ( ( C = zero_zero_real )
        | ( A = B ) ) ) ).

% mult_cancel_left
tff(fact_1451_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_1452_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_1453_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_1454_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_1455_mult__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( ( aa_real_real(times_times_real(A),C) = aa_real_real(times_times_real(B),C) )
    <=> ( ( C = zero_zero_real )
        | ( A = B ) ) ) ).

% mult_cancel_right
tff(fact_1456_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_1457_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_1458_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_1459_double__eq__0__iff,axiom,
    ! [A: real] :
      ( ( aa_real_real(plus_plus_real(A),A) = zero_zero_real )
    <=> ( A = zero_zero_real ) ) ).

% double_eq_0_iff
tff(fact_1460_double__eq__0__iff,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(plus_plus_rat(A),A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% double_eq_0_iff
tff(fact_1461_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_1462_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: complex] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(V)),aa_complex_complex(times_times_complex(numera6690914467698888265omplex(W)),Z)) = aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(times_times_num(V),W))),Z) ).

% mult_numeral_left_semiring_numeral
tff(fact_1463_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: real] : aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,V)),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,W)),Z)) = aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(times_times_num(V),W))),Z) ).

% mult_numeral_left_semiring_numeral
tff(fact_1464_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: rat] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,V)),aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W)),Z)) = aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(times_times_num(V),W))),Z) ).

% mult_numeral_left_semiring_numeral
tff(fact_1465_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,V)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,W)),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(times_times_num(V),W))),Z) ).

% mult_numeral_left_semiring_numeral
tff(fact_1466_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,V)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,W)),Z)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(times_times_num(V),W))),Z) ).

% mult_numeral_left_semiring_numeral
tff(fact_1467_numeral__times__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(Mb)),numera6690914467698888265omplex(Nb)) = numera6690914467698888265omplex(aa_num_num(times_times_num(Mb),Nb)) ).

% numeral_times_numeral
tff(fact_1468_numeral__times__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,Mb)),aa_num_real(numeral_numeral_real,Nb)) = aa_num_real(numeral_numeral_real,aa_num_num(times_times_num(Mb),Nb)) ).

% numeral_times_numeral
tff(fact_1469_numeral__times__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,Mb)),aa_num_rat(numeral_numeral_rat,Nb)) = aa_num_rat(numeral_numeral_rat,aa_num_num(times_times_num(Mb),Nb)) ).

% numeral_times_numeral
tff(fact_1470_numeral__times__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Nb)) = aa_num_nat(numeral_numeral_nat,aa_num_num(times_times_num(Mb),Nb)) ).

% numeral_times_numeral
tff(fact_1471_numeral__times__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,Nb)) = aa_num_int(numeral_numeral_int,aa_num_num(times_times_num(Mb),Nb)) ).

% numeral_times_numeral
tff(fact_1472_mult_Oright__neutral,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(A),one_one_complex) = A ).

% mult.right_neutral
tff(fact_1473_mult_Oright__neutral,axiom,
    ! [A: real] : aa_real_real(times_times_real(A),one_one_real) = A ).

% mult.right_neutral
tff(fact_1474_mult_Oright__neutral,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(A),one_one_rat) = A ).

% mult.right_neutral
tff(fact_1475_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_1476_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_1477_mult__1,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(one_one_complex),A) = A ).

% mult_1
tff(fact_1478_mult__1,axiom,
    ! [A: real] : aa_real_real(times_times_real(one_one_real),A) = A ).

% mult_1
tff(fact_1479_mult__1,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(one_one_rat),A) = A ).

% mult_1
tff(fact_1480_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_1481_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_1482_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_1483_times__divide__eq__left,axiom,
    ! [B: real,C: real,A: real] : aa_real_real(times_times_real(aa_real_real(divide_divide_real(B),C)),A) = aa_real_real(divide_divide_real(aa_real_real(times_times_real(B),A)),C) ).

% times_divide_eq_left
tff(fact_1484_times__divide__eq__left,axiom,
    ! [B: rat,C: rat,A: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(divide_divide_rat(B),C)),A) = aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(B),A)),C) ).

% times_divide_eq_left
tff(fact_1485_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_1486_divide__divide__eq__left,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(divide_divide_real(aa_real_real(divide_divide_real(A),B)),C) = aa_real_real(divide_divide_real(A),aa_real_real(times_times_real(B),C)) ).

% divide_divide_eq_left
tff(fact_1487_divide__divide__eq__left,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(divide_divide_rat(aa_rat_rat(divide_divide_rat(A),B)),C) = aa_rat_rat(divide_divide_rat(A),aa_rat_rat(times_times_rat(B),C)) ).

% divide_divide_eq_left
tff(fact_1488_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_1489_divide__divide__eq__right,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(divide_divide_real(A),aa_real_real(divide_divide_real(B),C)) = aa_real_real(divide_divide_real(aa_real_real(times_times_real(A),C)),B) ).

% divide_divide_eq_right
tff(fact_1490_divide__divide__eq__right,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(divide_divide_rat(A),aa_rat_rat(divide_divide_rat(B),C)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(A),C)),B) ).

% divide_divide_eq_right
tff(fact_1491_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_1492_times__divide__eq__right,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(times_times_real(A),aa_real_real(divide_divide_real(B),C)) = aa_real_real(divide_divide_real(aa_real_real(times_times_real(A),B)),C) ).

% times_divide_eq_right
tff(fact_1493_times__divide__eq__right,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(A),aa_rat_rat(divide_divide_rat(B),C)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(A),B)),C) ).

% times_divide_eq_right
tff(fact_1494_max_Oabsorb1,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( aa_rat_rat(ord_max_rat(A),B) = A ) ) ).

% max.absorb1
tff(fact_1495_max_Oabsorb1,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
     => ( aa_num_num(ord_max_num(A),B) = A ) ) ).

% max.absorb1
tff(fact_1496_max_Oabsorb1,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(ord_max_nat,A),B) = A ) ) ).

% max.absorb1
tff(fact_1497_max_Oabsorb1,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(ord_max_int,A),B) = A ) ) ).

% max.absorb1
tff(fact_1498_max_Oabsorb2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_rat_rat(ord_max_rat(A),B) = B ) ) ).

% max.absorb2
tff(fact_1499_max_Oabsorb2,axiom,
    ! [A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_num_num(ord_max_num(A),B) = B ) ) ).

% max.absorb2
tff(fact_1500_max_Oabsorb2,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(ord_max_nat,A),B) = B ) ) ).

% max.absorb2
tff(fact_1501_max_Oabsorb2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) = B ) ) ).

% max.absorb2
tff(fact_1502_max_Obounded__iff,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(ord_max_rat(B),C)),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_eq_rat,C),A) ) ) ).

% max.bounded_iff
tff(fact_1503_max_Obounded__iff,axiom,
    ! [B: num,C: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(ord_max_num(B),C)),A)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
        & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),A) ) ) ).

% max.bounded_iff
tff(fact_1504_max_Obounded__iff,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,B),C)),A)
    <=> ( 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,C),A) ) ) ).

% max.bounded_iff
tff(fact_1505_max_Obounded__iff,axiom,
    ! [B: int,C: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(ord_max_int,B),C)),A)
    <=> ( 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),A) ) ) ).

% max.bounded_iff
tff(fact_1506_mult__is__0,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb) = zero_zero_nat )
    <=> ( ( Mb = zero_zero_nat )
        | ( Nb = zero_zero_nat ) ) ) ).

% mult_is_0
tff(fact_1507_mult__0__right,axiom,
    ! [Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),zero_zero_nat) = zero_zero_nat ).

% mult_0_right
tff(fact_1508_mult__cancel1,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Mb) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb) )
    <=> ( ( Mb = Nb )
        | ( K = zero_zero_nat ) ) ) ).

% mult_cancel1
tff(fact_1509_mult__cancel2,axiom,
    ! [Mb: nat,K: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),K) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),K) )
    <=> ( ( Mb = Nb )
        | ( K = zero_zero_nat ) ) ) ).

% mult_cancel2
tff(fact_1510_nat__1__eq__mult__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( one_one_nat = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb) )
    <=> ( ( Mb = one_one_nat )
        & ( Nb = one_one_nat ) ) ) ).

% nat_1_eq_mult_iff
tff(fact_1511_nat__mult__eq__1__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb) = one_one_nat )
    <=> ( ( Mb = one_one_nat )
        & ( Nb = one_one_nat ) ) ) ).

% nat_mult_eq_1_iff
tff(fact_1512_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_1513_mult__cancel__left1,axiom,
    ! [C: real,B: real] :
      ( ( C = aa_real_real(times_times_real(C),B) )
    <=> ( ( C = zero_zero_real )
        | ( B = one_one_real ) ) ) ).

% mult_cancel_left1
tff(fact_1514_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_1515_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_1516_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_1517_mult__cancel__left2,axiom,
    ! [C: real,A: real] :
      ( ( aa_real_real(times_times_real(C),A) = C )
    <=> ( ( C = zero_zero_real )
        | ( A = one_one_real ) ) ) ).

% mult_cancel_left2
tff(fact_1518_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_1519_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_1520_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_1521_mult__cancel__right1,axiom,
    ! [C: real,B: real] :
      ( ( C = aa_real_real(times_times_real(B),C) )
    <=> ( ( C = zero_zero_real )
        | ( B = one_one_real ) ) ) ).

% mult_cancel_right1
tff(fact_1522_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_1523_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_1524_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_1525_mult__cancel__right2,axiom,
    ! [A: real,C: real] :
      ( ( aa_real_real(times_times_real(A),C) = C )
    <=> ( ( C = zero_zero_real )
        | ( A = one_one_real ) ) ) ).

% mult_cancel_right2
tff(fact_1526_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_1527_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_1528_sum__squares__eq__zero__iff,axiom,
    ! [Xb: real,Yb: real] :
      ( ( aa_real_real(plus_plus_real(aa_real_real(times_times_real(Xb),Xb)),aa_real_real(times_times_real(Yb),Yb)) = zero_zero_real )
    <=> ( ( Xb = zero_zero_real )
        & ( Yb = zero_zero_real ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_1529_sum__squares__eq__zero__iff,axiom,
    ! [Xb: rat,Yb: rat] :
      ( ( aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(Xb),Xb)),aa_rat_rat(times_times_rat(Yb),Yb)) = zero_zero_rat )
    <=> ( ( Xb = zero_zero_rat )
        & ( Yb = zero_zero_rat ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_1530_sum__squares__eq__zero__iff,axiom,
    ! [Xb: int,Yb: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Xb),Xb)),aa_int_int(aa_int_fun_int_int(times_times_int,Yb),Yb)) = zero_zero_int )
    <=> ( ( Xb = zero_zero_int )
        & ( Yb = zero_zero_int ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_1531_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_1532_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_1533_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_1534_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_1535_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_1536_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_1537_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_1538_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( aa_real_real(divide_divide_real(aa_real_real(times_times_real(A),C)),aa_real_real(times_times_real(C),B)) = aa_real_real(divide_divide_real(A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
tff(fact_1539_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(C),B)) = aa_rat_rat(divide_divide_rat(A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
tff(fact_1540_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_1541_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( aa_real_real(divide_divide_real(aa_real_real(times_times_real(A),C)),aa_real_real(times_times_real(B),C)) = aa_real_real(divide_divide_real(A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_right
tff(fact_1542_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C)) = aa_rat_rat(divide_divide_rat(A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_right
tff(fact_1543_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_1544_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( aa_real_real(divide_divide_real(aa_real_real(times_times_real(C),A)),aa_real_real(times_times_real(B),C)) = aa_real_real(divide_divide_real(A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
tff(fact_1545_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(B),C)) = aa_rat_rat(divide_divide_rat(A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
tff(fact_1546_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_1547_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( aa_real_real(divide_divide_real(aa_real_real(times_times_real(C),A)),aa_real_real(times_times_real(C),B)) = aa_real_real(divide_divide_real(A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_left
tff(fact_1548_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(C),A)),aa_rat_rat(times_times_rat(C),B)) = aa_rat_rat(divide_divide_rat(A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_left
tff(fact_1549_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_1550_mult__divide__mult__cancel__left__if,axiom,
    ! [C: real,A: real,B: real] :
      aa_real_real(divide_divide_real(aa_real_real(times_times_real(C),A)),aa_real_real(times_times_real(C),B)) = $ite(C = zero_zero_real,zero_zero_real,aa_real_real(divide_divide_real(A),B)) ).

% mult_divide_mult_cancel_left_if
tff(fact_1551_mult__divide__mult__cancel__left__if,axiom,
    ! [C: rat,A: rat,B: rat] :
      aa_rat_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,aa_rat_rat(divide_divide_rat(A),B)) ).

% mult_divide_mult_cancel_left_if
tff(fact_1552_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_1553_nonzero__mult__div__cancel__right,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( aa_real_real(divide_divide_real(aa_real_real(times_times_real(A),B)),B) = A ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_1554_nonzero__mult__div__cancel__right,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(A),B)),B) = A ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_1555_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_1556_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_1557_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_1558_nonzero__mult__div__cancel__left,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(divide_divide_real(aa_real_real(times_times_real(A),B)),A) = B ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_1559_nonzero__mult__div__cancel__left,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(A),B)),A) = B ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_1560_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_1561_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_1562_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_1563_distrib__left__numeral,axiom,
    ! [V: num,B: real,C: real] : aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,V)),aa_real_real(plus_plus_real(B),C)) = aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,V)),B)),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,V)),C)) ).

% distrib_left_numeral
tff(fact_1564_distrib__left__numeral,axiom,
    ! [V: num,B: rat,C: rat] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,V)),aa_rat_rat(plus_plus_rat(B),C)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,V)),B)),aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,V)),C)) ).

% distrib_left_numeral
tff(fact_1565_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_1566_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_1567_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_1568_distrib__right__numeral,axiom,
    ! [A: real,B: real,V: num] : aa_real_real(times_times_real(aa_real_real(plus_plus_real(A),B)),aa_num_real(numeral_numeral_real,V)) = aa_real_real(plus_plus_real(aa_real_real(times_times_real(A),aa_num_real(numeral_numeral_real,V))),aa_real_real(times_times_real(B),aa_num_real(numeral_numeral_real,V))) ).

% distrib_right_numeral
tff(fact_1569_distrib__right__numeral,axiom,
    ! [A: rat,B: rat,V: num] : aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(A),B)),aa_num_rat(numeral_numeral_rat,V)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),aa_num_rat(numeral_numeral_rat,V))),aa_rat_rat(times_times_rat(B),aa_num_rat(numeral_numeral_rat,V))) ).

% distrib_right_numeral
tff(fact_1570_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_1571_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_1572_mult__eq__1__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ( ( Mb = aa_nat_nat(suc,zero_zero_nat) )
        & ( Nb = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% mult_eq_1_iff
tff(fact_1573_one__eq__mult__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(suc,zero_zero_nat) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb) )
    <=> ( ( Mb = aa_nat_nat(suc,zero_zero_nat) )
        & ( Nb = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% one_eq_mult_iff
tff(fact_1574_mult__less__cancel2,axiom,
    ! [Mb: nat,K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),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,Mb),Nb) ) ) ).

% mult_less_cancel2
tff(fact_1575_nat__0__less__mult__iff,axiom,
    ! [Mb: nat,Nb: 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,Mb),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% nat_0_less_mult_iff
tff(fact_1576_nat__mult__less__cancel__disj,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb))
    <=> ( 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,Mb),Nb) ) ) ).

% nat_mult_less_cancel_disj
tff(fact_1577_mult__Suc__right,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)) ).

% mult_Suc_right
tff(fact_1578_nat__mult__div__cancel__disj,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb)) = $ite(K = zero_zero_nat,zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)) ).

% nat_mult_div_cancel_disj
tff(fact_1579_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(divide_divide_real(B),aa_num_real(numeral_numeral_real,W))),A)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(times_times_real(A),aa_num_real(numeral_numeral_real,W))) ) ).

% divide_le_eq_numeral1(1)
tff(fact_1580_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,aa_rat_rat(divide_divide_rat(B),aa_num_rat(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),aa_num_rat(numeral_numeral_rat,W))) ) ).

% divide_le_eq_numeral1(1)
tff(fact_1581_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(divide_divide_real(B),aa_num_real(numeral_numeral_real,W)))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A),aa_num_real(numeral_numeral_real,W))),B) ) ).

% le_divide_eq_numeral1(1)
tff(fact_1582_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),aa_rat_rat(divide_divide_rat(B),aa_num_rat(numeral_numeral_rat,W)))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A),aa_num_rat(numeral_numeral_rat,W))),B) ) ).

% le_divide_eq_numeral1(1)
tff(fact_1583_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_1584_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B: real,W: num,A: real] :
      ( ( aa_real_real(divide_divide_real(B),aa_num_real(numeral_numeral_real,W)) = A )
    <=> $ite(aa_num_real(numeral_numeral_real,W) != zero_zero_real,B = aa_real_real(times_times_real(A),aa_num_real(numeral_numeral_real,W)),A = zero_zero_real) ) ).

% divide_eq_eq_numeral1(1)
tff(fact_1585_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( ( aa_rat_rat(divide_divide_rat(B),aa_num_rat(numeral_numeral_rat,W)) = A )
    <=> $ite(aa_num_rat(numeral_numeral_rat,W) != zero_zero_rat,B = aa_rat_rat(times_times_rat(A),aa_num_rat(numeral_numeral_rat,W)),A = zero_zero_rat) ) ).

% divide_eq_eq_numeral1(1)
tff(fact_1586_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_1587_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: real,B: real,W: num] :
      ( ( A = aa_real_real(divide_divide_real(B),aa_num_real(numeral_numeral_real,W)) )
    <=> $ite(aa_num_real(numeral_numeral_real,W) != zero_zero_real,aa_real_real(times_times_real(A),aa_num_real(numeral_numeral_real,W)) = B,A = zero_zero_real) ) ).

% eq_divide_eq_numeral1(1)
tff(fact_1588_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( ( A = aa_rat_rat(divide_divide_rat(B),aa_num_rat(numeral_numeral_rat,W)) )
    <=> $ite(aa_num_rat(numeral_numeral_rat,W) != zero_zero_rat,aa_rat_rat(times_times_rat(A),aa_num_rat(numeral_numeral_rat,W)) = B,A = zero_zero_rat) ) ).

% eq_divide_eq_numeral1(1)
tff(fact_1589_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(divide_divide_real(B),aa_num_real(numeral_numeral_real,W))),A)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(times_times_real(A),aa_num_real(numeral_numeral_real,W))) ) ).

% divide_less_eq_numeral1(1)
tff(fact_1590_divide__less__eq__numeral1_I1_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(B),aa_num_rat(numeral_numeral_rat,W))),A)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(times_times_rat(A),aa_num_rat(numeral_numeral_rat,W))) ) ).

% divide_less_eq_numeral1(1)
tff(fact_1591_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(divide_divide_real(B),aa_num_real(numeral_numeral_real,W)))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(A),aa_num_real(numeral_numeral_real,W))),B) ) ).

% less_divide_eq_numeral1(1)
tff(fact_1592_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),aa_rat_rat(divide_divide_rat(B),aa_num_rat(numeral_numeral_rat,W)))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),aa_num_rat(numeral_numeral_rat,W))),B) ) ).

% less_divide_eq_numeral1(1)
tff(fact_1593_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_1594_nonzero__divide__mult__cancel__right,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( aa_real_real(divide_divide_real(B),aa_real_real(times_times_real(A),B)) = aa_real_real(divide_divide_real(one_one_real),A) ) ) ).

% nonzero_divide_mult_cancel_right
tff(fact_1595_nonzero__divide__mult__cancel__right,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(B),aa_rat_rat(times_times_rat(A),B)) = aa_rat_rat(divide_divide_rat(one_one_rat),A) ) ) ).

% nonzero_divide_mult_cancel_right
tff(fact_1596_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_1597_nonzero__divide__mult__cancel__left,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(divide_divide_real(A),aa_real_real(times_times_real(A),B)) = aa_real_real(divide_divide_real(one_one_real),B) ) ) ).

% nonzero_divide_mult_cancel_left
tff(fact_1598_nonzero__divide__mult__cancel__left,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(A),aa_rat_rat(times_times_rat(A),B)) = aa_rat_rat(divide_divide_rat(one_one_rat),B) ) ) ).

% nonzero_divide_mult_cancel_left
tff(fact_1599_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_1600_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_1601_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_1602_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_1603_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_1604_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_1605_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_1606_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_1607_one__le__mult__iff,axiom,
    ! [Mb: nat,Nb: 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,Mb),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),Mb)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),Nb) ) ) ).

% one_le_mult_iff
tff(fact_1608_mult__le__cancel2,axiom,
    ! [Mb: nat,K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),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,Mb),Nb) ) ) ).

% mult_le_cancel2
tff(fact_1609_nat__mult__le__cancel__disj,axiom,
    ! [K: nat,Mb: nat,Nb: 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),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb))
    <=> ( 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,Mb),Nb) ) ) ).

% nat_mult_le_cancel_disj
tff(fact_1610_div__mult__self__is__m,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)),Nb) = Mb ) ) ).

% div_mult_self_is_m
tff(fact_1611_div__mult__self1__is__m,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Mb)),Nb) = Mb ) ) ).

% div_mult_self1_is_m
tff(fact_1612_power__add__numeral,axiom,
    ! [A: complex,Mb: num,Nb: num] : aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,Mb))),aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,Nb))) = aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(Mb),Nb))) ).

% power_add_numeral
tff(fact_1613_power__add__numeral,axiom,
    ! [A: real,Mb: num,Nb: num] : aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,Mb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,Nb))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(Mb),Nb))) ).

% power_add_numeral
tff(fact_1614_power__add__numeral,axiom,
    ! [A: rat,Mb: num,Nb: num] : aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,Mb))),aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,Nb))) = aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(Mb),Nb))) ).

% power_add_numeral
tff(fact_1615_power__add__numeral,axiom,
    ! [A: nat,Mb: num,Nb: 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,Mb))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(Mb),Nb))) ).

% power_add_numeral
tff(fact_1616_power__add__numeral,axiom,
    ! [A: int,Mb: num,Nb: 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,Mb))),aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,Nb))) = aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(Mb),Nb))) ).

% power_add_numeral
tff(fact_1617_power__add__numeral2,axiom,
    ! [A: complex,Mb: num,Nb: num,B: complex] : aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,Mb))),aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,Nb))),B)) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(Mb),Nb)))),B) ).

% power_add_numeral2
tff(fact_1618_power__add__numeral2,axiom,
    ! [A: real,Mb: num,Nb: num,B: real] : aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,Mb))),aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,Nb))),B)) = aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(Mb),Nb)))),B) ).

% power_add_numeral2
tff(fact_1619_power__add__numeral2,axiom,
    ! [A: rat,Mb: num,Nb: num,B: rat] : aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,Mb))),aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,Nb))),B)) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(Mb),Nb)))),B) ).

% power_add_numeral2
tff(fact_1620_power__add__numeral2,axiom,
    ! [A: nat,Mb: num,Nb: 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,Mb))),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,Nb))),B)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(Mb),Nb)))),B) ).

% power_add_numeral2
tff(fact_1621_power__add__numeral2,axiom,
    ! [A: int,Mb: num,Nb: 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,Mb))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,Nb))),B)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(Mb),Nb)))),B) ).

% power_add_numeral2
tff(fact_1622_mult_Oleft__commute,axiom,
    ! [B: real,A: real,C: real] : aa_real_real(times_times_real(B),aa_real_real(times_times_real(A),C)) = aa_real_real(times_times_real(A),aa_real_real(times_times_real(B),C)) ).

% mult.left_commute
tff(fact_1623_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_1624_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_1625_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_1626_mult_Ocommute,axiom,
    ! [A: real,B: real] : aa_real_real(times_times_real(A),B) = aa_real_real(times_times_real(B),A) ).

% mult.commute
tff(fact_1627_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_1628_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_1629_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_1630_mult_Oassoc,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(times_times_real(aa_real_real(times_times_real(A),B)),C) = aa_real_real(times_times_real(A),aa_real_real(times_times_real(B),C)) ).

% mult.assoc
tff(fact_1631_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_1632_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_1633_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_1634_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(times_times_real(aa_real_real(times_times_real(A),B)),C) = aa_real_real(times_times_real(A),aa_real_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__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_1639_mult__not__zero,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(times_times_real(A),B) != zero_zero_real )
     => ( ( A != zero_zero_real )
        & ( B != zero_zero_real ) ) ) ).

% mult_not_zero
tff(fact_1640_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_1641_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_1642_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_1643_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_1644_divisors__zero,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(times_times_real(A),B) = zero_zero_real )
     => ( ( A = zero_zero_real )
        | ( B = zero_zero_real ) ) ) ).

% divisors_zero
tff(fact_1645_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_1646_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_1647_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_1648_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_1649_no__zero__divisors,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( ( B != zero_zero_real )
       => ( aa_real_real(times_times_real(A),B) != zero_zero_real ) ) ) ).

% no_zero_divisors
tff(fact_1650_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_1651_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_1652_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_1653_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_1654_mult__left__cancel,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( aa_real_real(times_times_real(C),A) = aa_real_real(times_times_real(C),B) )
      <=> ( A = B ) ) ) ).

% mult_left_cancel
tff(fact_1655_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_1656_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_1657_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_1658_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_1659_mult__right__cancel,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( aa_real_real(times_times_real(A),C) = aa_real_real(times_times_real(B),C) )
      <=> ( A = B ) ) ) ).

% mult_right_cancel
tff(fact_1660_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_1661_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_1662_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_1663_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_1664_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: real] : aa_real_real(times_times_real(one_one_real),A) = A ).

% comm_monoid_mult_class.mult_1
tff(fact_1665_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_1666_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_1667_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_1668_mult_Ocomm__neutral,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(A),one_one_complex) = A ).

% mult.comm_neutral
tff(fact_1669_mult_Ocomm__neutral,axiom,
    ! [A: real] : aa_real_real(times_times_real(A),one_one_real) = A ).

% mult.comm_neutral
tff(fact_1670_mult_Ocomm__neutral,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(A),one_one_rat) = A ).

% mult.comm_neutral
tff(fact_1671_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_1672_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_1673_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(times_times_real(aa_real_real(plus_plus_real(A),B)),C) = aa_real_real(plus_plus_real(aa_real_real(times_times_real(A),C)),aa_real_real(times_times_real(B),C)) ).

% ring_class.ring_distribs(2)
tff(fact_1674_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(A),B)),C) = aa_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_1675_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_1676_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(times_times_real(A),aa_real_real(plus_plus_real(B),C)) = aa_real_real(plus_plus_real(aa_real_real(times_times_real(A),B)),aa_real_real(times_times_real(A),C)) ).

% ring_class.ring_distribs(1)
tff(fact_1677_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(A),aa_rat_rat(plus_plus_rat(B),C)) = aa_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_1678_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_1679_comm__semiring__class_Odistrib,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(times_times_real(aa_real_real(plus_plus_real(A),B)),C) = aa_real_real(plus_plus_real(aa_real_real(times_times_real(A),C)),aa_real_real(times_times_real(B),C)) ).

% comm_semiring_class.distrib
tff(fact_1680_comm__semiring__class_Odistrib,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(A),B)),C) = aa_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_1681_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_1682_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_1683_distrib__left,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(times_times_real(A),aa_real_real(plus_plus_real(B),C)) = aa_real_real(plus_plus_real(aa_real_real(times_times_real(A),B)),aa_real_real(times_times_real(A),C)) ).

% distrib_left
tff(fact_1684_distrib__left,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(A),aa_rat_rat(plus_plus_rat(B),C)) = aa_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_1685_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_1686_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_1687_distrib__right,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(times_times_real(aa_real_real(plus_plus_real(A),B)),C) = aa_real_real(plus_plus_real(aa_real_real(times_times_real(A),C)),aa_real_real(times_times_real(B),C)) ).

% distrib_right
tff(fact_1688_distrib__right,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(A),B)),C) = aa_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_1689_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_1690_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_1691_combine__common__factor,axiom,
    ! [A: real,E2: real,B: real,C: real] : aa_real_real(plus_plus_real(aa_real_real(times_times_real(A),E2)),aa_real_real(plus_plus_real(aa_real_real(times_times_real(B),E2)),C)) = aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_real_real(plus_plus_real(A),B)),E2)),C) ).

% combine_common_factor
tff(fact_1692_combine__common__factor,axiom,
    ! [A: rat,E2: rat,B: rat,C: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),E2)),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(B),E2)),C)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(A),B)),E2)),C) ).

% combine_common_factor
tff(fact_1693_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_1694_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_1695_times__divide__times__eq,axiom,
    ! [Xb: complex,Yb: complex,Z: complex,W: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(divide1717551699836669952omplex(Xb),Yb)),aa_complex_complex(divide1717551699836669952omplex(Z),W)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(Xb),Z)),aa_complex_complex(times_times_complex(Yb),W)) ).

% times_divide_times_eq
tff(fact_1696_times__divide__times__eq,axiom,
    ! [Xb: real,Yb: real,Z: real,W: real] : aa_real_real(times_times_real(aa_real_real(divide_divide_real(Xb),Yb)),aa_real_real(divide_divide_real(Z),W)) = aa_real_real(divide_divide_real(aa_real_real(times_times_real(Xb),Z)),aa_real_real(times_times_real(Yb),W)) ).

% times_divide_times_eq
tff(fact_1697_times__divide__times__eq,axiom,
    ! [Xb: rat,Yb: rat,Z: rat,W: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(divide_divide_rat(Xb),Yb)),aa_rat_rat(divide_divide_rat(Z),W)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(Xb),Z)),aa_rat_rat(times_times_rat(Yb),W)) ).

% times_divide_times_eq
tff(fact_1698_divide__divide__times__eq,axiom,
    ! [Xb: complex,Yb: complex,Z: complex,W: complex] : aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(divide1717551699836669952omplex(Xb),Yb)),aa_complex_complex(divide1717551699836669952omplex(Z),W)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(Xb),W)),aa_complex_complex(times_times_complex(Yb),Z)) ).

% divide_divide_times_eq
tff(fact_1699_divide__divide__times__eq,axiom,
    ! [Xb: real,Yb: real,Z: real,W: real] : aa_real_real(divide_divide_real(aa_real_real(divide_divide_real(Xb),Yb)),aa_real_real(divide_divide_real(Z),W)) = aa_real_real(divide_divide_real(aa_real_real(times_times_real(Xb),W)),aa_real_real(times_times_real(Yb),Z)) ).

% divide_divide_times_eq
tff(fact_1700_divide__divide__times__eq,axiom,
    ! [Xb: rat,Yb: rat,Z: rat,W: rat] : aa_rat_rat(divide_divide_rat(aa_rat_rat(divide_divide_rat(Xb),Yb)),aa_rat_rat(divide_divide_rat(Z),W)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(Xb),W)),aa_rat_rat(times_times_rat(Yb),Z)) ).

% divide_divide_times_eq
tff(fact_1701_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_1702_divide__divide__eq__left_H,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(divide_divide_real(aa_real_real(divide_divide_real(A),B)),C) = aa_real_real(divide_divide_real(A),aa_real_real(times_times_real(C),B)) ).

% divide_divide_eq_left'
tff(fact_1703_divide__divide__eq__left_H,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(divide_divide_rat(aa_rat_rat(divide_divide_rat(A),B)),C) = aa_rat_rat(divide_divide_rat(A),aa_rat_rat(times_times_rat(C),B)) ).

% divide_divide_eq_left'
tff(fact_1704_power__commutes,axiom,
    ! [A: complex,Nb: nat] : aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),Nb)),A) = aa_complex_complex(times_times_complex(A),aa_nat_complex(power_power_complex(A),Nb)) ).

% power_commutes
tff(fact_1705_power__commutes,axiom,
    ! [A: real,Nb: nat] : aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),A) = aa_real_real(times_times_real(A),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ).

% power_commutes
tff(fact_1706_power__commutes,axiom,
    ! [A: rat,Nb: nat] : aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A),Nb)),A) = aa_rat_rat(times_times_rat(A),aa_nat_rat(power_power_rat(A),Nb)) ).

% power_commutes
tff(fact_1707_power__commutes,axiom,
    ! [A: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),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),Nb)) ).

% power_commutes
tff(fact_1708_power__commutes,axiom,
    ! [A: int,Nb: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),Nb)),A) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(A),Nb)) ).

% power_commutes
tff(fact_1709_power__mult__distrib,axiom,
    ! [A: complex,B: complex,Nb: nat] : aa_nat_complex(power_power_complex(aa_complex_complex(times_times_complex(A),B)),Nb) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),Nb)),aa_nat_complex(power_power_complex(B),Nb)) ).

% power_mult_distrib
tff(fact_1710_power__mult__distrib,axiom,
    ! [A: real,B: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(times_times_real(A),B)),Nb) = aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb)) ).

% power_mult_distrib
tff(fact_1711_power__mult__distrib,axiom,
    ! [A: rat,B: rat,Nb: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(times_times_rat(A),B)),Nb) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A),Nb)),aa_nat_rat(power_power_rat(B),Nb)) ).

% power_mult_distrib
tff(fact_1712_power__mult__distrib,axiom,
    ! [A: nat,B: nat,Nb: 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)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb)) ).

% power_mult_distrib
tff(fact_1713_power__mult__distrib,axiom,
    ! [A: int,B: int,Nb: nat] : aa_nat_int(power_power_int(aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),Nb) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),Nb)),aa_nat_int(power_power_int(B),Nb)) ).

% power_mult_distrib
tff(fact_1714_power__commuting__commutes,axiom,
    ! [Xb: complex,Yb: complex,Nb: nat] :
      ( ( aa_complex_complex(times_times_complex(Xb),Yb) = aa_complex_complex(times_times_complex(Yb),Xb) )
     => ( aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(Xb),Nb)),Yb) = aa_complex_complex(times_times_complex(Yb),aa_nat_complex(power_power_complex(Xb),Nb)) ) ) ).

% power_commuting_commutes
tff(fact_1715_power__commuting__commutes,axiom,
    ! [Xb: real,Yb: real,Nb: nat] :
      ( ( aa_real_real(times_times_real(Xb),Yb) = aa_real_real(times_times_real(Yb),Xb) )
     => ( aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)),Yb) = aa_real_real(times_times_real(Yb),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)) ) ) ).

% power_commuting_commutes
tff(fact_1716_power__commuting__commutes,axiom,
    ! [Xb: rat,Yb: rat,Nb: nat] :
      ( ( aa_rat_rat(times_times_rat(Xb),Yb) = aa_rat_rat(times_times_rat(Yb),Xb) )
     => ( aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(Xb),Nb)),Yb) = aa_rat_rat(times_times_rat(Yb),aa_nat_rat(power_power_rat(Xb),Nb)) ) ) ).

% power_commuting_commutes
tff(fact_1717_power__commuting__commutes,axiom,
    ! [Xb: nat,Yb: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Xb),Yb) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Yb),Xb) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),Nb)),Yb) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Yb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),Nb)) ) ) ).

% power_commuting_commutes
tff(fact_1718_power__commuting__commutes,axiom,
    ! [Xb: int,Yb: int,Nb: nat] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,Xb),Yb) = aa_int_int(aa_int_fun_int_int(times_times_int,Yb),Xb) )
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(Xb),Nb)),Yb) = aa_int_int(aa_int_fun_int_int(times_times_int,Yb),aa_nat_int(power_power_int(Xb),Nb)) ) ) ).

% power_commuting_commutes
tff(fact_1719_Suc__mult__cancel1,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),Mb) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),Nb) )
    <=> ( Mb = Nb ) ) ).

% Suc_mult_cancel1
tff(fact_1720_power__mult,axiom,
    ! [A: nat,Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Mb)),Nb) ).

% power_mult
tff(fact_1721_power__mult,axiom,
    ! [A: int,Mb: nat,Nb: nat] : aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)) = aa_nat_int(power_power_int(aa_nat_int(power_power_int(A),Mb)),Nb) ).

% power_mult
tff(fact_1722_power__mult,axiom,
    ! [A: real,Mb: nat,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Mb)),Nb) ).

% power_mult
tff(fact_1723_power__mult,axiom,
    ! [A: complex,Mb: nat,Nb: nat] : aa_nat_complex(power_power_complex(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)) = aa_nat_complex(power_power_complex(aa_nat_complex(power_power_complex(A),Mb)),Nb) ).

% power_mult
tff(fact_1724_mult__0,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,zero_zero_nat),Nb) = zero_zero_nat ).

% mult_0
tff(fact_1725_nat__mult__eq__cancel__disj,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Mb) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb) )
    <=> ( ( K = zero_zero_nat )
        | ( Mb = Nb ) ) ) ).

% nat_mult_eq_cancel_disj
tff(fact_1726_le__cube,axiom,
    ! [Mb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Mb))) ).

% le_cube
tff(fact_1727_le__square,axiom,
    ! [Mb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Mb)) ).

% le_square
tff(fact_1728_mult__le__mono,axiom,
    ! [I3: nat,J: nat,K: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I3),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,I3),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),L)) ) ) ).

% mult_le_mono
tff(fact_1729_mult__le__mono1,axiom,
    ! [I3: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I3),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I3),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),K)) ) ).

% mult_le_mono1
tff(fact_1730_mult__le__mono2,axiom,
    ! [I3: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I3),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),I3)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),J)) ) ).

% mult_le_mono2
tff(fact_1731_add__mult__distrib,axiom,
    ! [Mb: nat,Nb: 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,Mb),Nb)),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),K)) ).

% add_mult_distrib
tff(fact_1732_add__mult__distrib2,axiom,
    ! [K: nat,Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb)) ).

% add_mult_distrib2
tff(fact_1733_left__add__mult__distrib,axiom,
    ! [I3: 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,I3),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,I3),J)),U)),K) ).

% left_add_mult_distrib
tff(fact_1734_nat__mult__1,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,one_one_nat),Nb) = Nb ).

% nat_mult_1
tff(fact_1735_nat__mult__1__right,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),one_one_nat) = Nb ).

% nat_mult_1_right
tff(fact_1736_div__mult2__eq,axiom,
    ! [Mb: nat,Nb: nat,Q2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Q2)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)),Q2) ).

% div_mult2_eq
tff(fact_1737_nat__mult__max__left,axiom,
    ! [Mb: nat,Nb: nat,Q2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Mb),Nb)),Q2) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Q2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Q2)) ).

% nat_mult_max_left
tff(fact_1738_nat__mult__max__right,axiom,
    ! [Mb: nat,Nb: nat,Q2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Nb),Q2)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Q2)) ).

% nat_mult_max_right
tff(fact_1739_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_1740_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_1741_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(times_times_real(A),C)),aa_real_real(times_times_real(B),D)) ) ) ) ) ).

% mult_mono
tff(fact_1742_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_1743_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_1744_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_1745_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(times_times_real(A),C)),aa_real_real(times_times_real(B),D)) ) ) ) ) ).

% mult_mono'
tff(fact_1746_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_1747_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_1748_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_1749_zero__le__square,axiom,
    ! [A: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(times_times_real(A),A)) ).

% zero_le_square
tff(fact_1750_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_1751_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_1752_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(times_times_real(A),B)) ) ).

% split_mult_pos_le
tff(fact_1753_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_1754_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_1755_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(times_times_real(C),A)),aa_real_real(times_times_real(C),B)) ) ) ).

% mult_left_mono_neg
tff(fact_1756_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_1757_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_1758_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(times_times_real(A),B)) ) ) ).

% mult_nonpos_nonpos
tff(fact_1759_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_1760_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_1761_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(times_times_real(C),A)),aa_real_real(times_times_real(C),B)) ) ) ).

% mult_left_mono
tff(fact_1762_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_1763_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_1764_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_1765_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(times_times_real(A),C)),aa_real_real(times_times_real(B),C)) ) ) ).

% mult_right_mono_neg
tff(fact_1766_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_1767_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_1768_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(times_times_real(A),C)),aa_real_real(times_times_real(B),C)) ) ) ).

% mult_right_mono
tff(fact_1769_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_1770_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_1771_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_1772_mult__le__0__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_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_1773_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_1774_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_1775_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(times_times_real(A),B)),zero_zero_real) ) ).

% split_mult_neg_le
tff(fact_1776_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_1777_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_1778_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_1779_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(times_times_real(A),B)) ) ) ).

% mult_nonneg_nonneg
tff(fact_1780_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_1781_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_1782_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_1783_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(times_times_real(A),B)),zero_zero_real) ) ) ).

% mult_nonneg_nonpos
tff(fact_1784_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_1785_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_1786_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_1787_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(times_times_real(A),B)),zero_zero_real) ) ) ).

% mult_nonpos_nonneg
tff(fact_1788_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_1789_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_1790_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_1791_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(times_times_real(B),A)),zero_zero_real) ) ) ).

% mult_nonneg_nonpos2
tff(fact_1792_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_1793_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_1794_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_1795_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(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_1796_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_1797_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_1798_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(times_times_real(C),A)),aa_real_real(times_times_real(C),B)) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_1799_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_1800_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_1801_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_1802_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(times_times_real(A),B)) ) ) ).

% mult_neg_neg
tff(fact_1803_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_1804_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_1805_not__square__less__zero,axiom,
    ! [A: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(A),A)),zero_zero_real) ).

% not_square_less_zero
tff(fact_1806_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_1807_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_1808_mult__less__0__iff,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_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_1809_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_1810_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_1811_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(times_times_real(A),B)),zero_zero_real) ) ) ).

% mult_neg_pos
tff(fact_1812_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_1813_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_1814_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_1815_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(times_times_real(A),B)),zero_zero_real) ) ) ).

% mult_pos_neg
tff(fact_1816_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_1817_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_1818_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_1819_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(times_times_real(A),B)) ) ) ).

% mult_pos_pos
tff(fact_1820_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_1821_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_1822_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_1823_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(times_times_real(B),A)),zero_zero_real) ) ) ).

% mult_pos_neg2
tff(fact_1824_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_1825_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_1826_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_1827_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(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_1828_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_1829_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_1830_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(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_1831_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_1832_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_1833_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_1834_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(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_1835_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_1836_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_1837_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_1838_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(times_times_real(C),A)),aa_real_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_1839_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_1840_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_1841_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(times_times_real(C),A)),aa_real_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_1842_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_1843_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_1844_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(times_times_real(C),A)),aa_real_real(times_times_real(C),B)) ) ) ).

% mult_strict_left_mono_neg
tff(fact_1845_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_1846_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_1847_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(times_times_real(C),A)),aa_real_real(times_times_real(C),B)) ) ) ).

% mult_strict_left_mono
tff(fact_1848_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_1849_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_1850_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_1851_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(times_times_real(C),A)),aa_real_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_1852_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_1853_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_1854_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(times_times_real(A),C)),aa_real_real(times_times_real(B),C)) ) ) ).

% mult_strict_right_mono_neg
tff(fact_1855_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_1856_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_1857_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(times_times_real(A),C)),aa_real_real(times_times_real(B),C)) ) ) ).

% mult_strict_right_mono
tff(fact_1858_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_1859_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_1860_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_1861_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(times_times_real(A),C)),aa_real_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_1862_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_1863_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_1864_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(times_times_real(C),A)),aa_real_real(times_times_real(C),B)) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_1865_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_1866_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_1867_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_1868_less__1__mult,axiom,
    ! [Mb: real,Nb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Mb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Nb)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(times_times_real(Mb),Nb)) ) ) ).

% less_1_mult
tff(fact_1869_less__1__mult,axiom,
    ! [Mb: rat,Nb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),Mb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),Nb)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_rat_rat(times_times_rat(Mb),Nb)) ) ) ).

% less_1_mult
tff(fact_1870_less__1__mult,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),Nb)
       => 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,Mb),Nb)) ) ) ).

% less_1_mult
tff(fact_1871_less__1__mult,axiom,
    ! [Mb: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),Mb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),Nb)
       => 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,Mb),Nb)) ) ) ).

% less_1_mult
tff(fact_1872_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_1873_nonzero__eq__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( A = aa_real_real(divide_divide_real(B),C) )
      <=> ( aa_real_real(times_times_real(A),C) = B ) ) ) ).

% nonzero_eq_divide_eq
tff(fact_1874_nonzero__eq__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( A = aa_rat_rat(divide_divide_rat(B),C) )
      <=> ( aa_rat_rat(times_times_rat(A),C) = B ) ) ) ).

% nonzero_eq_divide_eq
tff(fact_1875_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_1876_nonzero__divide__eq__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( ( C != zero_zero_real )
     => ( ( aa_real_real(divide_divide_real(B),C) = A )
      <=> ( B = aa_real_real(times_times_real(A),C) ) ) ) ).

% nonzero_divide_eq_eq
tff(fact_1877_nonzero__divide__eq__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( C != zero_zero_rat )
     => ( ( aa_rat_rat(divide_divide_rat(B),C) = A )
      <=> ( B = aa_rat_rat(times_times_rat(A),C) ) ) ) ).

% nonzero_divide_eq_eq
tff(fact_1878_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_1879_eq__divide__imp,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( aa_real_real(times_times_real(A),C) = B )
       => ( A = aa_real_real(divide_divide_real(B),C) ) ) ) ).

% eq_divide_imp
tff(fact_1880_eq__divide__imp,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( aa_rat_rat(times_times_rat(A),C) = B )
       => ( A = aa_rat_rat(divide_divide_rat(B),C) ) ) ) ).

% eq_divide_imp
tff(fact_1881_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_1882_divide__eq__imp,axiom,
    ! [C: real,B: real,A: real] :
      ( ( C != zero_zero_real )
     => ( ( B = aa_real_real(times_times_real(A),C) )
       => ( aa_real_real(divide_divide_real(B),C) = A ) ) ) ).

% divide_eq_imp
tff(fact_1883_divide__eq__imp,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( C != zero_zero_rat )
     => ( ( B = aa_rat_rat(times_times_rat(A),C) )
       => ( aa_rat_rat(divide_divide_rat(B),C) = A ) ) ) ).

% divide_eq_imp
tff(fact_1884_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_1885_eq__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( A = aa_real_real(divide_divide_real(B),C) )
    <=> $ite(C != zero_zero_real,aa_real_real(times_times_real(A),C) = B,A = zero_zero_real) ) ).

% eq_divide_eq
tff(fact_1886_eq__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( A = aa_rat_rat(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_1887_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_1888_divide__eq__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( ( aa_real_real(divide_divide_real(B),C) = A )
    <=> $ite(C != zero_zero_real,B = aa_real_real(times_times_real(A),C),A = zero_zero_real) ) ).

% divide_eq_eq
tff(fact_1889_divide__eq__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( aa_rat_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_1890_frac__eq__eq,axiom,
    ! [Yb: complex,Z: complex,Xb: complex,W: complex] :
      ( ( Yb != zero_zero_complex )
     => ( ( Z != zero_zero_complex )
       => ( ( aa_complex_complex(divide1717551699836669952omplex(Xb),Yb) = aa_complex_complex(divide1717551699836669952omplex(W),Z) )
        <=> ( aa_complex_complex(times_times_complex(Xb),Z) = aa_complex_complex(times_times_complex(W),Yb) ) ) ) ) ).

% frac_eq_eq
tff(fact_1891_frac__eq__eq,axiom,
    ! [Yb: real,Z: real,Xb: real,W: real] :
      ( ( Yb != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( ( aa_real_real(divide_divide_real(Xb),Yb) = aa_real_real(divide_divide_real(W),Z) )
        <=> ( aa_real_real(times_times_real(Xb),Z) = aa_real_real(times_times_real(W),Yb) ) ) ) ) ).

% frac_eq_eq
tff(fact_1892_frac__eq__eq,axiom,
    ! [Yb: rat,Z: rat,Xb: rat,W: rat] :
      ( ( Yb != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( ( aa_rat_rat(divide_divide_rat(Xb),Yb) = aa_rat_rat(divide_divide_rat(W),Z) )
        <=> ( aa_rat_rat(times_times_rat(Xb),Z) = aa_rat_rat(times_times_rat(W),Yb) ) ) ) ) ).

% frac_eq_eq
tff(fact_1893_mult__numeral__1,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(one)),A) = A ).

% mult_numeral_1
tff(fact_1894_mult__numeral__1,axiom,
    ! [A: real] : aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,one)),A) = A ).

% mult_numeral_1
tff(fact_1895_mult__numeral__1,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,one)),A) = A ).

% mult_numeral_1
tff(fact_1896_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_1897_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_1898_mult__numeral__1__right,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(A),numera6690914467698888265omplex(one)) = A ).

% mult_numeral_1_right
tff(fact_1899_mult__numeral__1__right,axiom,
    ! [A: real] : aa_real_real(times_times_real(A),aa_num_real(numeral_numeral_real,one)) = A ).

% mult_numeral_1_right
tff(fact_1900_mult__numeral__1__right,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(A),aa_num_rat(numeral_numeral_rat,one)) = A ).

% mult_numeral_1_right
tff(fact_1901_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_1902_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_1903_left__right__inverse__power,axiom,
    ! [Xb: complex,Yb: complex,Nb: nat] :
      ( ( aa_complex_complex(times_times_complex(Xb),Yb) = one_one_complex )
     => ( aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(Xb),Nb)),aa_nat_complex(power_power_complex(Yb),Nb)) = one_one_complex ) ) ).

% left_right_inverse_power
tff(fact_1904_left__right__inverse__power,axiom,
    ! [Xb: real,Yb: real,Nb: nat] :
      ( ( aa_real_real(times_times_real(Xb),Yb) = one_one_real )
     => ( aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Yb),Nb)) = one_one_real ) ) ).

% left_right_inverse_power
tff(fact_1905_left__right__inverse__power,axiom,
    ! [Xb: rat,Yb: rat,Nb: nat] :
      ( ( aa_rat_rat(times_times_rat(Xb),Yb) = one_one_rat )
     => ( aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(Xb),Nb)),aa_nat_rat(power_power_rat(Yb),Nb)) = one_one_rat ) ) ).

% left_right_inverse_power
tff(fact_1906_left__right__inverse__power,axiom,
    ! [Xb: nat,Yb: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Xb),Yb) = 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,Xb),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Yb),Nb)) = one_one_nat ) ) ).

% left_right_inverse_power
tff(fact_1907_left__right__inverse__power,axiom,
    ! [Xb: int,Yb: int,Nb: nat] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,Xb),Yb) = one_one_int )
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(Xb),Nb)),aa_nat_int(power_power_int(Yb),Nb)) = one_one_int ) ) ).

% left_right_inverse_power
tff(fact_1908_power__Suc,axiom,
    ! [A: complex,Nb: nat] : aa_nat_complex(power_power_complex(A),aa_nat_nat(suc,Nb)) = aa_complex_complex(times_times_complex(A),aa_nat_complex(power_power_complex(A),Nb)) ).

% power_Suc
tff(fact_1909_power__Suc,axiom,
    ! [A: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,Nb)) = aa_real_real(times_times_real(A),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ).

% power_Suc
tff(fact_1910_power__Suc,axiom,
    ! [A: rat,Nb: nat] : aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,Nb)) = aa_rat_rat(times_times_rat(A),aa_nat_rat(power_power_rat(A),Nb)) ).

% power_Suc
tff(fact_1911_power__Suc,axiom,
    ! [A: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ).

% power_Suc
tff(fact_1912_power__Suc,axiom,
    ! [A: int,Nb: nat] : aa_nat_int(power_power_int(A),aa_nat_nat(suc,Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(A),Nb)) ).

% power_Suc
tff(fact_1913_power__Suc2,axiom,
    ! [A: complex,Nb: nat] : aa_nat_complex(power_power_complex(A),aa_nat_nat(suc,Nb)) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),Nb)),A) ).

% power_Suc2
tff(fact_1914_power__Suc2,axiom,
    ! [A: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,Nb)) = aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),A) ).

% power_Suc2
tff(fact_1915_power__Suc2,axiom,
    ! [A: rat,Nb: nat] : aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,Nb)) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A),Nb)),A) ).

% power_Suc2
tff(fact_1916_power__Suc2,axiom,
    ! [A: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),A) ).

% power_Suc2
tff(fact_1917_power__Suc2,axiom,
    ! [A: int,Nb: nat] : aa_nat_int(power_power_int(A),aa_nat_nat(suc,Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),Nb)),A) ).

% power_Suc2
tff(fact_1918_Suc__mult__less__cancel1,axiom,
    ! [K: nat,Mb: nat,Nb: 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)),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% Suc_mult_less_cancel1
tff(fact_1919_power__add,axiom,
    ! [A: complex,Mb: nat,Nb: nat] : aa_nat_complex(power_power_complex(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),Mb)),aa_nat_complex(power_power_complex(A),Nb)) ).

% power_add
tff(fact_1920_power__add,axiom,
    ! [A: real,Mb: nat,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) = aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Mb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ).

% power_add
tff(fact_1921_power__add,axiom,
    ! [A: rat,Mb: nat,Nb: nat] : aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A),Mb)),aa_nat_rat(power_power_rat(A),Nb)) ).

% power_add
tff(fact_1922_power__add,axiom,
    ! [A: nat,Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ).

% power_add
tff(fact_1923_power__add,axiom,
    ! [A: int,Mb: nat,Nb: nat] : aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),Mb)),aa_nat_int(power_power_int(A),Nb)) ).

% power_add
tff(fact_1924_mult__less__mono1,axiom,
    ! [I3: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),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,I3),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),K)) ) ) ).

% mult_less_mono1
tff(fact_1925_mult__less__mono2,axiom,
    ! [I3: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),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),I3)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),J)) ) ) ).

% mult_less_mono2
tff(fact_1926_nat__mult__eq__cancel1,axiom,
    ! [K: nat,Mb: nat,Nb: 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),Mb) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb) )
      <=> ( Mb = Nb ) ) ) ).

% nat_mult_eq_cancel1
tff(fact_1927_nat__mult__less__cancel1,axiom,
    ! [K: nat,Mb: nat,Nb: 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),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ) ).

% nat_mult_less_cancel1
tff(fact_1928_Suc__mult__le__cancel1,axiom,
    ! [K: nat,Mb: nat,Nb: 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)),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% Suc_mult_le_cancel1
tff(fact_1929_mult__Suc,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,Mb)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)) ).

% mult_Suc
tff(fact_1930_mult__eq__self__implies__10,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( Mb = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb) )
     => ( ( Nb = one_one_nat )
        | ( Mb = zero_zero_nat ) ) ) ).

% mult_eq_self_implies_10
tff(fact_1931_less__mult__imp__div__less,axiom,
    ! [Mb: nat,I3: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I3),Nb))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)),I3) ) ).

% less_mult_imp_div_less
tff(fact_1932_times__div__less__eq__dividend,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb))),Mb) ).

% times_div_less_eq_dividend
tff(fact_1933_div__times__less__eq__dividend,axiom,
    ! [Mb: nat,Nb: 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,Mb),Nb)),Nb)),Mb) ).

% div_times_less_eq_dividend
tff(fact_1934_power__odd__eq,axiom,
    ! [A: complex,Nb: 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))),Nb))) = aa_complex_complex(times_times_complex(A),aa_nat_complex(power_power_complex(aa_nat_complex(power_power_complex(A),Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_1935_power__odd__eq,axiom,
    ! [A: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) = aa_real_real(times_times_real(A),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_1936_power__odd__eq,axiom,
    ! [A: rat,Nb: 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))),Nb))) = aa_rat_rat(times_times_rat(A),aa_nat_rat(power_power_rat(aa_nat_rat(power_power_rat(A),Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_1937_power__odd__eq,axiom,
    ! [A: nat,Nb: 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))),Nb))) = 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),Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_1938_power__odd__eq,axiom,
    ! [A: int,Nb: 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))),Nb))) = 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),Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_1939_Suc__double__not__eq__double,axiom,
    ! [Mb: nat,Nb: 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))),Mb)) != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) ).

% Suc_double_not_eq_double
tff(fact_1940_double__not__eq__Suc__double,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb) != 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))),Nb)) ).

% double_not_eq_Suc_double
tff(fact_1941_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(times_times_real(C),A)),aa_real_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_1942_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_1943_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_1944_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(times_times_real(A),C)),aa_real_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_1945_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_1946_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_1947_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(times_times_real(C),A)),aa_real_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_1948_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_1949_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_1950_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_1951_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(times_times_real(A),C)),aa_real_real(times_times_real(B),D)) ) ) ) ) ).

% mult_strict_mono
tff(fact_1952_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_1953_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_1954_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_1955_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(times_times_real(C),A)),aa_real_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_1956_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_1957_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_1958_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(times_times_real(A),C)),aa_real_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_1959_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_1960_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_1961_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_1962_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(times_times_real(A),C)),aa_real_real(times_times_real(B),D)) ) ) ) ) ).

% mult_strict_mono'
tff(fact_1963_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_1964_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_1965_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_1966_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(times_times_real(A),C)),aa_real_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_1967_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_1968_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_1969_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(times_times_real(C),A)),aa_real_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_1970_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_1971_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_1972_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(times_times_real(C),A)),aa_real_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_1973_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_1974_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_1975_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(times_times_real(C),A)),aa_real_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_1976_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_1977_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_1978_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_1979_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(times_times_real(A),C)),aa_real_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_1980_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_1981_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_1982_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_1983_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(times_times_real(A),C)),aa_real_real(times_times_real(B),D)) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_1984_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_1985_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_1986_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_1987_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(times_times_real(A),C)),aa_real_real(times_times_real(B),D)) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_1988_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_1989_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_1990_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_1991_mult__left__le__one__le,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Yb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Yb),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(Yb),Xb)),Xb) ) ) ) ).

% mult_left_le_one_le
tff(fact_1992_mult__left__le__one__le,axiom,
    ! [Xb: rat,Yb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Yb)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Yb),one_one_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(Yb),Xb)),Xb) ) ) ) ).

% mult_left_le_one_le
tff(fact_1993_mult__left__le__one__le,axiom,
    ! [Xb: int,Yb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Yb)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Yb),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,Yb),Xb)),Xb) ) ) ) ).

% mult_left_le_one_le
tff(fact_1994_mult__right__le__one__le,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Yb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Yb),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(Xb),Yb)),Xb) ) ) ) ).

% mult_right_le_one_le
tff(fact_1995_mult__right__le__one__le,axiom,
    ! [Xb: rat,Yb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Yb)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Yb),one_one_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(Xb),Yb)),Xb) ) ) ) ).

% mult_right_le_one_le
tff(fact_1996_mult__right__le__one__le,axiom,
    ! [Xb: int,Yb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Yb)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Yb),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,Xb),Yb)),Xb) ) ) ) ).

% mult_right_le_one_le
tff(fact_1997_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(times_times_real(A),B)),one_one_real) ) ) ) ).

% mult_le_one
tff(fact_1998_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_1999_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_2000_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_2001_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(times_times_real(A),C)),A) ) ) ).

% mult_left_le
tff(fact_2002_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_2003_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_2004_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_2005_sum__squares__le__zero__iff,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(aa_real_real(times_times_real(Xb),Xb)),aa_real_real(times_times_real(Yb),Yb))),zero_zero_real)
    <=> ( ( Xb = zero_zero_real )
        & ( Yb = zero_zero_real ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_2006_sum__squares__le__zero__iff,axiom,
    ! [Xb: rat,Yb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(Xb),Xb)),aa_rat_rat(times_times_rat(Yb),Yb))),zero_zero_rat)
    <=> ( ( Xb = zero_zero_rat )
        & ( Yb = zero_zero_rat ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_2007_sum__squares__le__zero__iff,axiom,
    ! [Xb: int,Yb: 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,Xb),Xb)),aa_int_int(aa_int_fun_int_int(times_times_int,Yb),Yb))),zero_zero_int)
    <=> ( ( Xb = zero_zero_int )
        & ( Yb = zero_zero_int ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_2008_sum__squares__ge__zero,axiom,
    ! [Xb: real,Yb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(plus_plus_real(aa_real_real(times_times_real(Xb),Xb)),aa_real_real(times_times_real(Yb),Yb))) ).

% sum_squares_ge_zero
tff(fact_2009_sum__squares__ge__zero,axiom,
    ! [Xb: rat,Yb: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(Xb),Xb)),aa_rat_rat(times_times_rat(Yb),Yb))) ).

% sum_squares_ge_zero
tff(fact_2010_sum__squares__ge__zero,axiom,
    ! [Xb: int,Yb: 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,Xb),Xb)),aa_int_int(aa_int_fun_int_int(times_times_int,Yb),Yb))) ).

% sum_squares_ge_zero
tff(fact_2011_sum__squares__gt__zero__iff,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(plus_plus_real(aa_real_real(times_times_real(Xb),Xb)),aa_real_real(times_times_real(Yb),Yb)))
    <=> ( ( Xb != zero_zero_real )
        | ( Yb != zero_zero_real ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_2012_sum__squares__gt__zero__iff,axiom,
    ! [Xb: rat,Yb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(Xb),Xb)),aa_rat_rat(times_times_rat(Yb),Yb)))
    <=> ( ( Xb != zero_zero_rat )
        | ( Yb != zero_zero_rat ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_2013_sum__squares__gt__zero__iff,axiom,
    ! [Xb: int,Yb: 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,Xb),Xb)),aa_int_int(aa_int_fun_int_int(times_times_int,Yb),Yb)))
    <=> ( ( Xb != zero_zero_int )
        | ( Yb != zero_zero_int ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_2014_not__sum__squares__lt__zero,axiom,
    ! [Xb: real,Yb: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(aa_real_real(times_times_real(Xb),Xb)),aa_real_real(times_times_real(Yb),Yb))),zero_zero_real) ).

% not_sum_squares_lt_zero
tff(fact_2015_not__sum__squares__lt__zero,axiom,
    ! [Xb: rat,Yb: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(Xb),Xb)),aa_rat_rat(times_times_rat(Yb),Yb))),zero_zero_rat) ).

% not_sum_squares_lt_zero
tff(fact_2016_not__sum__squares__lt__zero,axiom,
    ! [Xb: int,Yb: 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,Xb),Xb)),aa_int_int(aa_int_fun_int_int(times_times_int,Yb),Yb))),zero_zero_int) ).

% not_sum_squares_lt_zero
tff(fact_2017_divide__less__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_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(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(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_2018_divide__less__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_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_2019_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(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(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(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_2020_less__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_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)),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_2021_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(divide_divide_real(B),C)),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(A),C)),B) ) ) ).

% neg_divide_less_eq
tff(fact_2022_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,aa_rat_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_2023_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(divide_divide_real(B),C))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(times_times_real(A),C)) ) ) ).

% neg_less_divide_eq
tff(fact_2024_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),aa_rat_rat(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_2025_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(divide_divide_real(B),C)),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(times_times_real(A),C)) ) ) ).

% pos_divide_less_eq
tff(fact_2026_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,aa_rat_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_2027_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(divide_divide_real(B),C))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(A),C)),B) ) ) ).

% pos_less_divide_eq
tff(fact_2028_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),aa_rat_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)),B) ) ) ).

% pos_less_divide_eq
tff(fact_2029_mult__imp__div__pos__less,axiom,
    ! [Yb: real,Xb: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Yb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(times_times_real(Z),Yb))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(Xb),Yb)),Z) ) ) ).

% mult_imp_div_pos_less
tff(fact_2030_mult__imp__div__pos__less,axiom,
    ! [Yb: rat,Xb: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Yb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),aa_rat_rat(times_times_rat(Z),Yb))
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(Xb),Yb)),Z) ) ) ).

% mult_imp_div_pos_less
tff(fact_2031_mult__imp__less__div__pos,axiom,
    ! [Yb: real,Z: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Yb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(Z),Yb)),Xb)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Z),aa_real_real(divide_divide_real(Xb),Yb)) ) ) ).

% mult_imp_less_div_pos
tff(fact_2032_mult__imp__less__div__pos,axiom,
    ! [Yb: rat,Z: rat,Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Yb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(Z),Yb)),Xb)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),aa_rat_rat(divide_divide_rat(Xb),Yb)) ) ) ).

% mult_imp_less_div_pos
tff(fact_2033_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(times_times_real(A),B))
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(C),A)),aa_real_real(divide_divide_real(C),B)) ) ) ) ).

% divide_strict_left_mono
tff(fact_2034_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,aa_rat_rat(divide_divide_rat(C),A)),aa_rat_rat(divide_divide_rat(C),B)) ) ) ) ).

% divide_strict_left_mono
tff(fact_2035_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(times_times_real(A),B))
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(C),A)),aa_real_real(divide_divide_real(C),B)) ) ) ) ).

% divide_strict_left_mono_neg
tff(fact_2036_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,aa_rat_rat(divide_divide_rat(C),A)),aa_rat_rat(divide_divide_rat(C),B)) ) ) ) ).

% divide_strict_left_mono_neg
tff(fact_2037_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_2038_divide__eq__eq__numeral_I1_J,axiom,
    ! [B: real,C: real,W: num] :
      ( ( aa_real_real(divide_divide_real(B),C) = aa_num_real(numeral_numeral_real,W) )
    <=> $ite(C != zero_zero_real,B = aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,W)),C),aa_num_real(numeral_numeral_real,W) = zero_zero_real) ) ).

% divide_eq_eq_numeral(1)
tff(fact_2039_divide__eq__eq__numeral_I1_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( ( aa_rat_rat(divide_divide_rat(B),C) = aa_num_rat(numeral_numeral_rat,W) )
    <=> $ite(C != zero_zero_rat,B = aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W)),C),aa_num_rat(numeral_numeral_rat,W) = zero_zero_rat) ) ).

% divide_eq_eq_numeral(1)
tff(fact_2040_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_2041_eq__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: real,C: real] :
      ( ( aa_num_real(numeral_numeral_real,W) = aa_real_real(divide_divide_real(B),C) )
    <=> $ite(C != zero_zero_real,aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,W)),C) = B,aa_num_real(numeral_numeral_real,W) = zero_zero_real) ) ).

% eq_divide_eq_numeral(1)
tff(fact_2042_eq__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( ( aa_num_rat(numeral_numeral_rat,W) = aa_rat_rat(divide_divide_rat(B),C) )
    <=> $ite(C != zero_zero_rat,aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W)),C) = B,aa_num_rat(numeral_numeral_rat,W) = zero_zero_rat) ) ).

% eq_divide_eq_numeral(1)
tff(fact_2043_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_2044_add__divide__eq__if__simps_I2_J,axiom,
    ! [A: real,Z: real,B: real] :
      aa_real_real(plus_plus_real(aa_real_real(divide_divide_real(A),Z)),B) = $ite(Z = zero_zero_real,B,aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(A),aa_real_real(times_times_real(B),Z))),Z)) ).

% add_divide_eq_if_simps(2)
tff(fact_2045_add__divide__eq__if__simps_I2_J,axiom,
    ! [A: rat,Z: rat,B: rat] :
      aa_rat_rat(plus_plus_rat(aa_rat_rat(divide_divide_rat(A),Z)),B) = $ite(Z = zero_zero_rat,B,aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(A),aa_rat_rat(times_times_rat(B),Z))),Z)) ).

% add_divide_eq_if_simps(2)
tff(fact_2046_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_2047_add__divide__eq__if__simps_I1_J,axiom,
    ! [A: real,B: real,Z: real] :
      aa_real_real(plus_plus_real(A),aa_real_real(divide_divide_real(B),Z)) = $ite(Z = zero_zero_real,A,aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(aa_real_real(times_times_real(A),Z)),B)),Z)) ).

% add_divide_eq_if_simps(1)
tff(fact_2048_add__divide__eq__if__simps_I1_J,axiom,
    ! [A: rat,B: rat,Z: rat] :
      aa_rat_rat(plus_plus_rat(A),aa_rat_rat(divide_divide_rat(B),Z)) = $ite(Z = zero_zero_rat,A,aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),Z)),B)),Z)) ).

% add_divide_eq_if_simps(1)
tff(fact_2049_add__frac__eq,axiom,
    ! [Yb: complex,Z: complex,Xb: complex,W: complex] :
      ( ( Yb != zero_zero_complex )
     => ( ( Z != zero_zero_complex )
       => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(divide1717551699836669952omplex(Xb),Yb)),aa_complex_complex(divide1717551699836669952omplex(W),Z)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(Xb),Z)),aa_complex_complex(times_times_complex(W),Yb))),aa_complex_complex(times_times_complex(Yb),Z)) ) ) ) ).

% add_frac_eq
tff(fact_2050_add__frac__eq,axiom,
    ! [Yb: real,Z: real,Xb: real,W: real] :
      ( ( Yb != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( aa_real_real(plus_plus_real(aa_real_real(divide_divide_real(Xb),Yb)),aa_real_real(divide_divide_real(W),Z)) = aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(aa_real_real(times_times_real(Xb),Z)),aa_real_real(times_times_real(W),Yb))),aa_real_real(times_times_real(Yb),Z)) ) ) ) ).

% add_frac_eq
tff(fact_2051_add__frac__eq,axiom,
    ! [Yb: rat,Z: rat,Xb: rat,W: rat] :
      ( ( Yb != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( aa_rat_rat(plus_plus_rat(aa_rat_rat(divide_divide_rat(Xb),Yb)),aa_rat_rat(divide_divide_rat(W),Z)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(Xb),Z)),aa_rat_rat(times_times_rat(W),Yb))),aa_rat_rat(times_times_rat(Yb),Z)) ) ) ) ).

% add_frac_eq
tff(fact_2052_add__frac__num,axiom,
    ! [Yb: complex,Xb: complex,Z: complex] :
      ( ( Yb != zero_zero_complex )
     => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(divide1717551699836669952omplex(Xb),Yb)),Z) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(Xb),aa_complex_complex(times_times_complex(Z),Yb))),Yb) ) ) ).

% add_frac_num
tff(fact_2053_add__frac__num,axiom,
    ! [Yb: real,Xb: real,Z: real] :
      ( ( Yb != zero_zero_real )
     => ( aa_real_real(plus_plus_real(aa_real_real(divide_divide_real(Xb),Yb)),Z) = aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(Xb),aa_real_real(times_times_real(Z),Yb))),Yb) ) ) ).

% add_frac_num
tff(fact_2054_add__frac__num,axiom,
    ! [Yb: rat,Xb: rat,Z: rat] :
      ( ( Yb != zero_zero_rat )
     => ( aa_rat_rat(plus_plus_rat(aa_rat_rat(divide_divide_rat(Xb),Yb)),Z) = aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(Xb),aa_rat_rat(times_times_rat(Z),Yb))),Yb) ) ) ).

% add_frac_num
tff(fact_2055_add__num__frac,axiom,
    ! [Yb: complex,Z: complex,Xb: complex] :
      ( ( Yb != zero_zero_complex )
     => ( aa_complex_complex(plus_plus_complex(Z),aa_complex_complex(divide1717551699836669952omplex(Xb),Yb)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(Xb),aa_complex_complex(times_times_complex(Z),Yb))),Yb) ) ) ).

% add_num_frac
tff(fact_2056_add__num__frac,axiom,
    ! [Yb: real,Z: real,Xb: real] :
      ( ( Yb != zero_zero_real )
     => ( aa_real_real(plus_plus_real(Z),aa_real_real(divide_divide_real(Xb),Yb)) = aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(Xb),aa_real_real(times_times_real(Z),Yb))),Yb) ) ) ).

% add_num_frac
tff(fact_2057_add__num__frac,axiom,
    ! [Yb: rat,Z: rat,Xb: rat] :
      ( ( Yb != zero_zero_rat )
     => ( aa_rat_rat(plus_plus_rat(Z),aa_rat_rat(divide_divide_rat(Xb),Yb)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(Xb),aa_rat_rat(times_times_rat(Z),Yb))),Yb) ) ) ).

% add_num_frac
tff(fact_2058_add__divide__eq__iff,axiom,
    ! [Z: complex,Xb: complex,Yb: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_complex_complex(plus_plus_complex(Xb),aa_complex_complex(divide1717551699836669952omplex(Yb),Z)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(Xb),Z)),Yb)),Z) ) ) ).

% add_divide_eq_iff
tff(fact_2059_add__divide__eq__iff,axiom,
    ! [Z: real,Xb: real,Yb: real] :
      ( ( Z != zero_zero_real )
     => ( aa_real_real(plus_plus_real(Xb),aa_real_real(divide_divide_real(Yb),Z)) = aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(aa_real_real(times_times_real(Xb),Z)),Yb)),Z) ) ) ).

% add_divide_eq_iff
tff(fact_2060_add__divide__eq__iff,axiom,
    ! [Z: rat,Xb: rat,Yb: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(plus_plus_rat(Xb),aa_rat_rat(divide_divide_rat(Yb),Z)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(Xb),Z)),Yb)),Z) ) ) ).

% add_divide_eq_iff
tff(fact_2061_divide__add__eq__iff,axiom,
    ! [Z: complex,Xb: complex,Yb: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(divide1717551699836669952omplex(Xb),Z)),Yb) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(Xb),aa_complex_complex(times_times_complex(Yb),Z))),Z) ) ) ).

% divide_add_eq_iff
tff(fact_2062_divide__add__eq__iff,axiom,
    ! [Z: real,Xb: real,Yb: real] :
      ( ( Z != zero_zero_real )
     => ( aa_real_real(plus_plus_real(aa_real_real(divide_divide_real(Xb),Z)),Yb) = aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(Xb),aa_real_real(times_times_real(Yb),Z))),Z) ) ) ).

% divide_add_eq_iff
tff(fact_2063_divide__add__eq__iff,axiom,
    ! [Z: rat,Xb: rat,Yb: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(plus_plus_rat(aa_rat_rat(divide_divide_rat(Xb),Z)),Yb) = aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(Xb),aa_rat_rat(times_times_rat(Yb),Z))),Z) ) ) ).

% divide_add_eq_iff
tff(fact_2064_power__gt1__lemma,axiom,
    ! [A: real,Nb: 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(times_times_real(A),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb))) ) ).

% power_gt1_lemma
tff(fact_2065_power__gt1__lemma,axiom,
    ! [A: rat,Nb: 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),Nb))) ) ).

% power_gt1_lemma
tff(fact_2066_power__gt1__lemma,axiom,
    ! [A: nat,Nb: 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),Nb))) ) ).

% power_gt1_lemma
tff(fact_2067_power__gt1__lemma,axiom,
    ! [A: int,Nb: 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),Nb))) ) ).

% power_gt1_lemma
tff(fact_2068_power__less__power__Suc,axiom,
    ! [A: real,Nb: 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(aa_real_fun_nat_real(power_power_real,A),Nb)),aa_real_real(times_times_real(A),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb))) ) ).

% power_less_power_Suc
tff(fact_2069_power__less__power__Suc,axiom,
    ! [A: rat,Nb: 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),Nb)),aa_rat_rat(times_times_rat(A),aa_nat_rat(power_power_rat(A),Nb))) ) ).

% power_less_power_Suc
tff(fact_2070_power__less__power__Suc,axiom,
    ! [A: nat,Nb: 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),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb))) ) ).

% power_less_power_Suc
tff(fact_2071_power__less__power__Suc,axiom,
    ! [A: int,Nb: 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),Nb)),aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(A),Nb))) ) ).

% power_less_power_Suc
tff(fact_2072_one__less__mult,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Mb)
       => 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,Mb),Nb)) ) ) ).

% one_less_mult
tff(fact_2073_n__less__m__mult__n,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Mb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)) ) ) ).

% n_less_m_mult_n
tff(fact_2074_n__less__n__mult__m,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Mb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Mb)) ) ) ).

% n_less_n_mult_m
tff(fact_2075_nat__mult__le__cancel1,axiom,
    ! [K: nat,Mb: nat,Nb: 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),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ).

% nat_mult_le_cancel1
tff(fact_2076_div__less__iff__less__mult,axiom,
    ! [Q2: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Q2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Q2)),Nb)
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Q2)) ) ) ).

% div_less_iff_less_mult
tff(fact_2077_nat__mult__div__cancel1,axiom,
    ! [K: nat,Mb: nat,Nb: 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),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb) ) ) ).

% nat_mult_div_cancel1
tff(fact_2078_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_2079_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_2080_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_2081_int__ge__induct,axiom,
    ! [K: int,I3: int,P: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),I3)
     => ( 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,I3) ) ) ) ).

% int_ge_induct
tff(fact_2082_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_2083_int__gr__induct,axiom,
    ! [K: int,I3: int,P: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),I3)
     => ( 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,I3) ) ) ) ).

% int_gr_induct
tff(fact_2084_max_Oassoc,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,B),C)) ).

% max.assoc
tff(fact_2085_max_Oassoc,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(aa_int_fun_int_int(ord_max_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(ord_max_int,A),aa_int_int(aa_int_fun_int_int(ord_max_int,B),C)) ).

% max.assoc
tff(fact_2086_max_Ocommute,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,B),A) ).

% max.commute
tff(fact_2087_max_Ocommute,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) = aa_int_int(aa_int_fun_int_int(ord_max_int,B),A) ).

% max.commute
tff(fact_2088_max_Oleft__commute,axiom,
    ! [B: nat,A: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),C)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,B),C)) ).

% max.left_commute
tff(fact_2089_max_Oleft__commute,axiom,
    ! [B: int,A: int,C: int] : aa_int_int(aa_int_fun_int_int(ord_max_int,B),aa_int_int(aa_int_fun_int_int(ord_max_int,A),C)) = aa_int_int(aa_int_fun_int_int(ord_max_int,A),aa_int_int(aa_int_fun_int_int(ord_max_int,B),C)) ).

% max.left_commute
tff(fact_2090_field__le__mult__one__interval,axiom,
    ! [Xb: real,Yb: 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(times_times_real(Z2),Xb)),Yb) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Yb) ) ).

% field_le_mult_one_interval
tff(fact_2091_field__le__mult__one__interval,axiom,
    ! [Xb: rat,Yb: 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),Xb)),Yb) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Yb) ) ).

% field_le_mult_one_interval
tff(fact_2092_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(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_2093_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_2094_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_2095_mult__le__cancel__left2,axiom,
    ! [C: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_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_2096_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_2097_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_2098_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(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_2099_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_2100_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_2101_mult__le__cancel__right2,axiom,
    ! [A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_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_2102_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_2103_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_2104_mult__less__cancel__left1,axiom,
    ! [C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),aa_real_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_2105_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_2106_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_2107_mult__less__cancel__left2,axiom,
    ! [C: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_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_2108_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_2109_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_2110_mult__less__cancel__right1,axiom,
    ! [C: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),aa_real_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_2111_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_2112_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_2113_mult__less__cancel__right2,axiom,
    ! [A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_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_2114_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_2115_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_2116_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(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(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(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_2117_divide__le__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_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_2118_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(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(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(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_2119_le__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_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)),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_2120_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(times_times_real(A),B))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(C),A)),aa_real_real(divide_divide_real(C),B)) ) ) ) ).

% divide_left_mono
tff(fact_2121_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,aa_rat_rat(divide_divide_rat(C),A)),aa_rat_rat(divide_divide_rat(C),B)) ) ) ) ).

% divide_left_mono
tff(fact_2122_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(divide_divide_real(B),C)),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A),C)),B) ) ) ).

% neg_divide_le_eq
tff(fact_2123_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,aa_rat_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_2124_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(divide_divide_real(B),C))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(times_times_real(A),C)) ) ) ).

% neg_le_divide_eq
tff(fact_2125_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),aa_rat_rat(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_2126_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(divide_divide_real(B),C)),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(times_times_real(A),C)) ) ) ).

% pos_divide_le_eq
tff(fact_2127_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,aa_rat_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_2128_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(divide_divide_real(B),C))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A),C)),B) ) ) ).

% pos_le_divide_eq
tff(fact_2129_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),aa_rat_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)),B) ) ) ).

% pos_le_divide_eq
tff(fact_2130_mult__imp__div__pos__le,axiom,
    ! [Yb: real,Xb: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Yb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(times_times_real(Z),Yb))
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(Xb),Yb)),Z) ) ) ).

% mult_imp_div_pos_le
tff(fact_2131_mult__imp__div__pos__le,axiom,
    ! [Yb: rat,Xb: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Yb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),aa_rat_rat(times_times_rat(Z),Yb))
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(Xb),Yb)),Z) ) ) ).

% mult_imp_div_pos_le
tff(fact_2132_mult__imp__le__div__pos,axiom,
    ! [Yb: real,Z: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Yb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(Z),Yb)),Xb)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Z),aa_real_real(divide_divide_real(Xb),Yb)) ) ) ).

% mult_imp_le_div_pos
tff(fact_2133_mult__imp__le__div__pos,axiom,
    ! [Yb: rat,Z: rat,Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Yb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(Z),Yb)),Xb)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z),aa_rat_rat(divide_divide_rat(Xb),Yb)) ) ) ).

% mult_imp_le_div_pos
tff(fact_2134_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(times_times_real(A),B))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(C),A)),aa_real_real(divide_divide_real(C),B)) ) ) ) ).

% divide_left_mono_neg
tff(fact_2135_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,aa_rat_rat(divide_divide_rat(C),A)),aa_rat_rat(divide_divide_rat(C),B)) ) ) ) ).

% divide_left_mono_neg
tff(fact_2136_convex__bound__le,axiom,
    ! [Xb: real,A: real,Yb: real,U: real,V: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Yb),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(plus_plus_real(U),V) = one_one_real )
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(aa_real_real(times_times_real(U),Xb)),aa_real_real(times_times_real(V),Yb))),A) ) ) ) ) ) ).

% convex_bound_le
tff(fact_2137_convex__bound__le,axiom,
    ! [Xb: rat,A: rat,Yb: rat,U: rat,V: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Yb),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(plus_plus_rat(U),V) = one_one_rat )
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(U),Xb)),aa_rat_rat(times_times_rat(V),Yb))),A) ) ) ) ) ) ).

% convex_bound_le
tff(fact_2138_convex__bound__le,axiom,
    ! [Xb: int,A: int,Yb: int,U: int,V: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Yb),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),Xb)),aa_int_int(aa_int_fun_int_int(times_times_int,V),Yb))),A) ) ) ) ) ) ).

% convex_bound_le
tff(fact_2139_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(divide_divide_real(B),C)),aa_num_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(times_times_real(aa_num_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(times_times_real(aa_num_real(numeral_numeral_real,W)),C)),B),aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_num_real(numeral_numeral_real,W))) ) ) ).

% divide_less_eq_numeral(1)
tff(fact_2140_divide__less__eq__numeral_I1_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(B),C)),aa_num_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(aa_num_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(aa_num_rat(numeral_numeral_rat,W)),C)),B),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_num_rat(numeral_numeral_rat,W))) ) ) ).

% divide_less_eq_numeral(1)
tff(fact_2141_less__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,W)),aa_real_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(times_times_real(aa_num_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(times_times_real(aa_num_real(numeral_numeral_real,W)),C)),aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,W)),zero_zero_real)) ) ) ).

% less_divide_eq_numeral(1)
tff(fact_2142_less__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,W)),aa_rat_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(aa_num_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(aa_num_rat(numeral_numeral_rat,W)),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,W)),zero_zero_rat)) ) ) ).

% less_divide_eq_numeral(1)
tff(fact_2143_power__Suc__less,axiom,
    ! [A: real,Nb: 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(times_times_real(A),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ) ) ).

% power_Suc_less
tff(fact_2144_power__Suc__less,axiom,
    ! [A: rat,Nb: 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),Nb))),aa_nat_rat(power_power_rat(A),Nb)) ) ) ).

% power_Suc_less
tff(fact_2145_power__Suc__less,axiom,
    ! [A: nat,Nb: 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),Nb))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ) ) ).

% power_Suc_less
tff(fact_2146_power__Suc__less,axiom,
    ! [A: int,Nb: 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),Nb))),aa_nat_int(power_power_int(A),Nb)) ) ) ).

% power_Suc_less
tff(fact_2147_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_2148_mult__2,axiom,
    ! [Z: real] : aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Z) = aa_real_real(plus_plus_real(Z),Z) ).

% mult_2
tff(fact_2149_mult__2,axiom,
    ! [Z: rat] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),Z) = aa_rat_rat(plus_plus_rat(Z),Z) ).

% mult_2
tff(fact_2150_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_2151_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_2152_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_2153_mult__2__right,axiom,
    ! [Z: real] : aa_real_real(times_times_real(Z),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) = aa_real_real(plus_plus_real(Z),Z) ).

% mult_2_right
tff(fact_2154_mult__2__right,axiom,
    ! [Z: rat] : aa_rat_rat(times_times_rat(Z),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))) = aa_rat_rat(plus_plus_rat(Z),Z) ).

% mult_2_right
tff(fact_2155_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_2156_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_2157_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_2158_left__add__twice,axiom,
    ! [A: real,B: real] : aa_real_real(plus_plus_real(A),aa_real_real(plus_plus_real(A),B)) = aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),A)),B) ).

% left_add_twice
tff(fact_2159_left__add__twice,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(plus_plus_rat(A),aa_rat_rat(plus_plus_rat(A),B)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),A)),B) ).

% left_add_twice
tff(fact_2160_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_2161_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_2162_power4__eq__xxxx,axiom,
    ! [Xb: complex] : aa_nat_complex(power_power_complex(Xb),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(Xb),Xb)),Xb)),Xb) ).

% power4_eq_xxxx
tff(fact_2163_power4__eq__xxxx,axiom,
    ! [Xb: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_real_real(times_times_real(aa_real_real(times_times_real(aa_real_real(times_times_real(Xb),Xb)),Xb)),Xb) ).

% power4_eq_xxxx
tff(fact_2164_power4__eq__xxxx,axiom,
    ! [Xb: rat] : aa_nat_rat(power_power_rat(Xb),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(Xb),Xb)),Xb)),Xb) ).

% power4_eq_xxxx
tff(fact_2165_power4__eq__xxxx,axiom,
    ! [Xb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),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,Xb),Xb)),Xb)),Xb) ).

% power4_eq_xxxx
tff(fact_2166_power4__eq__xxxx,axiom,
    ! [Xb: int] : aa_nat_int(power_power_int(Xb),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,Xb),Xb)),Xb)),Xb) ).

% power4_eq_xxxx
tff(fact_2167_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_2168_power2__eq__square,axiom,
    ! [A: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(times_times_real(A),A) ).

% power2_eq_square
tff(fact_2169_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_2170_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_2171_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_2172_power__even__eq,axiom,
    ! [A: nat,Nb: 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))),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power_even_eq
tff(fact_2173_power__even__eq,axiom,
    ! [A: int,Nb: 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))),Nb)) = aa_nat_int(power_power_int(aa_nat_int(power_power_int(A),Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power_even_eq
tff(fact_2174_power__even__eq,axiom,
    ! [A: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power_even_eq
tff(fact_2175_power__even__eq,axiom,
    ! [A: complex,Nb: 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))),Nb)) = aa_nat_complex(power_power_complex(aa_nat_complex(power_power_complex(A),Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power_even_eq
tff(fact_2176_num_Osize_I4_J,axiom,
    aa_num_nat(size_size_num,one) = zero_zero_nat ).

% num.size(4)
tff(fact_2177_div__nat__eqI,axiom,
    ! [Nb: nat,Q2: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Q2)),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(suc,Q2)))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb) = Q2 ) ) ) ).

% div_nat_eqI
tff(fact_2178_less__eq__div__iff__mult__less__eq,axiom,
    ! [Q2: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Q2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),Q2))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Q2)),Nb) ) ) ).

% less_eq_div_iff_mult_less_eq
tff(fact_2179_split__div,axiom,
    ! [P: fun_nat_o,Mb: nat,Nb: nat] :
      ( aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb))
    <=> ( ( ( Nb = zero_zero_nat )
         => aa_nat_o(P,zero_zero_nat) )
        & ( ( Nb != zero_zero_nat )
         => ! [I5: nat,J3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),Nb)
             => ( ( Mb = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),I5)),J3) )
               => aa_nat_o(P,I5) ) ) ) ) ) ).

% split_div
tff(fact_2180_dividend__less__div__times,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)),Nb))) ) ).

% dividend_less_div_times
tff(fact_2181_dividend__less__times__div,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)))) ) ).

% dividend_less_times_div
tff(fact_2182_convex__bound__lt,axiom,
    ! [Xb: real,A: real,Yb: real,U: real,V: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Yb),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(plus_plus_real(U),V) = one_one_real )
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(aa_real_real(times_times_real(U),Xb)),aa_real_real(times_times_real(V),Yb))),A) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_2183_convex__bound__lt,axiom,
    ! [Xb: rat,A: rat,Yb: rat,U: rat,V: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Yb),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(plus_plus_rat(U),V) = one_one_rat )
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(U),Xb)),aa_rat_rat(times_times_rat(V),Yb))),A) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_2184_convex__bound__lt,axiom,
    ! [Xb: int,A: int,Yb: int,U: int,V: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Yb),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),Xb)),aa_int_int(aa_int_fun_int_int(times_times_int,V),Yb))),A) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_2185_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(divide_divide_real(B),C)),aa_num_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(times_times_real(aa_num_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(times_times_real(aa_num_real(numeral_numeral_real,W)),C)),B),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_num_real(numeral_numeral_real,W))) ) ) ).

% divide_le_eq_numeral(1)
tff(fact_2186_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,aa_rat_rat(divide_divide_rat(B),C)),aa_num_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(aa_num_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(aa_num_rat(numeral_numeral_rat,W)),C)),B),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_num_rat(numeral_numeral_rat,W))) ) ) ).

% divide_le_eq_numeral(1)
tff(fact_2187_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,aa_num_real(numeral_numeral_real,W)),aa_real_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(times_times_real(aa_num_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(times_times_real(aa_num_real(numeral_numeral_real,W)),C)),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,W)),zero_zero_real)) ) ) ).

% le_divide_eq_numeral(1)
tff(fact_2188_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,aa_num_rat(numeral_numeral_rat,W)),aa_rat_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(aa_num_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(aa_num_rat(numeral_numeral_rat,W)),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,W)),zero_zero_rat)) ) ) ).

% le_divide_eq_numeral(1)
tff(fact_2189_split__div_H,axiom,
    ! [P: fun_nat_o,Mb: nat,Nb: nat] :
      ( aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb))
    <=> ( ( ( Nb = 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,Nb),Q4)),Mb)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(suc,Q4)))
            & aa_nat_o(P,Q4) ) ) ) ).

% split_div'
tff(fact_2190_power2__sum,axiom,
    ! [Xb: complex,Yb: complex] : aa_nat_complex(power_power_complex(aa_complex_complex(plus_plus_complex(Xb),Yb)),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(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_complex(power_power_complex(Yb),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))),Xb)),Yb)) ).

% power2_sum
tff(fact_2191_power2__sum,axiom,
    ! [Xb: real,Yb: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(plus_plus_real(Xb),Yb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(plus_plus_real(aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(times_times_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Xb)),Yb)) ).

% power2_sum
tff(fact_2192_power2__sum,axiom,
    ! [Xb: rat,Yb: rat] : aa_nat_rat(power_power_rat(aa_rat_rat(plus_plus_rat(Xb),Yb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),Xb)),Yb)) ).

% power2_sum
tff(fact_2193_power2__sum,axiom,
    ! [Xb: nat,Yb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xb),Yb)),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,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Yb),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))),Xb)),Yb)) ).

% power2_sum
tff(fact_2194_power2__sum,axiom,
    ! [Xb: int,Yb: int] : aa_nat_int(power_power_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Yb)),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(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Yb),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))),Xb)),Yb)) ).

% power2_sum
tff(fact_2195_zero__le__even__power_H,axiom,
    ! [A: real,Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) ).

% zero_le_even_power'
tff(fact_2196_zero__le__even__power_H,axiom,
    ! [A: rat,Nb: 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))),Nb))) ).

% zero_le_even_power'
tff(fact_2197_zero__le__even__power_H,axiom,
    ! [A: int,Nb: 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))),Nb))) ).

% zero_le_even_power'
tff(fact_2198_nat__bit__induct,axiom,
    ! [P: fun_nat_o,Nb: 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,Nb) ) ) ) ).

% nat_bit_induct
tff(fact_2199_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_2200_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_2201_two__realpow__ge__one,axiom,
    ! [Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Nb)) ).

% two_realpow_ge_one
tff(fact_2202_option_Osize_I4_J,axiom,
    ! [X2: product_prod_nat_nat] : size_s170228958280169651at_nat(some_P7363390416028606310at_nat(X2)) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(4)
tff(fact_2203_option_Osize_I4_J,axiom,
    ! [X2: num] : size_size_option_num(some_num(X2)) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(4)
tff(fact_2204_max_Omono,axiom,
    ! [C: rat,A: rat,D: rat,B: rat] :
      ( 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,D),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(ord_max_rat(C),D)),aa_rat_rat(ord_max_rat(A),B)) ) ) ).

% max.mono
tff(fact_2205_max_Omono,axiom,
    ! [C: num,A: num,D: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),A)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,D),B)
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(ord_max_num(C),D)),aa_num_num(ord_max_num(A),B)) ) ) ).

% max.mono
tff(fact_2206_max_Omono,axiom,
    ! [C: nat,A: nat,D: nat,B: nat] :
      ( 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,D),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,C),D)),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B)) ) ) ).

% max.mono
tff(fact_2207_max_Omono,axiom,
    ! [C: int,A: int,D: int,B: int] :
      ( 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,D),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(ord_max_int,C),D)),aa_int_int(aa_int_fun_int_int(ord_max_int,A),B)) ) ) ).

% max.mono
tff(fact_2208_max_OorderE,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
     => ( A = aa_rat_rat(ord_max_rat(A),B) ) ) ).

% max.orderE
tff(fact_2209_max_OorderE,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
     => ( A = aa_num_num(ord_max_num(A),B) ) ) ).

% max.orderE
tff(fact_2210_max_OorderE,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
     => ( A = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) ) ) ).

% max.orderE
tff(fact_2211_max_OorderE,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( A = aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) ) ) ).

% max.orderE
tff(fact_2212_max_OorderI,axiom,
    ! [A: rat,B: rat] :
      ( ( A = aa_rat_rat(ord_max_rat(A),B) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ).

% max.orderI
tff(fact_2213_max_OorderI,axiom,
    ! [A: num,B: num] :
      ( ( A = aa_num_num(ord_max_num(A),B) )
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A) ) ).

% max.orderI
tff(fact_2214_max_OorderI,axiom,
    ! [A: nat,B: nat] :
      ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A) ) ).

% max.orderI
tff(fact_2215_max_OorderI,axiom,
    ! [A: int,B: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ).

% max.orderI
tff(fact_2216_max_OboundedE,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(ord_max_rat(B),C)),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_eq_rat,C),A) ) ) ).

% max.boundedE
tff(fact_2217_max_OboundedE,axiom,
    ! [B: num,C: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(ord_max_num(B),C)),A)
     => ~ ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
         => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),A) ) ) ).

% max.boundedE
tff(fact_2218_max_OboundedE,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,B),C)),A)
     => ~ ( 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,C),A) ) ) ).

% max.boundedE
tff(fact_2219_max_OboundedE,axiom,
    ! [B: int,C: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(ord_max_int,B),C)),A)
     => ~ ( 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),A) ) ) ).

% max.boundedE
tff(fact_2220_max_OboundedI,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),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(ord_max_rat(B),C)),A) ) ) ).

% max.boundedI
tff(fact_2221_max_OboundedI,axiom,
    ! [B: num,A: num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),A)
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(ord_max_num(B),C)),A) ) ) ).

% max.boundedI
tff(fact_2222_max_OboundedI,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( 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,C),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,B),C)),A) ) ) ).

% max.boundedI
tff(fact_2223_max_OboundedI,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),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(ord_max_int,B),C)),A) ) ) ).

% max.boundedI
tff(fact_2224_max_Oorder__iff,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
    <=> ( A = aa_rat_rat(ord_max_rat(A),B) ) ) ).

% max.order_iff
tff(fact_2225_max_Oorder__iff,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
    <=> ( A = aa_num_num(ord_max_num(A),B) ) ) ).

% max.order_iff
tff(fact_2226_max_Oorder__iff,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
    <=> ( A = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) ) ) ).

% max.order_iff
tff(fact_2227_max_Oorder__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
    <=> ( A = aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) ) ) ).

% max.order_iff
tff(fact_2228_max_Ocobounded1,axiom,
    ! [A: rat,B: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(ord_max_rat(A),B)) ).

% max.cobounded1
tff(fact_2229_max_Ocobounded1,axiom,
    ! [A: num,B: num] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_num_num(ord_max_num(A),B)) ).

% max.cobounded1
tff(fact_2230_max_Ocobounded1,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(ord_max_nat,A),B)) ).

% max.cobounded1
tff(fact_2231_max_Ocobounded1,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(ord_max_int,A),B)) ).

% max.cobounded1
tff(fact_2232_max_Ocobounded2,axiom,
    ! [B: rat,A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(ord_max_rat(A),B)) ).

% max.cobounded2
tff(fact_2233_max_Ocobounded2,axiom,
    ! [B: num,A: num] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),aa_num_num(ord_max_num(A),B)) ).

% max.cobounded2
tff(fact_2234_max_Ocobounded2,axiom,
    ! [B: nat,A: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B)) ).

% max.cobounded2
tff(fact_2235_max_Ocobounded2,axiom,
    ! [B: int,A: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),aa_int_int(aa_int_fun_int_int(ord_max_int,A),B)) ).

% max.cobounded2
tff(fact_2236_le__max__iff__disj,axiom,
    ! [Z: rat,Xb: rat,Yb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z),aa_rat_rat(ord_max_rat(Xb),Yb))
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z),Xb)
        | aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z),Yb) ) ) ).

% le_max_iff_disj
tff(fact_2237_le__max__iff__disj,axiom,
    ! [Z: num,Xb: num,Yb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Z),aa_num_num(ord_max_num(Xb),Yb))
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Z),Xb)
        | aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Z),Yb) ) ) ).

% le_max_iff_disj
tff(fact_2238_le__max__iff__disj,axiom,
    ! [Z: nat,Xb: nat,Yb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xb),Yb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z),Xb)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z),Yb) ) ) ).

% le_max_iff_disj
tff(fact_2239_le__max__iff__disj,axiom,
    ! [Z: int,Xb: int,Yb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),aa_int_int(aa_int_fun_int_int(ord_max_int,Xb),Yb))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),Xb)
        | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),Yb) ) ) ).

% le_max_iff_disj
tff(fact_2240_max_Oabsorb__iff1,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A)
    <=> ( aa_rat_rat(ord_max_rat(A),B) = A ) ) ).

% max.absorb_iff1
tff(fact_2241_max_Oabsorb__iff1,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
    <=> ( aa_num_num(ord_max_num(A),B) = A ) ) ).

% max.absorb_iff1
tff(fact_2242_max_Oabsorb__iff1,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(ord_max_nat,A),B) = A ) ) ).

% max.absorb_iff1
tff(fact_2243_max_Oabsorb__iff1,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(ord_max_int,A),B) = A ) ) ).

% max.absorb_iff1
tff(fact_2244_max_Oabsorb__iff2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
    <=> ( aa_rat_rat(ord_max_rat(A),B) = B ) ) ).

% max.absorb_iff2
tff(fact_2245_max_Oabsorb__iff2,axiom,
    ! [A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
    <=> ( aa_num_num(ord_max_num(A),B) = B ) ) ).

% max.absorb_iff2
tff(fact_2246_max_Oabsorb__iff2,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(ord_max_nat,A),B) = B ) ) ).

% max.absorb_iff2
tff(fact_2247_max_Oabsorb__iff2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
    <=> ( aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) = B ) ) ).

% max.absorb_iff2
tff(fact_2248_max_OcoboundedI1,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( 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,C),aa_rat_rat(ord_max_rat(A),B)) ) ).

% max.coboundedI1
tff(fact_2249_max_OcoboundedI1,axiom,
    ! [C: num,A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),A)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),aa_num_num(ord_max_num(A),B)) ) ).

% max.coboundedI1
tff(fact_2250_max_OcoboundedI1,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( 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,C),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B)) ) ).

% max.coboundedI1
tff(fact_2251_max_OcoboundedI1,axiom,
    ! [C: int,A: int,B: int] :
      ( 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,C),aa_int_int(aa_int_fun_int_int(ord_max_int,A),B)) ) ).

% max.coboundedI1
tff(fact_2252_max_OcoboundedI2,axiom,
    ! [C: rat,B: rat,A: 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,C),aa_rat_rat(ord_max_rat(A),B)) ) ).

% max.coboundedI2
tff(fact_2253_max_OcoboundedI2,axiom,
    ! [C: num,B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),B)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),aa_num_num(ord_max_num(A),B)) ) ).

% max.coboundedI2
tff(fact_2254_max_OcoboundedI2,axiom,
    ! [C: nat,B: nat,A: 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,C),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B)) ) ).

% max.coboundedI2
tff(fact_2255_max_OcoboundedI2,axiom,
    ! [C: int,B: int,A: 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,C),aa_int_int(aa_int_fun_int_int(ord_max_int,A),B)) ) ).

% max.coboundedI2
tff(fact_2256_less__max__iff__disj,axiom,
    ! [Z: real,Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z),aa_real_real(ord_max_real(Xb),Yb))
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z),Xb)
        | aa_real_o(aa_real_fun_real_o(ord_less_real,Z),Yb) ) ) ).

% less_max_iff_disj
tff(fact_2257_less__max__iff__disj,axiom,
    ! [Z: rat,Xb: rat,Yb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),aa_rat_rat(ord_max_rat(Xb),Yb))
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),Xb)
        | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),Yb) ) ) ).

% less_max_iff_disj
tff(fact_2258_less__max__iff__disj,axiom,
    ! [Z: num,Xb: num,Yb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z),aa_num_num(ord_max_num(Xb),Yb))
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z),Xb)
        | aa_num_o(aa_num_fun_num_o(ord_less_num,Z),Yb) ) ) ).

% less_max_iff_disj
tff(fact_2259_less__max__iff__disj,axiom,
    ! [Z: nat,Xb: nat,Yb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xb),Yb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z),Xb)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z),Yb) ) ) ).

% less_max_iff_disj
tff(fact_2260_less__max__iff__disj,axiom,
    ! [Z: int,Xb: int,Yb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z),aa_int_int(aa_int_fun_int_int(ord_max_int,Xb),Yb))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z),Xb)
        | aa_int_o(aa_int_fun_int_o(ord_less_int,Z),Yb) ) ) ).

% less_max_iff_disj
tff(fact_2261_max_Ostrict__boundedE,axiom,
    ! [B: real,C: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(ord_max_real(B),C)),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,C),A) ) ) ).

% max.strict_boundedE
tff(fact_2262_max_Ostrict__boundedE,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(ord_max_rat(B),C)),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,C),A) ) ) ).

% max.strict_boundedE
tff(fact_2263_max_Ostrict__boundedE,axiom,
    ! [B: num,C: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(ord_max_num(B),C)),A)
     => ~ ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
         => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,C),A) ) ) ).

% max.strict_boundedE
tff(fact_2264_max_Ostrict__boundedE,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,B),C)),A)
     => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
         => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),A) ) ) ).

% max.strict_boundedE
tff(fact_2265_max_Ostrict__boundedE,axiom,
    ! [B: int,C: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(ord_max_int,B),C)),A)
     => ~ ( 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),A) ) ) ).

% max.strict_boundedE
tff(fact_2266_max_Ostrict__order__iff,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
    <=> ( ( A = aa_real_real(ord_max_real(A),B) )
        & ( A != B ) ) ) ).

% max.strict_order_iff
tff(fact_2267_max_Ostrict__order__iff,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
    <=> ( ( A = aa_rat_rat(ord_max_rat(A),B) )
        & ( A != B ) ) ) ).

% max.strict_order_iff
tff(fact_2268_max_Ostrict__order__iff,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
    <=> ( ( A = aa_num_num(ord_max_num(A),B) )
        & ( A != B ) ) ) ).

% max.strict_order_iff
tff(fact_2269_max_Ostrict__order__iff,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
    <=> ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) )
        & ( A != B ) ) ) ).

% max.strict_order_iff
tff(fact_2270_max_Ostrict__order__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
    <=> ( ( A = aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) )
        & ( A != B ) ) ) ).

% max.strict_order_iff
tff(fact_2271_max_Ostrict__coboundedI1,axiom,
    ! [C: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,C),aa_real_real(ord_max_real(A),B)) ) ).

% max.strict_coboundedI1
tff(fact_2272_max_Ostrict__coboundedI1,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),aa_rat_rat(ord_max_rat(A),B)) ) ).

% max.strict_coboundedI1
tff(fact_2273_max_Ostrict__coboundedI1,axiom,
    ! [C: num,A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,C),A)
     => aa_num_o(aa_num_fun_num_o(ord_less_num,C),aa_num_num(ord_max_num(A),B)) ) ).

% max.strict_coboundedI1
tff(fact_2274_max_Ostrict__coboundedI1,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B)) ) ).

% max.strict_coboundedI1
tff(fact_2275_max_Ostrict__coboundedI1,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,C),aa_int_int(aa_int_fun_int_int(ord_max_int,A),B)) ) ).

% max.strict_coboundedI1
tff(fact_2276_max_Ostrict__coboundedI2,axiom,
    ! [C: real,B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),B)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,C),aa_real_real(ord_max_real(A),B)) ) ).

% max.strict_coboundedI2
tff(fact_2277_max_Ostrict__coboundedI2,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),aa_rat_rat(ord_max_rat(A),B)) ) ).

% max.strict_coboundedI2
tff(fact_2278_max_Ostrict__coboundedI2,axiom,
    ! [C: num,B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,C),B)
     => aa_num_o(aa_num_fun_num_o(ord_less_num,C),aa_num_num(ord_max_num(A),B)) ) ).

% max.strict_coboundedI2
tff(fact_2279_max_Ostrict__coboundedI2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B)) ) ).

% max.strict_coboundedI2
tff(fact_2280_max_Ostrict__coboundedI2,axiom,
    ! [C: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,C),aa_int_int(aa_int_fun_int_int(ord_max_int,A),B)) ) ).

% max.strict_coboundedI2
tff(fact_2281_odd__0__le__power__imp__0__le,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))))
     => 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_2282_odd__0__le__power__imp__0__le,axiom,
    ! [A: rat,Nb: 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))),Nb))))
     => 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_2283_odd__0__le__power__imp__0__le,axiom,
    ! [A: int,Nb: 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))),Nb))))
     => 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_2284_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_2285_odd__power__less__zero,axiom,
    ! [A: real,Nb: 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(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))),zero_zero_real) ) ).

% odd_power_less_zero
tff(fact_2286_odd__power__less__zero,axiom,
    ! [A: rat,Nb: 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))),Nb)))),zero_zero_rat) ) ).

% odd_power_less_zero
tff(fact_2287_odd__power__less__zero,axiom,
    ! [A: int,Nb: 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))),Nb)))),zero_zero_int) ) ).

% odd_power_less_zero
tff(fact_2288_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_2289_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_2290_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_2291_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_2292_num_Osize_I5_J,axiom,
    ! [X2: num] : aa_num_nat(size_size_num,aa_num_num(bit0,X2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(size_size_num,X2)),aa_nat_nat(suc,zero_zero_nat)) ).

% num.size(5)
tff(fact_2293_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_2294_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_2295_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_2296_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_2297_int__div__less__self,axiom,
    ! [Xb: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Xb)
     => ( 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,Xb),K)),Xb) ) ) ).

% int_div_less_self
tff(fact_2298_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_2299_VEBT__internal_OminNull_Osimps_I5_J,axiom,
    ! [Uz: product_prod_nat_nat,Va2: nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT] : ~ aa_VEBT_VEBT_o(vEBT_VEBT_minNull,vEBT_Node(some_P7363390416028606310at_nat(Uz),Va2,Vb,Vc)) ).

% VEBT_internal.minNull.simps(5)
tff(fact_2300_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_2301_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_2302_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_2303_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_2304_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_2305_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_2306_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_2307_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_2308_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_2309_pos__imp__zdiv__pos__iff,axiom,
    ! [K: int,I3: 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,I3),K))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),I3) ) ) ).

% pos_imp_zdiv_pos_iff
tff(fact_2310_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_2311_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_2312_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_2313_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_2314_zdiv__mono2__neg,axiom,
    ! [A: int,B2: 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),B2)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),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),B2)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ) ).

% zdiv_mono2_neg
tff(fact_2315_zdiv__mono1__neg,axiom,
    ! [A: int,A2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),A2)
     => ( 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,A2),B)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ).

% zdiv_mono1_neg
tff(fact_2316_zdiv__eq__0__iff,axiom,
    ! [I3: int,K: int] :
      ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,I3),K) = zero_zero_int )
    <=> ( ( K = zero_zero_int )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),I3)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,I3),K) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I3),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,K),I3) ) ) ) ).

% zdiv_eq_0_iff
tff(fact_2317_zdiv__mono2,axiom,
    ! [A: int,B2: 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),B2)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),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),B2)) ) ) ) ).

% zdiv_mono2
tff(fact_2318_zdiv__mono1,axiom,
    ! [A: int,A2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),A2)
     => ( 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,A2),B)) ) ) ).

% zdiv_mono1
tff(fact_2319_vebt__member_Osimps_I3_J,axiom,
    ! [V: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,Xb: nat] : ~ aa_nat_o(vEBT_vebt_member(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Uy,Uz)),Xb) ).

% vebt_member.simps(3)
tff(fact_2320_field__sum__of__halves,axiom,
    ! [Xb: real] : aa_real_real(plus_plus_real(aa_real_real(divide_divide_real(Xb),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_real_real(divide_divide_real(Xb),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = Xb ).

% field_sum_of_halves
tff(fact_2321_field__sum__of__halves,axiom,
    ! [Xb: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(divide_divide_rat(Xb),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))),aa_rat_rat(divide_divide_rat(Xb),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))) = Xb ).

% field_sum_of_halves
tff(fact_2322_arith__geo__mean,axiom,
    ! [U: real,Xb: real,Yb: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,U),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(times_times_real(Xb),Yb) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Yb)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,U),aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(Xb),Yb)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ) ).

% arith_geo_mean
tff(fact_2323_arith__geo__mean,axiom,
    ! [U: rat,Xb: rat,Yb: 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(Xb),Yb) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xb)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Yb)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,U),aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(Xb),Yb)),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))) ) ) ) ).

% arith_geo_mean
tff(fact_2324_sum__squares__bound,axiom,
    ! [Xb: real,Yb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Xb)),Yb)),aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% sum_squares_bound
tff(fact_2325_sum__squares__bound,axiom,
    ! [Xb: rat,Yb: 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(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),Xb)),Yb)),aa_rat_rat(plus_plus_rat(aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% sum_squares_bound
tff(fact_2326_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_2327_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_2328_enat__ord__number_I1_J,axiom,
    ! [Mb: num,Nb: num] :
      ( ord_le2932123472753598470d_enat(numera1916890842035813515d_enat(Mb),numera1916890842035813515d_enat(Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Nb)) ) ).

% enat_ord_number(1)
tff(fact_2329_enat__ord__number_I2_J,axiom,
    ! [Mb: num,Nb: num] :
      ( ord_le72135733267957522d_enat(numera1916890842035813515d_enat(Mb),numera1916890842035813515d_enat(Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Nb)) ) ).

% enat_ord_number(2)
tff(fact_2330_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_2331_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_2332_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_2333_mult__le__cancel__iff1,axiom,
    ! [Z: real,Xb: real,Yb: 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(times_times_real(Xb),Z)),aa_real_real(times_times_real(Yb),Z))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Yb) ) ) ).

% mult_le_cancel_iff1
tff(fact_2334_mult__le__cancel__iff1,axiom,
    ! [Z: rat,Xb: rat,Yb: 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(Xb),Z)),aa_rat_rat(times_times_rat(Yb),Z))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Yb) ) ) ).

% mult_le_cancel_iff1
tff(fact_2335_mult__le__cancel__iff1,axiom,
    ! [Z: int,Xb: int,Yb: 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,Xb),Z)),aa_int_int(aa_int_fun_int_int(times_times_int,Yb),Z))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Yb) ) ) ).

% mult_le_cancel_iff1
tff(fact_2336_mult__le__cancel__iff2,axiom,
    ! [Z: real,Xb: real,Yb: 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(times_times_real(Z),Xb)),aa_real_real(times_times_real(Z),Yb))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Yb) ) ) ).

% mult_le_cancel_iff2
tff(fact_2337_mult__le__cancel__iff2,axiom,
    ! [Z: rat,Xb: rat,Yb: 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),Xb)),aa_rat_rat(times_times_rat(Z),Yb))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Yb) ) ) ).

% mult_le_cancel_iff2
tff(fact_2338_mult__le__cancel__iff2,axiom,
    ! [Z: int,Xb: int,Yb: 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),Xb)),aa_int_int(aa_int_fun_int_int(times_times_int,Z),Yb))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Yb) ) ) ).

% mult_le_cancel_iff2
tff(fact_2339_divides__aux__eq,axiom,
    ! [Q2: nat,R2: nat] :
      ( aa_Pro6150637604445964947_nat_o(unique6322359934112328802ux_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Q2),R2))
    <=> ( R2 = zero_zero_nat ) ) ).

% divides_aux_eq
tff(fact_2340_divides__aux__eq,axiom,
    ! [Q2: int,R2: int] :
      ( aa_Pro4592068669816069595_int_o(unique6319869463603278526ux_int,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q2),R2))
    <=> ( R2 = zero_zero_int ) ) ).

% divides_aux_eq
tff(fact_2341_unset__bit__nonnegative__int__iff,axiom,
    ! [Nb: 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,Nb),K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K) ) ).

% unset_bit_nonnegative_int_iff
tff(fact_2342_set__bit__nonnegative__int__iff,axiom,
    ! [Nb: 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,Nb),K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K) ) ).

% set_bit_nonnegative_int_iff
tff(fact_2343_unset__bit__negative__int__iff,axiom,
    ! [Nb: 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,Nb),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_2344_set__bit__negative__int__iff,axiom,
    ! [Nb: 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,Nb),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_2345_semiring__norm_I13_J,axiom,
    ! [Mb: num,Nb: num] : aa_num_num(times_times_num(aa_num_num(bit0,Mb)),aa_num_num(bit0,Nb)) = aa_num_num(bit0,aa_num_num(bit0,aa_num_num(times_times_num(Mb),Nb))) ).

% semiring_norm(13)
tff(fact_2346_semiring__norm_I11_J,axiom,
    ! [Mb: num] : aa_num_num(times_times_num(Mb),one) = Mb ).

% semiring_norm(11)
tff(fact_2347_semiring__norm_I12_J,axiom,
    ! [Nb: num] : aa_num_num(times_times_num(one),Nb) = Nb ).

% semiring_norm(12)
tff(fact_2348_num__double,axiom,
    ! [Nb: num] : aa_num_num(times_times_num(aa_num_num(bit0,one)),Nb) = aa_num_num(bit0,Nb) ).

% num_double
tff(fact_2349_power__mult__numeral,axiom,
    ! [A: nat,Mb: num,Nb: 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,Mb))),aa_num_nat(numeral_numeral_nat,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(times_times_num(Mb),Nb))) ).

% power_mult_numeral
tff(fact_2350_power__mult__numeral,axiom,
    ! [A: int,Mb: num,Nb: num] : aa_nat_int(power_power_int(aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,Mb))),aa_num_nat(numeral_numeral_nat,Nb)) = aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(times_times_num(Mb),Nb))) ).

% power_mult_numeral
tff(fact_2351_power__mult__numeral,axiom,
    ! [A: real,Mb: num,Nb: num] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,Mb))),aa_num_nat(numeral_numeral_nat,Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(times_times_num(Mb),Nb))) ).

% power_mult_numeral
tff(fact_2352_power__mult__numeral,axiom,
    ! [A: complex,Mb: num,Nb: num] : aa_nat_complex(power_power_complex(aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,Mb))),aa_num_nat(numeral_numeral_nat,Nb)) = aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,aa_num_num(times_times_num(Mb),Nb))) ).

% power_mult_numeral
tff(fact_2353_complete__real,axiom,
    ! [S3: set_real] :
      ( ? [X: real] : member_real(X,S3)
     => ( ? [Z3: real] :
          ! [X3: real] :
            ( member_real(X3,S3)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),Z3) )
       => ? [Y: real] :
            ( ! [X: real] :
                ( member_real(X,S3)
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Y) )
            & ! [Z3: real] :
                ( ! [X3: real] :
                    ( member_real(X3,S3)
                   => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),Z3) )
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Z3) ) ) ) ) ).

% complete_real
tff(fact_2354_real__arch__pow,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb)
     => ? [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,Yb),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),N)) ) ).

% real_arch_pow
tff(fact_2355_less__eq__real__def,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Yb)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Yb)
        | ( Xb = Yb ) ) ) ).

% less_eq_real_def
tff(fact_2356_real__arch__pow__inv,axiom,
    ! [Yb: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Yb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),one_one_real)
       => ? [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),N)),Yb) ) ) ).

% real_arch_pow_inv
tff(fact_2357_enat__0__less__mult__iff,axiom,
    ! [Mb: extended_enat,Nb: extended_enat] :
      ( ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat,times_7803423173614009249d_enat(Mb,Nb))
    <=> ( ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat,Mb)
        & ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat,Nb) ) ) ).

% enat_0_less_mult_iff
tff(fact_2358_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_2359_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_2360_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_2361_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_2362_unset__bit__less__eq,axiom,
    ! [Nb: 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,Nb),K)),K) ).

% unset_bit_less_eq
tff(fact_2363_set__bit__greater__eq,axiom,
    ! [K: int,Nb: 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,Nb),K)) ).

% set_bit_greater_eq
tff(fact_2364_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(times_times_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(times_times_real(A),C))),aa_real_real(times_times_real(B),D))),aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,D),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,B),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,C),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% L2_set_mult_ineq_lemma
tff(fact_2365_four__x__squared,axiom,
    ! [Xb: real] : aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% four_x_squared
tff(fact_2366_div__mult2__numeral__eq,axiom,
    ! [A: nat,K: num,L: num] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,K))),aa_num_nat(numeral_numeral_nat,L)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(times_times_num(K),L))) ).

% div_mult2_numeral_eq
tff(fact_2367_div__mult2__numeral__eq,axiom,
    ! [A: int,K: num,L: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,K))),aa_num_int(numeral_numeral_int,L)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(times_times_num(K),L))) ).

% div_mult2_numeral_eq
tff(fact_2368_zmult__zless__mono2,axiom,
    ! [I3: int,J: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,I3),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),I3)),aa_int_int(aa_int_fun_int_int(times_times_int,K),J)) ) ) ).

% zmult_zless_mono2
tff(fact_2369_pos__zmult__eq__1__iff,axiom,
    ! [Mb: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Mb)
     => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,Mb),Nb) = one_one_int )
      <=> ( ( Mb = one_one_int )
          & ( Nb = one_one_int ) ) ) ) ).

% pos_zmult_eq_1_iff
tff(fact_2370_realpow__pos__nth2,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ? [R3: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R3)
          & ( aa_nat_real(aa_real_fun_nat_real(power_power_real,R3),aa_nat_nat(suc,Nb)) = A ) ) ) ).

% realpow_pos_nth2
tff(fact_2371_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_2372_q__pos__lemma,axiom,
    ! [B2: int,Q5: int,R4: 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,B2),Q5)),R4))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R4),B2)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Q5) ) ) ) ).

% q_pos_lemma
tff(fact_2373_zdiv__mono2__lemma,axiom,
    ! [B: int,Q2: int,R2: int,B2: int,Q5: int,R4: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q2)),R2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),Q5)),R4) )
     => ( 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,B2),Q5)),R4))
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R4),B2)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),R2)
           => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
             => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),B)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Q2),Q5) ) ) ) ) ) ) ).

% zdiv_mono2_lemma
tff(fact_2374_zdiv__mono2__neg__lemma,axiom,
    ! [B: int,Q2: int,R2: int,B2: int,Q5: int,R4: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q2)),R2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),Q5)),R4) )
     => ( 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,B2),Q5)),R4)),zero_zero_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R2),B)
         => ( 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),B2)
             => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),B)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Q5),Q2) ) ) ) ) ) ) ).

% zdiv_mono2_neg_lemma
tff(fact_2375_unique__quotient__lemma,axiom,
    ! [B: int,Q5: int,R4: int,Q2: int,R2: 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),Q5)),R4)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q2)),R2))
     => ( 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_o(aa_int_fun_int_o(ord_less_int,R2),B)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Q5),Q2) ) ) ) ) ).

% unique_quotient_lemma
tff(fact_2376_unique__quotient__lemma__neg,axiom,
    ! [B: int,Q5: int,R4: int,Q2: int,R2: 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),Q5)),R4)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q2)),R2))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,R2),zero_zero_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),R2)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),R4)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Q2),Q5) ) ) ) ) ).

% unique_quotient_lemma_neg
tff(fact_2377_split__zdiv,axiom,
    ! [P: fun_int_o,Nb: int,K: int] :
      ( aa_int_o(P,aa_int_int(aa_int_fun_int_int(divide_divide_int,Nb),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)
         => ! [I5: 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)
                & ( Nb = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I5)),J3) ) )
             => aa_int_o(P,I5) ) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
         => ! [I5: 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)
                & ( Nb = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I5)),J3) ) )
             => aa_int_o(P,I5) ) ) ) ) ).

% split_zdiv
tff(fact_2378_int__div__neg__eq,axiom,
    ! [A: int,B: int,Q2: int,R2: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q2)),R2) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,R2),zero_zero_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),R2)
         => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) = Q2 ) ) ) ) ).

% int_div_neg_eq
tff(fact_2379_int__div__pos__eq,axiom,
    ! [A: int,B: int,Q2: int,R2: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q2)),R2) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),R2)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R2),B)
         => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) = Q2 ) ) ) ) ).

% int_div_pos_eq
tff(fact_2380_realpow__pos__nth__unique,axiom,
    ! [Nb: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => ? [X3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X3)
            & ( aa_nat_real(aa_real_fun_nat_real(power_power_real,X3),Nb) = A )
            & ! [Y3: real] :
                ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y3)
                  & ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Y3),Nb) = A ) )
               => ( Y3 = X3 ) ) ) ) ) ).

% realpow_pos_nth_unique
tff(fact_2381_realpow__pos__nth,axiom,
    ! [Nb: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => ? [R3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R3)
            & ( aa_nat_real(aa_real_fun_nat_real(power_power_real,R3),Nb) = A ) ) ) ) ).

% realpow_pos_nth
tff(fact_2382_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_2383_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_2384_mult__less__iff1,axiom,
    ! [Z: real,Xb: real,Yb: 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(times_times_real(Xb),Z)),aa_real_real(times_times_real(Yb),Z))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Yb) ) ) ).

% mult_less_iff1
tff(fact_2385_mult__less__iff1,axiom,
    ! [Z: rat,Xb: rat,Yb: 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(Xb),Z)),aa_rat_rat(times_times_rat(Yb),Z))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Yb) ) ) ).

% mult_less_iff1
tff(fact_2386_mult__less__iff1,axiom,
    ! [Z: int,Xb: int,Yb: 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,Xb),Z)),aa_int_int(aa_int_fun_int_int(times_times_int,Yb),Z))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Yb) ) ) ).

% mult_less_iff1
tff(fact_2387_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)
     => ( ! [X3: int] :
            ( aa_int_o(P,X3)
           => aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D)) )
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
         => ! [X: int] :
              ( aa_int_o(P,X)
             => aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),aa_int_int(aa_int_fun_int_int(times_times_int,K),D))) ) ) ) ) ).

% incr_mult_lemma
tff(fact_2388_low__def,axiom,
    ! [Xb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% low_def
tff(fact_2389_invar__vebt_Ointros_I3_J,axiom,
    ! [TreeLista: list_VEBT_VEBT,Nb: nat,Summarya: vEBT_VEBT,Mb: nat,Dega: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeLista))
         => aa_nat_o(vEBT_invar_vebt(X3),Nb) )
     => ( aa_nat_o(vEBT_invar_vebt(Summarya),Mb)
       => ( ( 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))),Mb) )
         => ( ( Mb = aa_nat_nat(suc,Nb) )
           => ( ( Dega = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb) )
             => ( ~ ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summarya),X_12)
               => ( ! [X3: vEBT_VEBT] :
                      ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeLista))
                     => ~ ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X3),X_12) )
                 => aa_nat_o(vEBT_invar_vebt(vEBT_Node(none_P5556105721700978146at_nat,Dega,TreeLista,Summarya)),Dega) ) ) ) ) ) ) ) ).

% invar_vebt.intros(3)
tff(fact_2390_dbl__simps_I3_J,axiom,
    aa_complex_complex(neg_nu7009210354673126013omplex,one_one_complex) = numera6690914467698888265omplex(aa_num_num(bit0,one)) ).

% dbl_simps(3)
tff(fact_2391_dbl__simps_I3_J,axiom,
    aa_real_real(neg_numeral_dbl_real,one_one_real) = aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)) ).

% dbl_simps(3)
tff(fact_2392_dbl__simps_I3_J,axiom,
    aa_rat_rat(neg_numeral_dbl_rat,one_one_rat) = aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)) ).

% dbl_simps(3)
tff(fact_2393_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_2394_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_2395_deg1Leaf,axiom,
    ! [Ta: vEBT_VEBT] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),one_one_nat)
    <=> ? [A5: $o,B5: $o] : Ta = vEBT_Leaf((A5),(B5)) ) ).

% deg1Leaf
tff(fact_2396_deg__1__Leaf,axiom,
    ! [Ta: vEBT_VEBT] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),one_one_nat)
     => ? [A4: $o,B3: $o] : Ta = vEBT_Leaf((A4),(B3)) ) ).

% deg_1_Leaf
tff(fact_2397_deg__1__Leafy,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( ( Nb = one_one_nat )
       => ? [A4: $o,B3: $o] : Ta = vEBT_Leaf((A4),(B3)) ) ) ).

% deg_1_Leafy
tff(fact_2398_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_2399_mod__mod__trivial,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ).

% mod_mod_trivial
tff(fact_2400_real__divide__square__eq,axiom,
    ! [R2: real,A: real] : aa_real_real(divide_divide_real(aa_real_real(times_times_real(R2),A)),aa_real_real(times_times_real(R2),R2)) = aa_real_real(divide_divide_real(A),R2) ).

% real_divide_square_eq
tff(fact_2401_VEBT_Oinject_I2_J,axiom,
    ! [X21: $o,X22: $o,Y21: $o,Y22: $o] :
      ( ( vEBT_Leaf((X21),(X22)) = vEBT_Leaf((Y21),(Y22)) )
    <=> ( ( (X21)
        <=> (Y21) )
        & ( (X22)
        <=> (Y22) ) ) ) ).

% VEBT.inject(2)
tff(fact_2402_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_2403_bits__mod__0,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,zero_zero_int),A) = zero_zero_int ).

% bits_mod_0
tff(fact_2404_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_2405_mod__0,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,zero_zero_int),A) = zero_zero_int ).

% mod_0
tff(fact_2406_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_2407_mod__by__0,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),zero_zero_int) = A ).

% mod_by_0
tff(fact_2408_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_2409_mod__self,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),A) = zero_zero_int ).

% mod_self
tff(fact_2410_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_2411_mod__add__self1,axiom,
    ! [B: int,A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A)),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ).

% mod_add_self1
tff(fact_2412_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_2413_mod__add__self2,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ).

% mod_add_self2
tff(fact_2414_mod__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb) = Mb ) ) ).

% mod_less
tff(fact_2415_not__Some__eq,axiom,
    ! [Xb: option4927543243414619207at_nat] :
      ( ! [Y4: product_prod_nat_nat] : Xb != some_P7363390416028606310at_nat(Y4)
    <=> ( Xb = none_P5556105721700978146at_nat ) ) ).

% not_Some_eq
tff(fact_2416_not__Some__eq,axiom,
    ! [Xb: option_num] :
      ( ! [Y4: num] : Xb != some_num(Y4)
    <=> ( Xb = none_num ) ) ).

% not_Some_eq
tff(fact_2417_not__None__eq,axiom,
    ! [Xb: option4927543243414619207at_nat] :
      ( ( Xb != none_P5556105721700978146at_nat )
    <=> ? [Y4: product_prod_nat_nat] : Xb = some_P7363390416028606310at_nat(Y4) ) ).

% not_None_eq
tff(fact_2418_not__None__eq,axiom,
    ! [Xb: option_num] :
      ( ( Xb != none_num )
    <=> ? [Y4: num] : Xb = some_num(Y4) ) ).

% not_None_eq
tff(fact_2419_dbl__simps_I2_J,axiom,
    aa_complex_complex(neg_nu7009210354673126013omplex,zero_zero_complex) = zero_zero_complex ).

% dbl_simps(2)
tff(fact_2420_dbl__simps_I2_J,axiom,
    aa_real_real(neg_numeral_dbl_real,zero_zero_real) = zero_zero_real ).

% dbl_simps(2)
tff(fact_2421_dbl__simps_I2_J,axiom,
    aa_rat_rat(neg_numeral_dbl_rat,zero_zero_rat) = zero_zero_rat ).

% dbl_simps(2)
tff(fact_2422_dbl__simps_I2_J,axiom,
    aa_int_int(neg_numeral_dbl_int,zero_zero_int) = zero_zero_int ).

% dbl_simps(2)
tff(fact_2423_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_2424_mod__mult__self1__is__0,axiom,
    ! [B: int,A: int] : aa_int_int(aa_int_fun_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_2425_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_2426_mod__mult__self2__is__0,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_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_2427_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_2428_bits__mod__by__1,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),one_one_int) = zero_zero_int ).

% bits_mod_by_1
tff(fact_2429_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_2430_mod__by__1,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),one_one_int) = zero_zero_int ).

% mod_by_1
tff(fact_2431_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_2432_mod__div__trivial,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),B) = zero_zero_int ).

% mod_div_trivial
tff(fact_2433_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_2434_bits__mod__div__trivial,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),B) = zero_zero_int ).

% bits_mod_div_trivial
tff(fact_2435_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_2436_mod__mult__self1,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(aa_int_fun_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(aa_int_fun_int_int(modulo_modulo_int,A),B) ).

% mod_mult_self1
tff(fact_2437_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_2438_mod__mult__self2,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_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(aa_int_fun_int_int(modulo_modulo_int,A),B) ).

% mod_mult_self2
tff(fact_2439_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_2440_mod__mult__self3,axiom,
    ! [C: int,B: int,A: int] : aa_int_int(aa_int_fun_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(aa_int_fun_int_int(modulo_modulo_int,A),B) ).

% mod_mult_self3
tff(fact_2441_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_2442_mod__mult__self4,axiom,
    ! [B: int,C: int,A: int] : aa_int_int(aa_int_fun_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(aa_int_fun_int_int(modulo_modulo_int,A),B) ).

% mod_mult_self4
tff(fact_2443_not__real__square__gt__zero,axiom,
    ! [Xb: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(times_times_real(Xb),Xb))
    <=> ( Xb = zero_zero_real ) ) ).

% not_real_square_gt_zero
tff(fact_2444_mod__by__Suc__0,axiom,
    ! [Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_nat_nat(suc,zero_zero_nat)) = zero_zero_nat ).

% mod_by_Suc_0
tff(fact_2445_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_2446_dbl__simps_I5_J,axiom,
    ! [K: num] : aa_real_real(neg_numeral_dbl_real,aa_num_real(numeral_numeral_real,K)) = aa_num_real(numeral_numeral_real,aa_num_num(bit0,K)) ).

% dbl_simps(5)
tff(fact_2447_dbl__simps_I5_J,axiom,
    ! [K: num] : aa_rat_rat(neg_numeral_dbl_rat,aa_num_rat(numeral_numeral_rat,K)) = aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,K)) ).

% dbl_simps(5)
tff(fact_2448_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_2449_Suc__mod__mult__self1,axiom,
    ! [Mb: nat,K: nat,Nb: 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,Mb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb)))),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,Mb)),Nb) ).

% Suc_mod_mult_self1
tff(fact_2450_Suc__mod__mult__self2,axiom,
    ! [Mb: nat,Nb: 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,Mb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),K)))),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,Mb)),Nb) ).

% Suc_mod_mult_self2
tff(fact_2451_Suc__mod__mult__self3,axiom,
    ! [K: nat,Nb: nat,Mb: 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),Nb)),Mb))),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,Mb)),Nb) ).

% Suc_mod_mult_self3
tff(fact_2452_Suc__mod__mult__self4,axiom,
    ! [Nb: nat,K: nat,Mb: 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,Nb),K)),Mb))),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,Mb)),Nb) ).

% Suc_mod_mult_self4
tff(fact_2453_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_2454_bits__one__mod__two__eq__one,axiom,
    aa_int_int(aa_int_fun_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_2455_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_2456_one__mod__two__eq__one,axiom,
    aa_int_int(aa_int_fun_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_2457_mod2__Suc__Suc,axiom,
    ! [Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,Mb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% mod2_Suc_Suc
tff(fact_2458_Suc__times__numeral__mod__eq,axiom,
    ! [K: num,Nb: 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)),Nb))),aa_num_nat(numeral_numeral_nat,K)) = one_one_nat ) ) ).

% Suc_times_numeral_mod_eq
tff(fact_2459_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_2460_not__mod__2__eq__0__eq__1,axiom,
    ! [A: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) != zero_zero_int )
    <=> ( aa_int_int(aa_int_fun_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_2461_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_2462_not__mod__2__eq__1__eq__0,axiom,
    ! [A: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) != one_one_int )
    <=> ( aa_int_int(aa_int_fun_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_2463_not__mod2__eq__Suc__0__eq__0,axiom,
    ! [Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),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,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ) ) ).

% not_mod2_eq_Suc_0_eq_0
tff(fact_2464_add__self__mod__2,axiom,
    ! [Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Mb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ).

% add_self_mod_2
tff(fact_2465_mod2__gr__0,axiom,
    ! [Mb: 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,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_nat ) ) ).

% mod2_gr_0
tff(fact_2466_zero__one__enat__neq_I1_J,axiom,
    zero_z5237406670263579293d_enat != one_on7984719198319812577d_enat ).

% zero_one_enat_neq(1)
tff(fact_2467_iadd__is__0,axiom,
    ! [Mb: extended_enat,Nb: extended_enat] :
      ( ( plus_p3455044024723400733d_enat(Mb,Nb) = zero_z5237406670263579293d_enat )
    <=> ( ( Mb = zero_z5237406670263579293d_enat )
        & ( Nb = zero_z5237406670263579293d_enat ) ) ) ).

% iadd_is_0
tff(fact_2468_imult__is__0,axiom,
    ! [Mb: extended_enat,Nb: extended_enat] :
      ( ( times_7803423173614009249d_enat(Mb,Nb) = zero_z5237406670263579293d_enat )
    <=> ( ( Mb = zero_z5237406670263579293d_enat )
        | ( Nb = zero_z5237406670263579293d_enat ) ) ) ).

% imult_is_0
tff(fact_2469_VEBT__internal_OminNull_Ocases,axiom,
    ! [Xb: vEBT_VEBT] :
      ( ( Xb != vEBT_Leaf($false,$false) )
     => ( ! [Uv2: $o] : Xb != vEBT_Leaf($true,(Uv2))
       => ( ! [Uu2: $o] : Xb != vEBT_Leaf((Uu2),$true)
         => ( ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : Xb != vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2,Uy2)
           => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : Xb != vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2,Vc2) ) ) ) ) ).

% VEBT_internal.minNull.cases
tff(fact_2470_VEBT_Osize_I4_J,axiom,
    ! [X21: $o,X22: $o] : aa_VEBT_VEBT_nat(size_size_VEBT_VEBT,vEBT_Leaf((X21),(X22))) = zero_zero_nat ).

% VEBT.size(4)
tff(fact_2471_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_2472_mod__mult__right__eq,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C))),C) = aa_int_int(aa_int_fun_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_2473_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_2474_mod__mult__left__eq,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C)),B)),C) = aa_int_int(aa_int_fun_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_2475_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_2476_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(aa_int_fun_int_int(modulo_modulo_int,A),B)) = aa_int_int(aa_int_fun_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_2477_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_2478_mod__mult__mult2,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(aa_int_fun_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(aa_int_fun_int_int(modulo_modulo_int,A),B)),C) ).

% mod_mult_mult2
tff(fact_2479_mod__mult__cong,axiom,
    ! [A: nat,C: nat,A2: nat,B: nat,B2: 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,A2),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,B2),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,A2),B2)),C) ) ) ) ).

% mod_mult_cong
tff(fact_2480_mod__mult__cong,axiom,
    ! [A: int,C: int,A2: int,B: int,B2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),C) )
     => ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B2),C) )
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),B2)),C) ) ) ) ).

% mod_mult_cong
tff(fact_2481_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_2482_mod__mult__eq,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C))),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C) ).

% mod_mult_eq
tff(fact_2483_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_2484_mod__add__eq,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C))),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) ).

% mod_add_eq
tff(fact_2485_mod__add__cong,axiom,
    ! [A: nat,C: nat,A2: nat,B: nat,B2: 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,A2),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,B2),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,A2),B2)),C) ) ) ) ).

% mod_add_cong
tff(fact_2486_mod__add__cong,axiom,
    ! [A: int,C: int,A2: int,B: int,B2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),C) )
     => ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B2),C) )
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B2)),C) ) ) ) ).

% mod_add_cong
tff(fact_2487_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_2488_mod__add__left__eq,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C)),B)),C) = aa_int_int(aa_int_fun_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_2489_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_2490_mod__add__right__eq,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_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(modulo_modulo_int,B),C))),C) = aa_int_int(aa_int_fun_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_2491_VEBT__internal_OminNull_Oelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Xb)
     => ( ( Xb != vEBT_Leaf($false,$false) )
       => ~ ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : Xb != vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2,Uy2) ) ) ).

% VEBT_internal.minNull.elims(2)
tff(fact_2492_power__mod,axiom,
    ! [A: nat,B: nat,Nb: 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)),Nb)),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),B) ).

% power_mod
tff(fact_2493_power__mod,axiom,
    ! [A: int,B: int,Nb: nat] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_nat_int(power_power_int(aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),Nb)),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_nat_int(power_power_int(A),Nb)),B) ).

% power_mod
tff(fact_2494_VEBT__internal_Ovalid_H_Ocases,axiom,
    ! [Xb: produc9072475918466114483BT_nat] :
      ( ! [Uu2: $o,Uv2: $o,D3: nat] : Xb != produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(Uv2)),D3)
     => ~ ! [Mima: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Deg2: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(Mima,Deg,TreeList,Summary),Deg2) ) ).

% VEBT_internal.valid'.cases
tff(fact_2495_VEBT_Oexhaust,axiom,
    ! [Yb: vEBT_VEBT] :
      ( ! [X112: option4927543243414619207at_nat,X122: nat,X132: list_VEBT_VEBT,X142: vEBT_VEBT] : Yb != vEBT_Node(X112,X122,X132,X142)
     => ~ ! [X212: $o,X222: $o] : Yb != vEBT_Leaf((X212),(X222)) ) ).

% VEBT.exhaust
tff(fact_2496_VEBT_Odistinct_I1_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT,X21: $o,X22: $o] : vEBT_Node(X11,X12,X13,X14) != vEBT_Leaf((X21),(X22)) ).

% VEBT.distinct(1)
tff(fact_2497_mod__Suc__eq,axiom,
    ! [Mb: nat,Nb: 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,Mb),Nb))),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,Mb)),Nb) ).

% mod_Suc_eq
tff(fact_2498_mod__Suc__Suc__eq,axiom,
    ! [Mb: nat,Nb: 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,Mb),Nb)))),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,Mb))),Nb) ).

% mod_Suc_Suc_eq
tff(fact_2499_mod__less__eq__dividend,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb)),Mb) ).

% mod_less_eq_dividend
tff(fact_2500_combine__options__cases,axiom,
    ! [Xb: option4927543243414619207at_nat,P: fun_op1221328669798608686_nat_o,Yb: option4927543243414619207at_nat] :
      ( ( ( Xb = none_P5556105721700978146at_nat )
       => aa_opt7112393969400857165_nat_o(aa_opt9039718665754746463_nat_o(P,Xb),Yb) )
     => ( ( ( Yb = none_P5556105721700978146at_nat )
         => aa_opt7112393969400857165_nat_o(aa_opt9039718665754746463_nat_o(P,Xb),Yb) )
       => ( ! [A4: product_prod_nat_nat,B3: product_prod_nat_nat] :
              ( ( Xb = some_P7363390416028606310at_nat(A4) )
             => ( ( Yb = some_P7363390416028606310at_nat(B3) )
               => aa_opt7112393969400857165_nat_o(aa_opt9039718665754746463_nat_o(P,Xb),Yb) ) )
         => aa_opt7112393969400857165_nat_o(aa_opt9039718665754746463_nat_o(P,Xb),Yb) ) ) ) ).

% combine_options_cases
tff(fact_2501_combine__options__cases,axiom,
    ! [Xb: option4927543243414619207at_nat,P: fun_op7653269905477231939_num_o,Yb: option_num] :
      ( ( ( Xb = none_P5556105721700978146at_nat )
       => aa_option_num_o(aa_opt7868583491048818492_num_o(P,Xb),Yb) )
     => ( ( ( Yb = none_num )
         => aa_option_num_o(aa_opt7868583491048818492_num_o(P,Xb),Yb) )
       => ( ! [A4: product_prod_nat_nat,B3: num] :
              ( ( Xb = some_P7363390416028606310at_nat(A4) )
             => ( ( Yb = some_num(B3) )
               => aa_option_num_o(aa_opt7868583491048818492_num_o(P,Xb),Yb) ) )
         => aa_option_num_o(aa_opt7868583491048818492_num_o(P,Xb),Yb) ) ) ) ).

% combine_options_cases
tff(fact_2502_combine__options__cases,axiom,
    ! [Xb: option_num,P: fun_op5373868528577402299_nat_o,Yb: option4927543243414619207at_nat] :
      ( ( ( Xb = none_num )
       => aa_opt7112393969400857165_nat_o(aa_opt1662640216596507900_nat_o(P,Xb),Yb) )
     => ( ( ( Yb = none_P5556105721700978146at_nat )
         => aa_opt7112393969400857165_nat_o(aa_opt1662640216596507900_nat_o(P,Xb),Yb) )
       => ( ! [A4: num,B3: product_prod_nat_nat] :
              ( ( Xb = some_num(A4) )
             => ( ( Yb = some_P7363390416028606310at_nat(B3) )
               => aa_opt7112393969400857165_nat_o(aa_opt1662640216596507900_nat_o(P,Xb),Yb) ) )
         => aa_opt7112393969400857165_nat_o(aa_opt1662640216596507900_nat_o(P,Xb),Yb) ) ) ) ).

% combine_options_cases
tff(fact_2503_combine__options__cases,axiom,
    ! [Xb: option_num,P: fun_op493924057549976310_num_o,Yb: option_num] :
      ( ( ( Xb = none_num )
       => aa_option_num_o(aa_opt8674010985164783839_num_o(P,Xb),Yb) )
     => ( ( ( Yb = none_num )
         => aa_option_num_o(aa_opt8674010985164783839_num_o(P,Xb),Yb) )
       => ( ! [A4: num,B3: num] :
              ( ( Xb = some_num(A4) )
             => ( ( Yb = some_num(B3) )
               => aa_option_num_o(aa_opt8674010985164783839_num_o(P,Xb),Yb) ) )
         => aa_option_num_o(aa_opt8674010985164783839_num_o(P,Xb),Yb) ) ) ) ).

% combine_options_cases
tff(fact_2504_split__option__all,axiom,
    ! [P: fun_op8028897214545316308_nat_o] :
      ( ! [X_13: option4927543243414619207at_nat] : aa_opt7112393969400857165_nat_o(P,X_13)
    <=> ( 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_2505_split__option__all,axiom,
    ! [P: fun_option_num_o] :
      ( ! [X_13: option_num] : aa_option_num_o(P,X_13)
    <=> ( aa_option_num_o(P,none_num)
        & ! [X4: num] : aa_option_num_o(P,some_num(X4)) ) ) ).

% split_option_all
tff(fact_2506_split__option__ex,axiom,
    ! [P: fun_op8028897214545316308_nat_o] :
      ( ? [X_13: option4927543243414619207at_nat] : aa_opt7112393969400857165_nat_o(P,X_13)
    <=> ( 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_2507_split__option__ex,axiom,
    ! [P: fun_option_num_o] :
      ( ? [X_13: option_num] : aa_option_num_o(P,X_13)
    <=> ( aa_option_num_o(P,none_num)
        | ? [X4: num] : aa_option_num_o(P,some_num(X4)) ) ) ).

% split_option_ex
tff(fact_2508_option_Oexhaust,axiom,
    ! [Yb: option4927543243414619207at_nat] :
      ( ( Yb != none_P5556105721700978146at_nat )
     => ~ ! [X23: product_prod_nat_nat] : Yb != some_P7363390416028606310at_nat(X23) ) ).

% option.exhaust
tff(fact_2509_option_Oexhaust,axiom,
    ! [Yb: option_num] :
      ( ( Yb != none_num )
     => ~ ! [X23: num] : Yb != some_num(X23) ) ).

% option.exhaust
tff(fact_2510_option_OdiscI,axiom,
    ! [Option: option4927543243414619207at_nat,X2: product_prod_nat_nat] :
      ( ( Option = some_P7363390416028606310at_nat(X2) )
     => ( Option != none_P5556105721700978146at_nat ) ) ).

% option.discI
tff(fact_2511_option_OdiscI,axiom,
    ! [Option: option_num,X2: num] :
      ( ( Option = some_num(X2) )
     => ( Option != none_num ) ) ).

% option.discI
tff(fact_2512_option_Odistinct_I1_J,axiom,
    ! [X2: product_prod_nat_nat] : none_P5556105721700978146at_nat != some_P7363390416028606310at_nat(X2) ).

% option.distinct(1)
tff(fact_2513_option_Odistinct_I1_J,axiom,
    ! [X2: num] : none_num != some_num(X2) ).

% option.distinct(1)
tff(fact_2514_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_2515_VEBT__internal_OminNull_Osimps_I3_J,axiom,
    ! [Uu: $o] : ~ aa_VEBT_VEBT_o(vEBT_VEBT_minNull,vEBT_Leaf((Uu),$true)) ).

% VEBT_internal.minNull.simps(3)
tff(fact_2516_VEBT__internal_OminNull_Osimps_I2_J,axiom,
    ! [Uv: $o] : ~ aa_VEBT_VEBT_o(vEBT_VEBT_minNull,vEBT_Leaf($true,(Uv))) ).

% VEBT_internal.minNull.simps(2)
tff(fact_2517_VEBT__internal_OminNull_Osimps_I1_J,axiom,
    aa_VEBT_VEBT_o(vEBT_VEBT_minNull,vEBT_Leaf($false,$false)) ).

% VEBT_internal.minNull.simps(1)
tff(fact_2518_VEBT__internal_OminNull_Oelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Yb: $o] :
      ( ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Xb)
      <=> (Yb) )
     => ( ( ( Xb = vEBT_Leaf($false,$false) )
         => ~ (Yb) )
       => ( ( ? [Uv2: $o] : Xb = vEBT_Leaf($true,(Uv2))
           => (Yb) )
         => ( ( ? [Uu2: $o] : Xb = vEBT_Leaf((Uu2),$true)
             => (Yb) )
           => ( ( ? [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : Xb = vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2,Uy2)
               => ~ (Yb) )
             => ~ ( ? [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2,Vc2)
                 => (Yb) ) ) ) ) ) ) ).

% VEBT_internal.minNull.elims(1)
tff(fact_2519_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_2520_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(aa_int_fun_int_int(modulo_modulo_int,A),B)),A) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
tff(fact_2521_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_2522_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(aa_int_fun_int_int(modulo_modulo_int,A),B)),B) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
tff(fact_2523_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_2524_mod__eq__self__iff__div__eq__0,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_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_2525_cong__exp__iff__simps_I9_J,axiom,
    ! [Mb: num,Q2: num,Nb: num] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Mb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Nb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Q2)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Nb)),aa_num_nat(numeral_numeral_nat,Q2)) ) ) ).

% cong_exp_iff_simps(9)
tff(fact_2526_cong__exp__iff__simps_I9_J,axiom,
    ! [Mb: num,Q2: num,Nb: num] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Mb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) )
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,Q2)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,Nb)),aa_num_int(numeral_numeral_int,Q2)) ) ) ).

% cong_exp_iff_simps(9)
tff(fact_2527_cong__exp__iff__simps_I4_J,axiom,
    ! [Mb: num,Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,one)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Nb)),aa_num_nat(numeral_numeral_nat,one)) ).

% cong_exp_iff_simps(4)
tff(fact_2528_cong__exp__iff__simps_I4_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,one)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,Nb)),aa_num_int(numeral_numeral_int,one)) ).

% cong_exp_iff_simps(4)
tff(fact_2529_mod__eqE,axiom,
    ! [A: int,C: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C) = aa_int_int(aa_int_fun_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_2530_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_2531_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(aa_int_fun_int_int(modulo_modulo_int,A),C)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C))),C)) ).

% div_add1_eq
tff(fact_2532_mod__Suc,axiom,
    ! [Mb: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,Mb)),Nb) = $ite(aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb)) = Nb,zero_zero_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb))) ).

% mod_Suc
tff(fact_2533_mod__induct,axiom,
    ! [P: fun_nat_o,Nb: nat,P2: nat,Mb: nat] :
      ( aa_nat_o(P,Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),P2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),P2)
         => ( ! [N: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),P2)
               => ( 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)),P2)) ) )
           => aa_nat_o(P,Mb) ) ) ) ) ).

% mod_induct
tff(fact_2534_mod__less__divisor,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb)),Nb) ) ).

% mod_less_divisor
tff(fact_2535_mod__Suc__le__divisor,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_nat_nat(suc,Nb))),Nb) ).

% mod_Suc_le_divisor
tff(fact_2536_mod__eq__0D,axiom,
    ! [Mb: nat,D: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),D) = zero_zero_nat )
     => ? [Q3: nat] : Mb = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,D),Q3) ) ).

% mod_eq_0D
tff(fact_2537_nat__mod__eq__iff,axiom,
    ! [Xb: nat,Nb: nat,Yb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Yb),Nb) )
    <=> ? [Q1: nat,Q22: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Q1)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Yb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Q22)) ) ).

% nat_mod_eq_iff
tff(fact_2538_vebt__buildup_Osimps_I1_J,axiom,
    vEBT_vebt_buildup(zero_zero_nat) = vEBT_Leaf($false,$false) ).

% vebt_buildup.simps(1)
tff(fact_2539_dbl__def,axiom,
    ! [Xb: real] : aa_real_real(neg_numeral_dbl_real,Xb) = aa_real_real(plus_plus_real(Xb),Xb) ).

% dbl_def
tff(fact_2540_dbl__def,axiom,
    ! [Xb: rat] : aa_rat_rat(neg_numeral_dbl_rat,Xb) = aa_rat_rat(plus_plus_rat(Xb),Xb) ).

% dbl_def
tff(fact_2541_dbl__def,axiom,
    ! [Xb: int] : aa_int_int(neg_numeral_dbl_int,Xb) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Xb) ).

% dbl_def
tff(fact_2542_vebt__member_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,Xb: nat] : ~ aa_nat_o(vEBT_vebt_member(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw)),Xb) ).

% vebt_member.simps(2)
tff(fact_2543_vebt__member_Ocases,axiom,
    ! [Xb: produc9072475918466114483BT_nat] :
      ( ! [A4: $o,B3: $o,X3: nat] : Xb != produc738532404422230701BT_nat(vEBT_Leaf((A4),(B3)),X3)
     => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,X3: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2),X3)
       => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,X3: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2),X3)
         => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT,X3: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2),X3)
           => ~ ! [Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X3: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary),X3) ) ) ) ) ).

% vebt_member.cases
tff(fact_2544_VEBT__internal_Omembermima_Ocases,axiom,
    ! [Xb: produc9072475918466114483BT_nat] :
      ( ! [Uu2: $o,Uv2: $o,Uw2: nat] : Xb != produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(Uv2)),Uw2)
     => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT,Uz2: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2),Uz2)
       => ( ! [Mi: nat,Ma: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT,X3: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,Va3,Vb2),X3)
         => ( ! [Mi: nat,Ma: nat,V2: nat,TreeList: list_VEBT_VEBT,Vc2: vEBT_VEBT,X3: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,V2),TreeList,Vc2),X3)
           => ~ ! [V2: nat,TreeList: list_VEBT_VEBT,Vd: vEBT_VEBT,X3: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList,Vd),X3) ) ) ) ) ).

% VEBT_internal.membermima.cases
tff(fact_2545_vebt__insert_Ocases,axiom,
    ! [Xb: produc9072475918466114483BT_nat] :
      ( ! [A4: $o,B3: $o,X3: nat] : Xb != produc738532404422230701BT_nat(vEBT_Leaf((A4),(B3)),X3)
     => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT,X3: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(Info2,zero_zero_nat,Ts2,S),X3)
       => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT,X3: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2,S),X3)
         => ( ! [V2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X3: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList,Summary),X3)
           => ~ ! [Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X3: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary),X3) ) ) ) ) ).

% vebt_insert.cases
tff(fact_2546_VEBT__internal_OminNull_Osimps_I4_J,axiom,
    ! [Uw: nat,Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] : aa_VEBT_VEBT_o(vEBT_VEBT_minNull,vEBT_Node(none_P5556105721700978146at_nat,Uw,Ux,Uy)) ).

% VEBT_internal.minNull.simps(4)
tff(fact_2547_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_2548_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(aa_int_fun_int_int(modulo_modulo_int,A),B) = A ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
tff(fact_2549_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_2550_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(aa_int_fun_int_int(modulo_modulo_int,A),B)) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
tff(fact_2551_cong__exp__iff__simps_I2_J,axiom,
    ! [Nb: num,Q2: num] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Nb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) = zero_zero_nat )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Nb)),aa_num_nat(numeral_numeral_nat,Q2)) = zero_zero_nat ) ) ).

% cong_exp_iff_simps(2)
tff(fact_2552_cong__exp__iff__simps_I2_J,axiom,
    ! [Nb: num,Q2: num] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) = zero_zero_int )
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,Nb)),aa_num_int(numeral_numeral_int,Q2)) = zero_zero_int ) ) ).

% cong_exp_iff_simps(2)
tff(fact_2553_cong__exp__iff__simps_I1_J,axiom,
    ! [Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Nb)),aa_num_nat(numeral_numeral_nat,one)) = zero_zero_nat ).

% cong_exp_iff_simps(1)
tff(fact_2554_cong__exp__iff__simps_I1_J,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,Nb)),aa_num_int(numeral_numeral_int,one)) = zero_zero_int ).

% cong_exp_iff_simps(1)
tff(fact_2555_VEBT__internal_Onaive__member_Ocases,axiom,
    ! [Xb: produc9072475918466114483BT_nat] :
      ( ! [A4: $o,B3: $o,X3: nat] : Xb != produc738532404422230701BT_nat(vEBT_Leaf((A4),(B3)),X3)
     => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,Ux2: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2),Ux2)
       => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList: list_VEBT_VEBT,S: vEBT_VEBT,X3: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList,S),X3) ) ) ).

% VEBT_internal.naive_member.cases
tff(fact_2556_cong__exp__iff__simps_I8_J,axiom,
    ! [Mb: num,Q2: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Mb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) != aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,one)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) ).

% cong_exp_iff_simps(8)
tff(fact_2557_cong__exp__iff__simps_I8_J,axiom,
    ! [Mb: num,Q2: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Mb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) != aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,one)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) ).

% cong_exp_iff_simps(8)
tff(fact_2558_cong__exp__iff__simps_I6_J,axiom,
    ! [Q2: num,Nb: 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,Q2))) != aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Nb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) ).

% cong_exp_iff_simps(6)
tff(fact_2559_cong__exp__iff__simps_I6_J,axiom,
    ! [Q2: num,Nb: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,one)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) != aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) ).

% cong_exp_iff_simps(6)
tff(fact_2560_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_2561_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(aa_int_fun_int_int(modulo_modulo_int,B),C))),C)) ).

% div_mult1_eq
tff(fact_2562_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_2563_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(aa_int_fun_int_int(modulo_modulo_int,A),B)) = A ).

% mult_div_mod_eq
tff(fact_2564_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_2565_mod__mult__div__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(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_2566_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_2567_mod__div__mult__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(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_2568_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_2569_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(aa_int_fun_int_int(modulo_modulo_int,A),B)) = A ).

% div_mult_mod_eq
tff(fact_2570_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_2571_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(aa_int_fun_int_int(modulo_modulo_int,A),B)) ).

% mod_div_decomp
tff(fact_2572_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_2573_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(aa_int_fun_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_2574_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_2575_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(aa_int_fun_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_2576_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_2577_mod__le__divisor,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb)),Nb) ) ).

% mod_le_divisor
tff(fact_2578_vebt__member_Osimps_I1_J,axiom,
    ! [A: $o,B: $o,Xb: nat] :
      ( aa_nat_o(vEBT_vebt_member(vEBT_Leaf((A),(B))),Xb)
    <=> $ite(
          Xb = zero_zero_nat,
          (A),
          $ite(Xb = one_one_nat,(B),$false) ) ) ).

% vebt_member.simps(1)
tff(fact_2579_mod__eq__nat1E,axiom,
    ! [Mb: nat,Q2: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Q2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),Q2) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => ~ ! [S: nat] : Mb != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Q2),S)) ) ) ).

% mod_eq_nat1E
tff(fact_2580_mod__eq__nat2E,axiom,
    ! [Mb: nat,Q2: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Q2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),Q2) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
       => ~ ! [S: nat] : Nb != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Q2),S)) ) ) ).

% mod_eq_nat2E
tff(fact_2581_nat__mod__eq__lemma,axiom,
    ! [Xb: nat,Nb: nat,Yb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Yb),Nb) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Yb),Xb)
       => ? [Q3: nat] : Xb = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Yb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Q3)) ) ) ).

% nat_mod_eq_lemma
tff(fact_2582_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_2583_vebt__insert_Osimps_I1_J,axiom,
    ! [A: $o,B: $o,Xb: nat] :
      vEBT_vebt_insert(vEBT_Leaf((A),(B)),Xb) = $ite(
        Xb = zero_zero_nat,
        vEBT_Leaf($true,(B)),
        $ite(Xb = one_one_nat,vEBT_Leaf((A),$true),vEBT_Leaf((A),(B))) ) ).

% vebt_insert.simps(1)
tff(fact_2584_mod__mult2__eq,axiom,
    ! [Mb: nat,Nb: nat,Q2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Q2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)),Q2))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb)) ).

% mod_mult2_eq
tff(fact_2585_VEBT__internal_Onaive__member_Osimps_I1_J,axiom,
    ! [A: $o,B: $o,Xb: nat] :
      ( aa_nat_o(vEBT_V5719532721284313246member(vEBT_Leaf((A),(B))),Xb)
    <=> $ite(
          Xb = zero_zero_nat,
          (A),
          $ite(Xb = one_one_nat,(B),$false) ) ) ).

% VEBT_internal.naive_member.simps(1)
tff(fact_2586_minf_I7_J,axiom,
    ! [Ta: real] :
    ? [Z2: real] :
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Z2)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Ta),X) ) ).

% minf(7)
tff(fact_2587_minf_I7_J,axiom,
    ! [Ta: rat] :
    ? [Z2: rat] :
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Z2)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ta),X) ) ).

% minf(7)
tff(fact_2588_minf_I7_J,axiom,
    ! [Ta: num] :
    ? [Z2: num] :
    ! [X: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Z2)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Ta),X) ) ).

% minf(7)
tff(fact_2589_minf_I7_J,axiom,
    ! [Ta: nat] :
    ? [Z2: nat] :
    ! [X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Z2)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ta),X) ) ).

% minf(7)
tff(fact_2590_minf_I7_J,axiom,
    ! [Ta: int] :
    ? [Z2: int] :
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Z2)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Ta),X) ) ).

% minf(7)
tff(fact_2591_minf_I5_J,axiom,
    ! [Ta: real] :
    ? [Z2: real] :
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Z2)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,X),Ta) ) ).

% minf(5)
tff(fact_2592_minf_I5_J,axiom,
    ! [Ta: rat] :
    ? [Z2: rat] :
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Z2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Ta) ) ).

% minf(5)
tff(fact_2593_minf_I5_J,axiom,
    ! [Ta: num] :
    ? [Z2: num] :
    ! [X: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Z2)
     => aa_num_o(aa_num_fun_num_o(ord_less_num,X),Ta) ) ).

% minf(5)
tff(fact_2594_minf_I5_J,axiom,
    ! [Ta: nat] :
    ? [Z2: nat] :
    ! [X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Z2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Ta) ) ).

% minf(5)
tff(fact_2595_minf_I5_J,axiom,
    ! [Ta: int] :
    ? [Z2: int] :
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Z2)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,X),Ta) ) ).

% minf(5)
tff(fact_2596_minf_I4_J,axiom,
    ! [Ta: real] :
    ? [Z2: real] :
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Z2)
     => ( X != Ta ) ) ).

% minf(4)
tff(fact_2597_minf_I4_J,axiom,
    ! [Ta: rat] :
    ? [Z2: rat] :
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Z2)
     => ( X != Ta ) ) ).

% minf(4)
tff(fact_2598_minf_I4_J,axiom,
    ! [Ta: num] :
    ? [Z2: num] :
    ! [X: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Z2)
     => ( X != Ta ) ) ).

% minf(4)
tff(fact_2599_minf_I4_J,axiom,
    ! [Ta: nat] :
    ? [Z2: nat] :
    ! [X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Z2)
     => ( X != Ta ) ) ).

% minf(4)
tff(fact_2600_minf_I4_J,axiom,
    ! [Ta: int] :
    ? [Z2: int] :
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Z2)
     => ( X != Ta ) ) ).

% minf(4)
tff(fact_2601_minf_I3_J,axiom,
    ! [Ta: real] :
    ? [Z2: real] :
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Z2)
     => ( X != Ta ) ) ).

% minf(3)
tff(fact_2602_minf_I3_J,axiom,
    ! [Ta: rat] :
    ? [Z2: rat] :
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Z2)
     => ( X != Ta ) ) ).

% minf(3)
tff(fact_2603_minf_I3_J,axiom,
    ! [Ta: num] :
    ? [Z2: num] :
    ! [X: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Z2)
     => ( X != Ta ) ) ).

% minf(3)
tff(fact_2604_minf_I3_J,axiom,
    ! [Ta: nat] :
    ? [Z2: nat] :
    ! [X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Z2)
     => ( X != Ta ) ) ).

% minf(3)
tff(fact_2605_minf_I3_J,axiom,
    ! [Ta: int] :
    ? [Z2: int] :
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Z2)
     => ( X != Ta ) ) ).

% minf(3)
tff(fact_2606_minf_I2_J,axiom,
    ! [P: fun_real_o,P3: fun_real_o,Q: fun_real_o,Q6: fun_real_o] :
      ( ? [Z3: real] :
        ! [X3: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Z3)
         => ( aa_real_o(P,X3)
          <=> aa_real_o(P3,X3) ) )
     => ( ? [Z3: real] :
          ! [X3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Z3)
           => ( aa_real_o(Q,X3)
            <=> aa_real_o(Q6,X3) ) )
       => ? [Z2: real] :
          ! [X: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Z2)
           => ( ( aa_real_o(P,X)
                | aa_real_o(Q,X) )
            <=> ( aa_real_o(P3,X)
                | aa_real_o(Q6,X) ) ) ) ) ) ).

% minf(2)
tff(fact_2607_minf_I2_J,axiom,
    ! [P: fun_rat_o,P3: fun_rat_o,Q: fun_rat_o,Q6: fun_rat_o] :
      ( ? [Z3: rat] :
        ! [X3: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Z3)
         => ( aa_rat_o(P,X3)
          <=> aa_rat_o(P3,X3) ) )
     => ( ? [Z3: rat] :
          ! [X3: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Z3)
           => ( aa_rat_o(Q,X3)
            <=> aa_rat_o(Q6,X3) ) )
       => ? [Z2: rat] :
          ! [X: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Z2)
           => ( ( aa_rat_o(P,X)
                | aa_rat_o(Q,X) )
            <=> ( aa_rat_o(P3,X)
                | aa_rat_o(Q6,X) ) ) ) ) ) ).

% minf(2)
tff(fact_2608_minf_I2_J,axiom,
    ! [P: fun_num_o,P3: fun_num_o,Q: fun_num_o,Q6: fun_num_o] :
      ( ? [Z3: num] :
        ! [X3: num] :
          ( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Z3)
         => ( aa_num_o(P,X3)
          <=> aa_num_o(P3,X3) ) )
     => ( ? [Z3: num] :
          ! [X3: num] :
            ( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Z3)
           => ( aa_num_o(Q,X3)
            <=> aa_num_o(Q6,X3) ) )
       => ? [Z2: num] :
          ! [X: num] :
            ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Z2)
           => ( ( aa_num_o(P,X)
                | aa_num_o(Q,X) )
            <=> ( aa_num_o(P3,X)
                | aa_num_o(Q6,X) ) ) ) ) ) ).

% minf(2)
tff(fact_2609_minf_I2_J,axiom,
    ! [P: fun_nat_o,P3: fun_nat_o,Q: fun_nat_o,Q6: fun_nat_o] :
      ( ? [Z3: nat] :
        ! [X3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Z3)
         => ( aa_nat_o(P,X3)
          <=> aa_nat_o(P3,X3) ) )
     => ( ? [Z3: nat] :
          ! [X3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Z3)
           => ( aa_nat_o(Q,X3)
            <=> aa_nat_o(Q6,X3) ) )
       => ? [Z2: nat] :
          ! [X: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Z2)
           => ( ( aa_nat_o(P,X)
                | aa_nat_o(Q,X) )
            <=> ( aa_nat_o(P3,X)
                | aa_nat_o(Q6,X) ) ) ) ) ) ).

% minf(2)
tff(fact_2610_minf_I2_J,axiom,
    ! [P: fun_int_o,P3: fun_int_o,Q: fun_int_o,Q6: fun_int_o] :
      ( ? [Z3: int] :
        ! [X3: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Z3)
         => ( aa_int_o(P,X3)
          <=> aa_int_o(P3,X3) ) )
     => ( ? [Z3: int] :
          ! [X3: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Z3)
           => ( aa_int_o(Q,X3)
            <=> aa_int_o(Q6,X3) ) )
       => ? [Z2: int] :
          ! [X: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Z2)
           => ( ( aa_int_o(P,X)
                | aa_int_o(Q,X) )
            <=> ( aa_int_o(P3,X)
                | aa_int_o(Q6,X) ) ) ) ) ) ).

% minf(2)
tff(fact_2611_minf_I1_J,axiom,
    ! [P: fun_real_o,P3: fun_real_o,Q: fun_real_o,Q6: fun_real_o] :
      ( ? [Z3: real] :
        ! [X3: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Z3)
         => ( aa_real_o(P,X3)
          <=> aa_real_o(P3,X3) ) )
     => ( ? [Z3: real] :
          ! [X3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Z3)
           => ( aa_real_o(Q,X3)
            <=> aa_real_o(Q6,X3) ) )
       => ? [Z2: real] :
          ! [X: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Z2)
           => ( ( aa_real_o(P,X)
                & aa_real_o(Q,X) )
            <=> ( aa_real_o(P3,X)
                & aa_real_o(Q6,X) ) ) ) ) ) ).

% minf(1)
tff(fact_2612_minf_I1_J,axiom,
    ! [P: fun_rat_o,P3: fun_rat_o,Q: fun_rat_o,Q6: fun_rat_o] :
      ( ? [Z3: rat] :
        ! [X3: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Z3)
         => ( aa_rat_o(P,X3)
          <=> aa_rat_o(P3,X3) ) )
     => ( ? [Z3: rat] :
          ! [X3: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Z3)
           => ( aa_rat_o(Q,X3)
            <=> aa_rat_o(Q6,X3) ) )
       => ? [Z2: rat] :
          ! [X: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Z2)
           => ( ( aa_rat_o(P,X)
                & aa_rat_o(Q,X) )
            <=> ( aa_rat_o(P3,X)
                & aa_rat_o(Q6,X) ) ) ) ) ) ).

% minf(1)
tff(fact_2613_minf_I1_J,axiom,
    ! [P: fun_num_o,P3: fun_num_o,Q: fun_num_o,Q6: fun_num_o] :
      ( ? [Z3: num] :
        ! [X3: num] :
          ( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Z3)
         => ( aa_num_o(P,X3)
          <=> aa_num_o(P3,X3) ) )
     => ( ? [Z3: num] :
          ! [X3: num] :
            ( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Z3)
           => ( aa_num_o(Q,X3)
            <=> aa_num_o(Q6,X3) ) )
       => ? [Z2: num] :
          ! [X: num] :
            ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Z2)
           => ( ( aa_num_o(P,X)
                & aa_num_o(Q,X) )
            <=> ( aa_num_o(P3,X)
                & aa_num_o(Q6,X) ) ) ) ) ) ).

% minf(1)
tff(fact_2614_minf_I1_J,axiom,
    ! [P: fun_nat_o,P3: fun_nat_o,Q: fun_nat_o,Q6: fun_nat_o] :
      ( ? [Z3: nat] :
        ! [X3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Z3)
         => ( aa_nat_o(P,X3)
          <=> aa_nat_o(P3,X3) ) )
     => ( ? [Z3: nat] :
          ! [X3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Z3)
           => ( aa_nat_o(Q,X3)
            <=> aa_nat_o(Q6,X3) ) )
       => ? [Z2: nat] :
          ! [X: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Z2)
           => ( ( aa_nat_o(P,X)
                & aa_nat_o(Q,X) )
            <=> ( aa_nat_o(P3,X)
                & aa_nat_o(Q6,X) ) ) ) ) ) ).

% minf(1)
tff(fact_2615_minf_I1_J,axiom,
    ! [P: fun_int_o,P3: fun_int_o,Q: fun_int_o,Q6: fun_int_o] :
      ( ? [Z3: int] :
        ! [X3: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Z3)
         => ( aa_int_o(P,X3)
          <=> aa_int_o(P3,X3) ) )
     => ( ? [Z3: int] :
          ! [X3: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Z3)
           => ( aa_int_o(Q,X3)
            <=> aa_int_o(Q6,X3) ) )
       => ? [Z2: int] :
          ! [X: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Z2)
           => ( ( aa_int_o(P,X)
                & aa_int_o(Q,X) )
            <=> ( aa_int_o(P3,X)
                & aa_int_o(Q6,X) ) ) ) ) ) ).

% minf(1)
tff(fact_2616_pinf_I7_J,axiom,
    ! [Ta: real] :
    ? [Z2: real] :
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),X)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,Ta),X) ) ).

% pinf(7)
tff(fact_2617_pinf_I7_J,axiom,
    ! [Ta: rat] :
    ? [Z2: rat] :
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),X)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ta),X) ) ).

% pinf(7)
tff(fact_2618_pinf_I7_J,axiom,
    ! [Ta: num] :
    ? [Z2: num] :
    ! [X: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z2),X)
     => aa_num_o(aa_num_fun_num_o(ord_less_num,Ta),X) ) ).

% pinf(7)
tff(fact_2619_pinf_I7_J,axiom,
    ! [Ta: nat] :
    ? [Z2: nat] :
    ! [X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z2),X)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ta),X) ) ).

% pinf(7)
tff(fact_2620_pinf_I7_J,axiom,
    ! [Ta: int] :
    ? [Z2: int] :
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),X)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,Ta),X) ) ).

% pinf(7)
tff(fact_2621_pinf_I5_J,axiom,
    ! [Ta: real] :
    ? [Z2: real] :
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),X)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,X),Ta) ) ).

% pinf(5)
tff(fact_2622_pinf_I5_J,axiom,
    ! [Ta: rat] :
    ? [Z2: rat] :
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),X)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Ta) ) ).

% pinf(5)
tff(fact_2623_pinf_I5_J,axiom,
    ! [Ta: num] :
    ? [Z2: num] :
    ! [X: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z2),X)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,X),Ta) ) ).

% pinf(5)
tff(fact_2624_pinf_I5_J,axiom,
    ! [Ta: nat] :
    ? [Z2: nat] :
    ! [X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z2),X)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Ta) ) ).

% pinf(5)
tff(fact_2625_pinf_I5_J,axiom,
    ! [Ta: int] :
    ? [Z2: int] :
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),X)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,X),Ta) ) ).

% pinf(5)
tff(fact_2626_pinf_I4_J,axiom,
    ! [Ta: real] :
    ? [Z2: real] :
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),X)
     => ( X != Ta ) ) ).

% pinf(4)
tff(fact_2627_pinf_I4_J,axiom,
    ! [Ta: rat] :
    ? [Z2: rat] :
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),X)
     => ( X != Ta ) ) ).

% pinf(4)
tff(fact_2628_pinf_I4_J,axiom,
    ! [Ta: num] :
    ? [Z2: num] :
    ! [X: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z2),X)
     => ( X != Ta ) ) ).

% pinf(4)
tff(fact_2629_pinf_I4_J,axiom,
    ! [Ta: nat] :
    ? [Z2: nat] :
    ! [X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z2),X)
     => ( X != Ta ) ) ).

% pinf(4)
tff(fact_2630_pinf_I4_J,axiom,
    ! [Ta: int] :
    ? [Z2: int] :
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),X)
     => ( X != Ta ) ) ).

% pinf(4)
tff(fact_2631_pinf_I3_J,axiom,
    ! [Ta: real] :
    ? [Z2: real] :
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),X)
     => ( X != Ta ) ) ).

% pinf(3)
tff(fact_2632_pinf_I3_J,axiom,
    ! [Ta: rat] :
    ? [Z2: rat] :
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),X)
     => ( X != Ta ) ) ).

% pinf(3)
tff(fact_2633_pinf_I3_J,axiom,
    ! [Ta: num] :
    ? [Z2: num] :
    ! [X: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z2),X)
     => ( X != Ta ) ) ).

% pinf(3)
tff(fact_2634_pinf_I3_J,axiom,
    ! [Ta: nat] :
    ? [Z2: nat] :
    ! [X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z2),X)
     => ( X != Ta ) ) ).

% pinf(3)
tff(fact_2635_pinf_I3_J,axiom,
    ! [Ta: int] :
    ? [Z2: int] :
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),X)
     => ( X != Ta ) ) ).

% pinf(3)
tff(fact_2636_pinf_I2_J,axiom,
    ! [P: fun_real_o,P3: fun_real_o,Q: fun_real_o,Q6: fun_real_o] :
      ( ? [Z3: real] :
        ! [X3: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),X3)
         => ( aa_real_o(P,X3)
          <=> aa_real_o(P3,X3) ) )
     => ( ? [Z3: real] :
          ! [X3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),X3)
           => ( aa_real_o(Q,X3)
            <=> aa_real_o(Q6,X3) ) )
       => ? [Z2: real] :
          ! [X: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),X)
           => ( ( aa_real_o(P,X)
                | aa_real_o(Q,X) )
            <=> ( aa_real_o(P3,X)
                | aa_real_o(Q6,X) ) ) ) ) ) ).

% pinf(2)
tff(fact_2637_pinf_I2_J,axiom,
    ! [P: fun_rat_o,P3: fun_rat_o,Q: fun_rat_o,Q6: fun_rat_o] :
      ( ? [Z3: rat] :
        ! [X3: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z3),X3)
         => ( aa_rat_o(P,X3)
          <=> aa_rat_o(P3,X3) ) )
     => ( ? [Z3: rat] :
          ! [X3: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z3),X3)
           => ( aa_rat_o(Q,X3)
            <=> aa_rat_o(Q6,X3) ) )
       => ? [Z2: rat] :
          ! [X: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),X)
           => ( ( aa_rat_o(P,X)
                | aa_rat_o(Q,X) )
            <=> ( aa_rat_o(P3,X)
                | aa_rat_o(Q6,X) ) ) ) ) ) ).

% pinf(2)
tff(fact_2638_pinf_I2_J,axiom,
    ! [P: fun_num_o,P3: fun_num_o,Q: fun_num_o,Q6: fun_num_o] :
      ( ? [Z3: num] :
        ! [X3: num] :
          ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z3),X3)
         => ( aa_num_o(P,X3)
          <=> aa_num_o(P3,X3) ) )
     => ( ? [Z3: num] :
          ! [X3: num] :
            ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z3),X3)
           => ( aa_num_o(Q,X3)
            <=> aa_num_o(Q6,X3) ) )
       => ? [Z2: num] :
          ! [X: num] :
            ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z2),X)
           => ( ( aa_num_o(P,X)
                | aa_num_o(Q,X) )
            <=> ( aa_num_o(P3,X)
                | aa_num_o(Q6,X) ) ) ) ) ) ).

% pinf(2)
tff(fact_2639_pinf_I2_J,axiom,
    ! [P: fun_nat_o,P3: fun_nat_o,Q: fun_nat_o,Q6: fun_nat_o] :
      ( ? [Z3: nat] :
        ! [X3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z3),X3)
         => ( aa_nat_o(P,X3)
          <=> aa_nat_o(P3,X3) ) )
     => ( ? [Z3: nat] :
          ! [X3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z3),X3)
           => ( aa_nat_o(Q,X3)
            <=> aa_nat_o(Q6,X3) ) )
       => ? [Z2: nat] :
          ! [X: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z2),X)
           => ( ( aa_nat_o(P,X)
                | aa_nat_o(Q,X) )
            <=> ( aa_nat_o(P3,X)
                | aa_nat_o(Q6,X) ) ) ) ) ) ).

% pinf(2)
tff(fact_2640_pinf_I2_J,axiom,
    ! [P: fun_int_o,P3: fun_int_o,Q: fun_int_o,Q6: fun_int_o] :
      ( ? [Z3: int] :
        ! [X3: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),X3)
         => ( aa_int_o(P,X3)
          <=> aa_int_o(P3,X3) ) )
     => ( ? [Z3: int] :
          ! [X3: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),X3)
           => ( aa_int_o(Q,X3)
            <=> aa_int_o(Q6,X3) ) )
       => ? [Z2: int] :
          ! [X: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),X)
           => ( ( aa_int_o(P,X)
                | aa_int_o(Q,X) )
            <=> ( aa_int_o(P3,X)
                | aa_int_o(Q6,X) ) ) ) ) ) ).

% pinf(2)
tff(fact_2641_pinf_I1_J,axiom,
    ! [P: fun_real_o,P3: fun_real_o,Q: fun_real_o,Q6: fun_real_o] :
      ( ? [Z3: real] :
        ! [X3: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),X3)
         => ( aa_real_o(P,X3)
          <=> aa_real_o(P3,X3) ) )
     => ( ? [Z3: real] :
          ! [X3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),X3)
           => ( aa_real_o(Q,X3)
            <=> aa_real_o(Q6,X3) ) )
       => ? [Z2: real] :
          ! [X: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),X)
           => ( ( aa_real_o(P,X)
                & aa_real_o(Q,X) )
            <=> ( aa_real_o(P3,X)
                & aa_real_o(Q6,X) ) ) ) ) ) ).

% pinf(1)
tff(fact_2642_pinf_I1_J,axiom,
    ! [P: fun_rat_o,P3: fun_rat_o,Q: fun_rat_o,Q6: fun_rat_o] :
      ( ? [Z3: rat] :
        ! [X3: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z3),X3)
         => ( aa_rat_o(P,X3)
          <=> aa_rat_o(P3,X3) ) )
     => ( ? [Z3: rat] :
          ! [X3: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z3),X3)
           => ( aa_rat_o(Q,X3)
            <=> aa_rat_o(Q6,X3) ) )
       => ? [Z2: rat] :
          ! [X: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),X)
           => ( ( aa_rat_o(P,X)
                & aa_rat_o(Q,X) )
            <=> ( aa_rat_o(P3,X)
                & aa_rat_o(Q6,X) ) ) ) ) ) ).

% pinf(1)
tff(fact_2643_pinf_I1_J,axiom,
    ! [P: fun_num_o,P3: fun_num_o,Q: fun_num_o,Q6: fun_num_o] :
      ( ? [Z3: num] :
        ! [X3: num] :
          ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z3),X3)
         => ( aa_num_o(P,X3)
          <=> aa_num_o(P3,X3) ) )
     => ( ? [Z3: num] :
          ! [X3: num] :
            ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z3),X3)
           => ( aa_num_o(Q,X3)
            <=> aa_num_o(Q6,X3) ) )
       => ? [Z2: num] :
          ! [X: num] :
            ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z2),X)
           => ( ( aa_num_o(P,X)
                & aa_num_o(Q,X) )
            <=> ( aa_num_o(P3,X)
                & aa_num_o(Q6,X) ) ) ) ) ) ).

% pinf(1)
tff(fact_2644_pinf_I1_J,axiom,
    ! [P: fun_nat_o,P3: fun_nat_o,Q: fun_nat_o,Q6: fun_nat_o] :
      ( ? [Z3: nat] :
        ! [X3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z3),X3)
         => ( aa_nat_o(P,X3)
          <=> aa_nat_o(P3,X3) ) )
     => ( ? [Z3: nat] :
          ! [X3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z3),X3)
           => ( aa_nat_o(Q,X3)
            <=> aa_nat_o(Q6,X3) ) )
       => ? [Z2: nat] :
          ! [X: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z2),X)
           => ( ( aa_nat_o(P,X)
                & aa_nat_o(Q,X) )
            <=> ( aa_nat_o(P3,X)
                & aa_nat_o(Q6,X) ) ) ) ) ) ).

% pinf(1)
tff(fact_2645_pinf_I1_J,axiom,
    ! [P: fun_int_o,P3: fun_int_o,Q: fun_int_o,Q6: fun_int_o] :
      ( ? [Z3: int] :
        ! [X3: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),X3)
         => ( aa_int_o(P,X3)
          <=> aa_int_o(P3,X3) ) )
     => ( ? [Z3: int] :
          ! [X3: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),X3)
           => ( aa_int_o(Q,X3)
            <=> aa_int_o(Q6,X3) ) )
       => ? [Z2: int] :
          ! [X: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),X)
           => ( ( aa_int_o(P,X)
                & aa_int_o(Q,X) )
            <=> ( aa_int_o(P3,X)
                & aa_int_o(Q6,X) ) ) ) ) ) ).

% pinf(1)
tff(fact_2646_VEBT__internal_OminNull_Oelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT] :
      ( ~ aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Xb)
     => ( ! [Uv2: $o] : Xb != vEBT_Leaf($true,(Uv2))
       => ( ! [Uu2: $o] : Xb != vEBT_Leaf((Uu2),$true)
         => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : Xb != vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2,Vc2) ) ) ) ).

% VEBT_internal.minNull.elims(3)
tff(fact_2647_bounded__Max__nat,axiom,
    ! [P: fun_nat_o,Xb: nat,M6: nat] :
      ( aa_nat_o(P,Xb)
     => ( ! [X3: nat] :
            ( aa_nat_o(P,X3)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),M6) )
       => ~ ! [M2: nat] :
              ( aa_nat_o(P,M2)
             => ~ ! [X: nat] :
                    ( aa_nat_o(P,X)
                   => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),M2) ) ) ) ) ).

% bounded_Max_nat
tff(fact_2648_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_2649_fold__atLeastAtMost__nat_Ocases,axiom,
    ! [Xb: produc8114805097830739791at_num] :
      ~ ! [F2: fun_nat_fun_num_num,A4: nat,B3: nat,Acc: num] : Xb != produc460779904311030791at_num(F2,produc1195630363706982562at_num(A4,product_Pair_nat_num(B3,Acc))) ).

% fold_atLeastAtMost_nat.cases
tff(fact_2650_fold__atLeastAtMost__nat_Ocases,axiom,
    ! [Xb: produc9217583074051984497at_nat] :
      ~ ! [F2: fun_nat_fun_nat_nat,A4: nat,B3: nat,Acc: nat] : Xb != produc2818902965508115497at_nat(F2,produc487386426758144856at_nat(A4,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B3),Acc))) ).

% fold_atLeastAtMost_nat.cases
tff(fact_2651_option_Osize_I3_J,axiom,
    size_s170228958280169651at_nat(none_P5556105721700978146at_nat) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(3)
tff(fact_2652_option_Osize_I3_J,axiom,
    size_size_option_num(none_num) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(3)
tff(fact_2653_split__mod,axiom,
    ! [P: fun_nat_o,Mb: nat,Nb: nat] :
      ( aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb))
    <=> ( ( ( Nb = zero_zero_nat )
         => aa_nat_o(P,Mb) )
        & ( ( Nb != zero_zero_nat )
         => ! [I5: nat,J3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),Nb)
             => ( ( Mb = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),I5)),J3) )
               => aa_nat_o(P,J3) ) ) ) ) ) ).

% split_mod
tff(fact_2654_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_2655_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(aa_int_fun_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(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_2656_Suc__times__mod__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Mb)
     => ( 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,Mb),Nb))),Mb) = one_one_nat ) ) ).

% Suc_times_mod_eq
tff(fact_2657_vebt__insert_Osimps_I4_J,axiom,
    ! [V: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xb: nat] : vEBT_vebt_insert(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V)),TreeLista,Summarya),Xb) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xb),Xb)),aa_nat_nat(suc,aa_nat_nat(suc,V)),TreeLista,Summarya) ).

% vebt_insert.simps(4)
tff(fact_2658_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_2659_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(aa_int_fun_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(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(modulo_modulo_int,A),B) ) ) ) ).

% divmod_digit_0(2)
tff(fact_2660_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_2661_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(aa_int_fun_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_2662_div__exp__mod__exp__eq,axiom,
    ! [A: nat,Nb: nat,Mb: 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))),Nb))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)) = 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,Nb),Mb)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% div_exp_mod_exp_eq
tff(fact_2663_div__exp__mod__exp__eq,axiom,
    ! [A: int,Nb: nat,Mb: nat] : aa_int_int(aa_int_fun_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))),Nb))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Mb)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_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,Nb),Mb)))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ).

% div_exp_mod_exp_eq
tff(fact_2664_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_2665_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(aa_int_fun_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_2666_minf_I8_J,axiom,
    ! [Ta: real] :
    ? [Z2: real] :
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Z2)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ta),X) ) ).

% minf(8)
tff(fact_2667_minf_I8_J,axiom,
    ! [Ta: rat] :
    ? [Z2: rat] :
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Z2)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ta),X) ) ).

% minf(8)
tff(fact_2668_minf_I8_J,axiom,
    ! [Ta: num] :
    ? [Z2: num] :
    ! [X: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Z2)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ta),X) ) ).

% minf(8)
tff(fact_2669_minf_I8_J,axiom,
    ! [Ta: nat] :
    ? [Z2: nat] :
    ! [X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Z2)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ta),X) ) ).

% minf(8)
tff(fact_2670_minf_I8_J,axiom,
    ! [Ta: int] :
    ? [Z2: int] :
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Z2)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ta),X) ) ).

% minf(8)
tff(fact_2671_minf_I6_J,axiom,
    ! [Ta: real] :
    ? [Z2: real] :
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Z2)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Ta) ) ).

% minf(6)
tff(fact_2672_minf_I6_J,axiom,
    ! [Ta: rat] :
    ? [Z2: rat] :
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Z2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Ta) ) ).

% minf(6)
tff(fact_2673_minf_I6_J,axiom,
    ! [Ta: num] :
    ? [Z2: num] :
    ! [X: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Z2)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Ta) ) ).

% minf(6)
tff(fact_2674_minf_I6_J,axiom,
    ! [Ta: nat] :
    ? [Z2: nat] :
    ! [X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Z2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Ta) ) ).

% minf(6)
tff(fact_2675_minf_I6_J,axiom,
    ! [Ta: int] :
    ? [Z2: int] :
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Z2)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Ta) ) ).

% minf(6)
tff(fact_2676_pinf_I8_J,axiom,
    ! [Ta: real] :
    ? [Z2: real] :
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),X)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ta),X) ) ).

% pinf(8)
tff(fact_2677_pinf_I8_J,axiom,
    ! [Ta: rat] :
    ? [Z2: rat] :
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),X)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ta),X) ) ).

% pinf(8)
tff(fact_2678_pinf_I8_J,axiom,
    ! [Ta: num] :
    ? [Z2: num] :
    ! [X: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z2),X)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ta),X) ) ).

% pinf(8)
tff(fact_2679_pinf_I8_J,axiom,
    ! [Ta: nat] :
    ? [Z2: nat] :
    ! [X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z2),X)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ta),X) ) ).

% pinf(8)
tff(fact_2680_pinf_I8_J,axiom,
    ! [Ta: int] :
    ? [Z2: int] :
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),X)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ta),X) ) ).

% pinf(8)
tff(fact_2681_pinf_I6_J,axiom,
    ! [Ta: real] :
    ? [Z2: real] :
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),X)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Ta) ) ).

% pinf(6)
tff(fact_2682_pinf_I6_J,axiom,
    ! [Ta: rat] :
    ? [Z2: rat] :
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),X)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Ta) ) ).

% pinf(6)
tff(fact_2683_pinf_I6_J,axiom,
    ! [Ta: num] :
    ? [Z2: num] :
    ! [X: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z2),X)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Ta) ) ).

% pinf(6)
tff(fact_2684_pinf_I6_J,axiom,
    ! [Ta: nat] :
    ? [Z2: nat] :
    ! [X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z2),X)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Ta) ) ).

% pinf(6)
tff(fact_2685_pinf_I6_J,axiom,
    ! [Ta: int] :
    ? [Z2: int] :
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),X)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Ta) ) ).

% pinf(6)
tff(fact_2686_conj__le__cong,axiom,
    ! [Xb: int,X5: int,P: $o,P3: $o] :
      ( ( Xb = X5 )
     => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X5)
         => ( (P)
          <=> (P3) ) )
       => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
            & (P) )
        <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X5)
            & (P3) ) ) ) ) ).

% conj_le_cong
tff(fact_2687_imp__le__cong,axiom,
    ! [Xb: int,X5: int,P: $o,P3: $o] :
      ( ( Xb = X5 )
     => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X5)
         => ( (P)
          <=> (P3) ) )
       => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
           => (P) )
        <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X5)
           => (P3) ) ) ) ) ).

% imp_le_cong
tff(fact_2688_mod__double__modulus,axiom,
    ! [Mb: nat,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Xb)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Mb) )
          | ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Mb)),Mb) ) ) ) ) ).

% mod_double_modulus
tff(fact_2689_mod__double__modulus,axiom,
    ! [Mb: int,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Mb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
       => ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Xb),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Mb)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Xb),Mb) )
          | ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Xb),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Mb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Xb),Mb)),Mb) ) ) ) ) ).

% mod_double_modulus
tff(fact_2690_unset__bit__Suc,axiom,
    ! [Nb: nat,A: int] : aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,aa_nat_nat(suc,Nb)),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_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,Nb),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_2691_unset__bit__Suc,axiom,
    ! [Nb: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,aa_nat_nat(suc,Nb)),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,Nb),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_2692_set__bit__Suc,axiom,
    ! [Nb: nat,A: int] : aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,aa_nat_nat(suc,Nb)),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_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,Nb),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_2693_set__bit__Suc,axiom,
    ! [Nb: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,aa_nat_nat(suc,Nb)),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,Nb),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_2694_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_2695_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(aa_int_fun_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_2696_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_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary2),X_13)
            & ! [X4: vEBT_VEBT] :
                ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList2))
               => ~ ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X4),X_13) ) )
        | ? [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_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary2),X_13)
            & ! [X4: vEBT_VEBT] :
                ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList2))
               => ~ ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X4),X_13) ) )
        | ? [TreeList2: list_VEBT_VEBT,N4: nat,Summary2: vEBT_VEBT,Mi2: nat,Ma2: nat] :
            ( ( A1 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),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) )
            & ! [I5: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N4))
               => ( ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList2,I5)),X_13)
                <=> aa_nat_o(vEBT_V8194947554948674370ptions(Summary2),I5) ) )
            & ( ( Mi2 = Ma2 )
             => ! [X4: vEBT_VEBT] :
                  ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList2))
                 => ~ ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X4),X_13) ) )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mi2),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))),A22))
            & ( ( Mi2 != Ma2 )
             => ! [I5: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),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,Ma2),N4) = I5 )
                     => aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList2,I5)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma2),N4)) )
                    & ! [X4: nat] :
                        ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X4),N4) = I5 )
                          & aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList2,I5)),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),Ma2) ) ) ) ) ) )
        | ? [TreeList2: list_VEBT_VEBT,N4: nat,Summary2: vEBT_VEBT,Mi2: nat,Ma2: nat] :
            ( ( A1 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),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)) )
            & ! [I5: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),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_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList2,I5)),X_13)
                <=> aa_nat_o(vEBT_V8194947554948674370ptions(Summary2),I5) ) )
            & ( ( Mi2 = Ma2 )
             => ! [X4: vEBT_VEBT] :
                  ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList2))
                 => ~ ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X4),X_13) ) )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mi2),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))),A22))
            & ( ( Mi2 != Ma2 )
             => ! [I5: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),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,Ma2),N4) = I5 )
                     => aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList2,I5)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma2),N4)) )
                    & ! [X4: nat] :
                        ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X4),N4) = I5 )
                          & aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList2,I5)),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),Ma2) ) ) ) ) ) ) ) ) ).

% invar_vebt.simps
tff(fact_2697_invar__vebt_Ocases,axiom,
    ! [A1: vEBT_VEBT,A22: nat] :
      ( aa_nat_o(vEBT_invar_vebt(A1),A22)
     => ( ( ? [A4: $o,B3: $o] : A1 = vEBT_Leaf((A4),(B3))
         => ( A22 != aa_nat_nat(suc,zero_zero_nat) ) )
       => ( ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M2: nat,Deg: nat] :
              ( ( A1 = vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList,Summary) )
             => ( ( A22 = Deg )
               => ( ! [X: vEBT_VEBT] :
                      ( member_VEBT_VEBT(X,set_VEBT_VEBT2(TreeList))
                     => aa_nat_o(vEBT_invar_vebt(X),N) )
                 => ( aa_nat_o(vEBT_invar_vebt(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) )
                     => ( ( M2 = N )
                       => ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M2) )
                         => ( ~ ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary),X_1)
                           => ~ ! [X: vEBT_VEBT] :
                                  ( member_VEBT_VEBT(X,set_VEBT_VEBT2(TreeList))
                                 => ~ ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X),X_1) ) ) ) ) ) ) ) ) )
         => ( ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M2: nat,Deg: nat] :
                ( ( A1 = vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList,Summary) )
               => ( ( A22 = Deg )
                 => ( ! [X: vEBT_VEBT] :
                        ( member_VEBT_VEBT(X,set_VEBT_VEBT2(TreeList))
                       => aa_nat_o(vEBT_invar_vebt(X),N) )
                   => ( aa_nat_o(vEBT_invar_vebt(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) )
                       => ( ( M2 = aa_nat_nat(suc,N) )
                         => ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M2) )
                           => ( ~ ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary),X_1)
                             => ~ ! [X: vEBT_VEBT] :
                                    ( member_VEBT_VEBT(X,set_VEBT_VEBT2(TreeList))
                                   => ~ ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X),X_1) ) ) ) ) ) ) ) ) )
           => ( ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M2: nat,Deg: nat,Mi: nat,Ma: nat] :
                  ( ( A1 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary) )
                 => ( ( A22 = Deg )
                   => ( ! [X: vEBT_VEBT] :
                          ( member_VEBT_VEBT(X,set_VEBT_VEBT2(TreeList))
                         => aa_nat_o(vEBT_invar_vebt(X),N) )
                     => ( aa_nat_o(vEBT_invar_vebt(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) )
                         => ( ( M2 = N )
                           => ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M2) )
                             => ( ! [I: nat] :
                                    ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M2))
                                   => ( ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList,I)),X_13)
                                    <=> aa_nat_o(vEBT_V8194947554948674370ptions(Summary),I) ) )
                               => ( ( ( Mi = Ma )
                                   => ! [X: vEBT_VEBT] :
                                        ( member_VEBT_VEBT(X,set_VEBT_VEBT2(TreeList))
                                       => ~ ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X),X_1) ) )
                                 => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mi),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))),Deg))
                                     => ~ ( ( Mi != Ma )
                                         => ! [I: nat] :
                                              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M2))
                                             => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma),N) = I )
                                                 => aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList,I)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma),N)) )
                                                & ! [X: nat] :
                                                    ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),N) = I )
                                                      & aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList,I)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),N)) )
                                                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi),X)
                                                      & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Ma) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
             => ~ ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M2: nat,Deg: nat,Mi: nat,Ma: nat] :
                    ( ( A1 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary) )
                   => ( ( A22 = Deg )
                     => ( ! [X: vEBT_VEBT] :
                            ( member_VEBT_VEBT(X,set_VEBT_VEBT2(TreeList))
                           => aa_nat_o(vEBT_invar_vebt(X),N) )
                       => ( aa_nat_o(vEBT_invar_vebt(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) )
                           => ( ( M2 = aa_nat_nat(suc,N) )
                             => ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M2) )
                               => ( ! [I: nat] :
                                      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M2))
                                     => ( ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList,I)),X_13)
                                      <=> aa_nat_o(vEBT_V8194947554948674370ptions(Summary),I) ) )
                                 => ( ( ( Mi = Ma )
                                     => ! [X: vEBT_VEBT] :
                                          ( member_VEBT_VEBT(X,set_VEBT_VEBT2(TreeList))
                                         => ~ ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X),X_1) ) )
                                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mi),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))),Deg))
                                       => ~ ( ( Mi != Ma )
                                           => ! [I: nat] :
                                                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M2))
                                               => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma),N) = I )
                                                   => aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList,I)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma),N)) )
                                                  & ! [X: nat] :
                                                      ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),N) = I )
                                                        & aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList,I)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),N)) )
                                                     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi),X)
                                                        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Ma) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.cases
tff(fact_2698_invar__vebt_Ointros_I2_J,axiom,
    ! [TreeLista: list_VEBT_VEBT,Nb: nat,Summarya: vEBT_VEBT,Mb: nat,Dega: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeLista))
         => aa_nat_o(vEBT_invar_vebt(X3),Nb) )
     => ( aa_nat_o(vEBT_invar_vebt(Summarya),Mb)
       => ( ( 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))),Mb) )
         => ( ( Mb = Nb )
           => ( ( Dega = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb) )
             => ( ~ ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summarya),X_12)
               => ( ! [X3: vEBT_VEBT] :
                      ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeLista))
                     => ~ ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X3),X_12) )
                 => aa_nat_o(vEBT_invar_vebt(vEBT_Node(none_P5556105721700978146at_nat,Dega,TreeLista,Summarya)),Dega) ) ) ) ) ) ) ) ).

% invar_vebt.intros(2)
tff(fact_2699_verit__le__mono__div,axiom,
    ! [A3: nat,B4: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A3),B4)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => 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,A3),Nb)),
                $ite(aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B4),Nb) = zero_zero_nat,one_one_nat,zero_zero_nat))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B4),Nb)) ) ) ).

% verit_le_mono_div
tff(fact_2700_div__mod__decomp,axiom,
    ! [A3: nat,Nb: nat] : A3 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A3),Nb)),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A3),Nb)) ).

% div_mod_decomp
tff(fact_2701_div__less__mono,axiom,
    ! [A3: nat,B4: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A3),B4)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A3),Nb) = zero_zero_nat )
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B4),Nb) = 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,A3),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B4),Nb)) ) ) ) ) ).

% div_less_mono
tff(fact_2702_flip__bit__Suc,axiom,
    ! [Nb: nat,A: int] : aa_int_int(bit_se2159334234014336723it_int(aa_nat_nat(suc,Nb)),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_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(Nb),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_2703_flip__bit__Suc,axiom,
    ! [Nb: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,aa_nat_nat(suc,Nb)),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,Nb),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_2704_product__nth,axiom,
    ! [Nb: nat,Xs: list_num,Ys: list_num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),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),Nb) = product_Pair_num_num(aa_nat_num(nth_num(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),size_size_list_num(Ys))),aa_nat_num(nth_num(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),size_size_list_num(Ys)))) ) ) ).

% product_nth
tff(fact_2705_product__nth,axiom,
    ! [Nb: nat,Xs: list_Code_integer,Ys: list_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),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),Nb) = produc6677183202524767010eger_o(aa_nat_Code_integer(nth_Code_integer(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),size_size_list_o(Ys))),aa_nat_o(nth_o(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),size_size_list_o(Ys)))) ) ) ).

% product_nth
tff(fact_2706_product__nth,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),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),Nb) = produc537772716801021591T_VEBT(nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),size_s6755466524823107622T_VEBT(Ys))),nth_VEBT_VEBT(Ys,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),size_s6755466524823107622T_VEBT(Ys)))) ) ) ).

% product_nth
tff(fact_2707_product__nth,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBT,Ys: list_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),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),Nb) = produc8721562602347293563VEBT_o(nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),size_size_list_o(Ys))),aa_nat_o(nth_o(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),size_size_list_o(Ys)))) ) ) ).

% product_nth
tff(fact_2708_product__nth,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBT,Ys: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),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),Nb) = produc738532404422230701BT_nat(nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),size_size_list_nat(Ys))),aa_nat_nat(nth_nat(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),size_size_list_nat(Ys)))) ) ) ).

% product_nth
tff(fact_2709_product__nth,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBT,Ys: list_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),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),Nb) = produc736041933913180425BT_int(nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),size_size_list_int(Ys))),aa_nat_int(nth_int(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),size_size_list_int(Ys)))) ) ) ).

% product_nth
tff(fact_2710_product__nth,axiom,
    ! [Nb: nat,Xs: list_o,Ys: list_VEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),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),Nb) = produc2982872950893828659T_VEBT(aa_nat_o(nth_o(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),size_s6755466524823107622T_VEBT(Ys))),nth_VEBT_VEBT(Ys,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),size_s6755466524823107622T_VEBT(Ys)))) ) ) ).

% product_nth
tff(fact_2711_product__nth,axiom,
    ! [Nb: nat,Xs: list_o,Ys: list_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),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),Nb) = product_Pair_o_o(aa_nat_o(nth_o(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),size_size_list_o(Ys))),aa_nat_o(nth_o(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),size_size_list_o(Ys)))) ) ) ).

% product_nth
tff(fact_2712_product__nth,axiom,
    ! [Nb: nat,Xs: list_o,Ys: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),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),Nb) = product_Pair_o_nat(aa_nat_o(nth_o(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),size_size_list_nat(Ys))),aa_nat_nat(nth_nat(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),size_size_list_nat(Ys)))) ) ) ).

% product_nth
tff(fact_2713_product__nth,axiom,
    ! [Nb: nat,Xs: list_o,Ys: list_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),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),Nb) = product_Pair_o_int(aa_nat_o(nth_o(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),size_size_list_int(Ys))),aa_nat_int(nth_int(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),size_size_list_int(Ys)))) ) ) ).

% product_nth
tff(fact_2714_pos__eucl__rel__int__mult__2,axiom,
    ! [B: int,A: int,Q2: int,R2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
     => ( aa_Pro4592068669816069595_int_o(eucl_rel_int(A,B),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q2),R2))
       => aa_Pro4592068669816069595_int_o(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)),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q2),aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),R2)))) ) ) ).

% pos_eucl_rel_int_mult_2
tff(fact_2715_gcd__nat__induct,axiom,
    ! [P: fun_nat_fun_nat_o,Mb: nat,Nb: nat] :
      ( ! [M2: nat] : aa_nat_o(aa_nat_fun_nat_o(P,M2),zero_zero_nat)
     => ( ! [M2: 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,M2),N))
             => aa_nat_o(aa_nat_fun_nat_o(P,M2),N) ) )
       => aa_nat_o(aa_nat_fun_nat_o(P,Mb),Nb) ) ) ).

% gcd_nat_induct
tff(fact_2716_option_Osize__gen_I2_J,axiom,
    ! [Xb: fun_Pr2602117396471556196at_nat,X2: product_prod_nat_nat] : size_o8335143837870341156at_nat(Xb,some_P7363390416028606310at_nat(X2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_Pro4243173770044397973at_nat(Xb,X2)),aa_nat_nat(suc,zero_zero_nat)) ).

% option.size_gen(2)
tff(fact_2717_option_Osize__gen_I2_J,axiom,
    ! [Xb: fun_num_nat,X2: num] : size_option_num(Xb,some_num(X2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(Xb,X2)),aa_nat_nat(suc,zero_zero_nat)) ).

% option.size_gen(2)
tff(fact_2718_verit__eq__simplify_I8_J,axiom,
    ! [X2: num,Y2: num] :
      ( ( aa_num_num(bit0,X2) = aa_num_num(bit0,Y2) )
    <=> ( X2 = Y2 ) ) ).

% verit_eq_simplify(8)
tff(fact_2719_flip__bit__nonnegative__int__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se2159334234014336723it_int(Nb),K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K) ) ).

% flip_bit_nonnegative_int_iff
tff(fact_2720_flip__bit__negative__int__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se2159334234014336723it_int(Nb),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_2721_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(aa_int_fun_int_int(modulo_modulo_int,K),L) = K ) ) ) ).

% mod_neg_neg_trivial
tff(fact_2722_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(aa_int_fun_int_int(modulo_modulo_int,K),L) = K ) ) ) ).

% mod_pos_pos_trivial
tff(fact_2723_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_2724_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_2725_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_2726_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_2727_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_2728_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_2729_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_2730_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_2731_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_2732_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_2733_zmod__numeral__Bit0,axiom,
    ! [V: num,W: num] : aa_int_int(aa_int_fun_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(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,V)),aa_num_int(numeral_numeral_int,W))) ).

% zmod_numeral_Bit0
tff(fact_2734_mod__int__unique,axiom,
    ! [K: int,L: int,Q2: int,R2: int] :
      ( aa_Pro4592068669816069595_int_o(eucl_rel_int(K,L),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q2),R2))
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L) = R2 ) ) ).

% mod_int_unique
tff(fact_2735_unique__quotient,axiom,
    ! [A: int,B: int,Q2: int,R2: int,Q5: int,R4: int] :
      ( aa_Pro4592068669816069595_int_o(eucl_rel_int(A,B),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q2),R2))
     => ( aa_Pro4592068669816069595_int_o(eucl_rel_int(A,B),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q5),R4))
       => ( Q2 = Q5 ) ) ) ).

% unique_quotient
tff(fact_2736_unique__remainder,axiom,
    ! [A: int,B: int,Q2: int,R2: int,Q5: int,R4: int] :
      ( aa_Pro4592068669816069595_int_o(eucl_rel_int(A,B),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q2),R2))
     => ( aa_Pro4592068669816069595_int_o(eucl_rel_int(A,B),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q5),R4))
       => ( R2 = R4 ) ) ) ).

% unique_remainder
tff(fact_2737_eucl__rel__int,axiom,
    ! [K: int,L: int] : aa_Pro4592068669816069595_int_o(eucl_rel_int(K,L),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L))) ).

% eucl_rel_int
tff(fact_2738_eucl__rel__int__by0,axiom,
    ! [K: int] : aa_Pro4592068669816069595_int_o(eucl_rel_int(K,zero_zero_int),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),K)) ).

% eucl_rel_int_by0
tff(fact_2739_div__int__unique,axiom,
    ! [K: int,L: int,Q2: int,R2: int] :
      ( aa_Pro4592068669816069595_int_o(eucl_rel_int(K,L),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q2),R2))
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = Q2 ) ) ).

% div_int_unique
tff(fact_2740_zmod__le__nonneg__dividend,axiom,
    ! [Mb: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Mb)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Mb),K)),Mb) ) ).

% zmod_le_nonneg_dividend
tff(fact_2741_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(aa_int_fun_int_int(modulo_modulo_int,K),L)) ) ).

% neg_mod_bound
tff(fact_2742_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(aa_int_fun_int_int(modulo_modulo_int,K),L)),L) ) ).

% Euclidean_Division.pos_mod_bound
tff(fact_2743_zmod__eq__0__iff,axiom,
    ! [Mb: int,D: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Mb),D) = zero_zero_int )
    <=> ? [Q4: int] : Mb = aa_int_int(aa_int_fun_int_int(times_times_int,D),Q4) ) ).

% zmod_eq_0_iff
tff(fact_2744_zmod__eq__0D,axiom,
    ! [Mb: int,D: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Mb),D) = zero_zero_int )
     => ? [Q3: int] : Mb = aa_int_int(aa_int_fun_int_int(times_times_int,D),Q3) ) ).

% zmod_eq_0D
tff(fact_2745_zdiv__mono__strict,axiom,
    ! [A3: int,B4: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A3),B4)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Nb)
       => ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A3),Nb) = zero_zero_int )
         => ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B4),Nb) = 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,A3),Nb)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B4),Nb)) ) ) ) ) ).

% zdiv_mono_strict
tff(fact_2746_div__mod__decomp__int,axiom,
    ! [A3: int,Nb: int] : A3 = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A3),Nb)),Nb)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A3),Nb)) ).

% div_mod_decomp_int
tff(fact_2747_eucl__rel__int__dividesI,axiom,
    ! [L: int,K: int,Q2: int] :
      ( ( L != zero_zero_int )
     => ( ( K = aa_int_int(aa_int_fun_int_int(times_times_int,Q2),L) )
       => aa_Pro4592068669816069595_int_o(eucl_rel_int(K,L),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q2),zero_zero_int)) ) ) ).

% eucl_rel_int_dividesI
tff(fact_2748_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(aa_int_fun_int_int(modulo_modulo_int,K),L)),zero_zero_int) ) ).

% neg_mod_sign
tff(fact_2749_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(aa_int_fun_int_int(modulo_modulo_int,K),L)) ) ).

% Euclidean_Division.pos_mod_sign
tff(fact_2750_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(aa_int_fun_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(aa_int_fun_int_int(modulo_modulo_int,A),B)) ) ) ).

% neg_mod_conj
tff(fact_2751_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(aa_int_fun_int_int(modulo_modulo_int,A),B))
        & aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),B) ) ) ).

% pos_mod_conj
tff(fact_2752_zmod__trivial__iff,axiom,
    ! [I3: int,K: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,I3),K) = I3 )
    <=> ( ( K = zero_zero_int )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),I3)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,I3),K) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I3),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,K),I3) ) ) ) ).

% zmod_trivial_iff
tff(fact_2753_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_2754_verit__la__disequality,axiom,
    ! [A: num,B: num] :
      ( ( A = B )
      | ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
      | ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A) ) ).

% verit_la_disequality
tff(fact_2755_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_2756_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_2757_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_2758_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_2759_verit__comp__simplify1_I2_J,axiom,
    ! [A: num] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),A) ).

% verit_comp_simplify1(2)
tff(fact_2760_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_2761_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_2762_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_2763_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_2764_verit__comp__simplify1_I1_J,axiom,
    ! [A: num] : ~ aa_num_o(aa_num_fun_num_o(ord_less_num,A),A) ).

% verit_comp_simplify1(1)
tff(fact_2765_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_2766_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_2767_verit__la__generic,axiom,
    ! [A: int,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),Xb)
      | ( A = Xb )
      | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),A) ) ).

% verit_la_generic
tff(fact_2768_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(aa_int_fun_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_2769_verit__le__mono__div__int,axiom,
    ! [A3: int,B4: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A3),B4)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Nb)
       => 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,A3),Nb)),
                $ite(aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B4),Nb) = zero_zero_int,one_one_int,zero_zero_int))),aa_int_int(aa_int_fun_int_int(divide_divide_int,B4),Nb)) ) ) ).

% verit_le_mono_div_int
tff(fact_2770_split__zmod,axiom,
    ! [P: fun_int_o,Nb: int,K: int] :
      ( aa_int_o(P,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Nb),K))
    <=> ( ( ( K = zero_zero_int )
         => aa_int_o(P,Nb) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
         => ! [I5: 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)
                & ( Nb = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I5)),J3) ) )
             => aa_int_o(P,J3) ) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
         => ! [I5: 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)
                & ( Nb = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I5)),J3) ) )
             => aa_int_o(P,J3) ) ) ) ) ).

% split_zmod
tff(fact_2771_int__mod__neg__eq,axiom,
    ! [A: int,B: int,Q2: int,R2: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q2)),R2) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,R2),zero_zero_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),R2)
         => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = R2 ) ) ) ) ).

% int_mod_neg_eq
tff(fact_2772_int__mod__pos__eq,axiom,
    ! [A: int,B: int,Q2: int,R2: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q2)),R2) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),R2)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R2),B)
         => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = R2 ) ) ) ) ).

% int_mod_pos_eq
tff(fact_2773_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(aa_int_fun_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(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)) ) ) ).

% zmod_zmult2_eq
tff(fact_2774_split__neg__lemma,axiom,
    ! [K: int,P: fun_int_fun_int_o,Nb: 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,Nb),K)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Nb),K))
      <=> ! [I5: 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)
              & ( Nb = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I5)),J3) ) )
           => aa_int_o(aa_int_fun_int_o(P,I5),J3) ) ) ) ).

% split_neg_lemma
tff(fact_2775_split__pos__lemma,axiom,
    ! [K: int,P: fun_int_fun_int_o,Nb: 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,Nb),K)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Nb),K))
      <=> ! [I5: 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)
              & ( Nb = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I5)),J3) ) )
           => aa_int_o(aa_int_fun_int_o(P,I5),J3) ) ) ) ).

% split_pos_lemma
tff(fact_2776_verit__comp__simplify1_I3_J,axiom,
    ! [B2: real,A2: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),A2)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2) ) ).

% verit_comp_simplify1(3)
tff(fact_2777_verit__comp__simplify1_I3_J,axiom,
    ! [B2: rat,A2: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A2)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2) ) ).

% verit_comp_simplify1(3)
tff(fact_2778_verit__comp__simplify1_I3_J,axiom,
    ! [B2: num,A2: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),A2)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,A2),B2) ) ).

% verit_comp_simplify1(3)
tff(fact_2779_verit__comp__simplify1_I3_J,axiom,
    ! [B2: nat,A2: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),A2)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B2) ) ).

% verit_comp_simplify1(3)
tff(fact_2780_verit__comp__simplify1_I3_J,axiom,
    ! [B2: int,A2: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A2)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B2) ) ).

% verit_comp_simplify1(3)
tff(fact_2781_verit__sum__simplify,axiom,
    ! [A: complex] : aa_complex_complex(plus_plus_complex(A),zero_zero_complex) = A ).

% verit_sum_simplify
tff(fact_2782_verit__sum__simplify,axiom,
    ! [A: real] : aa_real_real(plus_plus_real(A),zero_zero_real) = A ).

% verit_sum_simplify
tff(fact_2783_verit__sum__simplify,axiom,
    ! [A: rat] : aa_rat_rat(plus_plus_rat(A),zero_zero_rat) = A ).

% verit_sum_simplify
tff(fact_2784_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_2785_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_2786_verit__eq__simplify_I10_J,axiom,
    ! [X2: num] : one != aa_num_num(bit0,X2) ).

% verit_eq_simplify(10)
tff(fact_2787_Euclid__induct,axiom,
    ! [P: fun_nat_fun_nat_o,A: nat,B: nat] :
      ( ! [A4: nat,B3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(P,A4),B3)
        <=> aa_nat_o(aa_nat_fun_nat_o(P,B3),A4) )
     => ( ! [A4: nat] : aa_nat_o(aa_nat_fun_nat_o(P,A4),zero_zero_nat)
       => ( ! [A4: nat,B3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(P,A4),B3)
             => aa_nat_o(aa_nat_fun_nat_o(P,A4),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A4),B3)) )
         => aa_nat_o(aa_nat_fun_nat_o(P,A),B) ) ) ) ).

% Euclid_induct
tff(fact_2788_eucl__rel__int__iff,axiom,
    ! [K: int,L: int,Q2: int,R2: int] :
      ( aa_Pro4592068669816069595_int_o(eucl_rel_int(K,L),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q2),R2))
    <=> ( ( K = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,L),Q2)),R2) )
        & $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),R2)
            & aa_int_o(aa_int_fun_int_o(ord_less_int,R2),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),R2)
              & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,R2),zero_zero_int) ),
              Q2 = zero_zero_int ) ) ) ) ).

% eucl_rel_int_iff
tff(fact_2789_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(aa_int_fun_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(aa_int_fun_int_int(modulo_modulo_int,B),A))) ) ) ).

% pos_zmod_mult_2
tff(fact_2790_option_Osize__gen_I1_J,axiom,
    ! [Xb: fun_Pr2602117396471556196at_nat] : size_o8335143837870341156at_nat(Xb,none_P5556105721700978146at_nat) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size_gen(1)
tff(fact_2791_option_Osize__gen_I1_J,axiom,
    ! [Xb: fun_num_nat] : size_option_num(Xb,none_num) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size_gen(1)
tff(fact_2792_neg__eucl__rel__int__mult__2,axiom,
    ! [B: int,A: int,Q2: int,R2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int)
     => ( aa_Pro4592068669816069595_int_o(eucl_rel_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int),B),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q2),R2))
       => aa_Pro4592068669816069595_int_o(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)),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q2),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),R2)),one_one_int))) ) ) ).

% neg_eucl_rel_int_mult_2
tff(fact_2793_even__succ__mod__exp,axiom,
    ! [A: code_integer,Nb: 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),Nb)
       => ( modulo364778990260209775nteger(plus_p5714425477246183910nteger(one_one_Code_integer,A),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Nb)) = plus_p5714425477246183910nteger(one_one_Code_integer,modulo364778990260209775nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Nb))) ) ) ) ).

% even_succ_mod_exp
tff(fact_2794_even__succ__mod__exp,axiom,
    ! [A: nat,Nb: 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),Nb)
       => ( 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))),Nb)) = 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))),Nb))) ) ) ) ).

% even_succ_mod_exp
tff(fact_2795_even__succ__mod__exp,axiom,
    ! [A: int,Nb: 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),Nb)
       => ( aa_int_int(aa_int_fun_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))),Nb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))) ) ) ) ).

% even_succ_mod_exp
tff(fact_2796_even__succ__div__exp,axiom,
    ! [A: code_integer,Nb: 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),Nb)
       => ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(one_one_Code_integer,A),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Nb)) = divide6298287555418463151nteger(A,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Nb)) ) ) ) ).

% even_succ_div_exp
tff(fact_2797_even__succ__div__exp,axiom,
    ! [A: nat,Nb: 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),Nb)
       => ( 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))),Nb)) = 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))),Nb)) ) ) ) ).

% even_succ_div_exp
tff(fact_2798_even__succ__div__exp,axiom,
    ! [A: int,Nb: 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),Nb)
       => ( 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))),Nb)) = 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))),Nb)) ) ) ) ).

% even_succ_div_exp
tff(fact_2799_signed__take__bit__Suc,axiom,
    ! [Nb: nat,A: int] : aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(suc,Nb)),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_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(Nb),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_2800_vebt__insert_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Yb: vEBT_VEBT] :
      ( ( vEBT_vebt_insert(Xb,Xaa) = Yb )
     => ( ! [A4: $o,B3: $o] :
            ( ( Xb = vEBT_Leaf((A4),(B3)) )
           => ( Yb != $ite(
                  Xaa = zero_zero_nat,
                  vEBT_Leaf($true,(B3)),
                  $ite(Xaa = one_one_nat,vEBT_Leaf((A4),$true),vEBT_Leaf((A4),(B3))) ) ) )
       => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT] :
              ( ( Xb = vEBT_Node(Info2,zero_zero_nat,Ts2,S) )
             => ( Yb != vEBT_Node(Info2,zero_zero_nat,Ts2,S) ) )
         => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2,S) )
               => ( Yb != vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2,S) ) )
           => ( ! [V2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList,Summary) )
                 => ( Yb != vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xaa),Xaa)),aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList,Summary) ) )
             => ~ ! [Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary) )
                   => ( Yb != $let(
                          xn: nat,
                          xn:= 
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),Mi,Xaa),
                          $let(
                            h: nat,
                            h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xn),aa_nat_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)))),
                            $ite(
                              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList))
                              & ~ ( ( Xaa = Mi )
                                  | ( Xaa = Ma ) ) ),
                              vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat($ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),Xaa,Mi)),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,xn),Ma))),aa_nat_nat(suc,aa_nat_nat(suc,Va)),list_u1324408373059187874T_VEBT(TreeList,h,vEBT_vebt_insert(nth_VEBT_VEBT(TreeList,h),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xn),aa_nat_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_VEBT_VEBT_o(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeList,h)),vEBT_vebt_insert(Summary,h),Summary)),
                              vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary) ) ) ) ) ) ) ) ) ) ) ).

% vebt_insert.elims
tff(fact_2801_add__scale__eq__noteq,axiom,
    ! [R2: complex,A: complex,B: complex,C: complex,D: complex] :
      ( ( R2 != zero_zero_complex )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( aa_complex_complex(plus_plus_complex(A),aa_complex_complex(times_times_complex(R2),C)) != aa_complex_complex(plus_plus_complex(B),aa_complex_complex(times_times_complex(R2),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_2802_add__scale__eq__noteq,axiom,
    ! [R2: real,A: real,B: real,C: real,D: real] :
      ( ( R2 != zero_zero_real )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( aa_real_real(plus_plus_real(A),aa_real_real(times_times_real(R2),C)) != aa_real_real(plus_plus_real(B),aa_real_real(times_times_real(R2),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_2803_add__scale__eq__noteq,axiom,
    ! [R2: rat,A: rat,B: rat,C: rat,D: rat] :
      ( ( R2 != zero_zero_rat )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( aa_rat_rat(plus_plus_rat(A),aa_rat_rat(times_times_rat(R2),C)) != aa_rat_rat(plus_plus_rat(B),aa_rat_rat(times_times_rat(R2),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_2804_add__scale__eq__noteq,axiom,
    ! [R2: nat,A: nat,B: nat,C: nat,D: nat] :
      ( ( R2 != zero_zero_nat )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R2),C)) != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R2),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_2805_add__scale__eq__noteq,axiom,
    ! [R2: int,A: int,B: int,C: int,D: int] :
      ( ( R2 != zero_zero_int )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,R2),C)) != aa_int_int(aa_int_fun_int_int(plus_plus_int,B),aa_int_int(aa_int_fun_int_int(times_times_int,R2),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_2806_num_Osize__gen_I2_J,axiom,
    ! [X2: num] : aa_num_nat(size_num,aa_num_num(bit0,X2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(size_num,X2)),aa_nat_nat(suc,zero_zero_nat)) ).

% num.size_gen(2)
tff(fact_2807_power__numeral,axiom,
    ! [K: num,L: num] : aa_nat_complex(power_power_complex(numera6690914467698888265omplex(K)),aa_num_nat(numeral_numeral_nat,L)) = numera6690914467698888265omplex(aa_num_num(pow(K),L)) ).

% power_numeral
tff(fact_2808_power__numeral,axiom,
    ! [K: num,L: num] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,K)),aa_num_nat(numeral_numeral_nat,L)) = aa_num_real(numeral_numeral_real,aa_num_num(pow(K),L)) ).

% power_numeral
tff(fact_2809_power__numeral,axiom,
    ! [K: num,L: num] : aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,K)),aa_num_nat(numeral_numeral_nat,L)) = aa_num_rat(numeral_numeral_rat,aa_num_num(pow(K),L)) ).

% power_numeral
tff(fact_2810_power__numeral,axiom,
    ! [K: num,L: num] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,K)),aa_num_nat(numeral_numeral_nat,L)) = aa_num_nat(numeral_numeral_nat,aa_num_num(pow(K),L)) ).

% power_numeral
tff(fact_2811_power__numeral,axiom,
    ! [K: num,L: num] : aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,K)),aa_num_nat(numeral_numeral_nat,L)) = aa_num_int(numeral_numeral_int,aa_num_num(pow(K),L)) ).

% power_numeral
tff(fact_2812_buildup__gives__empty,axiom,
    ! [Nb: nat] : vEBT_VEBT_set_vebt(vEBT_vebt_buildup(Nb)) = bot_bot_set_nat ).

% buildup_gives_empty
tff(fact_2813_set__vebt_H__def,axiom,
    ! [Ta: vEBT_VEBT] : vEBT_VEBT_set_vebt(Ta) = collect_nat(vEBT_vebt_member(Ta)) ).

% set_vebt'_def
tff(fact_2814_nat__dvd__1__iff__1,axiom,
    ! [Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),one_one_nat)
    <=> ( Mb = one_one_nat ) ) ).

% nat_dvd_1_iff_1
tff(fact_2815_diff__self,axiom,
    ! [A: complex] : aa_complex_complex(minus_minus_complex(A),A) = zero_zero_complex ).

% diff_self
tff(fact_2816_diff__self,axiom,
    ! [A: real] : aa_real_real(minus_minus_real(A),A) = zero_zero_real ).

% diff_self
tff(fact_2817_diff__self,axiom,
    ! [A: rat] : aa_rat_rat(minus_minus_rat(A),A) = zero_zero_rat ).

% diff_self
tff(fact_2818_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_2819_diff__0__right,axiom,
    ! [A: complex] : aa_complex_complex(minus_minus_complex(A),zero_zero_complex) = A ).

% diff_0_right
tff(fact_2820_diff__0__right,axiom,
    ! [A: real] : aa_real_real(minus_minus_real(A),zero_zero_real) = A ).

% diff_0_right
tff(fact_2821_diff__0__right,axiom,
    ! [A: rat] : aa_rat_rat(minus_minus_rat(A),zero_zero_rat) = A ).

% diff_0_right
tff(fact_2822_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_2823_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_2824_diff__zero,axiom,
    ! [A: complex] : aa_complex_complex(minus_minus_complex(A),zero_zero_complex) = A ).

% diff_zero
tff(fact_2825_diff__zero,axiom,
    ! [A: real] : aa_real_real(minus_minus_real(A),zero_zero_real) = A ).

% diff_zero
tff(fact_2826_diff__zero,axiom,
    ! [A: rat] : aa_rat_rat(minus_minus_rat(A),zero_zero_rat) = A ).

% diff_zero
tff(fact_2827_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_2828_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_2829_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_2830_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: real] : aa_real_real(minus_minus_real(A),A) = zero_zero_real ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_2831_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: rat] : aa_rat_rat(minus_minus_rat(A),A) = zero_zero_rat ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_2832_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_2833_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_2834_add__diff__cancel,axiom,
    ! [A: real,B: real] : aa_real_real(minus_minus_real(aa_real_real(plus_plus_real(A),B)),B) = A ).

% add_diff_cancel
tff(fact_2835_add__diff__cancel,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(A),B)),B) = A ).

% add_diff_cancel
tff(fact_2836_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_2837_diff__add__cancel,axiom,
    ! [A: real,B: real] : aa_real_real(plus_plus_real(aa_real_real(minus_minus_real(A),B)),B) = A ).

% diff_add_cancel
tff(fact_2838_diff__add__cancel,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(minus_minus_rat(A),B)),B) = A ).

% diff_add_cancel
tff(fact_2839_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_2840_add__diff__cancel__left,axiom,
    ! [C: real,A: real,B: real] : aa_real_real(minus_minus_real(aa_real_real(plus_plus_real(C),A)),aa_real_real(plus_plus_real(C),B)) = aa_real_real(minus_minus_real(A),B) ).

% add_diff_cancel_left
tff(fact_2841_add__diff__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(C),A)),aa_rat_rat(plus_plus_rat(C),B)) = aa_rat_rat(minus_minus_rat(A),B) ).

% add_diff_cancel_left
tff(fact_2842_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_2843_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_2844_add__diff__cancel__left_H,axiom,
    ! [A: real,B: real] : aa_real_real(minus_minus_real(aa_real_real(plus_plus_real(A),B)),A) = B ).

% add_diff_cancel_left'
tff(fact_2845_add__diff__cancel__left_H,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(A),B)),A) = B ).

% add_diff_cancel_left'
tff(fact_2846_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_2847_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_2848_add__diff__cancel__right,axiom,
    ! [A: real,C: real,B: real] : aa_real_real(minus_minus_real(aa_real_real(plus_plus_real(A),C)),aa_real_real(plus_plus_real(B),C)) = aa_real_real(minus_minus_real(A),B) ).

% add_diff_cancel_right
tff(fact_2849_add__diff__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(A),C)),aa_rat_rat(plus_plus_rat(B),C)) = aa_rat_rat(minus_minus_rat(A),B) ).

% add_diff_cancel_right
tff(fact_2850_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_2851_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_2852_add__diff__cancel__right_H,axiom,
    ! [A: real,B: real] : aa_real_real(minus_minus_real(aa_real_real(plus_plus_real(A),B)),B) = A ).

% add_diff_cancel_right'
tff(fact_2853_add__diff__cancel__right_H,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(A),B)),B) = A ).

% add_diff_cancel_right'
tff(fact_2854_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_2855_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_2856_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_2857_dvd__0__right,axiom,
    ! [A: complex] : aa_complex_o(dvd_dvd_complex(A),zero_zero_complex) ).

% dvd_0_right
tff(fact_2858_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_2859_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_2860_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_2861_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_2862_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_2863_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_2864_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_2865_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_2866_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_2867_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_2868_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_2869_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(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_2870_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(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_2871_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_2872_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_2873_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_2874_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(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_2875_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(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_2876_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_2877_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_2878_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_2879_dvd__1__iff__1,axiom,
    ! [Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),aa_nat_nat(suc,zero_zero_nat))
    <=> ( Mb = aa_nat_nat(suc,zero_zero_nat) ) ) ).

% dvd_1_iff_1
tff(fact_2880_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_2881_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_2882_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_2883_minus__mod__self2,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ).

% minus_mod_self2
tff(fact_2884_nat__mult__dvd__cancel__disj,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb))
    <=> ( ( K = zero_zero_nat )
        | aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),Nb) ) ) ).

% nat_mult_dvd_cancel_disj
tff(fact_2885_signed__take__bit__of__0,axiom,
    ! [Nb: nat] : aa_int_int(bit_ri631733984087533419it_int(Nb),zero_zero_int) = zero_zero_int ).

% signed_take_bit_of_0
tff(fact_2886_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(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_2887_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),aa_rat_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_2888_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_2889_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(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_2890_diff__gt__0__iff__gt,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_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_2891_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_2892_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(plus_plus_real(B),aa_real_real(minus_minus_real(A),B)) = A ) ) ).

% le_add_diff_inverse
tff(fact_2893_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(plus_plus_rat(B),aa_rat_rat(minus_minus_rat(A),B)) = A ) ) ).

% le_add_diff_inverse
tff(fact_2894_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_2895_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_2896_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(plus_plus_real(aa_real_real(minus_minus_real(A),B)),B) = A ) ) ).

% le_add_diff_inverse2
tff(fact_2897_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(plus_plus_rat(aa_rat_rat(minus_minus_rat(A),B)),B) = A ) ) ).

% le_add_diff_inverse2
tff(fact_2898_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_2899_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_2900_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_2901_diff__numeral__special_I9_J,axiom,
    aa_real_real(minus_minus_real(one_one_real),one_one_real) = zero_zero_real ).

% diff_numeral_special(9)
tff(fact_2902_diff__numeral__special_I9_J,axiom,
    aa_rat_rat(minus_minus_rat(one_one_rat),one_one_rat) = zero_zero_rat ).

% diff_numeral_special(9)
tff(fact_2903_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_2904_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_2905_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_2906_right__diff__distrib__numeral,axiom,
    ! [V: num,B: real,C: real] : aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,V)),aa_real_real(minus_minus_real(B),C)) = aa_real_real(minus_minus_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,V)),B)),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,V)),C)) ).

% right_diff_distrib_numeral
tff(fact_2907_right__diff__distrib__numeral,axiom,
    ! [V: num,B: rat,C: rat] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,V)),aa_rat_rat(minus_minus_rat(B),C)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,V)),B)),aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,V)),C)) ).

% right_diff_distrib_numeral
tff(fact_2908_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_2909_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_2910_left__diff__distrib__numeral,axiom,
    ! [A: real,B: real,V: num] : aa_real_real(times_times_real(aa_real_real(minus_minus_real(A),B)),aa_num_real(numeral_numeral_real,V)) = aa_real_real(minus_minus_real(aa_real_real(times_times_real(A),aa_num_real(numeral_numeral_real,V))),aa_real_real(times_times_real(B),aa_num_real(numeral_numeral_real,V))) ).

% left_diff_distrib_numeral
tff(fact_2911_left__diff__distrib__numeral,axiom,
    ! [A: rat,B: rat,V: num] : aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(A),B)),aa_num_rat(numeral_numeral_rat,V)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(A),aa_num_rat(numeral_numeral_rat,V))),aa_rat_rat(times_times_rat(B),aa_num_rat(numeral_numeral_rat,V))) ).

% left_diff_distrib_numeral
tff(fact_2912_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_2913_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_2914_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_2915_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(times_times_real(C),A)),aa_real_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_2916_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_2917_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_2918_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_2919_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_2920_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(times_times_real(A),C)),aa_real_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_2921_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_2922_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_2923_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_2924_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_2925_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_2926_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_2927_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_2928_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_2929_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_2930_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_2931_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_2932_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_2933_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(plus_plus_real(B),aa_real_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_2934_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(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_2935_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_2936_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_2937_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_2938_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(plus_plus_real(aa_real_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_2939_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(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_2940_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_2941_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_2942_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_2943_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_2944_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_2945_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_2946_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_2947_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_2948_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_2949_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_2950_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_2951_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_2952_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_2953_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_2954_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_2955_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_2956_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_2957_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_2958_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_2959_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_2960_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_2961_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_2962_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_2963_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_2964_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(aa_int_fun_int_int(modulo_modulo_int,B),A) = zero_zero_int ) ) ).

% dvd_imp_mod_0
tff(fact_2965_signed__take__bit__Suc__1,axiom,
    ! [Nb: nat] : aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(suc,Nb)),one_one_int) = one_one_int ).

% signed_take_bit_Suc_1
tff(fact_2966_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_2967_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_2968_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_2969_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_2970_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_2971_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_2972_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_2973_even__Suc,axiom,
    ! [Nb: 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,Nb))
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) ) ).

% even_Suc
tff(fact_2974_even__Suc__Suc__iff,axiom,
    ! [Nb: 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,Nb)))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) ) ).

% even_Suc_Suc_iff
tff(fact_2975_pow__divides__pow__iff,axiom,
    ! [Nb: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B) ) ) ).

% pow_divides_pow_iff
tff(fact_2976_pow__divides__pow__iff,axiom,
    ! [Nb: nat,A: int,B: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(A),Nb)),aa_nat_int(power_power_int(B),Nb))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ) ).

% pow_divides_pow_iff
tff(fact_2977_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_2978_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_2979_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_2980_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_2981_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_2982_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_2983_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_2984_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_2985_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_2986_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_2987_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_2988_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_2989_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(aa_int_fun_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_2990_odd__Suc__div__two,axiom,
    ! [Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,Nb)),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,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ).

% odd_Suc_div_two
tff(fact_2991_even__Suc__div__two,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ) ) ).

% even_Suc_div_two
tff(fact_2992_signed__take__bit__Suc__bit0,axiom,
    ! [Nb: nat,K: num] : aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(suc,Nb)),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(Nb),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_2993_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(aa_real_fun_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_2994_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_2995_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_2996_power__less__zero__eq,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),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))),Nb)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ) ).

% power_less_zero_eq
tff(fact_2997_power__less__zero__eq,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),Nb)),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))),Nb)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ) ).

% power_less_zero_eq
tff(fact_2998_power__less__zero__eq,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),Nb)),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))),Nb)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ) ).

% power_less_zero_eq
tff(fact_2999_power__less__zero__eq__numeral,axiom,
    ! [A: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,W))),zero_zero_real)
    <=> ( ~ 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_3000_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_3001_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_3002_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_3003_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_3004_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_3005_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_3006_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_3007_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(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,W)))
    <=> ( ( aa_num_nat(numeral_numeral_nat,W) = zero_zero_nat )
        | ( 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_3008_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_3009_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_3010_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_3011_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_3012_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_3013_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_3014_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_3015_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_3016_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_3017_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_3018_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_3019_even__power,axiom,
    ! [A: code_integer,Nb: 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),Nb))
    <=> ( 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),Nb) ) ) ).

% even_power
tff(fact_3020_even__power,axiom,
    ! [A: nat,Nb: 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),Nb))
    <=> ( 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),Nb) ) ) ).

% even_power
tff(fact_3021_even__power,axiom,
    ! [A: int,Nb: 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),Nb))
    <=> ( 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),Nb) ) ) ).

% even_power
tff(fact_3022_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_3023_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_3024_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_3025_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(aa_real_fun_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_3026_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_3027_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_3028_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [Nb: 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))),Nb),one_one_Code_integer))
    <=> ( Nb = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
tff(fact_3029_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [Nb: 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))),Nb)),one_one_nat))
    <=> ( Nb = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
tff(fact_3030_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [Nb: 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))),Nb)),one_one_int))
    <=> ( Nb = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
tff(fact_3031_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_3032_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_3033_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(aa_int_fun_int_int(modulo_modulo_int,A),B))) ).

% dvd_minus_mod
tff(fact_3034_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_3035_mod__eq__dvd__iff,axiom,
    ! [A: int,C: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C) = aa_int_int(aa_int_fun_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_3036_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_3037_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_3038_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_3039_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_3040_diff__right__commute,axiom,
    ! [A: real,C: real,B: real] : aa_real_real(minus_minus_real(aa_real_real(minus_minus_real(A),C)),B) = aa_real_real(minus_minus_real(aa_real_real(minus_minus_real(A),B)),C) ).

% diff_right_commute
tff(fact_3041_diff__right__commute,axiom,
    ! [A: rat,C: rat,B: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(minus_minus_rat(A),C)),B) = aa_rat_rat(minus_minus_rat(aa_rat_rat(minus_minus_rat(A),B)),C) ).

% diff_right_commute
tff(fact_3042_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_3043_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_3044_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_3045_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_3046_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_3047_diff__eq__diff__eq,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( aa_real_real(minus_minus_real(A),B) = aa_real_real(minus_minus_real(C),D) )
     => ( ( A = B )
      <=> ( C = D ) ) ) ).

% diff_eq_diff_eq
tff(fact_3048_diff__eq__diff__eq,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( aa_rat_rat(minus_minus_rat(A),B) = aa_rat_rat(minus_minus_rat(C),D) )
     => ( ( A = B )
      <=> ( C = D ) ) ) ).

% diff_eq_diff_eq
tff(fact_3049_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_3050_dvd__refl,axiom,
    ! [A: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),A) ).

% dvd_refl
tff(fact_3051_dvd__refl,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),A) ).

% dvd_refl
tff(fact_3052_dvd__refl,axiom,
    ! [A: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),A) ).

% dvd_refl
tff(fact_3053_dvd__diff,axiom,
    ! [Xb: code_integer,Yb: code_integer,Z: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Xb),Yb)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Xb),Z)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Xb),minus_8373710615458151222nteger(Yb,Z)) ) ) ).

% dvd_diff
tff(fact_3054_dvd__diff,axiom,
    ! [Xb: real,Yb: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Xb),Yb)
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Xb),Z)
       => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Xb),aa_real_real(minus_minus_real(Yb),Z)) ) ) ).

% dvd_diff
tff(fact_3055_dvd__diff,axiom,
    ! [Xb: rat,Yb: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Xb),Yb)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Xb),Z)
       => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Xb),aa_rat_rat(minus_minus_rat(Yb),Z)) ) ) ).

% dvd_diff
tff(fact_3056_dvd__diff,axiom,
    ! [Xb: int,Yb: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xb),Yb)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xb),Z)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xb),aa_int_int(aa_int_fun_int_int(minus_minus_int,Yb),Z)) ) ) ).

% dvd_diff
tff(fact_3057_signed__take__bit__diff,axiom,
    ! [Nb: nat,K: int,L: int] : aa_int_int(bit_ri631733984087533419it_int(Nb),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(bit_ri631733984087533419it_int(Nb),K)),aa_int_int(bit_ri631733984087533419it_int(Nb),L))) = aa_int_int(bit_ri631733984087533419it_int(Nb),aa_int_int(aa_int_fun_int_int(minus_minus_int,K),L)) ).

% signed_take_bit_diff
tff(fact_3058_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_3059_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_3060_zdvd__zdiffD,axiom,
    ! [K: int,Mb: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,K),aa_int_int(aa_int_fun_int_int(minus_minus_int,Mb),Nb))
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,K),Nb)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,K),Mb) ) ) ).

% zdvd_zdiffD
tff(fact_3061_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_3062_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_3063_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_3064_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_3065_lambda__zero,axiom,
    aTP_Lamm_aj = times_times_complex(zero_zero_complex) ).

% lambda_zero
tff(fact_3066_lambda__zero,axiom,
    aTP_Lamm_ak = times_times_real(zero_zero_real) ).

% lambda_zero
tff(fact_3067_lambda__zero,axiom,
    aTP_Lamm_al = times_times_rat(zero_zero_rat) ).

% lambda_zero
tff(fact_3068_lambda__zero,axiom,
    aTP_Lamm_am = aa_nat_fun_nat_nat(times_times_nat,zero_zero_nat) ).

% lambda_zero
tff(fact_3069_lambda__zero,axiom,
    aTP_Lamm_an = aa_int_fun_int_int(times_times_int,zero_zero_int) ).

% lambda_zero
tff(fact_3070_lambda__one,axiom,
    aTP_Lamm_ao = times_times_complex(one_one_complex) ).

% lambda_one
tff(fact_3071_lambda__one,axiom,
    aTP_Lamm_ap = times_times_real(one_one_real) ).

% lambda_one
tff(fact_3072_lambda__one,axiom,
    aTP_Lamm_aq = times_times_rat(one_one_rat) ).

% lambda_one
tff(fact_3073_lambda__one,axiom,
    aTP_Lamm_ar = aa_nat_fun_nat_nat(times_times_nat,one_one_nat) ).

% lambda_one
tff(fact_3074_lambda__one,axiom,
    aTP_Lamm_as = aa_int_fun_int_int(times_times_int,one_one_int) ).

% lambda_one
tff(fact_3075_max__def__raw,axiom,
    ! [X: set_nat,Xa: set_nat] :
      ord_max_set_nat(X,Xa) = $ite(aa_set_nat_o(ord_less_eq_set_nat(X),Xa),Xa,X) ).

% max_def_raw
tff(fact_3076_max__def__raw,axiom,
    ! [X: rat,Xa: rat] :
      aa_rat_rat(ord_max_rat(X),Xa) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Xa),Xa,X) ).

% max_def_raw
tff(fact_3077_max__def__raw,axiom,
    ! [X: num,Xa: num] :
      aa_num_num(ord_max_num(X),Xa) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Xa),Xa,X) ).

% max_def_raw
tff(fact_3078_max__def__raw,axiom,
    ! [X: nat,Xa: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),Xa) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Xa),Xa,X) ).

% max_def_raw
tff(fact_3079_max__def__raw,axiom,
    ! [X: int,Xa: int] :
      aa_int_int(aa_int_fun_int_int(ord_max_int,X),Xa) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Xa),Xa,X) ).

% max_def_raw
tff(fact_3080_inf__period_I3_J,axiom,
    ! [D: code_integer,D4: code_integer,Ta: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),D4)
     => ! [X: code_integer,K4: code_integer] :
          ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X,Ta))
        <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(minus_8373710615458151222nteger(X,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,K4),D4)),Ta)) ) ) ).

% inf_period(3)
tff(fact_3081_inf__period_I3_J,axiom,
    ! [D: real,D4: real,Ta: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),D4)
     => ! [X: real,K4: real] :
          ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(plus_plus_real(X),Ta))
        <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(plus_plus_real(aa_real_real(minus_minus_real(X),aa_real_real(times_times_real(K4),D4))),Ta)) ) ) ).

% inf_period(3)
tff(fact_3082_inf__period_I3_J,axiom,
    ! [D: rat,D4: rat,Ta: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),D4)
     => ! [X: rat,K4: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X),Ta))
        <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(aa_rat_rat(minus_minus_rat(X),aa_rat_rat(times_times_rat(K4),D4))),Ta)) ) ) ).

% inf_period(3)
tff(fact_3083_inf__period_I3_J,axiom,
    ! [D: int,D4: int,Ta: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),D4)
     => ! [X: 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,X),Ta))
        <=> 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,X),aa_int_int(aa_int_fun_int_int(times_times_int,K4),D4))),Ta)) ) ) ).

% inf_period(3)
tff(fact_3084_inf__period_I4_J,axiom,
    ! [D: code_integer,D4: code_integer,Ta: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),D4)
     => ! [X: code_integer,K4: code_integer] :
          ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X,Ta))
        <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(minus_8373710615458151222nteger(X,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,K4),D4)),Ta)) ) ) ).

% inf_period(4)
tff(fact_3085_inf__period_I4_J,axiom,
    ! [D: real,D4: real,Ta: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),D4)
     => ! [X: real,K4: real] :
          ( ~ aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(plus_plus_real(X),Ta))
        <=> ~ aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(plus_plus_real(aa_real_real(minus_minus_real(X),aa_real_real(times_times_real(K4),D4))),Ta)) ) ) ).

% inf_period(4)
tff(fact_3086_inf__period_I4_J,axiom,
    ! [D: rat,D4: rat,Ta: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),D4)
     => ! [X: rat,K4: rat] :
          ( ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X),Ta))
        <=> ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(aa_rat_rat(minus_minus_rat(X),aa_rat_rat(times_times_rat(K4),D4))),Ta)) ) ) ).

% inf_period(4)
tff(fact_3087_inf__period_I4_J,axiom,
    ! [D: int,D4: int,Ta: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),D4)
     => ! [X: 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,X),Ta))
        <=> ~ 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,X),aa_int_int(aa_int_fun_int_int(times_times_int,K4),D4))),Ta)) ) ) ).

% inf_period(4)
tff(fact_3088_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_3089_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_3090_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_3091_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_3092_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_3093_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_3094_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_3095_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_3096_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_3097_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(minus_minus_real(A),C)),aa_real_real(minus_minus_real(B),D)) ) ) ).

% diff_mono
tff(fact_3098_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,aa_rat_rat(minus_minus_rat(A),C)),aa_rat_rat(minus_minus_rat(B),D)) ) ) ).

% diff_mono
tff(fact_3099_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_3100_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(minus_minus_real(C),A)),aa_real_real(minus_minus_real(C),B)) ) ).

% diff_left_mono
tff(fact_3101_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,aa_rat_rat(minus_minus_rat(C),A)),aa_rat_rat(minus_minus_rat(C),B)) ) ).

% diff_left_mono
tff(fact_3102_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_3103_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(minus_minus_real(A),C)),aa_real_real(minus_minus_real(B),C)) ) ).

% diff_right_mono
tff(fact_3104_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,aa_rat_rat(minus_minus_rat(A),C)),aa_rat_rat(minus_minus_rat(B),C)) ) ).

% diff_right_mono
tff(fact_3105_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_3106_diff__eq__diff__less__eq,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( aa_real_real(minus_minus_real(A),B) = aa_real_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_3107_diff__eq__diff__less__eq,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( aa_rat_rat(minus_minus_rat(A),B) = aa_rat_rat(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_3108_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_3109_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_3110_eq__iff__diff__eq__0,axiom,
    ! [A: real,B: real] :
      ( ( A = B )
    <=> ( aa_real_real(minus_minus_real(A),B) = zero_zero_real ) ) ).

% eq_iff_diff_eq_0
tff(fact_3111_eq__iff__diff__eq__0,axiom,
    ! [A: rat,B: rat] :
      ( ( A = B )
    <=> ( aa_rat_rat(minus_minus_rat(A),B) = zero_zero_rat ) ) ).

% eq_iff_diff_eq_0
tff(fact_3112_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_3113_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_3114_dvd__triv__right,axiom,
    ! [A: real,B: real] : aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(times_times_real(B),A)) ).

% dvd_triv_right
tff(fact_3115_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_3116_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_3117_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_3118_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_3119_dvd__mult__right,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_real_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_3120_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_3121_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_3122_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_3123_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_3124_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(times_times_real(A),C)),aa_real_real(times_times_real(B),D)) ) ) ).

% mult_dvd_mono
tff(fact_3125_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_3126_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_3127_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_3128_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_3129_dvd__triv__left,axiom,
    ! [A: real,B: real] : aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(times_times_real(A),B)) ).

% dvd_triv_left
tff(fact_3130_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_3131_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_3132_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_3133_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_3134_dvd__mult__left,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_real_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_3135_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_3136_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_3137_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_3138_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_3139_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(times_times_real(B),C)) ) ).

% dvd_mult2
tff(fact_3140_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_3141_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_3142_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_3143_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_3144_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(times_times_real(B),C)) ) ).

% dvd_mult
tff(fact_3145_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_3146_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_3147_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_3148_dvd__def,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) ) ).

% dvd_def
tff(fact_3149_dvd__def,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,B),A)
    <=> ? [K3: real] : A = aa_real_real(times_times_real(B),K3) ) ).

% dvd_def
tff(fact_3150_dvd__def,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) ) ).

% dvd_def
tff(fact_3151_dvd__def,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) ) ).

% dvd_def
tff(fact_3152_dvd__def,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) ) ).

% dvd_def
tff(fact_3153_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_3154_dvdI,axiom,
    ! [A: real,B: real,K: real] :
      ( ( A = aa_real_real(times_times_real(B),K) )
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,B),A) ) ).

% dvdI
tff(fact_3155_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_3156_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_3157_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_3158_dvdE,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) ) ).

% dvdE
tff(fact_3159_dvdE,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,B),A)
     => ~ ! [K2: real] : A != aa_real_real(times_times_real(B),K2) ) ).

% dvdE
tff(fact_3160_dvdE,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) ) ).

% dvdE
tff(fact_3161_dvdE,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) ) ).

% dvdE
tff(fact_3162_dvdE,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) ) ).

% dvdE
tff(fact_3163_dvd__productE,axiom,
    ! [P2: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,P2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B))
     => ~ ! [X3: nat,Y: nat] :
            ( ( P2 = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X3),Y) )
           => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,X3),A)
             => ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Y),B) ) ) ) ).

% dvd_productE
tff(fact_3164_dvd__productE,axiom,
    ! [P2: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,P2),aa_int_int(aa_int_fun_int_int(times_times_int,A),B))
     => ~ ! [X3: int,Y: int] :
            ( ( P2 = aa_int_int(aa_int_fun_int_int(times_times_int,X3),Y) )
           => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,X3),A)
             => ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Y),B) ) ) ) ).

% dvd_productE
tff(fact_3165_division__decomp,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(times_times_nat,B),C))
     => ? [B6: nat,C4: nat] :
          ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B6),C4) )
          & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B6),B)
          & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C4),C) ) ) ).

% division_decomp
tff(fact_3166_division__decomp,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(times_times_int,B),C))
     => ? [B6: int,C4: int] :
          ( ( A = aa_int_int(aa_int_fun_int_int(times_times_int,B6),C4) )
          & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B6),B)
          & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C4),C) ) ) ).

% division_decomp
tff(fact_3167_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(minus_minus_real(A),C)),aa_real_real(minus_minus_real(B),D)) ) ) ).

% diff_strict_mono
tff(fact_3168_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,aa_rat_rat(minus_minus_rat(A),C)),aa_rat_rat(minus_minus_rat(B),D)) ) ) ).

% diff_strict_mono
tff(fact_3169_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_3170_diff__eq__diff__less,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( aa_real_real(minus_minus_real(A),B) = aa_real_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_3171_diff__eq__diff__less,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( aa_rat_rat(minus_minus_rat(A),B) = aa_rat_rat(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_3172_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_3173_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(minus_minus_real(C),A)),aa_real_real(minus_minus_real(C),B)) ) ).

% diff_strict_left_mono
tff(fact_3174_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,aa_rat_rat(minus_minus_rat(C),A)),aa_rat_rat(minus_minus_rat(C),B)) ) ).

% diff_strict_left_mono
tff(fact_3175_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_3176_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(minus_minus_real(A),C)),aa_real_real(minus_minus_real(B),C)) ) ).

% diff_strict_right_mono
tff(fact_3177_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,aa_rat_rat(minus_minus_rat(A),C)),aa_rat_rat(minus_minus_rat(B),C)) ) ).

% diff_strict_right_mono
tff(fact_3178_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_3179_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_3180_one__dvd,axiom,
    ! [A: complex] : aa_complex_o(dvd_dvd_complex(one_one_complex),A) ).

% one_dvd
tff(fact_3181_one__dvd,axiom,
    ! [A: real] : aa_real_o(aa_real_fun_real_o(dvd_dvd_real,one_one_real),A) ).

% one_dvd
tff(fact_3182_one__dvd,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,one_one_rat),A) ).

% one_dvd
tff(fact_3183_one__dvd,axiom,
    ! [A: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,one_one_nat),A) ).

% one_dvd
tff(fact_3184_one__dvd,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,one_one_int),A) ).

% one_dvd
tff(fact_3185_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_3186_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_3187_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_3188_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_3189_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_3190_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_3191_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_3192_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(plus_plus_real(B),C)) ) ) ).

% dvd_add
tff(fact_3193_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(plus_plus_rat(B),C)) ) ) ).

% dvd_add
tff(fact_3194_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_3195_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_3196_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_3197_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(plus_plus_real(B),C))
      <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B) ) ) ).

% dvd_add_left_iff
tff(fact_3198_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(plus_plus_rat(B),C))
      <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B) ) ) ).

% dvd_add_left_iff
tff(fact_3199_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_3200_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_3201_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_3202_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(plus_plus_real(B),C))
      <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),C) ) ) ).

% dvd_add_right_iff
tff(fact_3203_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(plus_plus_rat(B),C))
      <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),C) ) ) ).

% dvd_add_right_iff
tff(fact_3204_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_3205_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_3206_inf__period_I1_J,axiom,
    ! [P: fun_real_o,D4: real,Q: fun_real_o] :
      ( ! [X3: real,K2: real] :
          ( aa_real_o(P,X3)
        <=> aa_real_o(P,aa_real_real(minus_minus_real(X3),aa_real_real(times_times_real(K2),D4))) )
     => ( ! [X3: real,K2: real] :
            ( aa_real_o(Q,X3)
          <=> aa_real_o(Q,aa_real_real(minus_minus_real(X3),aa_real_real(times_times_real(K2),D4))) )
       => ! [X: real,K4: real] :
            ( ( aa_real_o(P,X)
              & aa_real_o(Q,X) )
          <=> ( aa_real_o(P,aa_real_real(minus_minus_real(X),aa_real_real(times_times_real(K4),D4)))
              & aa_real_o(Q,aa_real_real(minus_minus_real(X),aa_real_real(times_times_real(K4),D4))) ) ) ) ) ).

% inf_period(1)
tff(fact_3207_inf__period_I1_J,axiom,
    ! [P: fun_rat_o,D4: rat,Q: fun_rat_o] :
      ( ! [X3: rat,K2: rat] :
          ( aa_rat_o(P,X3)
        <=> aa_rat_o(P,aa_rat_rat(minus_minus_rat(X3),aa_rat_rat(times_times_rat(K2),D4))) )
     => ( ! [X3: rat,K2: rat] :
            ( aa_rat_o(Q,X3)
          <=> aa_rat_o(Q,aa_rat_rat(minus_minus_rat(X3),aa_rat_rat(times_times_rat(K2),D4))) )
       => ! [X: rat,K4: rat] :
            ( ( aa_rat_o(P,X)
              & aa_rat_o(Q,X) )
          <=> ( aa_rat_o(P,aa_rat_rat(minus_minus_rat(X),aa_rat_rat(times_times_rat(K4),D4)))
              & aa_rat_o(Q,aa_rat_rat(minus_minus_rat(X),aa_rat_rat(times_times_rat(K4),D4))) ) ) ) ) ).

% inf_period(1)
tff(fact_3208_inf__period_I1_J,axiom,
    ! [P: fun_int_o,D4: int,Q: fun_int_o] :
      ( ! [X3: int,K2: 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,K2),D4))) )
     => ( ! [X3: int,K2: int] :
            ( aa_int_o(Q,X3)
          <=> aa_int_o(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D4))) )
       => ! [X: int,K4: int] :
            ( ( aa_int_o(P,X)
              & aa_int_o(Q,X) )
          <=> ( aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),aa_int_int(aa_int_fun_int_int(times_times_int,K4),D4)))
              & aa_int_o(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),aa_int_int(aa_int_fun_int_int(times_times_int,K4),D4))) ) ) ) ) ).

% inf_period(1)
tff(fact_3209_inf__period_I2_J,axiom,
    ! [P: fun_real_o,D4: real,Q: fun_real_o] :
      ( ! [X3: real,K2: real] :
          ( aa_real_o(P,X3)
        <=> aa_real_o(P,aa_real_real(minus_minus_real(X3),aa_real_real(times_times_real(K2),D4))) )
     => ( ! [X3: real,K2: real] :
            ( aa_real_o(Q,X3)
          <=> aa_real_o(Q,aa_real_real(minus_minus_real(X3),aa_real_real(times_times_real(K2),D4))) )
       => ! [X: real,K4: real] :
            ( ( aa_real_o(P,X)
              | aa_real_o(Q,X) )
          <=> ( aa_real_o(P,aa_real_real(minus_minus_real(X),aa_real_real(times_times_real(K4),D4)))
              | aa_real_o(Q,aa_real_real(minus_minus_real(X),aa_real_real(times_times_real(K4),D4))) ) ) ) ) ).

% inf_period(2)
tff(fact_3210_inf__period_I2_J,axiom,
    ! [P: fun_rat_o,D4: rat,Q: fun_rat_o] :
      ( ! [X3: rat,K2: rat] :
          ( aa_rat_o(P,X3)
        <=> aa_rat_o(P,aa_rat_rat(minus_minus_rat(X3),aa_rat_rat(times_times_rat(K2),D4))) )
     => ( ! [X3: rat,K2: rat] :
            ( aa_rat_o(Q,X3)
          <=> aa_rat_o(Q,aa_rat_rat(minus_minus_rat(X3),aa_rat_rat(times_times_rat(K2),D4))) )
       => ! [X: rat,K4: rat] :
            ( ( aa_rat_o(P,X)
              | aa_rat_o(Q,X) )
          <=> ( aa_rat_o(P,aa_rat_rat(minus_minus_rat(X),aa_rat_rat(times_times_rat(K4),D4)))
              | aa_rat_o(Q,aa_rat_rat(minus_minus_rat(X),aa_rat_rat(times_times_rat(K4),D4))) ) ) ) ) ).

% inf_period(2)
tff(fact_3211_inf__period_I2_J,axiom,
    ! [P: fun_int_o,D4: int,Q: fun_int_o] :
      ( ! [X3: int,K2: 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,K2),D4))) )
     => ( ! [X3: int,K2: int] :
            ( aa_int_o(Q,X3)
          <=> aa_int_o(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D4))) )
       => ! [X: int,K4: int] :
            ( ( aa_int_o(P,X)
              | aa_int_o(Q,X) )
          <=> ( aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),aa_int_int(aa_int_fun_int_int(times_times_int,K4),D4)))
              | aa_int_o(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),aa_int_int(aa_int_fun_int_int(times_times_int,K4),D4))) ) ) ) ) ).

% inf_period(2)
tff(fact_3212_right__diff__distrib_H,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(times_times_real(A),aa_real_real(minus_minus_real(B),C)) = aa_real_real(minus_minus_real(aa_real_real(times_times_real(A),B)),aa_real_real(times_times_real(A),C)) ).

% right_diff_distrib'
tff(fact_3213_right__diff__distrib_H,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(A),aa_rat_rat(minus_minus_rat(B),C)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(A),B)),aa_rat_rat(times_times_rat(A),C)) ).

% right_diff_distrib'
tff(fact_3214_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_3215_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_3216_left__diff__distrib_H,axiom,
    ! [B: real,C: real,A: real] : aa_real_real(times_times_real(aa_real_real(minus_minus_real(B),C)),A) = aa_real_real(minus_minus_real(aa_real_real(times_times_real(B),A)),aa_real_real(times_times_real(C),A)) ).

% left_diff_distrib'
tff(fact_3217_left__diff__distrib_H,axiom,
    ! [B: rat,C: rat,A: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(B),C)),A) = aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(B),A)),aa_rat_rat(times_times_rat(C),A)) ).

% left_diff_distrib'
tff(fact_3218_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_3219_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_3220_right__diff__distrib,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(times_times_real(A),aa_real_real(minus_minus_real(B),C)) = aa_real_real(minus_minus_real(aa_real_real(times_times_real(A),B)),aa_real_real(times_times_real(A),C)) ).

% right_diff_distrib
tff(fact_3221_right__diff__distrib,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(A),aa_rat_rat(minus_minus_rat(B),C)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(A),B)),aa_rat_rat(times_times_rat(A),C)) ).

% right_diff_distrib
tff(fact_3222_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_3223_left__diff__distrib,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(times_times_real(aa_real_real(minus_minus_real(A),B)),C) = aa_real_real(minus_minus_real(aa_real_real(times_times_real(A),C)),aa_real_real(times_times_real(B),C)) ).

% left_diff_distrib
tff(fact_3224_left__diff__distrib,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(A),B)),C) = aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),C)) ).

% left_diff_distrib
tff(fact_3225_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_3226_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_3227_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_3228_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(divide_divide_real(A),C) = aa_real_real(divide_divide_real(B),C) )
        <=> ( A = B ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_3229_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)
       => ( ( aa_rat_rat(divide_divide_rat(A),C) = aa_rat_rat(divide_divide_rat(B),C) )
        <=> ( A = B ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_3230_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_3231_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_3232_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_3233_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_3234_dvd__div__eq__cancel,axiom,
    ! [A: real,C: real,B: real] :
      ( ( aa_real_real(divide_divide_real(A),C) = aa_real_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_3235_dvd__div__eq__cancel,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( aa_rat_rat(divide_divide_rat(A),C) = aa_rat_rat(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_3236_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_3237_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_3238_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_3239_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_3240_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_3241_add__diff__add,axiom,
    ! [A: real,C: real,B: real,D: real] : aa_real_real(minus_minus_real(aa_real_real(plus_plus_real(A),C)),aa_real_real(plus_plus_real(B),D)) = aa_real_real(plus_plus_real(aa_real_real(minus_minus_real(A),B)),aa_real_real(minus_minus_real(C),D)) ).

% add_diff_add
tff(fact_3242_add__diff__add,axiom,
    ! [A: rat,C: rat,B: rat,D: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(A),C)),aa_rat_rat(plus_plus_rat(B),D)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(minus_minus_rat(A),B)),aa_rat_rat(minus_minus_rat(C),D)) ).

% add_diff_add
tff(fact_3243_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_3244_group__cancel_Osub1,axiom,
    ! [A3: real,K: real,A: real,B: real] :
      ( ( A3 = aa_real_real(plus_plus_real(K),A) )
     => ( aa_real_real(minus_minus_real(A3),B) = aa_real_real(plus_plus_real(K),aa_real_real(minus_minus_real(A),B)) ) ) ).

% group_cancel.sub1
tff(fact_3245_group__cancel_Osub1,axiom,
    ! [A3: rat,K: rat,A: rat,B: rat] :
      ( ( A3 = aa_rat_rat(plus_plus_rat(K),A) )
     => ( aa_rat_rat(minus_minus_rat(A3),B) = aa_rat_rat(plus_plus_rat(K),aa_rat_rat(minus_minus_rat(A),B)) ) ) ).

% group_cancel.sub1
tff(fact_3246_group__cancel_Osub1,axiom,
    ! [A3: int,K: int,A: int,B: int] :
      ( ( A3 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),A) )
     => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A3),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)) ) ) ).

% group_cancel.sub1
tff(fact_3247_diff__eq__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( aa_real_real(minus_minus_real(A),B) = C )
    <=> ( A = aa_real_real(plus_plus_real(C),B) ) ) ).

% diff_eq_eq
tff(fact_3248_diff__eq__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( aa_rat_rat(minus_minus_rat(A),B) = C )
    <=> ( A = aa_rat_rat(plus_plus_rat(C),B) ) ) ).

% diff_eq_eq
tff(fact_3249_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_3250_eq__diff__eq,axiom,
    ! [A: real,C: real,B: real] :
      ( ( A = aa_real_real(minus_minus_real(C),B) )
    <=> ( aa_real_real(plus_plus_real(A),B) = C ) ) ).

% eq_diff_eq
tff(fact_3251_eq__diff__eq,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( A = aa_rat_rat(minus_minus_rat(C),B) )
    <=> ( aa_rat_rat(plus_plus_rat(A),B) = C ) ) ).

% eq_diff_eq
tff(fact_3252_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_3253_add__diff__eq,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(plus_plus_real(A),aa_real_real(minus_minus_real(B),C)) = aa_real_real(minus_minus_real(aa_real_real(plus_plus_real(A),B)),C) ).

% add_diff_eq
tff(fact_3254_add__diff__eq,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(plus_plus_rat(A),aa_rat_rat(minus_minus_rat(B),C)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(A),B)),C) ).

% add_diff_eq
tff(fact_3255_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_3256_diff__diff__eq2,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(minus_minus_real(A),aa_real_real(minus_minus_real(B),C)) = aa_real_real(minus_minus_real(aa_real_real(plus_plus_real(A),C)),B) ).

% diff_diff_eq2
tff(fact_3257_diff__diff__eq2,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(minus_minus_rat(A),aa_rat_rat(minus_minus_rat(B),C)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(A),C)),B) ).

% diff_diff_eq2
tff(fact_3258_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_3259_diff__add__eq,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(plus_plus_real(aa_real_real(minus_minus_real(A),B)),C) = aa_real_real(minus_minus_real(aa_real_real(plus_plus_real(A),C)),B) ).

% diff_add_eq
tff(fact_3260_diff__add__eq,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(minus_minus_rat(A),B)),C) = aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(A),C)),B) ).

% diff_add_eq
tff(fact_3261_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_3262_diff__add__eq__diff__diff__swap,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(minus_minus_real(A),aa_real_real(plus_plus_real(B),C)) = aa_real_real(minus_minus_real(aa_real_real(minus_minus_real(A),C)),B) ).

% diff_add_eq_diff_diff_swap
tff(fact_3263_diff__add__eq__diff__diff__swap,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(minus_minus_rat(A),aa_rat_rat(plus_plus_rat(B),C)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(minus_minus_rat(A),C)),B) ).

% diff_add_eq_diff_diff_swap
tff(fact_3264_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_3265_add__implies__diff,axiom,
    ! [C: real,B: real,A: real] :
      ( ( aa_real_real(plus_plus_real(C),B) = A )
     => ( C = aa_real_real(minus_minus_real(A),B) ) ) ).

% add_implies_diff
tff(fact_3266_add__implies__diff,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( aa_rat_rat(plus_plus_rat(C),B) = A )
     => ( C = aa_rat_rat(minus_minus_rat(A),B) ) ) ).

% add_implies_diff
tff(fact_3267_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_3268_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_3269_diff__diff__eq,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(minus_minus_real(aa_real_real(minus_minus_real(A),B)),C) = aa_real_real(minus_minus_real(A),aa_real_real(plus_plus_real(B),C)) ).

% diff_diff_eq
tff(fact_3270_diff__diff__eq,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(minus_minus_rat(A),B)),C) = aa_rat_rat(minus_minus_rat(A),aa_rat_rat(plus_plus_rat(B),C)) ).

% diff_diff_eq
tff(fact_3271_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_3272_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_3273_dvd__power__same,axiom,
    ! [Xb: code_integer,Yb: code_integer,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Xb),Yb)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(Xb),Nb)),aa_nat_Code_integer(power_8256067586552552935nteger(Yb),Nb)) ) ).

% dvd_power_same
tff(fact_3274_dvd__power__same,axiom,
    ! [Xb: nat,Yb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Xb),Yb)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Yb),Nb)) ) ).

% dvd_power_same
tff(fact_3275_dvd__power__same,axiom,
    ! [Xb: int,Yb: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xb),Yb)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(Xb),Nb)),aa_nat_int(power_power_int(Yb),Nb)) ) ).

% dvd_power_same
tff(fact_3276_dvd__power__same,axiom,
    ! [Xb: real,Yb: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Xb),Yb)
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Yb),Nb)) ) ).

% dvd_power_same
tff(fact_3277_dvd__power__same,axiom,
    ! [Xb: complex,Yb: complex,Nb: nat] :
      ( aa_complex_o(dvd_dvd_complex(Xb),Yb)
     => aa_complex_o(dvd_dvd_complex(aa_nat_complex(power_power_complex(Xb),Nb)),aa_nat_complex(power_power_complex(Yb),Nb)) ) ).

% dvd_power_same
tff(fact_3278_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_3279_diff__divide__distrib,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(A),B)),C) = aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(A),C)),aa_real_real(divide_divide_real(B),C)) ).

% diff_divide_distrib
tff(fact_3280_diff__divide__distrib,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(A),B)),C) = aa_rat_rat(minus_minus_rat(aa_rat_rat(divide_divide_rat(A),C)),aa_rat_rat(divide_divide_rat(B),C)) ).

% diff_divide_distrib
tff(fact_3281_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_3282_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_3283_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_3284_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_3285_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_3286_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_3287_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_3288_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(aa_int_fun_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_3289_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_3290_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_3291_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(aa_int_fun_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_3292_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_3293_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_3294_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(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C) ) ) ).

% mod_mod_cancel
tff(fact_3295_dvd__mod,axiom,
    ! [K: code_integer,Mb: code_integer,Nb: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,K),Mb)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,K),Nb)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,K),modulo364778990260209775nteger(Mb,Nb)) ) ) ).

% dvd_mod
tff(fact_3296_dvd__mod,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb)) ) ) ).

% dvd_mod
tff(fact_3297_dvd__mod,axiom,
    ! [K: int,Mb: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,K),Mb)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,K),Nb)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,K),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Mb),Nb)) ) ) ).

% dvd_mod
tff(fact_3298_mod__diff__right__eq,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C))),C) = aa_int_int(aa_int_fun_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_3299_mod__diff__left__eq,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C)),B)),C) = aa_int_int(aa_int_fun_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_3300_mod__diff__cong,axiom,
    ! [A: int,C: int,A2: int,B: int,B2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),C) )
     => ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B2),C) )
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B2)),C) ) ) ) ).

% mod_diff_cong
tff(fact_3301_mod__diff__eq,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C))),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C) ).

% mod_diff_eq
tff(fact_3302_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_3303_signed__take__bit__mult,axiom,
    ! [Nb: nat,K: int,L: int] : aa_int_int(bit_ri631733984087533419it_int(Nb),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_ri631733984087533419it_int(Nb),K)),aa_int_int(bit_ri631733984087533419it_int(Nb),L))) = aa_int_int(bit_ri631733984087533419it_int(Nb),aa_int_int(aa_int_fun_int_int(times_times_int,K),L)) ).

% signed_take_bit_mult
tff(fact_3304_signed__take__bit__add,axiom,
    ! [Nb: nat,K: int,L: int] : aa_int_int(bit_ri631733984087533419it_int(Nb),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(bit_ri631733984087533419it_int(Nb),K)),aa_int_int(bit_ri631733984087533419it_int(Nb),L))) = aa_int_int(bit_ri631733984087533419it_int(Nb),aa_int_int(aa_int_fun_int_int(plus_plus_int,K),L)) ).

% signed_take_bit_add
tff(fact_3305_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_3306_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_3307_max__diff__distrib__left,axiom,
    ! [Xb: real,Yb: real,Z: real] : aa_real_real(minus_minus_real(aa_real_real(ord_max_real(Xb),Yb)),Z) = aa_real_real(ord_max_real(aa_real_real(minus_minus_real(Xb),Z)),aa_real_real(minus_minus_real(Yb),Z)) ).

% max_diff_distrib_left
tff(fact_3308_max__diff__distrib__left,axiom,
    ! [Xb: rat,Yb: rat,Z: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(ord_max_rat(Xb),Yb)),Z) = aa_rat_rat(ord_max_rat(aa_rat_rat(minus_minus_rat(Xb),Z)),aa_rat_rat(minus_minus_rat(Yb),Z)) ).

% max_diff_distrib_left
tff(fact_3309_max__diff__distrib__left,axiom,
    ! [Xb: int,Yb: int,Z: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(ord_max_int,Xb),Yb)),Z) = aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),Z)),aa_int_int(aa_int_fun_int_int(minus_minus_int,Yb),Z)) ).

% max_diff_distrib_left
tff(fact_3310_numeral__code_I2_J,axiom,
    ! [Nb: num] :
      numera6690914467698888265omplex(aa_num_num(bit0,Nb)) = $let(
        m2: complex,
        m2:= numera6690914467698888265omplex(Nb),
        aa_complex_complex(plus_plus_complex(m2),m2) ) ).

% numeral_code(2)
tff(fact_3311_numeral__code_I2_J,axiom,
    ! [Nb: num] :
      aa_num_real(numeral_numeral_real,aa_num_num(bit0,Nb)) = $let(
        m2: real,
        m2:= aa_num_real(numeral_numeral_real,Nb),
        aa_real_real(plus_plus_real(m2),m2) ) ).

% numeral_code(2)
tff(fact_3312_numeral__code_I2_J,axiom,
    ! [Nb: num] :
      aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,Nb)) = $let(
        m2: rat,
        m2:= aa_num_rat(numeral_numeral_rat,Nb),
        aa_rat_rat(plus_plus_rat(m2),m2) ) ).

% numeral_code(2)
tff(fact_3313_numeral__code_I2_J,axiom,
    ! [Nb: num] :
      aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Nb)) = $let(
        m2: nat,
        m2:= aa_num_nat(numeral_numeral_nat,Nb),
        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,m2),m2) ) ).

% numeral_code(2)
tff(fact_3314_numeral__code_I2_J,axiom,
    ! [Nb: num] :
      aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)) = $let(
        m2: int,
        m2:= aa_num_int(numeral_numeral_int,Nb),
        aa_int_int(aa_int_fun_int_int(plus_plus_int,m2),m2) ) ).

% numeral_code(2)
tff(fact_3315_set__vebt__def,axiom,
    ! [Ta: vEBT_VEBT] : vEBT_set_vebt(Ta) = collect_nat(vEBT_V8194947554948674370ptions(Ta)) ).

% set_vebt_def
tff(fact_3316_even__signed__take__bit__iff,axiom,
    ! [Mb: nat,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_ri6519982836138164636nteger(Mb,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_3317_even__signed__take__bit__iff,axiom,
    ! [Mb: 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(Mb),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_3318_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_3319_power__numeral__even,axiom,
    ! [Z: real,W: num] :
      aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,W))) = $let(
        w: real,
        w:= aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),aa_num_nat(numeral_numeral_nat,W)),
        aa_real_real(times_times_real(w),w) ) ).

% power_numeral_even
tff(fact_3320_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_3321_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_3322_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_3323_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_3324_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(minus_minus_real(A),B)),zero_zero_real) ) ).

% le_iff_diff_le_0
tff(fact_3325_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,aa_rat_rat(minus_minus_rat(A),B)),zero_zero_rat) ) ).

% le_iff_diff_le_0
tff(fact_3326_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_3327_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_3328_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_3329_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_3330_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(minus_minus_real(A),B)),zero_zero_real) ) ).

% less_iff_diff_less_0
tff(fact_3331_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,aa_rat_rat(minus_minus_rat(A),B)),zero_zero_rat) ) ).

% less_iff_diff_less_0
tff(fact_3332_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_3333_minf_I10_J,axiom,
    ! [D: code_integer,S2: code_integer] :
    ? [Z2: code_integer] :
    ! [X: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),Z2)
     => ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X,S2))
      <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X,S2)) ) ) ).

% minf(10)
tff(fact_3334_minf_I10_J,axiom,
    ! [D: real,S2: real] :
    ? [Z2: real] :
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Z2)
     => ( ~ aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(plus_plus_real(X),S2))
      <=> ~ aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(plus_plus_real(X),S2)) ) ) ).

% minf(10)
tff(fact_3335_minf_I10_J,axiom,
    ! [D: rat,S2: rat] :
    ? [Z2: rat] :
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Z2)
     => ( ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X),S2))
      <=> ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X),S2)) ) ) ).

% minf(10)
tff(fact_3336_minf_I10_J,axiom,
    ! [D: nat,S2: nat] :
    ? [Z2: nat] :
    ! [X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Z2)
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),S2))
      <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),S2)) ) ) ).

% minf(10)
tff(fact_3337_minf_I10_J,axiom,
    ! [D: int,S2: int] :
    ? [Z2: int] :
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Z2)
     => ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),S2))
      <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),S2)) ) ) ).

% minf(10)
tff(fact_3338_minf_I9_J,axiom,
    ! [D: code_integer,S2: code_integer] :
    ? [Z2: code_integer] :
    ! [X: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),Z2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X,S2))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X,S2)) ) ) ).

% minf(9)
tff(fact_3339_minf_I9_J,axiom,
    ! [D: real,S2: real] :
    ? [Z2: real] :
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Z2)
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(plus_plus_real(X),S2))
      <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(plus_plus_real(X),S2)) ) ) ).

% minf(9)
tff(fact_3340_minf_I9_J,axiom,
    ! [D: rat,S2: rat] :
    ? [Z2: rat] :
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Z2)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X),S2))
      <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X),S2)) ) ) ).

% minf(9)
tff(fact_3341_minf_I9_J,axiom,
    ! [D: nat,S2: nat] :
    ? [Z2: nat] :
    ! [X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Z2)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),S2))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),S2)) ) ) ).

% minf(9)
tff(fact_3342_minf_I9_J,axiom,
    ! [D: int,S2: int] :
    ? [Z2: int] :
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Z2)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),S2))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),S2)) ) ) ).

% minf(9)
tff(fact_3343_pinf_I10_J,axiom,
    ! [D: code_integer,S2: code_integer] :
    ? [Z2: code_integer] :
    ! [X: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Z2),X)
     => ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X,S2))
      <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X,S2)) ) ) ).

% pinf(10)
tff(fact_3344_pinf_I10_J,axiom,
    ! [D: real,S2: real] :
    ? [Z2: real] :
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),X)
     => ( ~ aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(plus_plus_real(X),S2))
      <=> ~ aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(plus_plus_real(X),S2)) ) ) ).

% pinf(10)
tff(fact_3345_pinf_I10_J,axiom,
    ! [D: rat,S2: rat] :
    ? [Z2: rat] :
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),X)
     => ( ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X),S2))
      <=> ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X),S2)) ) ) ).

% pinf(10)
tff(fact_3346_pinf_I10_J,axiom,
    ! [D: nat,S2: nat] :
    ? [Z2: nat] :
    ! [X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z2),X)
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),S2))
      <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),S2)) ) ) ).

% pinf(10)
tff(fact_3347_pinf_I10_J,axiom,
    ! [D: int,S2: int] :
    ? [Z2: int] :
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),X)
     => ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),S2))
      <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),S2)) ) ) ).

% pinf(10)
tff(fact_3348_pinf_I9_J,axiom,
    ! [D: code_integer,S2: code_integer] :
    ? [Z2: code_integer] :
    ! [X: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Z2),X)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X,S2))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X,S2)) ) ) ).

% pinf(9)
tff(fact_3349_pinf_I9_J,axiom,
    ! [D: real,S2: real] :
    ? [Z2: real] :
    ! [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),X)
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(plus_plus_real(X),S2))
      <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(plus_plus_real(X),S2)) ) ) ).

% pinf(9)
tff(fact_3350_pinf_I9_J,axiom,
    ! [D: rat,S2: rat] :
    ? [Z2: rat] :
    ! [X: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),X)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X),S2))
      <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X),S2)) ) ) ).

% pinf(9)
tff(fact_3351_pinf_I9_J,axiom,
    ! [D: nat,S2: nat] :
    ? [Z2: nat] :
    ! [X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z2),X)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),S2))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),S2)) ) ) ).

% pinf(9)
tff(fact_3352_pinf_I9_J,axiom,
    ! [D: int,S2: int] :
    ? [Z2: int] :
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),X)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),S2))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),S2)) ) ) ).

% pinf(9)
tff(fact_3353_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_3354_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_3355_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(divide_divide_real(A),B) = zero_zero_real )
      <=> ( A = zero_zero_real ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_3356_dvd__div__eq__0__iff,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,B),A)
     => ( ( aa_rat_rat(divide_divide_rat(A),B) = zero_zero_rat )
      <=> ( A = zero_zero_rat ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_3357_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_3358_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_3359_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_3360_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_3361_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_3362_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_3363_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_3364_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_3365_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_3366_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_3367_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_3368_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_3369_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_3370_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_3371_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_3372_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_3373_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_3374_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_3375_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_3376_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_3377_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_3378_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_3379_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_3380_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(minus_minus_real(A),B)),C)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(plus_plus_real(C),B)) ) ).

% diff_le_eq
tff(fact_3381_diff__le__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(A),B)),C)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(plus_plus_rat(C),B)) ) ).

% diff_le_eq
tff(fact_3382_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_3383_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(minus_minus_real(C),B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(A),B)),C) ) ).

% le_diff_eq
tff(fact_3384_le__diff__eq,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(minus_minus_rat(C),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A),B)),C) ) ).

% le_diff_eq
tff(fact_3385_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_3386_add__le__imp__le__diff,axiom,
    ! [I3: real,K: real,Nb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(I3),K)),Nb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,I3),aa_real_real(minus_minus_real(Nb),K)) ) ).

% add_le_imp_le_diff
tff(fact_3387_add__le__imp__le__diff,axiom,
    ! [I3: rat,K: rat,Nb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(I3),K)),Nb)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,I3),aa_rat_rat(minus_minus_rat(Nb),K)) ) ).

% add_le_imp_le_diff
tff(fact_3388_add__le__imp__le__diff,axiom,
    ! [I3: nat,K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I3),K)),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I3),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)) ) ).

% add_le_imp_le_diff
tff(fact_3389_add__le__imp__le__diff,axiom,
    ! [I3: int,K: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I3),K)),Nb)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I3),aa_int_int(aa_int_fun_int_int(minus_minus_int,Nb),K)) ) ).

% add_le_imp_le_diff
tff(fact_3390_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_3391_add__le__add__imp__diff__le,axiom,
    ! [I3: real,K: real,Nb: real,J: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(I3),K)),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Nb),aa_real_real(plus_plus_real(J),K))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(I3),K)),Nb)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Nb),aa_real_real(plus_plus_real(J),K))
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(Nb),K)),J) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_3392_add__le__add__imp__diff__le,axiom,
    ! [I3: rat,K: rat,Nb: rat,J: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(I3),K)),Nb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Nb),aa_rat_rat(plus_plus_rat(J),K))
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(I3),K)),Nb)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Nb),aa_rat_rat(plus_plus_rat(J),K))
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(Nb),K)),J) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_3393_add__le__add__imp__diff__le,axiom,
    ! [I3: nat,K: nat,Nb: 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,I3),K)),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),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,I3),K)),Nb)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),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,Nb),K)),J) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_3394_add__le__add__imp__diff__le,axiom,
    ! [I3: int,K: int,Nb: 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,I3),K)),Nb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Nb),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,I3),K)),Nb)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Nb),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,Nb),K)),J) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_3395_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_3396_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_3397_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_3398_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_3399_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_3400_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_3401_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_3402_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_3403_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_3404_diff__less__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(minus_minus_real(A),B)),C)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(plus_plus_real(C),B)) ) ).

% diff_less_eq
tff(fact_3405_diff__less__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(A),B)),C)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(plus_plus_rat(C),B)) ) ).

% diff_less_eq
tff(fact_3406_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_3407_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(minus_minus_real(C),B))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(A),B)),C) ) ).

% less_diff_eq
tff(fact_3408_less__diff__eq,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(minus_minus_rat(C),B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(A),B)),C) ) ).

% less_diff_eq
tff(fact_3409_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_3410_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(plus_plus_real(B),aa_real_real(minus_minus_real(A),B)) = A ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_3411_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(plus_plus_rat(B),aa_rat_rat(minus_minus_rat(A),B)) = A ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_3412_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_3413_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_3414_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_3415_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_3416_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_3417_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_3418_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_3419_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_3420_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_3421_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_3422_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_3423_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_3424_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_3425_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_3426_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_3427_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_3428_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_3429_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_3430_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_3431_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_3432_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_3433_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_3434_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_3435_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_3436_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_3437_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_3438_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_3439_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_3440_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_3441_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_3442_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_3443_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_3444_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_3445_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_3446_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_3447_mult__diff__mult,axiom,
    ! [Xb: real,Yb: real,A: real,B: real] : aa_real_real(minus_minus_real(aa_real_real(times_times_real(Xb),Yb)),aa_real_real(times_times_real(A),B)) = aa_real_real(plus_plus_real(aa_real_real(times_times_real(Xb),aa_real_real(minus_minus_real(Yb),B))),aa_real_real(times_times_real(aa_real_real(minus_minus_real(Xb),A)),B)) ).

% mult_diff_mult
tff(fact_3448_mult__diff__mult,axiom,
    ! [Xb: rat,Yb: rat,A: rat,B: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(Xb),Yb)),aa_rat_rat(times_times_rat(A),B)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(Xb),aa_rat_rat(minus_minus_rat(Yb),B))),aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(Xb),A)),B)) ).

% mult_diff_mult
tff(fact_3449_mult__diff__mult,axiom,
    ! [Xb: int,Yb: 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,Xb),Yb)),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,Xb),aa_int_int(aa_int_fun_int_int(minus_minus_int,Yb),B))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),A)),B)) ).

% mult_diff_mult
tff(fact_3450_square__diff__square__factored,axiom,
    ! [Xb: real,Yb: real] : aa_real_real(minus_minus_real(aa_real_real(times_times_real(Xb),Xb)),aa_real_real(times_times_real(Yb),Yb)) = aa_real_real(times_times_real(aa_real_real(plus_plus_real(Xb),Yb)),aa_real_real(minus_minus_real(Xb),Yb)) ).

% square_diff_square_factored
tff(fact_3451_square__diff__square__factored,axiom,
    ! [Xb: rat,Yb: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(Xb),Xb)),aa_rat_rat(times_times_rat(Yb),Yb)) = aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(Xb),Yb)),aa_rat_rat(minus_minus_rat(Xb),Yb)) ).

% square_diff_square_factored
tff(fact_3452_square__diff__square__factored,axiom,
    ! [Xb: int,Yb: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Xb),Xb)),aa_int_int(aa_int_fun_int_int(times_times_int,Yb),Yb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Yb)),aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),Yb)) ).

% square_diff_square_factored
tff(fact_3453_eq__add__iff2,axiom,
    ! [A: real,E2: real,C: real,B: real,D: real] :
      ( ( aa_real_real(plus_plus_real(aa_real_real(times_times_real(A),E2)),C) = aa_real_real(plus_plus_real(aa_real_real(times_times_real(B),E2)),D) )
    <=> ( C = aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_real_real(minus_minus_real(B),A)),E2)),D) ) ) ).

% eq_add_iff2
tff(fact_3454_eq__add__iff2,axiom,
    ! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
      ( ( aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),E2)),C) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(B),E2)),D) )
    <=> ( C = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(B),A)),E2)),D) ) ) ).

% eq_add_iff2
tff(fact_3455_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_3456_eq__add__iff1,axiom,
    ! [A: real,E2: real,C: real,B: real,D: real] :
      ( ( aa_real_real(plus_plus_real(aa_real_real(times_times_real(A),E2)),C) = aa_real_real(plus_plus_real(aa_real_real(times_times_real(B),E2)),D) )
    <=> ( aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_real_real(minus_minus_real(A),B)),E2)),C) = D ) ) ).

% eq_add_iff1
tff(fact_3457_eq__add__iff1,axiom,
    ! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
      ( ( aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),E2)),C) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(B),E2)),D) )
    <=> ( aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(A),B)),E2)),C) = D ) ) ).

% eq_add_iff1
tff(fact_3458_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_3459_div__power,axiom,
    ! [B: code_integer,A: code_integer,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B),A)
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(divide6298287555418463151nteger(A,B)),Nb) = divide6298287555418463151nteger(aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb),aa_nat_Code_integer(power_8256067586552552935nteger(B),Nb)) ) ) ).

% div_power
tff(fact_3460_div__power,axiom,
    ! [B: nat,A: nat,Nb: 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)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb)) ) ) ).

% div_power
tff(fact_3461_div__power,axiom,
    ! [B: int,A: int,Nb: 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)),Nb) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(power_power_int(A),Nb)),aa_nat_int(power_power_int(B),Nb)) ) ) ).

% div_power
tff(fact_3462_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_3463_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_3464_mod__eq__0__iff__dvd,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_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_3465_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_3466_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_3467_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(aa_int_fun_int_int(modulo_modulo_int,B),A) = zero_zero_int ) ) ).

% dvd_eq_mod_eq_0
tff(fact_3468_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_3469_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_3470_mod__0__imp__dvd,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_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_3471_dvd__power__le,axiom,
    ! [Xb: code_integer,Yb: code_integer,Nb: nat,Mb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Xb),Yb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(Xb),Nb)),aa_nat_Code_integer(power_8256067586552552935nteger(Yb),Mb)) ) ) ).

% dvd_power_le
tff(fact_3472_dvd__power__le,axiom,
    ! [Xb: nat,Yb: nat,Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Xb),Yb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Yb),Mb)) ) ) ).

% dvd_power_le
tff(fact_3473_dvd__power__le,axiom,
    ! [Xb: int,Yb: int,Nb: nat,Mb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xb),Yb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(Xb),Nb)),aa_nat_int(power_power_int(Yb),Mb)) ) ) ).

% dvd_power_le
tff(fact_3474_dvd__power__le,axiom,
    ! [Xb: real,Yb: real,Nb: nat,Mb: nat] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Xb),Yb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Yb),Mb)) ) ) ).

% dvd_power_le
tff(fact_3475_dvd__power__le,axiom,
    ! [Xb: complex,Yb: complex,Nb: nat,Mb: nat] :
      ( aa_complex_o(dvd_dvd_complex(Xb),Yb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => aa_complex_o(dvd_dvd_complex(aa_nat_complex(power_power_complex(Xb),Nb)),aa_nat_complex(power_power_complex(Yb),Mb)) ) ) ).

% dvd_power_le
tff(fact_3476_power__le__dvd,axiom,
    ! [A: code_integer,Nb: nat,B: code_integer,Mb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb)),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(A),Mb)),B) ) ) ).

% power_le_dvd
tff(fact_3477_power__le__dvd,axiom,
    ! [A: nat,Nb: nat,B: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Mb)),B) ) ) ).

% power_le_dvd
tff(fact_3478_power__le__dvd,axiom,
    ! [A: int,Nb: nat,B: int,Mb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(A),Nb)),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(A),Mb)),B) ) ) ).

% power_le_dvd
tff(fact_3479_power__le__dvd,axiom,
    ! [A: real,Nb: nat,B: real,Mb: nat] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
       => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Mb)),B) ) ) ).

% power_le_dvd
tff(fact_3480_power__le__dvd,axiom,
    ! [A: complex,Nb: nat,B: complex,Mb: nat] :
      ( aa_complex_o(dvd_dvd_complex(aa_nat_complex(power_power_complex(A),Nb)),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
       => aa_complex_o(dvd_dvd_complex(aa_nat_complex(power_power_complex(A),Mb)),B) ) ) ).

% power_le_dvd
tff(fact_3481_le__imp__power__dvd,axiom,
    ! [Mb: nat,Nb: nat,A: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(A),Mb)),aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb)) ) ).

% le_imp_power_dvd
tff(fact_3482_le__imp__power__dvd,axiom,
    ! [Mb: nat,Nb: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ) ).

% le_imp_power_dvd
tff(fact_3483_le__imp__power__dvd,axiom,
    ! [Mb: nat,Nb: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(A),Mb)),aa_nat_int(power_power_int(A),Nb)) ) ).

% le_imp_power_dvd
tff(fact_3484_le__imp__power__dvd,axiom,
    ! [Mb: nat,Nb: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Mb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ) ).

% le_imp_power_dvd
tff(fact_3485_le__imp__power__dvd,axiom,
    ! [Mb: nat,Nb: nat,A: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_complex_o(dvd_dvd_complex(aa_nat_complex(power_power_complex(A),Mb)),aa_nat_complex(power_power_complex(A),Nb)) ) ).

% le_imp_power_dvd
tff(fact_3486_nat__dvd__not__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
       => ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Nb),Mb) ) ) ).

% nat_dvd_not_less
tff(fact_3487_dvd__pos__nat,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb) ) ) ).

% dvd_pos_nat
tff(fact_3488_zdvd__antisym__nonneg,axiom,
    ! [Mb: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Mb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Nb)
       => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Mb),Nb)
         => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Nb),Mb)
           => ( Mb = Nb ) ) ) ) ) ).

% zdvd_antisym_nonneg
tff(fact_3489_zdvd__not__zless,axiom,
    ! [Mb: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Mb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Mb),Nb)
       => ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Nb),Mb) ) ) ).

% zdvd_not_zless
tff(fact_3490_zdvd__mono,axiom,
    ! [K: int,Mb: int,Ta: int] :
      ( ( K != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Mb),Ta)
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),Mb)),aa_int_int(aa_int_fun_int_int(times_times_int,K),Ta)) ) ) ).

% zdvd_mono
tff(fact_3491_zdvd__mult__cancel,axiom,
    ! [K: int,Mb: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),Mb)),aa_int_int(aa_int_fun_int_int(times_times_int,K),Nb))
     => ( ( K != zero_zero_int )
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Mb),Nb) ) ) ).

% zdvd_mult_cancel
tff(fact_3492_bezout__lemma__nat,axiom,
    ! [D: nat,A: nat,B: nat,Xb: nat,Yb: 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),Xb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Yb)),D) )
            | ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Xb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Yb)),D) ) )
         => ? [X3: 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),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),X3) = 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)),Y)),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)),X3) = 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) ) ) ) ) ) ) ).

% bezout_lemma_nat
tff(fact_3493_bezout__add__nat,axiom,
    ! [A: nat,B: nat] :
    ? [D3: nat,X3: nat,Y: 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),X3) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y)),D3) )
        | ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),X3) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y)),D3) ) ) ) ).

% bezout_add_nat
tff(fact_3494_int__le__induct,axiom,
    ! [I3: int,K: int,P: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I3),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,I3) ) ) ) ).

% int_le_induct
tff(fact_3495_int__less__induct,axiom,
    ! [I3: int,K: int,P: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,I3),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,I3) ) ) ) ).

% int_less_induct
tff(fact_3496_zdvd__reduce,axiom,
    ! [K: int,Nb: int,Mb: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,K),aa_int_int(aa_int_fun_int_int(plus_plus_int,Nb),aa_int_int(aa_int_fun_int_int(times_times_int,K),Mb)))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,K),Nb) ) ).

% zdvd_reduce
tff(fact_3497_zdvd__period,axiom,
    ! [A: int,D: int,Xb: int,Ta: 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,Xb),Ta))
      <=> 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,Xb),aa_int_int(aa_int_fun_int_int(times_times_int,C),D))),Ta)) ) ) ).

% zdvd_period
tff(fact_3498_pow_Osimps_I1_J,axiom,
    ! [Xb: num] : aa_num_num(pow(Xb),one) = Xb ).

% pow.simps(1)
tff(fact_3499_signed__take__bit__int__less__eq,axiom,
    ! [Nb: 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))),Nb)),K)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(bit_ri631733984087533419it_int(Nb),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,Nb)))) ) ).

% signed_take_bit_int_less_eq
tff(fact_3500_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_3501_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_3502_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_3503_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_3504_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_3505_unity__coeff__ex,axiom,
    ! [P: fun_real_o,L: real] :
      ( ? [X4: real] : aa_real_o(P,aa_real_real(times_times_real(L),X4))
    <=> ? [X4: real] :
          ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,L),aa_real_real(plus_plus_real(X4),zero_zero_real))
          & aa_real_o(P,X4) ) ) ).

% unity_coeff_ex
tff(fact_3506_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(plus_plus_rat(X4),zero_zero_rat))
          & aa_rat_o(P,X4) ) ) ).

% unity_coeff_ex
tff(fact_3507_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_3508_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_3509_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_3510_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_3511_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_3512_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_3513_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_3514_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_3515_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_3516_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_3517_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_3518_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_3519_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_3520_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_3521_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_3522_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_3523_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_3524_even__numeral,axiom,
    ! [Nb: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),numera6620942414471956472nteger(aa_num_num(bit0,Nb))) ).

% even_numeral
tff(fact_3525_even__numeral,axiom,
    ! [Nb: 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,Nb))) ).

% even_numeral
tff(fact_3526_even__numeral,axiom,
    ! [Nb: 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,Nb))) ).

% even_numeral
tff(fact_3527_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(plus_plus_real(aa_real_real(times_times_real(A),E2)),C)),aa_real_real(plus_plus_real(aa_real_real(times_times_real(B),E2)),D))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_real_real(minus_minus_real(A),B)),E2)),C)),D) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_3528_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(plus_plus_rat(aa_rat_rat(times_times_rat(A),E2)),C)),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(B),E2)),D))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(A),B)),E2)),C)),D) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_3529_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_3530_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(plus_plus_real(aa_real_real(times_times_real(A),E2)),C)),aa_real_real(plus_plus_real(aa_real_real(times_times_real(B),E2)),D))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_real_real(minus_minus_real(B),A)),E2)),D)) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_3531_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(plus_plus_rat(aa_rat_rat(times_times_rat(A),E2)),C)),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(B),E2)),D))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(B),A)),E2)),D)) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_3532_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_3533_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_3534_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_3535_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_3536_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_3537_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_3538_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_3539_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_3540_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_3541_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_3542_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_3543_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_3544_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_3545_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_3546_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_3547_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_3548_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_3549_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_3550_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_3551_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(plus_plus_real(aa_real_real(times_times_real(A),E2)),C)),aa_real_real(plus_plus_real(aa_real_real(times_times_real(B),E2)),D))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_real_real(minus_minus_real(A),B)),E2)),C)),D) ) ).

% less_add_iff1
tff(fact_3552_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(plus_plus_rat(aa_rat_rat(times_times_rat(A),E2)),C)),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(B),E2)),D))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(A),B)),E2)),C)),D) ) ).

% less_add_iff1
tff(fact_3553_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_3554_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(plus_plus_real(aa_real_real(times_times_real(A),E2)),C)),aa_real_real(plus_plus_real(aa_real_real(times_times_real(B),E2)),D))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,C),aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_real_real(minus_minus_real(B),A)),E2)),D)) ) ).

% less_add_iff2
tff(fact_3555_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(plus_plus_rat(aa_rat_rat(times_times_rat(A),E2)),C)),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(B),E2)),D))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(B),A)),E2)),D)) ) ).

% less_add_iff2
tff(fact_3556_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_3557_divide__diff__eq__iff,axiom,
    ! [Z: complex,Xb: complex,Yb: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(Xb),Z)),Yb) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(Xb),aa_complex_complex(times_times_complex(Yb),Z))),Z) ) ) ).

% divide_diff_eq_iff
tff(fact_3558_divide__diff__eq__iff,axiom,
    ! [Z: real,Xb: real,Yb: real] :
      ( ( Z != zero_zero_real )
     => ( aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(Xb),Z)),Yb) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(Xb),aa_real_real(times_times_real(Yb),Z))),Z) ) ) ).

% divide_diff_eq_iff
tff(fact_3559_divide__diff__eq__iff,axiom,
    ! [Z: rat,Xb: rat,Yb: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(minus_minus_rat(aa_rat_rat(divide_divide_rat(Xb),Z)),Yb) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(Xb),aa_rat_rat(times_times_rat(Yb),Z))),Z) ) ) ).

% divide_diff_eq_iff
tff(fact_3560_diff__divide__eq__iff,axiom,
    ! [Z: complex,Xb: complex,Yb: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_complex_complex(minus_minus_complex(Xb),aa_complex_complex(divide1717551699836669952omplex(Yb),Z)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(Xb),Z)),Yb)),Z) ) ) ).

% diff_divide_eq_iff
tff(fact_3561_diff__divide__eq__iff,axiom,
    ! [Z: real,Xb: real,Yb: real] :
      ( ( Z != zero_zero_real )
     => ( aa_real_real(minus_minus_real(Xb),aa_real_real(divide_divide_real(Yb),Z)) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(times_times_real(Xb),Z)),Yb)),Z) ) ) ).

% diff_divide_eq_iff
tff(fact_3562_diff__divide__eq__iff,axiom,
    ! [Z: rat,Xb: rat,Yb: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(minus_minus_rat(Xb),aa_rat_rat(divide_divide_rat(Yb),Z)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(Xb),Z)),Yb)),Z) ) ) ).

% diff_divide_eq_iff
tff(fact_3563_diff__frac__eq,axiom,
    ! [Yb: complex,Z: complex,Xb: complex,W: complex] :
      ( ( Yb != zero_zero_complex )
     => ( ( Z != zero_zero_complex )
       => ( aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(Xb),Yb)),aa_complex_complex(divide1717551699836669952omplex(W),Z)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(Xb),Z)),aa_complex_complex(times_times_complex(W),Yb))),aa_complex_complex(times_times_complex(Yb),Z)) ) ) ) ).

% diff_frac_eq
tff(fact_3564_diff__frac__eq,axiom,
    ! [Yb: real,Z: real,Xb: real,W: real] :
      ( ( Yb != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(Xb),Yb)),aa_real_real(divide_divide_real(W),Z)) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(times_times_real(Xb),Z)),aa_real_real(times_times_real(W),Yb))),aa_real_real(times_times_real(Yb),Z)) ) ) ) ).

% diff_frac_eq
tff(fact_3565_diff__frac__eq,axiom,
    ! [Yb: rat,Z: rat,Xb: rat,W: rat] :
      ( ( Yb != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( aa_rat_rat(minus_minus_rat(aa_rat_rat(divide_divide_rat(Xb),Yb)),aa_rat_rat(divide_divide_rat(W),Z)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(Xb),Z)),aa_rat_rat(times_times_rat(W),Yb))),aa_rat_rat(times_times_rat(Yb),Z)) ) ) ) ).

% diff_frac_eq
tff(fact_3566_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_3567_add__divide__eq__if__simps_I4_J,axiom,
    ! [A: real,B: real,Z: real] :
      aa_real_real(minus_minus_real(A),aa_real_real(divide_divide_real(B),Z)) = $ite(Z = zero_zero_real,A,aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(times_times_real(A),Z)),B)),Z)) ).

% add_divide_eq_if_simps(4)
tff(fact_3568_add__divide__eq__if__simps_I4_J,axiom,
    ! [A: rat,B: rat,Z: rat] :
      aa_rat_rat(minus_minus_rat(A),aa_rat_rat(divide_divide_rat(B),Z)) = $ite(Z = zero_zero_rat,A,aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(A),Z)),B)),Z)) ).

% add_divide_eq_if_simps(4)
tff(fact_3569_square__diff__one__factored,axiom,
    ! [Xb: complex] : aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(Xb),Xb)),one_one_complex) = aa_complex_complex(times_times_complex(aa_complex_complex(plus_plus_complex(Xb),one_one_complex)),aa_complex_complex(minus_minus_complex(Xb),one_one_complex)) ).

% square_diff_one_factored
tff(fact_3570_square__diff__one__factored,axiom,
    ! [Xb: real] : aa_real_real(minus_minus_real(aa_real_real(times_times_real(Xb),Xb)),one_one_real) = aa_real_real(times_times_real(aa_real_real(plus_plus_real(Xb),one_one_real)),aa_real_real(minus_minus_real(Xb),one_one_real)) ).

% square_diff_one_factored
tff(fact_3571_square__diff__one__factored,axiom,
    ! [Xb: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(Xb),Xb)),one_one_rat) = aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(Xb),one_one_rat)),aa_rat_rat(minus_minus_rat(Xb),one_one_rat)) ).

% square_diff_one_factored
tff(fact_3572_square__diff__one__factored,axiom,
    ! [Xb: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Xb),Xb)),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,Xb),one_one_int)),aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),one_one_int)) ).

% square_diff_one_factored
tff(fact_3573_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_3574_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_3575_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(aa_int_fun_int_int(modulo_modulo_int,A),B) = zero_zero_int ) ) ).

% unit_imp_mod_eq_0
tff(fact_3576_is__unit__power__iff,axiom,
    ! [A: code_integer,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb)),one_one_Code_integer)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
        | ( Nb = zero_zero_nat ) ) ) ).

% is_unit_power_iff
tff(fact_3577_is__unit__power__iff,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),one_one_nat)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
        | ( Nb = zero_zero_nat ) ) ) ).

% is_unit_power_iff
tff(fact_3578_is__unit__power__iff,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(A),Nb)),one_one_int)
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
        | ( Nb = zero_zero_nat ) ) ) ).

% is_unit_power_iff
tff(fact_3579_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_3580_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(aa_int_fun_int_int(modulo_modulo_int,A),B) ).

% minus_mult_div_eq_mod
tff(fact_3581_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_3582_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(aa_int_fun_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_3583_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_3584_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(aa_int_fun_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_3585_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_3586_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(aa_int_fun_int_int(modulo_modulo_int,A),B) ).

% minus_div_mult_eq_mod
tff(fact_3587_dvd__imp__le,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb) ) ) ).

% dvd_imp_le
tff(fact_3588_dvd__mult__cancel,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb))
     => ( 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,Mb),Nb) ) ) ).

% dvd_mult_cancel
tff(fact_3589_nat__mult__dvd__cancel1,axiom,
    ! [K: nat,Mb: nat,Nb: 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),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),Nb) ) ) ).

% nat_mult_dvd_cancel1
tff(fact_3590_bezout__add__strong__nat,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ? [D3: nat,X3: nat,Y: 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),X3) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y)),D3) ) ) ) ).

% bezout_add_strong_nat
tff(fact_3591_zdvd__imp__le,axiom,
    ! [Z: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Z),Nb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Nb)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),Nb) ) ) ).

% zdvd_imp_le
tff(fact_3592_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(aa_int_fun_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_3593_mod__greater__zero__iff__not__dvd,axiom,
    ! [Mb: nat,Nb: 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,Mb),Nb))
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Nb),Mb) ) ).

% mod_greater_zero_iff_not_dvd
tff(fact_3594_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(aa_int_fun_int_int(modulo_modulo_int,K),L) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,K),L)),L) ) ) ) ).

% mod_pos_geq
tff(fact_3595_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)
     => ( ! [X3: int,K2: int] :
            ( aa_int_o(P1,X3)
          <=> aa_int_o(P1,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D))) )
       => ( ? [Z3: int] :
            ! [X3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Z3)
             => ( aa_int_o(P,X3)
              <=> aa_int_o(P1,X3) ) )
         => ( ? [X_1: int] : aa_int_o(P1,X_1)
           => ? [X_12: int] : aa_int_o(P,X_12) ) ) ) ) ).

% minusinfinity
tff(fact_3596_plusinfinity,axiom,
    ! [D: int,P3: fun_int_o,P: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
     => ( ! [X3: int,K2: int] :
            ( aa_int_o(P3,X3)
          <=> aa_int_o(P3,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D))) )
       => ( ? [Z3: int] :
            ! [X3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),X3)
             => ( aa_int_o(P,X3)
              <=> aa_int_o(P3,X3) ) )
         => ( ? [X_1: int] : aa_int_o(P3,X_1)
           => ? [X_12: int] : aa_int_o(P,X_12) ) ) ) ) ).

% plusinfinity
tff(fact_3597_int__induct,axiom,
    ! [P: fun_int_o,K: int,I3: 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,I3) ) ) ) ).

% int_induct
tff(fact_3598_even__mask__div__iff_H,axiom,
    ! [Mb: nat,Nb: 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))),Mb),one_one_Code_integer),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Nb)))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% even_mask_div_iff'
tff(fact_3599_even__mask__div__iff_H,axiom,
    ! [Mb: nat,Nb: 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))),Mb)),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))),Nb)))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% even_mask_div_iff'
tff(fact_3600_even__mask__div__iff_H,axiom,
    ! [Mb: nat,Nb: 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))),Mb)),one_one_int)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% even_mask_div_iff'
tff(fact_3601_even__mask__div__iff,axiom,
    ! [Mb: nat,Nb: 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))),Mb),one_one_Code_integer),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Nb)))
    <=> ( ( aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Nb) = zero_z3403309356797280102nteger )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ).

% even_mask_div_iff
tff(fact_3602_even__mask__div__iff,axiom,
    ! [Mb: nat,Nb: 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))),Mb)),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))),Nb)))
    <=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) = zero_zero_nat )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ).

% even_mask_div_iff
tff(fact_3603_even__mask__div__iff,axiom,
    ! [Mb: nat,Nb: 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))),Mb)),one_one_int)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)))
    <=> ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb) = zero_zero_int )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ).

% even_mask_div_iff
tff(fact_3604_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_3605_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_3606_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_3607_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_3608_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_3609_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_3610_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_3611_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_3612_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_3613_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 )
         => ! [B3: code_integer] :
              ( ( B3 != zero_z3403309356797280102nteger )
             => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B3),one_one_Code_integer)
               => ( ( divide6298287555418463151nteger(one_one_Code_integer,A) = B3 )
                 => ( ( divide6298287555418463151nteger(one_one_Code_integer,B3) = A )
                   => ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B3) = one_one_Code_integer )
                     => ( divide6298287555418463151nteger(C,A) != aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B3) ) ) ) ) ) ) ) ) ).

% is_unitE
tff(fact_3614_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 )
         => ! [B3: nat] :
              ( ( B3 != zero_zero_nat )
             => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B3),one_one_nat)
               => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),A) = B3 )
                 => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),B3) = A )
                   => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B3) = one_one_nat )
                     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),A) != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B3) ) ) ) ) ) ) ) ) ).

% is_unitE
tff(fact_3615_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 )
         => ! [B3: int] :
              ( ( B3 != zero_zero_int )
             => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B3),one_one_int)
               => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),A) = B3 )
                 => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),B3) = A )
                   => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),B3) = one_one_int )
                     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,C),A) != aa_int_int(aa_int_fun_int_int(times_times_int,C),B3) ) ) ) ) ) ) ) ) ).

% is_unitE
tff(fact_3616_evenE,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
     => ~ ! [B3: code_integer] : A != aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),B3) ) ).

% evenE
tff(fact_3617_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)
     => ~ ! [B3: nat] : A != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B3) ) ).

% evenE
tff(fact_3618_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)
     => ~ ! [B3: int] : A != aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B3) ) ).

% evenE
tff(fact_3619_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_3620_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_3621_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_3622_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_3623_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_3624_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_3625_frac__le__eq,axiom,
    ! [Yb: real,Z: real,Xb: real,W: real] :
      ( ( Yb != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(Xb),Yb)),aa_real_real(divide_divide_real(W),Z))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(times_times_real(Xb),Z)),aa_real_real(times_times_real(W),Yb))),aa_real_real(times_times_real(Yb),Z))),zero_zero_real) ) ) ) ).

% frac_le_eq
tff(fact_3626_frac__le__eq,axiom,
    ! [Yb: rat,Z: rat,Xb: rat,W: rat] :
      ( ( Yb != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(Xb),Yb)),aa_rat_rat(divide_divide_rat(W),Z))
        <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(Xb),Z)),aa_rat_rat(times_times_rat(W),Yb))),aa_rat_rat(times_times_rat(Yb),Z))),zero_zero_rat) ) ) ) ).

% frac_le_eq
tff(fact_3627_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_3628_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_3629_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_3630_frac__less__eq,axiom,
    ! [Yb: real,Z: real,Xb: real,W: real] :
      ( ( Yb != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(Xb),Yb)),aa_real_real(divide_divide_real(W),Z))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(times_times_real(Xb),Z)),aa_real_real(times_times_real(W),Yb))),aa_real_real(times_times_real(Yb),Z))),zero_zero_real) ) ) ) ).

% frac_less_eq
tff(fact_3631_frac__less__eq,axiom,
    ! [Yb: rat,Z: rat,Xb: rat,W: rat] :
      ( ( Yb != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(Xb),Yb)),aa_rat_rat(divide_divide_rat(W),Z))
        <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(Xb),Z)),aa_rat_rat(times_times_rat(W),Yb))),aa_rat_rat(times_times_rat(Yb),Z))),zero_zero_rat) ) ) ) ).

% frac_less_eq
tff(fact_3632_dvd__power__iff,axiom,
    ! [Xb: code_integer,Mb: nat,Nb: nat] :
      ( ( Xb != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(Xb),Mb)),aa_nat_Code_integer(power_8256067586552552935nteger(Xb),Nb))
      <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Xb),one_one_Code_integer)
          | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ) ).

% dvd_power_iff
tff(fact_3633_dvd__power__iff,axiom,
    ! [Xb: nat,Mb: nat,Nb: nat] :
      ( ( Xb != 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,Xb),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),Nb))
      <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Xb),one_one_nat)
          | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ) ).

% dvd_power_iff
tff(fact_3634_dvd__power__iff,axiom,
    ! [Xb: int,Mb: nat,Nb: nat] :
      ( ( Xb != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(Xb),Mb)),aa_nat_int(power_power_int(Xb),Nb))
      <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xb),one_one_int)
          | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ) ).

% dvd_power_iff
tff(fact_3635_dvd__power,axiom,
    ! [Nb: nat,Xb: code_integer] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
        | ( Xb = one_one_Code_integer ) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Xb),aa_nat_Code_integer(power_8256067586552552935nteger(Xb),Nb)) ) ).

% dvd_power
tff(fact_3636_dvd__power,axiom,
    ! [Nb: nat,Xb: rat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
        | ( Xb = one_one_rat ) )
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Xb),aa_nat_rat(power_power_rat(Xb),Nb)) ) ).

% dvd_power
tff(fact_3637_dvd__power,axiom,
    ! [Nb: nat,Xb: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
        | ( Xb = one_one_nat ) )
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),Nb)) ) ).

% dvd_power
tff(fact_3638_dvd__power,axiom,
    ! [Nb: nat,Xb: int] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
        | ( Xb = one_one_int ) )
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xb),aa_nat_int(power_power_int(Xb),Nb)) ) ).

% dvd_power
tff(fact_3639_dvd__power,axiom,
    ! [Nb: nat,Xb: real] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
        | ( Xb = one_one_real ) )
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Xb),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)) ) ).

% dvd_power
tff(fact_3640_dvd__power,axiom,
    ! [Nb: nat,Xb: complex] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
        | ( Xb = one_one_complex ) )
     => aa_complex_o(dvd_dvd_complex(Xb),aa_nat_complex(power_power_complex(Xb),Nb)) ) ).

% dvd_power
tff(fact_3641_power2__commute,axiom,
    ! [Xb: complex,Yb: complex] : aa_nat_complex(power_power_complex(aa_complex_complex(minus_minus_complex(Xb),Yb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_complex(power_power_complex(aa_complex_complex(minus_minus_complex(Yb),Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_commute
tff(fact_3642_power2__commute,axiom,
    ! [Xb: real,Yb: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(minus_minus_real(Xb),Yb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(minus_minus_real(Yb),Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_commute
tff(fact_3643_power2__commute,axiom,
    ! [Xb: rat,Yb: rat] : aa_nat_rat(power_power_rat(aa_rat_rat(minus_minus_rat(Xb),Yb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_rat(power_power_rat(aa_rat_rat(minus_minus_rat(Yb),Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_commute
tff(fact_3644_power2__commute,axiom,
    ! [Xb: int,Yb: int] : aa_nat_int(power_power_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),Yb)),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,Yb),Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_commute
tff(fact_3645_even__even__mod__4__iff,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
    <=> 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,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one))))) ) ).

% even_even_mod_4_iff
tff(fact_3646_dvd__mult__cancel1,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)),Mb)
      <=> ( Nb = one_one_nat ) ) ) ).

% dvd_mult_cancel1
tff(fact_3647_dvd__mult__cancel2,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Mb)),Mb)
      <=> ( Nb = one_one_nat ) ) ) ).

% dvd_mult_cancel2
tff(fact_3648_power__dvd__imp__le,axiom,
    ! [I3: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I3),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I3),Nb))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),I3)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ).

% power_dvd_imp_le
tff(fact_3649_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)
     => ( ! [X3: int] :
            ( aa_int_o(P,X3)
           => aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D)) )
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
         => ! [X: int] :
              ( aa_int_o(P,X)
             => aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),aa_int_int(aa_int_fun_int_int(times_times_int,K),D))) ) ) ) ) ).

% decr_mult_lemma
tff(fact_3650_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_3651_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_3652_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_3653_scaling__mono,axiom,
    ! [U: real,V: real,R2: real,S2: 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),R2)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,R2),S2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(U),aa_real_real(divide_divide_real(aa_real_real(times_times_real(R2),aa_real_real(minus_minus_real(V),U))),S2))),V) ) ) ) ).

% scaling_mono
tff(fact_3654_scaling__mono,axiom,
    ! [U: rat,V: rat,R2: rat,S2: 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),R2)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,R2),S2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(U),aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(R2),aa_rat_rat(minus_minus_rat(V),U))),S2))),V) ) ) ) ).

% scaling_mono
tff(fact_3655_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_3656_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_3657_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(aa_int_fun_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_3658_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_3659_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_3660_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(aa_int_fun_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_3661_power__mono__odd,axiom,
    ! [Nb: 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))),Nb)
     => ( 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(aa_real_fun_nat_real(power_power_real,A),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb)) ) ) ).

% power_mono_odd
tff(fact_3662_power__mono__odd,axiom,
    ! [Nb: 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))),Nb)
     => ( 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),Nb)),aa_nat_rat(power_power_rat(B),Nb)) ) ) ).

% power_mono_odd
tff(fact_3663_power__mono__odd,axiom,
    ! [Nb: 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))),Nb)
     => ( 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),Nb)),aa_nat_int(power_power_int(B),Nb)) ) ) ).

% power_mono_odd
tff(fact_3664_odd__pos,axiom,
    ! [Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).

% odd_pos
tff(fact_3665_dvd__power__iff__le,axiom,
    ! [K: nat,Mb: nat,Nb: 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),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K),Nb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ).

% dvd_power_iff_le
tff(fact_3666_signed__take__bit__int__less__exp,axiom,
    ! [Nb: nat,K: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_ri631733984087533419it_int(Nb),K)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ).

% signed_take_bit_int_less_exp
tff(fact_3667_even__unset__bit__iff,axiom,
    ! [Mb: nat,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_se8260200283734997820nteger(Mb,A))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
        | ( Mb = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
tff(fact_3668_even__unset__bit__iff,axiom,
    ! [Mb: 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,Mb),A))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
        | ( Mb = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
tff(fact_3669_even__unset__bit__iff,axiom,
    ! [Mb: 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,Mb),A))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
        | ( Mb = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
tff(fact_3670_even__set__bit__iff,axiom,
    ! [Mb: nat,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_se2793503036327961859nteger(Mb,A))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
        & ( Mb != zero_zero_nat ) ) ) ).

% even_set_bit_iff
tff(fact_3671_even__set__bit__iff,axiom,
    ! [Mb: 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,Mb),A))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
        & ( Mb != zero_zero_nat ) ) ) ).

% even_set_bit_iff
tff(fact_3672_even__set__bit__iff,axiom,
    ! [Mb: 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,Mb),A))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
        & ( Mb != zero_zero_nat ) ) ) ).

% even_set_bit_iff
tff(fact_3673_even__flip__bit__iff,axiom,
    ! [Mb: nat,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_se1345352211410354436nteger(Mb,A))
    <=> ~ ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
        <=> ( Mb = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
tff(fact_3674_even__flip__bit__iff,axiom,
    ! [Mb: 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(Mb),A))
    <=> ~ ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
        <=> ( Mb = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
tff(fact_3675_even__flip__bit__iff,axiom,
    ! [Mb: 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,Mb),A))
    <=> ~ ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
        <=> ( Mb = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
tff(fact_3676_num_Osize__gen_I1_J,axiom,
    aa_num_nat(size_num,one) = zero_zero_nat ).

% num.size_gen(1)
tff(fact_3677_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
    ! [Uy: option4927543243414619207at_nat,V: nat,TreeLista: list_VEBT_VEBT,S2: vEBT_VEBT,Xb: nat] :
      ( aa_nat_o(vEBT_V5719532721284313246member(vEBT_Node(Uy,aa_nat_nat(suc,V),TreeLista,S2)),Xb)
    <=> $let(
          pos: nat,
          pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),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)))),
          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeLista)),aa_nat_o(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeLista,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ).

% VEBT_internal.naive_member.simps(3)
tff(fact_3678_oddE,axiom,
    ! [A: code_integer] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
     => ~ ! [B3: code_integer] : A != plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),B3),one_one_Code_integer) ) ).

% oddE
tff(fact_3679_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)
     => ~ ! [B3: nat] : A != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B3)),one_one_nat) ) ).

% oddE
tff(fact_3680_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)
     => ~ ! [B3: int] : A != aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B3)),one_one_int) ) ).

% oddE
tff(fact_3681_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_3682_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_3683_mod2__eq__if,axiom,
    ! [A: int] :
      aa_int_int(aa_int_fun_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_3684_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_3685_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_3686_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(aa_int_fun_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(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) != one_one_int ) ) ) ).

% parity_cases
tff(fact_3687_zero__le__power__eq,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A) ) ) ) ).

% zero_le_power_eq
tff(fact_3688_zero__le__power__eq,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ) ) ).

% zero_le_power_eq
tff(fact_3689_zero__le__power__eq,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ) ) ).

% zero_le_power_eq
tff(fact_3690_zero__le__odd__power,axiom,
    ! [Nb: 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))),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A) ) ) ).

% zero_le_odd_power
tff(fact_3691_zero__le__odd__power,axiom,
    ! [Nb: 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))),Nb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),Nb))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ) ).

% zero_le_odd_power
tff(fact_3692_zero__le__odd__power,axiom,
    ! [Nb: 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))),Nb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),Nb))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ) ).

% zero_le_odd_power
tff(fact_3693_zero__le__even__power,axiom,
    ! [Nb: 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))),Nb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ) ).

% zero_le_even_power
tff(fact_3694_zero__le__even__power,axiom,
    ! [Nb: 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))),Nb)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),Nb)) ) ).

% zero_le_even_power
tff(fact_3695_zero__le__even__power,axiom,
    ! [Nb: 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))),Nb)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A),Nb)) ) ).

% zero_le_even_power
tff(fact_3696_signed__take__bit__int__greater__eq__self__iff,axiom,
    ! [K: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),aa_int_int(bit_ri631733984087533419it_int(Nb),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))),Nb)) ) ).

% signed_take_bit_int_greater_eq_self_iff
tff(fact_3697_signed__take__bit__int__less__self__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_ri631733984087533419it_int(Nb),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))),Nb)),K) ) ).

% signed_take_bit_int_less_self_iff
tff(fact_3698_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
    ! [V: nat,TreeLista: list_VEBT_VEBT,Vd2: vEBT_VEBT,Xb: nat] :
      ( aa_nat_o(vEBT_VEBT_membermima(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V),TreeLista,Vd2)),Xb)
    <=> $let(
          pos: nat,
          pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),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)))),
          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeLista)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeLista,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ).

% VEBT_internal.membermima.simps(5)
tff(fact_3699_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_3700_vebt__member_Osimps_I5_J,axiom,
    ! [Mia: nat,Maa: nat,Va2: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xb: nat] :
      ( aa_nat_o(vEBT_vebt_member(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeLista,Summarya)),Xb)
    <=> $ite(
          Xb = Mia,
          $true,
          $ite(
            Xb = Maa,
            $true,
            $ite(
              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Mia),
              $false,
              $ite(
                aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Maa),Xb),
                $false,
                $let(
                  h: nat,
                  h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_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)))),
                  $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeLista)),aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(TreeLista,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ).

% vebt_member.simps(5)
tff(fact_3701_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
    ! [Mia: nat,Maa: nat,V: nat,TreeLista: list_VEBT_VEBT,Vc: vEBT_VEBT,Xb: nat] :
      ( aa_nat_o(vEBT_VEBT_membermima(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),aa_nat_nat(suc,V),TreeLista,Vc)),Xb)
    <=> ( ( Xb = Mia )
        | ( Xb = Maa )
        | $let(
            pos: nat,
            pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),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)))),
            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeLista)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeLista,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ).

% VEBT_internal.membermima.simps(4)
tff(fact_3702_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_3703_add__0__iff,axiom,
    ! [B: real,A: real] :
      ( ( B = aa_real_real(plus_plus_real(B),A) )
    <=> ( A = zero_zero_real ) ) ).

% add_0_iff
tff(fact_3704_add__0__iff,axiom,
    ! [B: rat,A: rat] :
      ( ( B = aa_rat_rat(plus_plus_rat(B),A) )
    <=> ( A = zero_zero_rat ) ) ).

% add_0_iff
tff(fact_3705_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_3706_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_3707_crossproduct__eq,axiom,
    ! [W: real,Yb: real,Xb: real,Z: real] :
      ( ( aa_real_real(plus_plus_real(aa_real_real(times_times_real(W),Yb)),aa_real_real(times_times_real(Xb),Z)) = aa_real_real(plus_plus_real(aa_real_real(times_times_real(W),Z)),aa_real_real(times_times_real(Xb),Yb)) )
    <=> ( ( W = Xb )
        | ( Yb = Z ) ) ) ).

% crossproduct_eq
tff(fact_3708_crossproduct__eq,axiom,
    ! [W: rat,Yb: rat,Xb: rat,Z: rat] :
      ( ( aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(W),Yb)),aa_rat_rat(times_times_rat(Xb),Z)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(W),Z)),aa_rat_rat(times_times_rat(Xb),Yb)) )
    <=> ( ( W = Xb )
        | ( Yb = Z ) ) ) ).

% crossproduct_eq
tff(fact_3709_crossproduct__eq,axiom,
    ! [W: nat,Yb: nat,Xb: 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),Yb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Xb),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,Xb),Yb)) )
    <=> ( ( W = Xb )
        | ( Yb = Z ) ) ) ).

% crossproduct_eq
tff(fact_3710_crossproduct__eq,axiom,
    ! [W: int,Yb: int,Xb: 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),Yb)),aa_int_int(aa_int_fun_int_int(times_times_int,Xb),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,Xb),Yb)) )
    <=> ( ( W = Xb )
        | ( Yb = Z ) ) ) ).

% crossproduct_eq
tff(fact_3711_crossproduct__noteq,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ( A != B )
        & ( C != D ) )
    <=> ( aa_real_real(plus_plus_real(aa_real_real(times_times_real(A),C)),aa_real_real(times_times_real(B),D)) != aa_real_real(plus_plus_real(aa_real_real(times_times_real(A),D)),aa_real_real(times_times_real(B),C)) ) ) ).

% crossproduct_noteq
tff(fact_3712_crossproduct__noteq,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ( A != B )
        & ( C != D ) )
    <=> ( aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A),C)),aa_rat_rat(times_times_rat(B),D)) != aa_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_3713_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_3714_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_3715_power2__diff,axiom,
    ! [Xb: complex,Yb: complex] : aa_nat_complex(power_power_complex(aa_complex_complex(minus_minus_complex(Xb),Yb)),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(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_complex(power_power_complex(Yb),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))),Xb)),Yb)) ).

% power2_diff
tff(fact_3716_power2__diff,axiom,
    ! [Xb: real,Yb: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(minus_minus_real(Xb),Yb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(minus_minus_real(aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(times_times_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Xb)),Yb)) ).

% power2_diff
tff(fact_3717_power2__diff,axiom,
    ! [Xb: rat,Yb: rat] : aa_nat_rat(power_power_rat(aa_rat_rat(minus_minus_rat(Xb),Yb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),Xb)),Yb)) ).

% power2_diff
tff(fact_3718_power2__diff,axiom,
    ! [Xb: int,Yb: int] : aa_nat_int(power_power_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),Yb)),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(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Yb),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))),Xb)),Yb)) ).

% power2_diff
tff(fact_3719_zero__less__power__eq,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb))
    <=> ( ( Nb = 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))),Nb)
          & ( 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))),Nb)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A) ) ) ) ).

% zero_less_power_eq
tff(fact_3720_zero__less__power__eq,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A),Nb))
    <=> ( ( Nb = 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))),Nb)
          & ( 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))),Nb)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ) ) ).

% zero_less_power_eq
tff(fact_3721_zero__less__power__eq,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(A),Nb))
    <=> ( ( Nb = 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))),Nb)
          & ( 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))),Nb)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A) ) ) ) ).

% zero_less_power_eq
tff(fact_3722_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(vEBT_V5719532721284313246member(Xb),Xaa)
     => ( ! [A4: $o,B3: $o] :
            ( ( Xb = vEBT_Leaf((A4),(B3)) )
           => $ite(
                Xaa = zero_zero_nat,
                (A4),
                $ite(Xaa = one_one_nat,(B3),$false) ) )
       => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : Xb != vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2)
         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList: list_VEBT_VEBT] :
                ( ? [S: vEBT_VEBT] : Xb = vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList,S)
               => $let(
                    pos: nat,
                    pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),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)))),
                    $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(3)
tff(fact_3723_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(vEBT_V5719532721284313246member(Xb),Xaa)
     => ( ! [A4: $o,B3: $o] :
            ( ( Xb = vEBT_Leaf((A4),(B3)) )
           => ~ $ite(
                  Xaa = zero_zero_nat,
                  (A4),
                  $ite(Xaa = one_one_nat,(B3),$false) ) )
       => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList: list_VEBT_VEBT] :
              ( ? [S: vEBT_VEBT] : Xb = vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList,S)
             => ~ $let(
                    pos: nat,
                    pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),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)))),
                    $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ).

% VEBT_internal.naive_member.elims(2)
tff(fact_3724_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Yb: $o] :
      ( ( aa_nat_o(vEBT_V5719532721284313246member(Xb),Xaa)
      <=> (Yb) )
     => ( ! [A4: $o,B3: $o] :
            ( ( Xb = vEBT_Leaf((A4),(B3)) )
           => ( (Yb)
            <=> ~ $ite(
                    Xaa = zero_zero_nat,
                    (A4),
                    $ite(Xaa = one_one_nat,(B3),$false) ) ) )
       => ( ( ? [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : Xb = vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2)
           => (Yb) )
         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList: list_VEBT_VEBT] :
                ( ? [S: vEBT_VEBT] : Xb = vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList,S)
               => ( (Yb)
                <=> ~ $let(
                        pos: nat,
                        pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),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)))),
                        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(1)
tff(fact_3725_VEBT__internal_Omembermima_Oelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(vEBT_VEBT_membermima(Xb),Xaa)
     => ( ! [Mi: nat,Ma: nat] :
            ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,Va3,Vb2)
           => ~ ( ( Xaa = Mi )
                | ( Xaa = Ma ) ) )
       => ( ! [Mi: nat,Ma: nat,V2: nat,TreeList: list_VEBT_VEBT] :
              ( ? [Vc2: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,V2),TreeList,Vc2)
             => ~ ( ( Xaa = Mi )
                  | ( Xaa = Ma )
                  | $let(
                      pos: nat,
                      pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),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)))),
                      $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) )
         => ~ ! [V2: nat,TreeList: list_VEBT_VEBT] :
                ( ? [Vd: vEBT_VEBT] : Xb = 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,Xaa),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)))),
                      $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(2)
tff(fact_3726_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_3727_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(aa_int_fun_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(aa_int_fun_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(modulo_modulo_int,A),B) ) ) ) ) ).

% divmod_digit_1(2)
tff(fact_3728_power__le__zero__eq,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),zero_zero_real)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
        & ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
            & 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))),Nb)
            & ( A = zero_zero_real ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_3729_power__le__zero__eq,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),Nb)),zero_zero_rat)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
        & ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
            & 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))),Nb)
            & ( A = zero_zero_rat ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_3730_power__le__zero__eq,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),Nb)),zero_zero_int)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
        & ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
            & 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))),Nb)
            & ( A = zero_zero_int ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_3731_vebt__member_Oelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(vEBT_vebt_member(Xb),Xaa)
     => ( ! [A4: $o,B3: $o] :
            ( ( Xb = vEBT_Leaf((A4),(B3)) )
           => ~ $ite(
                  Xaa = zero_zero_nat,
                  (A4),
                  $ite(Xaa = one_one_nat,(B3),$false) ) )
       => ~ ! [Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT] :
              ( ? [Summary: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary)
             => ~ $ite(
                    Xaa = Mi,
                    $true,
                    $ite(
                      Xaa = Ma,
                      $true,
                      $ite(
                        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),
                        $false,
                        $ite(
                          aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xaa),
                          $false,
                          $let(
                            h: nat,
                            h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_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)))),
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(TreeList,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(2)
tff(fact_3732_VEBT__internal_Omembermima_Oelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(vEBT_VEBT_membermima(Xb),Xaa)
     => ( ! [Uu2: $o,Uv2: $o] : Xb != vEBT_Leaf((Uu2),(Uv2))
       => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : Xb != vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2)
         => ( ! [Mi: nat,Ma: nat] :
                ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,Va3,Vb2)
               => ( ( Xaa = Mi )
                  | ( Xaa = Ma ) ) )
           => ( ! [Mi: nat,Ma: nat,V2: nat,TreeList: list_VEBT_VEBT] :
                  ( ? [Vc2: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,V2),TreeList,Vc2)
                 => ( ( Xaa = Mi )
                    | ( Xaa = Ma )
                    | $let(
                        pos: nat,
                        pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),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)))),
                        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) )
             => ~ ! [V2: nat,TreeList: list_VEBT_VEBT] :
                    ( ? [Vd: vEBT_VEBT] : Xb = 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,Xaa),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)))),
                        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(3)
tff(fact_3733_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Yb: $o] :
      ( ( aa_nat_o(vEBT_VEBT_membermima(Xb),Xaa)
      <=> (Yb) )
     => ( ( ? [Uu2: $o,Uv2: $o] : Xb = vEBT_Leaf((Uu2),(Uv2))
         => (Yb) )
       => ( ( ? [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : Xb = vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2)
           => (Yb) )
         => ( ! [Mi: nat,Ma: nat] :
                ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,Va3,Vb2)
               => ( (Yb)
                <=> ~ ( ( Xaa = Mi )
                      | ( Xaa = Ma ) ) ) )
           => ( ! [Mi: nat,Ma: nat,V2: nat,TreeList: list_VEBT_VEBT] :
                  ( ? [Vc2: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,V2),TreeList,Vc2)
                 => ( (Yb)
                  <=> ~ ( ( Xaa = Mi )
                        | ( Xaa = Ma )
                        | $let(
                            pos: nat,
                            pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),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)))),
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) )
             => ~ ! [V2: nat,TreeList: list_VEBT_VEBT] :
                    ( ? [Vd: vEBT_VEBT] : Xb = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList,Vd)
                   => ( (Yb)
                    <=> ~ $let(
                            pos: nat,
                            pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),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)))),
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(1)
tff(fact_3734_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(aa_int_fun_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(aa_int_fun_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_3735_vebt__insert_Osimps_I5_J,axiom,
    ! [Mia: nat,Maa: nat,Va2: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xb: nat] :
      vEBT_vebt_insert(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeLista,Summarya),Xb) = $let(
        xn: nat,
        xn:= 
          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Mia),Mia,Xb),
        $let(
          h: nat,
          h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xn),aa_nat_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)))),
          $ite(
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeLista))
            & ~ ( ( Xb = Mia )
                | ( Xb = Maa ) ) ),
            vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat($ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Mia),Xb,Mia)),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,xn),Maa))),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),list_u1324408373059187874T_VEBT(TreeLista,h,vEBT_vebt_insert(nth_VEBT_VEBT(TreeLista,h),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xn),aa_nat_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_VEBT_VEBT_o(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeLista,h)),vEBT_vebt_insert(Summarya,h),Summarya)),
            vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeLista,Summarya) ) ) ) ).

% vebt_insert.simps(5)
tff(fact_3736_vebt__member_Oelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(vEBT_vebt_member(Xb),Xaa)
     => ( ! [A4: $o,B3: $o] :
            ( ( Xb = vEBT_Leaf((A4),(B3)) )
           => $ite(
                Xaa = zero_zero_nat,
                (A4),
                $ite(Xaa = one_one_nat,(B3),$false) ) )
       => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : Xb != vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2)
         => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : Xb != vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2)
           => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : Xb != vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2)
             => ~ ! [Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT] :
                    ( ? [Summary: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary)
                   => $ite(
                        Xaa = Mi,
                        $true,
                        $ite(
                          Xaa = Ma,
                          $true,
                          $ite(
                            aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),
                            $false,
                            $ite(
                              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xaa),
                              $false,
                              $let(
                                h: nat,
                                h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_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)))),
                                $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(TreeList,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(3)
tff(fact_3737_vebt__member_Oelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Yb: $o] :
      ( ( aa_nat_o(vEBT_vebt_member(Xb),Xaa)
      <=> (Yb) )
     => ( ! [A4: $o,B3: $o] :
            ( ( Xb = vEBT_Leaf((A4),(B3)) )
           => ( (Yb)
            <=> ~ $ite(
                    Xaa = zero_zero_nat,
                    (A4),
                    $ite(Xaa = one_one_nat,(B3),$false) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : Xb = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2)
           => (Yb) )
         => ( ( ? [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2)
             => (Yb) )
           => ( ( ? [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2)
               => (Yb) )
             => ~ ! [Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT] :
                    ( ? [Summary: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary)
                   => ( (Yb)
                    <=> ~ $ite(
                            Xaa = Mi,
                            $true,
                            $ite(
                              Xaa = Ma,
                              $true,
                              $ite(
                                aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),
                                $false,
                                $ite(
                                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xaa),
                                  $false,
                                  $let(
                                    h: nat,
                                    h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_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)))),
                                    $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(TreeList,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(1)
tff(fact_3738_divmod__step__eq,axiom,
    ! [L: num,Q2: nat,R2: nat] :
      unique5026877609467782581ep_nat(L,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Q2),R2)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,L)),R2),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Q2)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,R2),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))),Q2)),R2)) ).

% divmod_step_eq
tff(fact_3739_divmod__step__eq,axiom,
    ! [L: num,Q2: int,R2: int] :
      aa_Pro6970504127699319548nt_int(unique5024387138958732305ep_int(L),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q2),R2)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,L)),R2),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(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))),Q2)),one_one_int)),aa_int_int(aa_int_fun_int_int(minus_minus_int,R2),aa_num_int(numeral_numeral_int,L))),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(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))),Q2)),R2)) ).

% divmod_step_eq
tff(fact_3740_divmod__step__eq,axiom,
    ! [L: num,Q2: code_integer,R2: code_integer] :
      unique4921790084139445826nteger(L,produc1086072967326762835nteger(Q2,R2)) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,numera6620942414471956472nteger(L)),R2),produc1086072967326762835nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),Q2),one_one_Code_integer),minus_8373710615458151222nteger(R2,numera6620942414471956472nteger(L))),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),Q2),R2)) ).

% divmod_step_eq
tff(fact_3741_max__bot,axiom,
    ! [Xb: set_nat] : ord_max_set_nat(bot_bot_set_nat,Xb) = Xb ).

% max_bot
tff(fact_3742_max__bot,axiom,
    ! [Xb: set_int] : ord_max_set_int(bot_bot_set_int,Xb) = Xb ).

% max_bot
tff(fact_3743_max__bot,axiom,
    ! [Xb: set_real] : ord_max_set_real(bot_bot_set_real,Xb) = Xb ).

% max_bot
tff(fact_3744_max__bot,axiom,
    ! [Xb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,bot_bot_nat),Xb) = Xb ).

% max_bot
tff(fact_3745_max__bot2,axiom,
    ! [Xb: set_nat] : ord_max_set_nat(Xb,bot_bot_set_nat) = Xb ).

% max_bot2
tff(fact_3746_max__bot2,axiom,
    ! [Xb: set_int] : ord_max_set_int(Xb,bot_bot_set_int) = Xb ).

% max_bot2
tff(fact_3747_max__bot2,axiom,
    ! [Xb: set_real] : ord_max_set_real(Xb,bot_bot_set_real) = Xb ).

% max_bot2
tff(fact_3748_max__bot2,axiom,
    ! [Xb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xb),bot_bot_nat) = Xb ).

% max_bot2
tff(fact_3749_empty__subsetI,axiom,
    ! [A3: set_int] : ord_less_eq_set_int(bot_bot_set_int,A3) ).

% empty_subsetI
tff(fact_3750_empty__subsetI,axiom,
    ! [A3: set_real] : ord_less_eq_set_real(bot_bot_set_real,A3) ).

% empty_subsetI
tff(fact_3751_empty__subsetI,axiom,
    ! [A3: set_nat] : aa_set_nat_o(ord_less_eq_set_nat(bot_bot_set_nat),A3) ).

% empty_subsetI
tff(fact_3752_subset__empty,axiom,
    ! [A3: set_int] :
      ( ord_less_eq_set_int(A3,bot_bot_set_int)
    <=> ( A3 = bot_bot_set_int ) ) ).

% subset_empty
tff(fact_3753_subset__empty,axiom,
    ! [A3: set_real] :
      ( ord_less_eq_set_real(A3,bot_bot_set_real)
    <=> ( A3 = bot_bot_set_real ) ) ).

% subset_empty
tff(fact_3754_subset__empty,axiom,
    ! [A3: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A3),bot_bot_set_nat)
    <=> ( A3 = bot_bot_set_nat ) ) ).

% subset_empty
tff(fact_3755_vebt__insert_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Yb: vEBT_VEBT] :
      ( ( vEBT_vebt_insert(Xb,Xaa) = Yb )
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(Xb,Xaa))
       => ( ! [A4: $o,B3: $o] :
              ( ( Xb = vEBT_Leaf((A4),(B3)) )
             => ( ( Yb = $ite(
                      Xaa = zero_zero_nat,
                      vEBT_Leaf($true,(B3)),
                      $ite(Xaa = one_one_nat,vEBT_Leaf((A4),$true),vEBT_Leaf((A4),(B3))) ) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(vEBT_Leaf((A4),(B3)),Xaa)) ) )
         => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Info2,zero_zero_nat,Ts2,S) )
               => ( ( Yb = vEBT_Node(Info2,zero_zero_nat,Ts2,S) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(vEBT_Node(Info2,zero_zero_nat,Ts2,S),Xaa)) ) )
           => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2,S) )
                 => ( ( Yb = vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2,S) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2,S),Xaa)) ) )
             => ( ! [V2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList,Summary) )
                   => ( ( Yb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xaa),Xaa)),aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList,Summary) )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList,Summary),Xaa)) ) )
               => ~ ! [Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary) )
                     => ( ( Yb = $let(
                              xn: nat,
                              xn:= 
                                $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),Mi,Xaa),
                              $let(
                                h: nat,
                                h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xn),aa_nat_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)))),
                                $ite(
                                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList))
                                  & ~ ( ( Xaa = Mi )
                                      | ( Xaa = Ma ) ) ),
                                  vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat($ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),Xaa,Mi)),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,xn),Ma))),aa_nat_nat(suc,aa_nat_nat(suc,Va)),list_u1324408373059187874T_VEBT(TreeList,h,vEBT_vebt_insert(nth_VEBT_VEBT(TreeList,h),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xn),aa_nat_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_VEBT_VEBT_o(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeList,h)),vEBT_vebt_insert(Summary,h),Summary)),
                                  vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary) ) ) ) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary),Xaa)) ) ) ) ) ) ) ) ) ).

% vebt_insert.pelims
tff(fact_3756_div2__even__ext__nat,axiom,
    ! [Xb: nat,Yb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Yb),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))),Xb)
        <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Yb) )
       => ( Xb = Yb ) ) ) ).

% div2_even_ext_nat
tff(fact_3757_dual__order_Orefl,axiom,
    ! [A: set_nat] : aa_set_nat_o(ord_less_eq_set_nat(A),A) ).

% dual_order.refl
tff(fact_3758_dual__order_Orefl,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),A) ).

% dual_order.refl
tff(fact_3759_dual__order_Orefl,axiom,
    ! [A: num] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),A) ).

% dual_order.refl
tff(fact_3760_dual__order_Orefl,axiom,
    ! [A: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),A) ).

% dual_order.refl
tff(fact_3761_dual__order_Orefl,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),A) ).

% dual_order.refl
tff(fact_3762_order__refl,axiom,
    ! [Xb: set_nat] : aa_set_nat_o(ord_less_eq_set_nat(Xb),Xb) ).

% order_refl
tff(fact_3763_order__refl,axiom,
    ! [Xb: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Xb) ).

% order_refl
tff(fact_3764_order__refl,axiom,
    ! [Xb: num] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Xb) ).

% order_refl
tff(fact_3765_order__refl,axiom,
    ! [Xb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Xb) ).

% order_refl
tff(fact_3766_order__refl,axiom,
    ! [Xb: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Xb) ).

% order_refl
tff(fact_3767_Suc__diff__diff,axiom,
    ! [Mb: nat,Nb: 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,Mb)),Nb)),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,Mb),Nb)),K) ).

% Suc_diff_diff
tff(fact_3768_diff__Suc__Suc,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,Mb)),aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb) ).

% diff_Suc_Suc
tff(fact_3769_diff__self__eq__0,axiom,
    ! [Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Mb) = zero_zero_nat ).

% diff_self_eq_0
tff(fact_3770_diff__0__eq__0,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,zero_zero_nat),Nb) = zero_zero_nat ).

% diff_0_eq_0
tff(fact_3771_subsetI,axiom,
    ! [A3: set_complex,B4: set_complex] :
      ( ! [X3: complex] :
          ( member_complex(X3,A3)
         => member_complex(X3,B4) )
     => ord_le211207098394363844omplex(A3,B4) ) ).

% subsetI
tff(fact_3772_subsetI,axiom,
    ! [A3: set_real,B4: set_real] :
      ( ! [X3: real] :
          ( member_real(X3,A3)
         => member_real(X3,B4) )
     => ord_less_eq_set_real(A3,B4) ) ).

% subsetI
tff(fact_3773_subsetI,axiom,
    ! [A3: set_set_nat,B4: set_set_nat] :
      ( ! [X3: set_nat] :
          ( member_set_nat(X3,A3)
         => member_set_nat(X3,B4) )
     => ord_le6893508408891458716et_nat(A3,B4) ) ).

% subsetI
tff(fact_3774_subsetI,axiom,
    ! [A3: set_int,B4: set_int] :
      ( ! [X3: int] :
          ( member_int(X3,A3)
         => member_int(X3,B4) )
     => ord_less_eq_set_int(A3,B4) ) ).

% subsetI
tff(fact_3775_subsetI,axiom,
    ! [A3: set_nat,B4: set_nat] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A3)
         => aa_set_nat_o(member_nat(X3),B4) )
     => aa_set_nat_o(ord_less_eq_set_nat(A3),B4) ) ).

% subsetI
tff(fact_3776_subset__antisym,axiom,
    ! [A3: set_nat,B4: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A3),B4)
     => ( aa_set_nat_o(ord_less_eq_set_nat(B4),A3)
       => ( A3 = B4 ) ) ) ).

% subset_antisym
tff(fact_3777_diff__diff__cancel,axiom,
    ! [I3: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I3),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),I3)) = I3 ) ) ).

% diff_diff_cancel
tff(fact_3778_diff__diff__left,axiom,
    ! [I3: 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,I3),J)),K) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I3),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K)) ).

% diff_diff_left
tff(fact_3779_Diff__eq__empty__iff,axiom,
    ! [A3: set_int,B4: set_int] :
      ( ( minus_minus_set_int(A3,B4) = bot_bot_set_int )
    <=> ord_less_eq_set_int(A3,B4) ) ).

% Diff_eq_empty_iff
tff(fact_3780_Diff__eq__empty__iff,axiom,
    ! [A3: set_real,B4: set_real] :
      ( ( minus_minus_set_real(A3,B4) = bot_bot_set_real )
    <=> ord_less_eq_set_real(A3,B4) ) ).

% Diff_eq_empty_iff
tff(fact_3781_Diff__eq__empty__iff,axiom,
    ! [A3: set_nat,B4: set_nat] :
      ( ( minus_minus_set_nat(A3,B4) = bot_bot_set_nat )
    <=> aa_set_nat_o(ord_less_eq_set_nat(A3),B4) ) ).

% Diff_eq_empty_iff
tff(fact_3782_zero__less__diff,axiom,
    ! [Nb: nat,Mb: 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,Nb),Mb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% zero_less_diff
tff(fact_3783_diff__is__0__eq_H,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb) = zero_zero_nat ) ) ).

% diff_is_0_eq'
tff(fact_3784_diff__is__0__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb) = zero_zero_nat )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% diff_is_0_eq
tff(fact_3785_Nat_Odiff__diff__right,axiom,
    ! [K: nat,J: nat,I3: 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,I3),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,I3),K)),J) ) ) ).

% Nat.diff_diff_right
tff(fact_3786_Nat_Oadd__diff__assoc2,axiom,
    ! [K: nat,J: nat,I3: 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)),I3) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),I3)),K) ) ) ).

% Nat.add_diff_assoc2
tff(fact_3787_Nat_Oadd__diff__assoc,axiom,
    ! [K: nat,J: nat,I3: 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,I3),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,I3),J)),K) ) ) ).

% Nat.add_diff_assoc
tff(fact_3788_diff__Suc__1,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,Nb)),one_one_nat) = Nb ).

% diff_Suc_1
tff(fact_3789_psubsetI,axiom,
    ! [A3: set_nat,B4: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A3),B4)
     => ( ( A3 != B4 )
       => aa_set_nat_o(ord_less_set_nat(A3),B4) ) ) ).

% psubsetI
tff(fact_3790_Suc__pred,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))) = Nb ) ) ).

% Suc_pred
tff(fact_3791_diff__Suc__diff__eq2,axiom,
    ! [K: nat,J: nat,I3: 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))),I3) = 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),I3)) ) ) ).

% diff_Suc_diff_eq2
tff(fact_3792_diff__Suc__diff__eq1,axiom,
    ! [K: nat,J: nat,I3: 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,I3),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,I3),K)),aa_nat_nat(suc,J)) ) ) ).

% diff_Suc_diff_eq1
tff(fact_3793_Suc__diff__1,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)) = Nb ) ) ).

% Suc_diff_1
tff(fact_3794_odd__Suc__minus__one,axiom,
    ! [Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))) = Nb ) ) ).

% odd_Suc_minus_one
tff(fact_3795_even__diff__nat,axiom,
    ! [Mb: nat,Nb: 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,Mb),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
        | 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,Mb),Nb)) ) ) ).

% even_diff_nat
tff(fact_3796_odd__two__times__div__two__nat,axiom,
    ! [Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( 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,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat) ) ) ).

% odd_two_times_div_two_nat
tff(fact_3797_dvd__diff__nat,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) ) ) ).

% dvd_diff_nat
tff(fact_3798_dvd__antisym,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Nb),Mb)
       => ( Mb = Nb ) ) ) ).

% dvd_antisym
tff(fact_3799_Diff__mono,axiom,
    ! [A3: set_nat,C5: set_nat,D4: set_nat,B4: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A3),C5)
     => ( aa_set_nat_o(ord_less_eq_set_nat(D4),B4)
       => aa_set_nat_o(ord_less_eq_set_nat(minus_minus_set_nat(A3,B4)),minus_minus_set_nat(C5,D4)) ) ) ).

% Diff_mono
tff(fact_3800_Diff__subset,axiom,
    ! [A3: set_nat,B4: set_nat] : aa_set_nat_o(ord_less_eq_set_nat(minus_minus_set_nat(A3,B4)),A3) ).

% Diff_subset
tff(fact_3801_double__diff,axiom,
    ! [A3: set_nat,B4: set_nat,C5: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A3),B4)
     => ( aa_set_nat_o(ord_less_eq_set_nat(B4),C5)
       => ( minus_minus_set_nat(B4,minus_minus_set_nat(C5,A3)) = A3 ) ) ) ).

% double_diff
tff(fact_3802_bot__nat__def,axiom,
    bot_bot_nat = zero_zero_nat ).

% bot_nat_def
tff(fact_3803_zero__induct__lemma,axiom,
    ! [P: fun_nat_o,K: nat,I3: 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),I3)) ) ) ).

% zero_induct_lemma
tff(fact_3804_diffs0__imp__equal,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb) = zero_zero_nat )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb) = zero_zero_nat )
       => ( Mb = Nb ) ) ) ).

% diffs0_imp_equal
tff(fact_3805_minus__nat_Odiff__0,axiom,
    ! [Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),zero_zero_nat) = Mb ).

% minus_nat.diff_0
tff(fact_3806_less__imp__diff__less,axiom,
    ! [J: nat,K: nat,Nb: 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),Nb)),K) ) ).

% less_imp_diff_less
tff(fact_3807_diff__less__mono2,axiom,
    ! [Mb: nat,Nb: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),L)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),Mb)) ) ) ).

% diff_less_mono2
tff(fact_3808_dvd__minus__self,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb)
        | aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),Nb) ) ) ).

% dvd_minus_self
tff(fact_3809_diff__le__mono2,axiom,
    ! [Mb: nat,Nb: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),Mb)) ) ).

% diff_le_mono2
tff(fact_3810_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_3811_diff__le__self,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),Mb) ).

% diff_le_self
tff(fact_3812_diff__le__mono,axiom,
    ! [Mb: nat,Nb: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),L)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),L)) ) ).

% diff_le_mono
tff(fact_3813_Nat_Odiff__diff__eq,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),K)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb) ) ) ) ).

% Nat.diff_diff_eq
tff(fact_3814_le__diff__iff,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),K)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ) ).

% le_diff_iff
tff(fact_3815_eq__diff__iff,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),K) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K) )
        <=> ( Mb = Nb ) ) ) ) ).

% eq_diff_iff
tff(fact_3816_dvd__diffD,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb))
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),Nb)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),Mb) ) ) ) ).

% dvd_diffD
tff(fact_3817_dvd__diffD1,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb))
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),Mb)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),Nb) ) ) ) ).

% dvd_diffD1
tff(fact_3818_less__eq__dvd__minus,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),Nb)
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)) ) ) ).

% less_eq_dvd_minus
tff(fact_3819_diff__add__inverse2,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)),Nb) = Mb ).

% diff_add_inverse2
tff(fact_3820_diff__add__inverse,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)),Nb) = Mb ).

% diff_add_inverse
tff(fact_3821_diff__cancel2,axiom,
    ! [Mb: nat,K: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb) ).

% diff_cancel2
tff(fact_3822_Nat_Odiff__cancel,axiom,
    ! [K: nat,Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb) ).

% Nat.diff_cancel
tff(fact_3823_diff__mult__distrib,axiom,
    ! [Mb: nat,Nb: 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,Mb),Nb)),K) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),K)) ).

% diff_mult_distrib
tff(fact_3824_diff__mult__distrib2,axiom,
    ! [K: nat,Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb)) ).

% diff_mult_distrib2
tff(fact_3825_bezout1__nat,axiom,
    ! [A: nat,B: nat] :
    ? [D3: nat,X3: nat,Y: 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(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X3)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y)) = D3 )
        | ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),X3)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y)) = D3 ) ) ) ).

% bezout1_nat
tff(fact_3826_diff__less__Suc,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),aa_nat_nat(suc,Mb)) ).

% diff_less_Suc
tff(fact_3827_Suc__diff__Suc,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb)
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb) ) ) ).

% Suc_diff_Suc
tff(fact_3828_diff__less,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),Mb) ) ) ).

% diff_less
tff(fact_3829_Suc__diff__le,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,Mb)),Nb) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) ) ) ).

% Suc_diff_le
tff(fact_3830_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_3831_less__diff__iff,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),K)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ) ) ).

% less_diff_iff
tff(fact_3832_diff__add__0,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)) = zero_zero_nat ).

% diff_add_0
tff(fact_3833_add__diff__inverse__nat,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) = Mb ) ) ).

% add_diff_inverse_nat
tff(fact_3834_less__diff__conv,axiom,
    ! [I3: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),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,I3),K)),J) ) ).

% less_diff_conv
tff(fact_3835_Nat_Ole__imp__diff__is__add,axiom,
    ! [I3: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I3),J)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I3) = K )
      <=> ( J = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),I3) ) ) ) ).

% Nat.le_imp_diff_is_add
tff(fact_3836_Nat_Odiff__add__assoc2,axiom,
    ! [K: nat,J: nat,I3: 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),I3)),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)),I3) ) ) ).

% Nat.diff_add_assoc2
tff(fact_3837_Nat_Odiff__add__assoc,axiom,
    ! [K: nat,J: nat,I3: 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,I3),J)),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I3),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)) ) ) ).

% Nat.diff_add_assoc
tff(fact_3838_Nat_Ole__diff__conv2,axiom,
    ! [K: nat,J: nat,I3: 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,I3),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,I3),K)),J) ) ) ).

% Nat.le_diff_conv2
tff(fact_3839_le__diff__conv,axiom,
    ! [J: nat,K: nat,I3: 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)),I3)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I3),K)) ) ).

% le_diff_conv
tff(fact_3840_diff__Suc__eq__diff__pred,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)),Nb) ).

% diff_Suc_eq_diff_pred
tff(fact_3841_mod__if,axiom,
    ! [Mb: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb),Mb,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),Nb)) ).

% mod_if
tff(fact_3842_mod__geq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),Nb) ) ) ).

% mod_geq
tff(fact_3843_le__mod__geq,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),Nb) ) ) ).

% le_mod_geq
tff(fact_3844_mod__eq__dvd__iff__nat,axiom,
    ! [Nb: nat,Mb: nat,Q2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Q2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),Q2) )
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Q2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) ) ) ).

% mod_eq_dvd_iff_nat
tff(fact_3845_nat__minus__add__max,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)),Mb) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Nb),Mb) ).

% nat_minus_add_max
tff(fact_3846_add__diff__assoc__enat,axiom,
    ! [Z: extended_enat,Yb: extended_enat,Xb: extended_enat] :
      ( ord_le2932123472753598470d_enat(Z,Yb)
     => ( plus_p3455044024723400733d_enat(Xb,minus_3235023915231533773d_enat(Yb,Z)) = minus_3235023915231533773d_enat(plus_p3455044024723400733d_enat(Xb,Yb),Z) ) ) ).

% add_diff_assoc_enat
tff(fact_3847_diff__Suc__less,axiom,
    ! [Nb: nat,I3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,I3))),Nb) ) ).

% diff_Suc_less
tff(fact_3848_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_3849_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_3850_less__diff__conv2,axiom,
    ! [K: nat,J: nat,I3: 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)),I3)
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I3),K)) ) ) ).

% less_diff_conv2
tff(fact_3851_order__antisym__conv,axiom,
    ! [Yb: set_nat,Xb: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(Yb),Xb)
     => ( aa_set_nat_o(ord_less_eq_set_nat(Xb),Yb)
      <=> ( Xb = Yb ) ) ) ).

% order_antisym_conv
tff(fact_3852_order__antisym__conv,axiom,
    ! [Yb: rat,Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Yb),Xb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Yb)
      <=> ( Xb = Yb ) ) ) ).

% order_antisym_conv
tff(fact_3853_order__antisym__conv,axiom,
    ! [Yb: num,Xb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Yb),Xb)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Yb)
      <=> ( Xb = Yb ) ) ) ).

% order_antisym_conv
tff(fact_3854_order__antisym__conv,axiom,
    ! [Yb: nat,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Yb),Xb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Yb)
      <=> ( Xb = Yb ) ) ) ).

% order_antisym_conv
tff(fact_3855_order__antisym__conv,axiom,
    ! [Yb: int,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Yb),Xb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Yb)
      <=> ( Xb = Yb ) ) ) ).

% order_antisym_conv
tff(fact_3856_linorder__le__cases,axiom,
    ! [Xb: rat,Yb: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Yb)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Yb),Xb) ) ).

% linorder_le_cases
tff(fact_3857_linorder__le__cases,axiom,
    ! [Xb: num,Yb: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Yb)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Yb),Xb) ) ).

% linorder_le_cases
tff(fact_3858_linorder__le__cases,axiom,
    ! [Xb: nat,Yb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Yb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Yb),Xb) ) ).

% linorder_le_cases
tff(fact_3859_linorder__le__cases,axiom,
    ! [Xb: int,Yb: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Yb)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Yb),Xb) ) ).

% linorder_le_cases
tff(fact_3860_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( ( aa_rat_rat(F,B) = C )
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,X3)),aa_rat_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,A)),C) ) ) ) ).

% ord_le_eq_subst
tff(fact_3861_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_num,C: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( ( aa_rat_num(F,B) = C )
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,X3)),aa_rat_num(F,Y)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,A)),C) ) ) ) ).

% ord_le_eq_subst
tff(fact_3862_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_nat,C: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( ( aa_rat_nat(F,B) = C )
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,X3)),aa_rat_nat(F,Y)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,A)),C) ) ) ) ).

% ord_le_eq_subst
tff(fact_3863_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_int,C: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( ( aa_rat_int(F,B) = C )
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,X3)),aa_rat_int(F,Y)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,A)),C) ) ) ) ).

% ord_le_eq_subst
tff(fact_3864_ord__le__eq__subst,axiom,
    ! [A: num,B: num,F: fun_num_rat,C: rat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( ( aa_num_rat(F,B) = C )
       => ( ! [X3: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,X3)),aa_num_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,A)),C) ) ) ) ).

% ord_le_eq_subst
tff(fact_3865_ord__le__eq__subst,axiom,
    ! [A: num,B: num,F: fun_num_num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( ( aa_num_num(F,B) = C )
       => ( ! [X3: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,X3)),aa_num_num(F,Y)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,A)),C) ) ) ) ).

% ord_le_eq_subst
tff(fact_3866_ord__le__eq__subst,axiom,
    ! [A: num,B: num,F: fun_num_nat,C: nat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( ( aa_num_nat(F,B) = C )
       => ( ! [X3: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,X3)),aa_num_nat(F,Y)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,A)),C) ) ) ) ).

% ord_le_eq_subst
tff(fact_3867_ord__le__eq__subst,axiom,
    ! [A: num,B: num,F: fun_num_int,C: int] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( ( aa_num_int(F,B) = C )
       => ( ! [X3: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(F,X3)),aa_num_int(F,Y)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(F,A)),C) ) ) ) ).

% ord_le_eq_subst
tff(fact_3868_ord__le__eq__subst,axiom,
    ! [A: nat,B: nat,F: fun_nat_rat,C: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( ( aa_nat_rat(F,B) = C )
       => ( ! [X3: nat,Y: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X3)),aa_nat_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,A)),C) ) ) ) ).

% ord_le_eq_subst
tff(fact_3869_ord__le__eq__subst,axiom,
    ! [A: nat,B: nat,F: fun_nat_num,C: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( ( aa_nat_num(F,B) = C )
       => ( ! [X3: nat,Y: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Y)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,X3)),aa_nat_num(F,Y)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,A)),C) ) ) ) ).

% ord_le_eq_subst
tff(fact_3870_ord__eq__le__subst,axiom,
    ! [A: rat,F: fun_rat_rat,B: rat,C: rat] :
      ( ( A = aa_rat_rat(F,B) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,X3)),aa_rat_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(F,C)) ) ) ) ).

% ord_eq_le_subst
tff(fact_3871_ord__eq__le__subst,axiom,
    ! [A: num,F: fun_rat_num,B: rat,C: rat] :
      ( ( A = aa_rat_num(F,B) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,X3)),aa_rat_num(F,Y)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_rat_num(F,C)) ) ) ) ).

% ord_eq_le_subst
tff(fact_3872_ord__eq__le__subst,axiom,
    ! [A: nat,F: fun_rat_nat,B: rat,C: rat] :
      ( ( A = aa_rat_nat(F,B) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,X3)),aa_rat_nat(F,Y)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_rat_nat(F,C)) ) ) ) ).

% ord_eq_le_subst
tff(fact_3873_ord__eq__le__subst,axiom,
    ! [A: int,F: fun_rat_int,B: rat,C: rat] :
      ( ( A = aa_rat_int(F,B) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,X3)),aa_rat_int(F,Y)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_rat_int(F,C)) ) ) ) ).

% ord_eq_le_subst
tff(fact_3874_ord__eq__le__subst,axiom,
    ! [A: rat,F: fun_num_rat,B: num,C: num] :
      ( ( A = aa_num_rat(F,B) )
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C)
       => ( ! [X3: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,X3)),aa_num_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_num_rat(F,C)) ) ) ) ).

% ord_eq_le_subst
tff(fact_3875_ord__eq__le__subst,axiom,
    ! [A: num,F: fun_num_num,B: num,C: num] :
      ( ( A = aa_num_num(F,B) )
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C)
       => ( ! [X3: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,X3)),aa_num_num(F,Y)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_num_num(F,C)) ) ) ) ).

% ord_eq_le_subst
tff(fact_3876_ord__eq__le__subst,axiom,
    ! [A: nat,F: fun_num_nat,B: num,C: num] :
      ( ( A = aa_num_nat(F,B) )
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C)
       => ( ! [X3: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,X3)),aa_num_nat(F,Y)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_num_nat(F,C)) ) ) ) ).

% ord_eq_le_subst
tff(fact_3877_ord__eq__le__subst,axiom,
    ! [A: int,F: fun_num_int,B: num,C: num] :
      ( ( A = aa_num_int(F,B) )
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C)
       => ( ! [X3: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(F,X3)),aa_num_int(F,Y)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_num_int(F,C)) ) ) ) ).

% ord_eq_le_subst
tff(fact_3878_ord__eq__le__subst,axiom,
    ! [A: rat,F: fun_nat_rat,B: nat,C: nat] :
      ( ( A = aa_nat_rat(F,B) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
       => ( ! [X3: nat,Y: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X3)),aa_nat_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_nat_rat(F,C)) ) ) ) ).

% ord_eq_le_subst
tff(fact_3879_ord__eq__le__subst,axiom,
    ! [A: num,F: fun_nat_num,B: nat,C: nat] :
      ( ( A = aa_nat_num(F,B) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
       => ( ! [X3: nat,Y: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Y)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,X3)),aa_nat_num(F,Y)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_nat_num(F,C)) ) ) ) ).

% ord_eq_le_subst
tff(fact_3880_linorder__linear,axiom,
    ! [Xb: rat,Yb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Yb)
      | aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Yb),Xb) ) ).

% linorder_linear
tff(fact_3881_linorder__linear,axiom,
    ! [Xb: num,Yb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Yb)
      | aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Yb),Xb) ) ).

% linorder_linear
tff(fact_3882_linorder__linear,axiom,
    ! [Xb: nat,Yb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Yb)
      | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Yb),Xb) ) ).

% linorder_linear
tff(fact_3883_linorder__linear,axiom,
    ! [Xb: int,Yb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Yb)
      | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Yb),Xb) ) ).

% linorder_linear
tff(fact_3884_order__eq__refl,axiom,
    ! [Xb: set_nat,Yb: set_nat] :
      ( ( Xb = Yb )
     => aa_set_nat_o(ord_less_eq_set_nat(Xb),Yb) ) ).

% order_eq_refl
tff(fact_3885_order__eq__refl,axiom,
    ! [Xb: rat,Yb: rat] :
      ( ( Xb = Yb )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Yb) ) ).

% order_eq_refl
tff(fact_3886_order__eq__refl,axiom,
    ! [Xb: num,Yb: num] :
      ( ( Xb = Yb )
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Yb) ) ).

% order_eq_refl
tff(fact_3887_order__eq__refl,axiom,
    ! [Xb: nat,Yb: nat] :
      ( ( Xb = Yb )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Yb) ) ).

% order_eq_refl
tff(fact_3888_order__eq__refl,axiom,
    ! [Xb: int,Yb: int] :
      ( ( Xb = Yb )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Yb) ) ).

% order_eq_refl
tff(fact_3889_order__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_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(F,B)),C)
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,X3)),aa_rat_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,A)),C) ) ) ) ).

% order_subst2
tff(fact_3890_order__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_num,C: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,B)),C)
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,X3)),aa_rat_num(F,Y)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,A)),C) ) ) ) ).

% order_subst2
tff(fact_3891_order__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_nat,C: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,B)),C)
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,X3)),aa_rat_nat(F,Y)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,A)),C) ) ) ) ).

% order_subst2
tff(fact_3892_order__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_int,C: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,B)),C)
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,X3)),aa_rat_int(F,Y)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,A)),C) ) ) ) ).

% order_subst2
tff(fact_3893_order__subst2,axiom,
    ! [A: num,B: num,F: fun_num_rat,C: rat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,B)),C)
       => ( ! [X3: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,X3)),aa_num_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,A)),C) ) ) ) ).

% order_subst2
tff(fact_3894_order__subst2,axiom,
    ! [A: num,B: num,F: fun_num_num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,B)),C)
       => ( ! [X3: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,X3)),aa_num_num(F,Y)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,A)),C) ) ) ) ).

% order_subst2
tff(fact_3895_order__subst2,axiom,
    ! [A: num,B: num,F: fun_num_nat,C: nat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,B)),C)
       => ( ! [X3: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,X3)),aa_num_nat(F,Y)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,A)),C) ) ) ) ).

% order_subst2
tff(fact_3896_order__subst2,axiom,
    ! [A: num,B: num,F: fun_num_int,C: int] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(F,B)),C)
       => ( ! [X3: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(F,X3)),aa_num_int(F,Y)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(F,A)),C) ) ) ) ).

% order_subst2
tff(fact_3897_order__subst2,axiom,
    ! [A: nat,B: nat,F: fun_nat_rat,C: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,B)),C)
       => ( ! [X3: nat,Y: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X3)),aa_nat_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,A)),C) ) ) ) ).

% order_subst2
tff(fact_3898_order__subst2,axiom,
    ! [A: nat,B: nat,F: fun_nat_num,C: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,B)),C)
       => ( ! [X3: nat,Y: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Y)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,X3)),aa_nat_num(F,Y)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,A)),C) ) ) ) ).

% order_subst2
tff(fact_3899_order__subst1,axiom,
    ! [A: rat,F: fun_rat_rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,X3)),aa_rat_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(F,C)) ) ) ) ).

% order_subst1
tff(fact_3900_order__subst1,axiom,
    ! [A: rat,F: fun_num_rat,B: num,C: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_num_rat(F,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C)
       => ( ! [X3: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,X3)),aa_num_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_num_rat(F,C)) ) ) ) ).

% order_subst1
tff(fact_3901_order__subst1,axiom,
    ! [A: rat,F: fun_nat_rat,B: nat,C: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_nat_rat(F,B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
       => ( ! [X3: nat,Y: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X3)),aa_nat_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_nat_rat(F,C)) ) ) ) ).

% order_subst1
tff(fact_3902_order__subst1,axiom,
    ! [A: rat,F: fun_int_rat,B: int,C: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_int_rat(F,B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C)
       => ( ! [X3: int,Y: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,X3)),aa_int_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_int_rat(F,C)) ) ) ) ).

% order_subst1
tff(fact_3903_order__subst1,axiom,
    ! [A: num,F: fun_rat_num,B: rat,C: rat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_rat_num(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,X3)),aa_rat_num(F,Y)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_rat_num(F,C)) ) ) ) ).

% order_subst1
tff(fact_3904_order__subst1,axiom,
    ! [A: num,F: fun_num_num,B: num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_num_num(F,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C)
       => ( ! [X3: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,X3)),aa_num_num(F,Y)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_num_num(F,C)) ) ) ) ).

% order_subst1
tff(fact_3905_order__subst1,axiom,
    ! [A: num,F: fun_nat_num,B: nat,C: nat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_nat_num(F,B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
       => ( ! [X3: nat,Y: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Y)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,X3)),aa_nat_num(F,Y)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_nat_num(F,C)) ) ) ) ).

% order_subst1
tff(fact_3906_order__subst1,axiom,
    ! [A: num,F: fun_int_num,B: int,C: int] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_int_num(F,B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C)
       => ( ! [X3: int,Y: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),Y)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_int_num(F,X3)),aa_int_num(F,Y)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_int_num(F,C)) ) ) ) ).

% order_subst1
tff(fact_3907_order__subst1,axiom,
    ! [A: nat,F: fun_rat_nat,B: rat,C: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_rat_nat(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,X3)),aa_rat_nat(F,Y)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_rat_nat(F,C)) ) ) ) ).

% order_subst1
tff(fact_3908_order__subst1,axiom,
    ! [A: nat,F: fun_num_nat,B: num,C: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_num_nat(F,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C)
       => ( ! [X3: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,X3)),aa_num_nat(F,Y)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_num_nat(F,C)) ) ) ) ).

% order_subst1
tff(fact_3909_Orderings_Oorder__eq__iff,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( A = B )
    <=> ( aa_set_nat_o(ord_less_eq_set_nat(A),B)
        & aa_set_nat_o(ord_less_eq_set_nat(B),A) ) ) ).

% Orderings.order_eq_iff
tff(fact_3910_Orderings_Oorder__eq__iff,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) ) ) ).

% Orderings.order_eq_iff
tff(fact_3911_Orderings_Oorder__eq__iff,axiom,
    ! [A: num,B: num] :
      ( ( A = B )
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
        & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A) ) ) ).

% Orderings.order_eq_iff
tff(fact_3912_Orderings_Oorder__eq__iff,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) ) ) ).

% Orderings.order_eq_iff
tff(fact_3913_Orderings_Oorder__eq__iff,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) ) ) ).

% Orderings.order_eq_iff
tff(fact_3914_antisym,axiom,
    ! [A: set_nat,B: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A),B)
     => ( aa_set_nat_o(ord_less_eq_set_nat(B),A)
       => ( A = B ) ) ) ).

% antisym
tff(fact_3915_antisym,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,B),A)
       => ( A = B ) ) ) ).

% antisym
tff(fact_3916_antisym,axiom,
    ! [A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
       => ( A = B ) ) ) ).

% antisym
tff(fact_3917_antisym,axiom,
    ! [A: nat,B: 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,B),A)
       => ( A = B ) ) ) ).

% antisym
tff(fact_3918_antisym,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,B),A)
       => ( A = B ) ) ) ).

% antisym
tff(fact_3919_dual__order_Otrans,axiom,
    ! [B: set_nat,A: set_nat,C: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(B),A)
     => ( aa_set_nat_o(ord_less_eq_set_nat(C),B)
       => aa_set_nat_o(ord_less_eq_set_nat(C),A) ) ) ).

% dual_order.trans
tff(fact_3920_dual__order_Otrans,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),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),A) ) ) ).

% dual_order.trans
tff(fact_3921_dual__order_Otrans,axiom,
    ! [B: num,A: num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),B)
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),A) ) ) ).

% dual_order.trans
tff(fact_3922_dual__order_Otrans,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( 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,C),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),A) ) ) ).

% dual_order.trans
tff(fact_3923_dual__order_Otrans,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),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),A) ) ) ).

% dual_order.trans
tff(fact_3924_dual__order_Oantisym,axiom,
    ! [B: set_nat,A: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(B),A)
     => ( aa_set_nat_o(ord_less_eq_set_nat(A),B)
       => ( A = B ) ) ) ).

% dual_order.antisym
tff(fact_3925_dual__order_Oantisym,axiom,
    ! [B: rat,A: 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,A),B)
       => ( A = B ) ) ) ).

% dual_order.antisym
tff(fact_3926_dual__order_Oantisym,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
       => ( A = B ) ) ) ).

% dual_order.antisym
tff(fact_3927_dual__order_Oantisym,axiom,
    ! [B: nat,A: nat] :
      ( 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,A),B)
       => ( A = B ) ) ) ).

% dual_order.antisym
tff(fact_3928_dual__order_Oantisym,axiom,
    ! [B: int,A: 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,A),B)
       => ( A = B ) ) ) ).

% dual_order.antisym
tff(fact_3929_dual__order_Oeq__iff,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( A = B )
    <=> ( aa_set_nat_o(ord_less_eq_set_nat(B),A)
        & aa_set_nat_o(ord_less_eq_set_nat(A),B) ) ) ).

% dual_order.eq_iff
tff(fact_3930_dual__order_Oeq__iff,axiom,
    ! [A: rat,B: 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,A),B) ) ) ).

% dual_order.eq_iff
tff(fact_3931_dual__order_Oeq__iff,axiom,
    ! [A: num,B: num] :
      ( ( A = B )
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
        & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B) ) ) ).

% dual_order.eq_iff
tff(fact_3932_dual__order_Oeq__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( A = 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_eq_nat,A),B) ) ) ).

% dual_order.eq_iff
tff(fact_3933_dual__order_Oeq__iff,axiom,
    ! [A: int,B: 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_eq_int,A),B) ) ) ).

% dual_order.eq_iff
tff(fact_3934_linorder__wlog,axiom,
    ! [P: fun_rat_fun_rat_o,A: rat,B: rat] :
      ( ! [A4: rat,B3: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A4),B3)
         => aa_rat_o(aa_rat_fun_rat_o(P,A4),B3) )
     => ( ! [A4: rat,B3: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(P,B3),A4)
           => aa_rat_o(aa_rat_fun_rat_o(P,A4),B3) )
       => aa_rat_o(aa_rat_fun_rat_o(P,A),B) ) ) ).

% linorder_wlog
tff(fact_3935_linorder__wlog,axiom,
    ! [P: fun_num_fun_num_o,A: num,B: num] :
      ( ! [A4: num,B3: num] :
          ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A4),B3)
         => aa_num_o(aa_num_fun_num_o(P,A4),B3) )
     => ( ! [A4: num,B3: num] :
            ( aa_num_o(aa_num_fun_num_o(P,B3),A4)
           => aa_num_o(aa_num_fun_num_o(P,A4),B3) )
       => aa_num_o(aa_num_fun_num_o(P,A),B) ) ) ).

% linorder_wlog
tff(fact_3936_linorder__wlog,axiom,
    ! [P: fun_nat_fun_nat_o,A: nat,B: nat] :
      ( ! [A4: nat,B3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A4),B3)
         => aa_nat_o(aa_nat_fun_nat_o(P,A4),B3) )
     => ( ! [A4: nat,B3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(P,B3),A4)
           => aa_nat_o(aa_nat_fun_nat_o(P,A4),B3) )
       => aa_nat_o(aa_nat_fun_nat_o(P,A),B) ) ) ).

% linorder_wlog
tff(fact_3937_linorder__wlog,axiom,
    ! [P: fun_int_fun_int_o,A: int,B: int] :
      ( ! [A4: int,B3: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A4),B3)
         => aa_int_o(aa_int_fun_int_o(P,A4),B3) )
     => ( ! [A4: int,B3: int] :
            ( aa_int_o(aa_int_fun_int_o(P,B3),A4)
           => aa_int_o(aa_int_fun_int_o(P,A4),B3) )
       => aa_int_o(aa_int_fun_int_o(P,A),B) ) ) ).

% linorder_wlog
tff(fact_3938_order__trans,axiom,
    ! [Xb: set_nat,Yb: set_nat,Z: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(Xb),Yb)
     => ( aa_set_nat_o(ord_less_eq_set_nat(Yb),Z)
       => aa_set_nat_o(ord_less_eq_set_nat(Xb),Z) ) ) ).

% order_trans
tff(fact_3939_order__trans,axiom,
    ! [Xb: rat,Yb: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Yb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Yb),Z)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Z) ) ) ).

% order_trans
tff(fact_3940_order__trans,axiom,
    ! [Xb: num,Yb: num,Z: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Yb)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Yb),Z)
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Z) ) ) ).

% order_trans
tff(fact_3941_order__trans,axiom,
    ! [Xb: nat,Yb: nat,Z: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Yb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Yb),Z)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Z) ) ) ).

% order_trans
tff(fact_3942_order__trans,axiom,
    ! [Xb: int,Yb: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Yb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Yb),Z)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Z) ) ) ).

% order_trans
tff(fact_3943_order_Otrans,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A),B)
     => ( aa_set_nat_o(ord_less_eq_set_nat(B),C)
       => aa_set_nat_o(ord_less_eq_set_nat(A),C) ) ) ).

% order.trans
tff(fact_3944_order_Otrans,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,B),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),C) ) ) ).

% order.trans
tff(fact_3945_order_Otrans,axiom,
    ! [A: num,B: num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C)
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),C) ) ) ).

% order.trans
tff(fact_3946_order_Otrans,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,B),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),C) ) ) ).

% order.trans
tff(fact_3947_order_Otrans,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,B),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),C) ) ) ).

% order.trans
tff(fact_3948_order__antisym,axiom,
    ! [Xb: set_nat,Yb: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(Xb),Yb)
     => ( aa_set_nat_o(ord_less_eq_set_nat(Yb),Xb)
       => ( Xb = Yb ) ) ) ).

% order_antisym
tff(fact_3949_order__antisym,axiom,
    ! [Xb: rat,Yb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Yb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Yb),Xb)
       => ( Xb = Yb ) ) ) ).

% order_antisym
tff(fact_3950_order__antisym,axiom,
    ! [Xb: num,Yb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Yb)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Yb),Xb)
       => ( Xb = Yb ) ) ) ).

% order_antisym
tff(fact_3951_order__antisym,axiom,
    ! [Xb: nat,Yb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Yb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Yb),Xb)
       => ( Xb = Yb ) ) ) ).

% order_antisym
tff(fact_3952_order__antisym,axiom,
    ! [Xb: int,Yb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Yb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Yb),Xb)
       => ( Xb = Yb ) ) ) ).

% order_antisym
tff(fact_3953_ord__le__eq__trans,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A),B)
     => ( ( B = C )
       => aa_set_nat_o(ord_less_eq_set_nat(A),C) ) ) ).

% ord_le_eq_trans
tff(fact_3954_ord__le__eq__trans,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( ( B = C )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),C) ) ) ).

% ord_le_eq_trans
tff(fact_3955_ord__le__eq__trans,axiom,
    ! [A: num,B: num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( ( B = C )
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),C) ) ) ).

% ord_le_eq_trans
tff(fact_3956_ord__le__eq__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( ( B = C )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),C) ) ) ).

% ord_le_eq_trans
tff(fact_3957_ord__le__eq__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( ( B = C )
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),C) ) ) ).

% ord_le_eq_trans
tff(fact_3958_ord__eq__le__trans,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat] :
      ( ( A = B )
     => ( aa_set_nat_o(ord_less_eq_set_nat(B),C)
       => aa_set_nat_o(ord_less_eq_set_nat(A),C) ) ) ).

% ord_eq_le_trans
tff(fact_3959_ord__eq__le__trans,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( A = B )
     => ( 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,A),C) ) ) ).

% ord_eq_le_trans
tff(fact_3960_ord__eq__le__trans,axiom,
    ! [A: num,B: num,C: num] :
      ( ( A = B )
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C)
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),C) ) ) ).

% ord_eq_le_trans
tff(fact_3961_ord__eq__le__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A = B )
     => ( 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,A),C) ) ) ).

% ord_eq_le_trans
tff(fact_3962_ord__eq__le__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A = B )
     => ( 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,A),C) ) ) ).

% ord_eq_le_trans
tff(fact_3963_order__class_Oorder__eq__iff,axiom,
    ! [Xb: set_nat,Yb: set_nat] :
      ( ( Xb = Yb )
    <=> ( aa_set_nat_o(ord_less_eq_set_nat(Xb),Yb)
        & aa_set_nat_o(ord_less_eq_set_nat(Yb),Xb) ) ) ).

% order_class.order_eq_iff
tff(fact_3964_order__class_Oorder__eq__iff,axiom,
    ! [Xb: rat,Yb: rat] :
      ( ( Xb = Yb )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Yb)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Yb),Xb) ) ) ).

% order_class.order_eq_iff
tff(fact_3965_order__class_Oorder__eq__iff,axiom,
    ! [Xb: num,Yb: num] :
      ( ( Xb = Yb )
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Yb)
        & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Yb),Xb) ) ) ).

% order_class.order_eq_iff
tff(fact_3966_order__class_Oorder__eq__iff,axiom,
    ! [Xb: nat,Yb: nat] :
      ( ( Xb = Yb )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Yb)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Yb),Xb) ) ) ).

% order_class.order_eq_iff
tff(fact_3967_order__class_Oorder__eq__iff,axiom,
    ! [Xb: int,Yb: int] :
      ( ( Xb = Yb )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Yb)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Yb),Xb) ) ) ).

% order_class.order_eq_iff
tff(fact_3968_le__cases3,axiom,
    ! [Xb: rat,Yb: rat,Z: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Yb)
       => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Yb),Z) )
     => ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Yb),Xb)
         => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Z) )
       => ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Z)
           => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z),Yb) )
         => ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z),Yb)
             => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Yb),Xb) )
           => ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Yb),Z)
               => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z),Xb) )
             => ~ ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z),Xb)
                 => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Yb) ) ) ) ) ) ) ).

% le_cases3
tff(fact_3969_le__cases3,axiom,
    ! [Xb: num,Yb: num,Z: num] :
      ( ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Yb)
       => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Yb),Z) )
     => ( ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Yb),Xb)
         => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Z) )
       => ( ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Z)
           => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Z),Yb) )
         => ( ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Z),Yb)
             => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Yb),Xb) )
           => ( ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Yb),Z)
               => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Z),Xb) )
             => ~ ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Z),Xb)
                 => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Yb) ) ) ) ) ) ) ).

% le_cases3
tff(fact_3970_le__cases3,axiom,
    ! [Xb: nat,Yb: nat,Z: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Yb)
       => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Yb),Z) )
     => ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Yb),Xb)
         => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Z) )
       => ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Z)
           => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z),Yb) )
         => ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z),Yb)
             => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Yb),Xb) )
           => ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Yb),Z)
               => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z),Xb) )
             => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z),Xb)
                 => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Yb) ) ) ) ) ) ) ).

% le_cases3
tff(fact_3971_le__cases3,axiom,
    ! [Xb: int,Yb: int,Z: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Yb)
       => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Yb),Z) )
     => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Yb),Xb)
         => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Z) )
       => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Z)
           => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),Yb) )
         => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),Yb)
             => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Yb),Xb) )
           => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Yb),Z)
               => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),Xb) )
             => ~ ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),Xb)
                 => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Yb) ) ) ) ) ) ) ).

% le_cases3
tff(fact_3972_nle__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,B),A)
        & ( B != A ) ) ) ).

% nle_le
tff(fact_3973_nle__le,axiom,
    ! [A: num,B: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
        & ( B != A ) ) ) ).

% nle_le
tff(fact_3974_nle__le,axiom,
    ! [A: nat,B: 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,B),A)
        & ( B != A ) ) ) ).

% nle_le
tff(fact_3975_nle__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,B),A)
        & ( B != A ) ) ) ).

% nle_le
tff(fact_3976_order__less__imp__not__less,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Yb)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Yb),Xb) ) ).

% order_less_imp_not_less
tff(fact_3977_order__less__imp__not__less,axiom,
    ! [Xb: rat,Yb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Yb)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Yb),Xb) ) ).

% order_less_imp_not_less
tff(fact_3978_order__less__imp__not__less,axiom,
    ! [Xb: num,Yb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Yb)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Yb),Xb) ) ).

% order_less_imp_not_less
tff(fact_3979_order__less__imp__not__less,axiom,
    ! [Xb: nat,Yb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Yb)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Yb),Xb) ) ).

% order_less_imp_not_less
tff(fact_3980_order__less__imp__not__less,axiom,
    ! [Xb: int,Yb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Yb)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Yb),Xb) ) ).

% order_less_imp_not_less
tff(fact_3981_order__less__imp__not__eq2,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Yb)
     => ( Yb != Xb ) ) ).

% order_less_imp_not_eq2
tff(fact_3982_order__less__imp__not__eq2,axiom,
    ! [Xb: rat,Yb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Yb)
     => ( Yb != Xb ) ) ).

% order_less_imp_not_eq2
tff(fact_3983_order__less__imp__not__eq2,axiom,
    ! [Xb: num,Yb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Yb)
     => ( Yb != Xb ) ) ).

% order_less_imp_not_eq2
tff(fact_3984_order__less__imp__not__eq2,axiom,
    ! [Xb: nat,Yb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Yb)
     => ( Yb != Xb ) ) ).

% order_less_imp_not_eq2
tff(fact_3985_order__less__imp__not__eq2,axiom,
    ! [Xb: int,Yb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Yb)
     => ( Yb != Xb ) ) ).

% order_less_imp_not_eq2
tff(fact_3986_order__less__imp__not__eq,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Yb)
     => ( Xb != Yb ) ) ).

% order_less_imp_not_eq
tff(fact_3987_order__less__imp__not__eq,axiom,
    ! [Xb: rat,Yb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Yb)
     => ( Xb != Yb ) ) ).

% order_less_imp_not_eq
tff(fact_3988_order__less__imp__not__eq,axiom,
    ! [Xb: num,Yb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Yb)
     => ( Xb != Yb ) ) ).

% order_less_imp_not_eq
tff(fact_3989_order__less__imp__not__eq,axiom,
    ! [Xb: nat,Yb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Yb)
     => ( Xb != Yb ) ) ).

% order_less_imp_not_eq
tff(fact_3990_order__less__imp__not__eq,axiom,
    ! [Xb: int,Yb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Yb)
     => ( Xb != Yb ) ) ).

% order_less_imp_not_eq
tff(fact_3991_linorder__less__linear,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Yb)
      | ( Xb = Yb )
      | aa_real_o(aa_real_fun_real_o(ord_less_real,Yb),Xb) ) ).

% linorder_less_linear
tff(fact_3992_linorder__less__linear,axiom,
    ! [Xb: rat,Yb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Yb)
      | ( Xb = Yb )
      | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Yb),Xb) ) ).

% linorder_less_linear
tff(fact_3993_linorder__less__linear,axiom,
    ! [Xb: num,Yb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Yb)
      | ( Xb = Yb )
      | aa_num_o(aa_num_fun_num_o(ord_less_num,Yb),Xb) ) ).

% linorder_less_linear
tff(fact_3994_linorder__less__linear,axiom,
    ! [Xb: nat,Yb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Yb)
      | ( Xb = Yb )
      | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Yb),Xb) ) ).

% linorder_less_linear
tff(fact_3995_linorder__less__linear,axiom,
    ! [Xb: int,Yb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Yb)
      | ( Xb = Yb )
      | aa_int_o(aa_int_fun_int_o(ord_less_int,Yb),Xb) ) ).

% linorder_less_linear
tff(fact_3996_order__less__imp__triv,axiom,
    ! [Xb: real,Yb: real,P: $o] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Yb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Yb),Xb)
       => (P) ) ) ).

% order_less_imp_triv
tff(fact_3997_order__less__imp__triv,axiom,
    ! [Xb: rat,Yb: rat,P: $o] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Yb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Yb),Xb)
       => (P) ) ) ).

% order_less_imp_triv
tff(fact_3998_order__less__imp__triv,axiom,
    ! [Xb: num,Yb: num,P: $o] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Yb)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,Yb),Xb)
       => (P) ) ) ).

% order_less_imp_triv
tff(fact_3999_order__less__imp__triv,axiom,
    ! [Xb: nat,Yb: nat,P: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Yb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Yb),Xb)
       => (P) ) ) ).

% order_less_imp_triv
tff(fact_4000_order__less__imp__triv,axiom,
    ! [Xb: int,Yb: int,P: $o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Yb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Yb),Xb)
       => (P) ) ) ).

% order_less_imp_triv
tff(fact_4001_order__less__not__sym,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Yb)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Yb),Xb) ) ).

% order_less_not_sym
tff(fact_4002_order__less__not__sym,axiom,
    ! [Xb: rat,Yb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Yb)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Yb),Xb) ) ).

% order_less_not_sym
tff(fact_4003_order__less__not__sym,axiom,
    ! [Xb: num,Yb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Yb)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Yb),Xb) ) ).

% order_less_not_sym
tff(fact_4004_order__less__not__sym,axiom,
    ! [Xb: nat,Yb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Yb)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Yb),Xb) ) ).

% order_less_not_sym
tff(fact_4005_order__less__not__sym,axiom,
    ! [Xb: int,Yb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Yb)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Yb),Xb) ) ).

% order_less_not_sym
tff(fact_4006_order__less__subst2,axiom,
    ! [A: real,B: real,F: fun_real_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(F,B)),C)
       => ( ! [X3: real,Y: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,X3)),aa_real_real(F,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,A)),C) ) ) ) ).

% order_less_subst2
tff(fact_4007_order__less__subst2,axiom,
    ! [A: real,B: real,F: fun_real_rat,C: rat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,B)),C)
       => ( ! [X3: real,Y: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,X3)),aa_real_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,A)),C) ) ) ) ).

% order_less_subst2
tff(fact_4008_order__less__subst2,axiom,
    ! [A: real,B: real,F: fun_real_num,C: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_real_num(F,B)),C)
       => ( ! [X3: real,Y: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y)
             => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_real_num(F,X3)),aa_real_num(F,Y)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_real_num(F,A)),C) ) ) ) ).

% order_less_subst2
tff(fact_4009_order__less__subst2,axiom,
    ! [A: real,B: real,F: fun_real_nat,C: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_real_nat(F,B)),C)
       => ( ! [X3: real,Y: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_real_nat(F,X3)),aa_real_nat(F,Y)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_real_nat(F,A)),C) ) ) ) ).

% order_less_subst2
tff(fact_4010_order__less__subst2,axiom,
    ! [A: real,B: real,F: fun_real_int,C: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(F,B)),C)
       => ( ! [X3: real,Y: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(F,X3)),aa_real_int(F,Y)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(F,A)),C) ) ) ) ).

% order_less_subst2
tff(fact_4011_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_real,C: real] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,B)),C)
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,X3)),aa_rat_real(F,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,A)),C) ) ) ) ).

% order_less_subst2
tff(fact_4012_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_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(F,B)),C)
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,X3)),aa_rat_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,A)),C) ) ) ) ).

% order_less_subst2
tff(fact_4013_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_num,C: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F,B)),C)
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y)
             => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F,X3)),aa_rat_num(F,Y)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F,A)),C) ) ) ) ).

% order_less_subst2
tff(fact_4014_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_nat,C: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,B)),C)
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,X3)),aa_rat_nat(F,Y)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,A)),C) ) ) ) ).

% order_less_subst2
tff(fact_4015_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_int,C: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(F,B)),C)
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(F,X3)),aa_rat_int(F,Y)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(F,A)),C) ) ) ) ).

% order_less_subst2
tff(fact_4016_order__less__subst1,axiom,
    ! [A: real,F: fun_real_real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(F,B))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
       => ( ! [X3: real,Y: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,X3)),aa_real_real(F,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(F,C)) ) ) ) ).

% order_less_subst1
tff(fact_4017_order__less__subst1,axiom,
    ! [A: real,F: fun_rat_real,B: rat,C: rat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_rat_real(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,X3)),aa_rat_real(F,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_rat_real(F,C)) ) ) ) ).

% order_less_subst1
tff(fact_4018_order__less__subst1,axiom,
    ! [A: real,F: fun_num_real,B: num,C: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_num_real(F,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),C)
       => ( ! [X3: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(F,X3)),aa_num_real(F,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_num_real(F,C)) ) ) ) ).

% order_less_subst1
tff(fact_4019_order__less__subst1,axiom,
    ! [A: real,F: fun_nat_real,B: nat,C: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_nat_real(F,B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
       => ( ! [X3: nat,Y: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,X3)),aa_nat_real(F,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_nat_real(F,C)) ) ) ) ).

% order_less_subst1
tff(fact_4020_order__less__subst1,axiom,
    ! [A: real,F: fun_int_real,B: int,C: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_int_real(F,B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C)
       => ( ! [X3: int,Y: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(F,X3)),aa_int_real(F,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_int_real(F,C)) ) ) ) ).

% order_less_subst1
tff(fact_4021_order__less__subst1,axiom,
    ! [A: rat,F: fun_real_rat,B: real,C: real] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_real_rat(F,B))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
       => ( ! [X3: real,Y: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,X3)),aa_real_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_real_rat(F,C)) ) ) ) ).

% order_less_subst1
tff(fact_4022_order__less__subst1,axiom,
    ! [A: rat,F: fun_rat_rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,X3)),aa_rat_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(F,C)) ) ) ) ).

% order_less_subst1
tff(fact_4023_order__less__subst1,axiom,
    ! [A: rat,F: fun_num_rat,B: num,C: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_num_rat(F,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),C)
       => ( ! [X3: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(F,X3)),aa_num_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_num_rat(F,C)) ) ) ) ).

% order_less_subst1
tff(fact_4024_order__less__subst1,axiom,
    ! [A: rat,F: fun_nat_rat,B: nat,C: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_nat_rat(F,B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
       => ( ! [X3: nat,Y: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,X3)),aa_nat_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_nat_rat(F,C)) ) ) ) ).

% order_less_subst1
tff(fact_4025_order__less__subst1,axiom,
    ! [A: rat,F: fun_int_rat,B: int,C: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_int_rat(F,B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C)
       => ( ! [X3: int,Y: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(F,X3)),aa_int_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_int_rat(F,C)) ) ) ) ).

% order_less_subst1
tff(fact_4026_order__less__irrefl,axiom,
    ! [Xb: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Xb) ).

% order_less_irrefl
tff(fact_4027_order__less__irrefl,axiom,
    ! [Xb: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Xb) ).

% order_less_irrefl
tff(fact_4028_order__less__irrefl,axiom,
    ! [Xb: num] : ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Xb) ).

% order_less_irrefl
tff(fact_4029_order__less__irrefl,axiom,
    ! [Xb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Xb) ).

% order_less_irrefl
tff(fact_4030_order__less__irrefl,axiom,
    ! [Xb: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Xb) ).

% order_less_irrefl
tff(fact_4031_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: fun_real_real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( ( aa_real_real(F,B) = C )
       => ( ! [X3: real,Y: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,X3)),aa_real_real(F,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,A)),C) ) ) ) ).

% ord_less_eq_subst
tff(fact_4032_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: fun_real_rat,C: rat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( ( aa_real_rat(F,B) = C )
       => ( ! [X3: real,Y: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,X3)),aa_real_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,A)),C) ) ) ) ).

% ord_less_eq_subst
tff(fact_4033_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: fun_real_num,C: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( ( aa_real_num(F,B) = C )
       => ( ! [X3: real,Y: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y)
             => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_real_num(F,X3)),aa_real_num(F,Y)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_real_num(F,A)),C) ) ) ) ).

% ord_less_eq_subst
tff(fact_4034_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: fun_real_nat,C: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( ( aa_real_nat(F,B) = C )
       => ( ! [X3: real,Y: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_real_nat(F,X3)),aa_real_nat(F,Y)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_real_nat(F,A)),C) ) ) ) ).

% ord_less_eq_subst
tff(fact_4035_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: fun_real_int,C: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( ( aa_real_int(F,B) = C )
       => ( ! [X3: real,Y: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(F,X3)),aa_real_int(F,Y)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(F,A)),C) ) ) ) ).

% ord_less_eq_subst
tff(fact_4036_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_real,C: real] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( ( aa_rat_real(F,B) = C )
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,X3)),aa_rat_real(F,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,A)),C) ) ) ) ).

% ord_less_eq_subst
tff(fact_4037_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( ( aa_rat_rat(F,B) = C )
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,X3)),aa_rat_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,A)),C) ) ) ) ).

% ord_less_eq_subst
tff(fact_4038_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_num,C: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( ( aa_rat_num(F,B) = C )
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y)
             => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F,X3)),aa_rat_num(F,Y)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F,A)),C) ) ) ) ).

% ord_less_eq_subst
tff(fact_4039_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_nat,C: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( ( aa_rat_nat(F,B) = C )
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,X3)),aa_rat_nat(F,Y)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,A)),C) ) ) ) ).

% ord_less_eq_subst
tff(fact_4040_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: fun_rat_int,C: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( ( aa_rat_int(F,B) = C )
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(F,X3)),aa_rat_int(F,Y)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(F,A)),C) ) ) ) ).

% ord_less_eq_subst
tff(fact_4041_ord__eq__less__subst,axiom,
    ! [A: real,F: fun_real_real,B: real,C: real] :
      ( ( A = aa_real_real(F,B) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
       => ( ! [X3: real,Y: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,X3)),aa_real_real(F,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(F,C)) ) ) ) ).

% ord_eq_less_subst
tff(fact_4042_ord__eq__less__subst,axiom,
    ! [A: rat,F: fun_real_rat,B: real,C: real] :
      ( ( A = aa_real_rat(F,B) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
       => ( ! [X3: real,Y: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,X3)),aa_real_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_real_rat(F,C)) ) ) ) ).

% ord_eq_less_subst
tff(fact_4043_ord__eq__less__subst,axiom,
    ! [A: num,F: fun_real_num,B: real,C: real] :
      ( ( A = aa_real_num(F,B) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
       => ( ! [X3: real,Y: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y)
             => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_real_num(F,X3)),aa_real_num(F,Y)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,A),aa_real_num(F,C)) ) ) ) ).

% ord_eq_less_subst
tff(fact_4044_ord__eq__less__subst,axiom,
    ! [A: nat,F: fun_real_nat,B: real,C: real] :
      ( ( A = aa_real_nat(F,B) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
       => ( ! [X3: real,Y: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_real_nat(F,X3)),aa_real_nat(F,Y)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_real_nat(F,C)) ) ) ) ).

% ord_eq_less_subst
tff(fact_4045_ord__eq__less__subst,axiom,
    ! [A: int,F: fun_real_int,B: real,C: real] :
      ( ( A = aa_real_int(F,B) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
       => ( ! [X3: real,Y: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(F,X3)),aa_real_int(F,Y)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_real_int(F,C)) ) ) ) ).

% ord_eq_less_subst
tff(fact_4046_ord__eq__less__subst,axiom,
    ! [A: real,F: fun_rat_real,B: rat,C: rat] :
      ( ( A = aa_rat_real(F,B) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,X3)),aa_rat_real(F,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_rat_real(F,C)) ) ) ) ).

% ord_eq_less_subst
tff(fact_4047_ord__eq__less__subst,axiom,
    ! [A: rat,F: fun_rat_rat,B: rat,C: rat] :
      ( ( A = aa_rat_rat(F,B) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,X3)),aa_rat_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(F,C)) ) ) ) ).

% ord_eq_less_subst
tff(fact_4048_ord__eq__less__subst,axiom,
    ! [A: num,F: fun_rat_num,B: rat,C: rat] :
      ( ( A = aa_rat_num(F,B) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y)
             => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F,X3)),aa_rat_num(F,Y)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,A),aa_rat_num(F,C)) ) ) ) ).

% ord_eq_less_subst
tff(fact_4049_ord__eq__less__subst,axiom,
    ! [A: nat,F: fun_rat_nat,B: rat,C: rat] :
      ( ( A = aa_rat_nat(F,B) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,X3)),aa_rat_nat(F,Y)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_rat_nat(F,C)) ) ) ) ).

% ord_eq_less_subst
tff(fact_4050_ord__eq__less__subst,axiom,
    ! [A: int,F: fun_rat_int,B: rat,C: rat] :
      ( ( A = aa_rat_int(F,B) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(F,X3)),aa_rat_int(F,Y)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_rat_int(F,C)) ) ) ) ).

% ord_eq_less_subst
tff(fact_4051_order__less__trans,axiom,
    ! [Xb: real,Yb: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Yb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Yb),Z)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Z) ) ) ).

% order_less_trans
tff(fact_4052_order__less__trans,axiom,
    ! [Xb: rat,Yb: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Yb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Yb),Z)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Z) ) ) ).

% order_less_trans
tff(fact_4053_order__less__trans,axiom,
    ! [Xb: num,Yb: num,Z: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Yb)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,Yb),Z)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Z) ) ) ).

% order_less_trans
tff(fact_4054_order__less__trans,axiom,
    ! [Xb: nat,Yb: nat,Z: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Yb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Yb),Z)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Z) ) ) ).

% order_less_trans
tff(fact_4055_order__less__trans,axiom,
    ! [Xb: int,Yb: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Yb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Yb),Z)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Z) ) ) ).

% order_less_trans
tff(fact_4056_order__less__asym_H,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),A) ) ).

% order_less_asym'
tff(fact_4057_order__less__asym_H,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),A) ) ).

% order_less_asym'
tff(fact_4058_order__less__asym_H,axiom,
    ! [A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,B),A) ) ).

% order_less_asym'
tff(fact_4059_order__less__asym_H,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,B),A) ) ).

% order_less_asym'
tff(fact_4060_order__less__asym_H,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,B),A) ) ).

% order_less_asym'
tff(fact_4061_linorder__neq__iff,axiom,
    ! [Xb: real,Yb: real] :
      ( ( Xb != Yb )
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Yb)
        | aa_real_o(aa_real_fun_real_o(ord_less_real,Yb),Xb) ) ) ).

% linorder_neq_iff
tff(fact_4062_linorder__neq__iff,axiom,
    ! [Xb: rat,Yb: rat] :
      ( ( Xb != Yb )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Yb)
        | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Yb),Xb) ) ) ).

% linorder_neq_iff
tff(fact_4063_linorder__neq__iff,axiom,
    ! [Xb: num,Yb: num] :
      ( ( Xb != Yb )
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Yb)
        | aa_num_o(aa_num_fun_num_o(ord_less_num,Yb),Xb) ) ) ).

% linorder_neq_iff
tff(fact_4064_linorder__neq__iff,axiom,
    ! [Xb: nat,Yb: nat] :
      ( ( Xb != Yb )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Yb)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Yb),Xb) ) ) ).

% linorder_neq_iff
tff(fact_4065_linorder__neq__iff,axiom,
    ! [Xb: int,Yb: int] :
      ( ( Xb != Yb )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Yb)
        | aa_int_o(aa_int_fun_int_o(ord_less_int,Yb),Xb) ) ) ).

% linorder_neq_iff
tff(fact_4066_order__less__asym,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Yb)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Yb),Xb) ) ).

% order_less_asym
tff(fact_4067_order__less__asym,axiom,
    ! [Xb: rat,Yb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Yb)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Yb),Xb) ) ).

% order_less_asym
tff(fact_4068_order__less__asym,axiom,
    ! [Xb: num,Yb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Yb)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Yb),Xb) ) ).

% order_less_asym
tff(fact_4069_order__less__asym,axiom,
    ! [Xb: nat,Yb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Yb)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Yb),Xb) ) ).

% order_less_asym
tff(fact_4070_order__less__asym,axiom,
    ! [Xb: int,Yb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Yb)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Yb),Xb) ) ).

% order_less_asym
tff(fact_4071_linorder__neqE,axiom,
    ! [Xb: real,Yb: real] :
      ( ( Xb != Yb )
     => ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Yb)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Yb),Xb) ) ) ).

% linorder_neqE
tff(fact_4072_linorder__neqE,axiom,
    ! [Xb: rat,Yb: rat] :
      ( ( Xb != Yb )
     => ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Yb)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Yb),Xb) ) ) ).

% linorder_neqE
tff(fact_4073_linorder__neqE,axiom,
    ! [Xb: num,Yb: num] :
      ( ( Xb != Yb )
     => ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Yb)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,Yb),Xb) ) ) ).

% linorder_neqE
tff(fact_4074_linorder__neqE,axiom,
    ! [Xb: nat,Yb: nat] :
      ( ( Xb != Yb )
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Yb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Yb),Xb) ) ) ).

% linorder_neqE
tff(fact_4075_linorder__neqE,axiom,
    ! [Xb: int,Yb: int] :
      ( ( Xb != Yb )
     => ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Yb)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,Yb),Xb) ) ) ).

% linorder_neqE
tff(fact_4076_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_4077_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_4078_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_4079_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_4080_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_4081_order_Ostrict__implies__not__eq,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( A != B ) ) ).

% order.strict_implies_not_eq
tff(fact_4082_order_Ostrict__implies__not__eq,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( A != B ) ) ).

% order.strict_implies_not_eq
tff(fact_4083_order_Ostrict__implies__not__eq,axiom,
    ! [A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
     => ( A != B ) ) ).

% order.strict_implies_not_eq
tff(fact_4084_order_Ostrict__implies__not__eq,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( A != B ) ) ).

% order.strict_implies_not_eq
tff(fact_4085_order_Ostrict__implies__not__eq,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( A != B ) ) ).

% order.strict_implies_not_eq
tff(fact_4086_dual__order_Ostrict__trans,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),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,C),A) ) ) ).

% dual_order.strict_trans
tff(fact_4087_dual__order_Ostrict__trans,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),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),A) ) ) ).

% dual_order.strict_trans
tff(fact_4088_dual__order_Ostrict__trans,axiom,
    ! [B: num,A: num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,C),B)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,C),A) ) ) ).

% dual_order.strict_trans
tff(fact_4089_dual__order_Ostrict__trans,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),A) ) ) ).

% dual_order.strict_trans
tff(fact_4090_dual__order_Ostrict__trans,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),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,C),A) ) ) ).

% dual_order.strict_trans
tff(fact_4091_not__less__iff__gr__or__eq,axiom,
    ! [Xb: real,Yb: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Yb)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Yb),Xb)
        | ( Xb = Yb ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_4092_not__less__iff__gr__or__eq,axiom,
    ! [Xb: rat,Yb: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Yb)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Yb),Xb)
        | ( Xb = Yb ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_4093_not__less__iff__gr__or__eq,axiom,
    ! [Xb: num,Yb: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Yb)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,Yb),Xb)
        | ( Xb = Yb ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_4094_not__less__iff__gr__or__eq,axiom,
    ! [Xb: nat,Yb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Yb)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Yb),Xb)
        | ( Xb = Yb ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_4095_not__less__iff__gr__or__eq,axiom,
    ! [Xb: int,Yb: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Yb)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Yb),Xb)
        | ( Xb = Yb ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_4096_order_Ostrict__trans,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,B),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A),C) ) ) ).

% order.strict_trans
tff(fact_4097_order_Ostrict__trans,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,B),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),C) ) ) ).

% order.strict_trans
tff(fact_4098_order_Ostrict__trans,axiom,
    ! [A: num,B: num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),C)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,A),C) ) ) ).

% order.strict_trans
tff(fact_4099_order_Ostrict__trans,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,B),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),C) ) ) ).

% order.strict_trans
tff(fact_4100_order_Ostrict__trans,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,B),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),C) ) ) ).

% order.strict_trans
tff(fact_4101_linorder__less__wlog,axiom,
    ! [P: fun_real_fun_real_o,A: real,B: real] :
      ( ! [A4: real,B3: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,A4),B3)
         => aa_real_o(aa_real_fun_real_o(P,A4),B3) )
     => ( ! [A4: real] : aa_real_o(aa_real_fun_real_o(P,A4),A4)
       => ( ! [A4: real,B3: real] :
              ( aa_real_o(aa_real_fun_real_o(P,B3),A4)
             => aa_real_o(aa_real_fun_real_o(P,A4),B3) )
         => aa_real_o(aa_real_fun_real_o(P,A),B) ) ) ) ).

% linorder_less_wlog
tff(fact_4102_linorder__less__wlog,axiom,
    ! [P: fun_rat_fun_rat_o,A: rat,B: rat] :
      ( ! [A4: rat,B3: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A4),B3)
         => aa_rat_o(aa_rat_fun_rat_o(P,A4),B3) )
     => ( ! [A4: rat] : aa_rat_o(aa_rat_fun_rat_o(P,A4),A4)
       => ( ! [A4: rat,B3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(P,B3),A4)
             => aa_rat_o(aa_rat_fun_rat_o(P,A4),B3) )
         => aa_rat_o(aa_rat_fun_rat_o(P,A),B) ) ) ) ).

% linorder_less_wlog
tff(fact_4103_linorder__less__wlog,axiom,
    ! [P: fun_num_fun_num_o,A: num,B: num] :
      ( ! [A4: num,B3: num] :
          ( aa_num_o(aa_num_fun_num_o(ord_less_num,A4),B3)
         => aa_num_o(aa_num_fun_num_o(P,A4),B3) )
     => ( ! [A4: num] : aa_num_o(aa_num_fun_num_o(P,A4),A4)
       => ( ! [A4: num,B3: num] :
              ( aa_num_o(aa_num_fun_num_o(P,B3),A4)
             => aa_num_o(aa_num_fun_num_o(P,A4),B3) )
         => aa_num_o(aa_num_fun_num_o(P,A),B) ) ) ) ).

% linorder_less_wlog
tff(fact_4104_linorder__less__wlog,axiom,
    ! [P: fun_nat_fun_nat_o,A: nat,B: nat] :
      ( ! [A4: nat,B3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A4),B3)
         => aa_nat_o(aa_nat_fun_nat_o(P,A4),B3) )
     => ( ! [A4: nat] : aa_nat_o(aa_nat_fun_nat_o(P,A4),A4)
       => ( ! [A4: nat,B3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(P,B3),A4)
             => aa_nat_o(aa_nat_fun_nat_o(P,A4),B3) )
         => aa_nat_o(aa_nat_fun_nat_o(P,A),B) ) ) ) ).

% linorder_less_wlog
tff(fact_4105_linorder__less__wlog,axiom,
    ! [P: fun_int_fun_int_o,A: int,B: int] :
      ( ! [A4: int,B3: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,A4),B3)
         => aa_int_o(aa_int_fun_int_o(P,A4),B3) )
     => ( ! [A4: int] : aa_int_o(aa_int_fun_int_o(P,A4),A4)
       => ( ! [A4: int,B3: int] :
              ( aa_int_o(aa_int_fun_int_o(P,B3),A4)
             => aa_int_o(aa_int_fun_int_o(P,A4),B3) )
         => aa_int_o(aa_int_fun_int_o(P,A),B) ) ) ) ).

% linorder_less_wlog
tff(fact_4106_exists__least__iff,axiom,
    ! [P: fun_nat_o] :
      ( ? [X_13: nat] : aa_nat_o(P,X_13)
    <=> ? [N4: nat] :
          ( aa_nat_o(P,N4)
          & ! [M5: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M5),N4)
             => ~ aa_nat_o(P,M5) ) ) ) ).

% exists_least_iff
tff(fact_4107_dual__order_Oirrefl,axiom,
    ! [A: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,A),A) ).

% dual_order.irrefl
tff(fact_4108_dual__order_Oirrefl,axiom,
    ! [A: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),A) ).

% dual_order.irrefl
tff(fact_4109_dual__order_Oirrefl,axiom,
    ! [A: num] : ~ aa_num_o(aa_num_fun_num_o(ord_less_num,A),A) ).

% dual_order.irrefl
tff(fact_4110_dual__order_Oirrefl,axiom,
    ! [A: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),A) ).

% dual_order.irrefl
tff(fact_4111_dual__order_Oirrefl,axiom,
    ! [A: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,A),A) ).

% dual_order.irrefl
tff(fact_4112_dual__order_Oasym,axiom,
    ! [B: real,A: 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,A),B) ) ).

% dual_order.asym
tff(fact_4113_dual__order_Oasym,axiom,
    ! [B: rat,A: 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,A),B) ) ).

% dual_order.asym
tff(fact_4114_dual__order_Oasym,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,A),B) ) ).

% dual_order.asym
tff(fact_4115_dual__order_Oasym,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ).

% dual_order.asym
tff(fact_4116_dual__order_Oasym,axiom,
    ! [B: int,A: 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,A),B) ) ).

% dual_order.asym
tff(fact_4117_linorder__cases,axiom,
    ! [Xb: real,Yb: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Yb)
     => ( ( Xb != Yb )
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Yb),Xb) ) ) ).

% linorder_cases
tff(fact_4118_linorder__cases,axiom,
    ! [Xb: rat,Yb: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Yb)
     => ( ( Xb != Yb )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Yb),Xb) ) ) ).

% linorder_cases
tff(fact_4119_linorder__cases,axiom,
    ! [Xb: num,Yb: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Yb)
     => ( ( Xb != Yb )
       => aa_num_o(aa_num_fun_num_o(ord_less_num,Yb),Xb) ) ) ).

% linorder_cases
tff(fact_4120_linorder__cases,axiom,
    ! [Xb: nat,Yb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Yb)
     => ( ( Xb != Yb )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Yb),Xb) ) ) ).

% linorder_cases
tff(fact_4121_linorder__cases,axiom,
    ! [Xb: int,Yb: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Yb)
     => ( ( Xb != Yb )
       => aa_int_o(aa_int_fun_int_o(ord_less_int,Yb),Xb) ) ) ).

% linorder_cases
tff(fact_4122_antisym__conv3,axiom,
    ! [Yb: real,Xb: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Yb),Xb)
     => ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Yb)
      <=> ( Xb = Yb ) ) ) ).

% antisym_conv3
tff(fact_4123_antisym__conv3,axiom,
    ! [Yb: rat,Xb: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Yb),Xb)
     => ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Yb)
      <=> ( Xb = Yb ) ) ) ).

% antisym_conv3
tff(fact_4124_antisym__conv3,axiom,
    ! [Yb: num,Xb: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Yb),Xb)
     => ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Yb)
      <=> ( Xb = Yb ) ) ) ).

% antisym_conv3
tff(fact_4125_antisym__conv3,axiom,
    ! [Yb: nat,Xb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Yb),Xb)
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Yb)
      <=> ( Xb = Yb ) ) ) ).

% antisym_conv3
tff(fact_4126_antisym__conv3,axiom,
    ! [Yb: int,Xb: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Yb),Xb)
     => ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Yb)
      <=> ( Xb = Yb ) ) ) ).

% antisym_conv3
tff(fact_4127_less__induct,axiom,
    ! [P: fun_nat_o,A: nat] :
      ( ! [X3: nat] :
          ( ! [Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y3),X3)
             => aa_nat_o(P,Y3) )
         => aa_nat_o(P,X3) )
     => aa_nat_o(P,A) ) ).

% less_induct
tff(fact_4128_ord__less__eq__trans,axiom,
    ! [A: real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( ( B = C )
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A),C) ) ) ).

% ord_less_eq_trans
tff(fact_4129_ord__less__eq__trans,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( ( B = C )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),C) ) ) ).

% ord_less_eq_trans
tff(fact_4130_ord__less__eq__trans,axiom,
    ! [A: num,B: num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
     => ( ( B = C )
       => aa_num_o(aa_num_fun_num_o(ord_less_num,A),C) ) ) ).

% ord_less_eq_trans
tff(fact_4131_ord__less__eq__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( ( B = C )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),C) ) ) ).

% ord_less_eq_trans
tff(fact_4132_ord__less__eq__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( ( B = C )
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),C) ) ) ).

% ord_less_eq_trans
tff(fact_4133_ord__eq__less__trans,axiom,
    ! [A: real,B: real,C: real] :
      ( ( A = B )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A),C) ) ) ).

% ord_eq_less_trans
tff(fact_4134_ord__eq__less__trans,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( A = B )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),C) ) ) ).

% ord_eq_less_trans
tff(fact_4135_ord__eq__less__trans,axiom,
    ! [A: num,B: num,C: num] :
      ( ( A = B )
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),C)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,A),C) ) ) ).

% ord_eq_less_trans
tff(fact_4136_ord__eq__less__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A = B )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),C) ) ) ).

% ord_eq_less_trans
tff(fact_4137_ord__eq__less__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A = B )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),C) ) ) ).

% ord_eq_less_trans
tff(fact_4138_order_Oasym,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),A) ) ).

% order.asym
tff(fact_4139_order_Oasym,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),A) ) ).

% order.asym
tff(fact_4140_order_Oasym,axiom,
    ! [A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,B),A) ) ).

% order.asym
tff(fact_4141_order_Oasym,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,B),A) ) ).

% order.asym
tff(fact_4142_order_Oasym,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,B),A) ) ).

% order.asym
tff(fact_4143_less__imp__neq,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Yb)
     => ( Xb != Yb ) ) ).

% less_imp_neq
tff(fact_4144_less__imp__neq,axiom,
    ! [Xb: rat,Yb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Yb)
     => ( Xb != Yb ) ) ).

% less_imp_neq
tff(fact_4145_less__imp__neq,axiom,
    ! [Xb: num,Yb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Yb)
     => ( Xb != Yb ) ) ).

% less_imp_neq
tff(fact_4146_less__imp__neq,axiom,
    ! [Xb: nat,Yb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Yb)
     => ( Xb != Yb ) ) ).

% less_imp_neq
tff(fact_4147_less__imp__neq,axiom,
    ! [Xb: int,Yb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Yb)
     => ( Xb != Yb ) ) ).

% less_imp_neq
tff(fact_4148_dense,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Yb)
     => ? [Z2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Z2)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),Yb) ) ) ).

% dense
tff(fact_4149_dense,axiom,
    ! [Xb: rat,Yb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Yb)
     => ? [Z2: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Z2)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),Yb) ) ) ).

% dense
tff(fact_4150_gt__ex,axiom,
    ! [Xb: real] :
    ? [X_12: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),X_12) ).

% gt_ex
tff(fact_4151_gt__ex,axiom,
    ! [Xb: rat] :
    ? [X_12: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),X_12) ).

% gt_ex
tff(fact_4152_gt__ex,axiom,
    ! [Xb: nat] :
    ? [X_12: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),X_12) ).

% gt_ex
tff(fact_4153_gt__ex,axiom,
    ! [Xb: int] :
    ? [X_12: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),X_12) ).

% gt_ex
tff(fact_4154_lt__ex,axiom,
    ! [Xb: real] :
    ? [Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xb) ).

% lt_ex
tff(fact_4155_lt__ex,axiom,
    ! [Xb: rat] :
    ? [Y: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Xb) ).

% lt_ex
tff(fact_4156_lt__ex,axiom,
    ! [Xb: int] :
    ? [Y: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Xb) ).

% lt_ex
tff(fact_4157_nat__diff__add__eq2,axiom,
    ! [I3: nat,J: nat,U: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I3),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,I3),U)),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),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),I3)),U)),Nb)) ) ) ).

% nat_diff_add_eq2
tff(fact_4158_nat__diff__add__eq1,axiom,
    ! [J: nat,I3: nat,U: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),I3)
     => ( 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,I3),U)),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Nb)) = 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,I3),J)),U)),Mb)),Nb) ) ) ).

% nat_diff_add_eq1
tff(fact_4159_nat__le__add__iff2,axiom,
    ! [I3: nat,J: nat,U: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I3),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,I3),U)),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Nb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),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),I3)),U)),Nb)) ) ) ).

% nat_le_add_iff2
tff(fact_4160_nat__le__add__iff1,axiom,
    ! [J: nat,I3: nat,U: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),I3)
     => ( 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,I3),U)),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Nb))
      <=> 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,I3),J)),U)),Mb)),Nb) ) ) ).

% nat_le_add_iff1
tff(fact_4161_nat__eq__add__iff2,axiom,
    ! [I3: nat,J: nat,U: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I3),J)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I3),U)),Mb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Nb) )
      <=> ( Mb = 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),I3)),U)),Nb) ) ) ) ).

% nat_eq_add_iff2
tff(fact_4162_nat__eq__add__iff1,axiom,
    ! [J: nat,I3: nat,U: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),I3)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I3),U)),Mb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Nb) )
      <=> ( 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,I3),J)),U)),Mb) = Nb ) ) ) ).

% nat_eq_add_iff1
tff(fact_4163_dvd__minus__add,axiom,
    ! [Q2: nat,Nb: nat,R2: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Q2),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Q2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R2),Mb))
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Q2))
        <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R2),Mb)),Q2))) ) ) ) ).

% dvd_minus_add
tff(fact_4164_mod__nat__eqI,axiom,
    ! [R2: nat,Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,R2),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,R2),Mb)
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),R2))
         => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb) = R2 ) ) ) ) ).

% mod_nat_eqI
tff(fact_4165_modulo__nat__def,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)),Nb)) ).

% modulo_nat_def
tff(fact_4166_in__mono,axiom,
    ! [A3: set_complex,B4: set_complex,Xb: complex] :
      ( ord_le211207098394363844omplex(A3,B4)
     => ( member_complex(Xb,A3)
       => member_complex(Xb,B4) ) ) ).

% in_mono
tff(fact_4167_in__mono,axiom,
    ! [A3: set_real,B4: set_real,Xb: real] :
      ( ord_less_eq_set_real(A3,B4)
     => ( member_real(Xb,A3)
       => member_real(Xb,B4) ) ) ).

% in_mono
tff(fact_4168_in__mono,axiom,
    ! [A3: set_set_nat,B4: set_set_nat,Xb: set_nat] :
      ( ord_le6893508408891458716et_nat(A3,B4)
     => ( member_set_nat(Xb,A3)
       => member_set_nat(Xb,B4) ) ) ).

% in_mono
tff(fact_4169_in__mono,axiom,
    ! [A3: set_int,B4: set_int,Xb: int] :
      ( ord_less_eq_set_int(A3,B4)
     => ( member_int(Xb,A3)
       => member_int(Xb,B4) ) ) ).

% in_mono
tff(fact_4170_in__mono,axiom,
    ! [A3: set_nat,B4: set_nat,Xb: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A3),B4)
     => ( aa_set_nat_o(member_nat(Xb),A3)
       => aa_set_nat_o(member_nat(Xb),B4) ) ) ).

% in_mono
tff(fact_4171_subsetD,axiom,
    ! [A3: set_complex,B4: set_complex,C: complex] :
      ( ord_le211207098394363844omplex(A3,B4)
     => ( member_complex(C,A3)
       => member_complex(C,B4) ) ) ).

% subsetD
tff(fact_4172_subsetD,axiom,
    ! [A3: set_real,B4: set_real,C: real] :
      ( ord_less_eq_set_real(A3,B4)
     => ( member_real(C,A3)
       => member_real(C,B4) ) ) ).

% subsetD
tff(fact_4173_subsetD,axiom,
    ! [A3: set_set_nat,B4: set_set_nat,C: set_nat] :
      ( ord_le6893508408891458716et_nat(A3,B4)
     => ( member_set_nat(C,A3)
       => member_set_nat(C,B4) ) ) ).

% subsetD
tff(fact_4174_subsetD,axiom,
    ! [A3: set_int,B4: set_int,C: int] :
      ( ord_less_eq_set_int(A3,B4)
     => ( member_int(C,A3)
       => member_int(C,B4) ) ) ).

% subsetD
tff(fact_4175_subsetD,axiom,
    ! [A3: set_nat,B4: set_nat,C: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A3),B4)
     => ( aa_set_nat_o(member_nat(C),A3)
       => aa_set_nat_o(member_nat(C),B4) ) ) ).

% subsetD
tff(fact_4176_psubsetE,axiom,
    ! [A3: set_nat,B4: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(A3),B4)
     => ~ ( aa_set_nat_o(ord_less_eq_set_nat(A3),B4)
         => aa_set_nat_o(ord_less_eq_set_nat(B4),A3) ) ) ).

% psubsetE
tff(fact_4177_equalityE,axiom,
    ! [A3: set_nat,B4: set_nat] :
      ( ( A3 = B4 )
     => ~ ( aa_set_nat_o(ord_less_eq_set_nat(A3),B4)
         => ~ aa_set_nat_o(ord_less_eq_set_nat(B4),A3) ) ) ).

% equalityE
tff(fact_4178_subset__eq,axiom,
    ! [A3: set_complex,B4: set_complex] :
      ( ord_le211207098394363844omplex(A3,B4)
    <=> ! [X4: complex] :
          ( member_complex(X4,A3)
         => member_complex(X4,B4) ) ) ).

% subset_eq
tff(fact_4179_subset__eq,axiom,
    ! [A3: set_real,B4: set_real] :
      ( ord_less_eq_set_real(A3,B4)
    <=> ! [X4: real] :
          ( member_real(X4,A3)
         => member_real(X4,B4) ) ) ).

% subset_eq
tff(fact_4180_subset__eq,axiom,
    ! [A3: set_set_nat,B4: set_set_nat] :
      ( ord_le6893508408891458716et_nat(A3,B4)
    <=> ! [X4: set_nat] :
          ( member_set_nat(X4,A3)
         => member_set_nat(X4,B4) ) ) ).

% subset_eq
tff(fact_4181_subset__eq,axiom,
    ! [A3: set_int,B4: set_int] :
      ( ord_less_eq_set_int(A3,B4)
    <=> ! [X4: int] :
          ( member_int(X4,A3)
         => member_int(X4,B4) ) ) ).

% subset_eq
tff(fact_4182_subset__eq,axiom,
    ! [A3: set_nat,B4: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A3),B4)
    <=> ! [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),A3)
         => aa_set_nat_o(member_nat(X4),B4) ) ) ).

% subset_eq
tff(fact_4183_equalityD1,axiom,
    ! [A3: set_nat,B4: set_nat] :
      ( ( A3 = B4 )
     => aa_set_nat_o(ord_less_eq_set_nat(A3),B4) ) ).

% equalityD1
tff(fact_4184_equalityD2,axiom,
    ! [A3: set_nat,B4: set_nat] :
      ( ( A3 = B4 )
     => aa_set_nat_o(ord_less_eq_set_nat(B4),A3) ) ).

% equalityD2
tff(fact_4185_psubset__eq,axiom,
    ! [A3: set_nat,B4: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(A3),B4)
    <=> ( aa_set_nat_o(ord_less_eq_set_nat(A3),B4)
        & ( A3 != B4 ) ) ) ).

% psubset_eq
tff(fact_4186_subset__iff,axiom,
    ! [A3: set_complex,B4: set_complex] :
      ( ord_le211207098394363844omplex(A3,B4)
    <=> ! [T: complex] :
          ( member_complex(T,A3)
         => member_complex(T,B4) ) ) ).

% subset_iff
tff(fact_4187_subset__iff,axiom,
    ! [A3: set_real,B4: set_real] :
      ( ord_less_eq_set_real(A3,B4)
    <=> ! [T: real] :
          ( member_real(T,A3)
         => member_real(T,B4) ) ) ).

% subset_iff
tff(fact_4188_subset__iff,axiom,
    ! [A3: set_set_nat,B4: set_set_nat] :
      ( ord_le6893508408891458716et_nat(A3,B4)
    <=> ! [T: set_nat] :
          ( member_set_nat(T,A3)
         => member_set_nat(T,B4) ) ) ).

% subset_iff
tff(fact_4189_subset__iff,axiom,
    ! [A3: set_int,B4: set_int] :
      ( ord_less_eq_set_int(A3,B4)
    <=> ! [T: int] :
          ( member_int(T,A3)
         => member_int(T,B4) ) ) ).

% subset_iff
tff(fact_4190_subset__iff,axiom,
    ! [A3: set_nat,B4: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A3),B4)
    <=> ! [T: nat] :
          ( aa_set_nat_o(member_nat(T),A3)
         => aa_set_nat_o(member_nat(T),B4) ) ) ).

% subset_iff
tff(fact_4191_subset__refl,axiom,
    ! [A3: set_nat] : aa_set_nat_o(ord_less_eq_set_nat(A3),A3) ).

% subset_refl
tff(fact_4192_Collect__mono,axiom,
    ! [P: fun_real_o,Q: fun_real_o] :
      ( ! [X3: real] :
          ( aa_real_o(P,X3)
         => aa_real_o(Q,X3) )
     => ord_less_eq_set_real(collect_real(P),collect_real(Q)) ) ).

% Collect_mono
tff(fact_4193_Collect__mono,axiom,
    ! [P: fun_list_nat_o,Q: fun_list_nat_o] :
      ( ! [X3: list_nat] :
          ( aa_list_nat_o(P,X3)
         => aa_list_nat_o(Q,X3) )
     => ord_le6045566169113846134st_nat(collect_list_nat(P),collect_list_nat(Q)) ) ).

% Collect_mono
tff(fact_4194_Collect__mono,axiom,
    ! [P: fun_set_nat_o,Q: fun_set_nat_o] :
      ( ! [X3: set_nat] :
          ( aa_set_nat_o(P,X3)
         => aa_set_nat_o(Q,X3) )
     => ord_le6893508408891458716et_nat(collect_set_nat(P),collect_set_nat(Q)) ) ).

% Collect_mono
tff(fact_4195_Collect__mono,axiom,
    ! [P: fun_int_o,Q: fun_int_o] :
      ( ! [X3: int] :
          ( aa_int_o(P,X3)
         => aa_int_o(Q,X3) )
     => ord_less_eq_set_int(collect_int(P),collect_int(Q)) ) ).

% Collect_mono
tff(fact_4196_Collect__mono,axiom,
    ! [P: fun_nat_o,Q: fun_nat_o] :
      ( ! [X3: nat] :
          ( aa_nat_o(P,X3)
         => aa_nat_o(Q,X3) )
     => aa_set_nat_o(ord_less_eq_set_nat(collect_nat(P)),collect_nat(Q)) ) ).

% Collect_mono
tff(fact_4197_subset__trans,axiom,
    ! [A3: set_nat,B4: set_nat,C5: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A3),B4)
     => ( aa_set_nat_o(ord_less_eq_set_nat(B4),C5)
       => aa_set_nat_o(ord_less_eq_set_nat(A3),C5) ) ) ).

% subset_trans
tff(fact_4198_set__eq__subset,axiom,
    ! [A3: set_nat,B4: set_nat] :
      ( ( A3 = B4 )
    <=> ( aa_set_nat_o(ord_less_eq_set_nat(A3),B4)
        & aa_set_nat_o(ord_less_eq_set_nat(B4),A3) ) ) ).

% set_eq_subset
tff(fact_4199_Collect__mono__iff,axiom,
    ! [P: fun_real_o,Q: fun_real_o] :
      ( ord_less_eq_set_real(collect_real(P),collect_real(Q))
    <=> ! [X4: real] :
          ( aa_real_o(P,X4)
         => aa_real_o(Q,X4) ) ) ).

% Collect_mono_iff
tff(fact_4200_Collect__mono__iff,axiom,
    ! [P: fun_list_nat_o,Q: fun_list_nat_o] :
      ( ord_le6045566169113846134st_nat(collect_list_nat(P),collect_list_nat(Q))
    <=> ! [X4: list_nat] :
          ( aa_list_nat_o(P,X4)
         => aa_list_nat_o(Q,X4) ) ) ).

% Collect_mono_iff
tff(fact_4201_Collect__mono__iff,axiom,
    ! [P: fun_set_nat_o,Q: fun_set_nat_o] :
      ( ord_le6893508408891458716et_nat(collect_set_nat(P),collect_set_nat(Q))
    <=> ! [X4: set_nat] :
          ( aa_set_nat_o(P,X4)
         => aa_set_nat_o(Q,X4) ) ) ).

% Collect_mono_iff
tff(fact_4202_Collect__mono__iff,axiom,
    ! [P: fun_int_o,Q: fun_int_o] :
      ( ord_less_eq_set_int(collect_int(P),collect_int(Q))
    <=> ! [X4: int] :
          ( aa_int_o(P,X4)
         => aa_int_o(Q,X4) ) ) ).

% Collect_mono_iff
tff(fact_4203_Collect__mono__iff,axiom,
    ! [P: fun_nat_o,Q: fun_nat_o] :
      ( aa_set_nat_o(ord_less_eq_set_nat(collect_nat(P)),collect_nat(Q))
    <=> ! [X4: nat] :
          ( aa_nat_o(P,X4)
         => aa_nat_o(Q,X4) ) ) ).

% Collect_mono_iff
tff(fact_4204_psubset__imp__subset,axiom,
    ! [A3: set_nat,B4: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(A3),B4)
     => aa_set_nat_o(ord_less_eq_set_nat(A3),B4) ) ).

% psubset_imp_subset
tff(fact_4205_psubset__subset__trans,axiom,
    ! [A3: set_nat,B4: set_nat,C5: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(A3),B4)
     => ( aa_set_nat_o(ord_less_eq_set_nat(B4),C5)
       => aa_set_nat_o(ord_less_set_nat(A3),C5) ) ) ).

% psubset_subset_trans
tff(fact_4206_subset__not__subset__eq,axiom,
    ! [A3: set_nat,B4: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(A3),B4)
    <=> ( aa_set_nat_o(ord_less_eq_set_nat(A3),B4)
        & ~ aa_set_nat_o(ord_less_eq_set_nat(B4),A3) ) ) ).

% subset_not_subset_eq
tff(fact_4207_subset__psubset__trans,axiom,
    ! [A3: set_nat,B4: set_nat,C5: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A3),B4)
     => ( aa_set_nat_o(ord_less_set_nat(B4),C5)
       => aa_set_nat_o(ord_less_set_nat(A3),C5) ) ) ).

% subset_psubset_trans
tff(fact_4208_subset__iff__psubset__eq,axiom,
    ! [A3: set_nat,B4: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A3),B4)
    <=> ( aa_set_nat_o(ord_less_set_nat(A3),B4)
        | ( A3 = B4 ) ) ) ).

% subset_iff_psubset_eq
tff(fact_4209_prod__decode__aux_Ocases,axiom,
    ! [Xb: product_prod_nat_nat] :
      ~ ! [K2: nat,M2: nat] : Xb != aa_nat7603221163087113075at_nat(product_Pair_nat_nat(K2),M2) ).

% prod_decode_aux.cases
tff(fact_4210_power__diff,axiom,
    ! [A: complex,Nb: nat,Mb: nat] :
      ( ( A != zero_zero_complex )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => ( aa_nat_complex(power_power_complex(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(power_power_complex(A),Mb)),aa_nat_complex(power_power_complex(A),Nb)) ) ) ) ).

% power_diff
tff(fact_4211_power__diff,axiom,
    ! [A: real,Nb: nat,Mb: nat] :
      ( ( A != zero_zero_real )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) = aa_real_real(divide_divide_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Mb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ) ) ) ).

% power_diff
tff(fact_4212_power__diff,axiom,
    ! [A: rat,Nb: nat,Mb: nat] :
      ( ( A != zero_zero_rat )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => ( aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) = aa_rat_rat(divide_divide_rat(aa_nat_rat(power_power_rat(A),Mb)),aa_nat_rat(power_power_rat(A),Nb)) ) ) ) ).

% power_diff
tff(fact_4213_power__diff,axiom,
    ! [A: nat,Nb: nat,Mb: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ) ) ) ).

% power_diff
tff(fact_4214_power__diff,axiom,
    ! [A: int,Nb: nat,Mb: nat] :
      ( ( A != zero_zero_int )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => ( aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(power_power_int(A),Mb)),aa_nat_int(power_power_int(A),Nb)) ) ) ) ).

% power_diff
tff(fact_4215_Suc__diff__eq__diff__pred,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,Mb)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)) ) ) ).

% Suc_diff_eq_diff_pred
tff(fact_4216_Suc__pred_H,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( Nb = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)) ) ) ).

% Suc_pred'
tff(fact_4217_div__if,axiom,
    ! [Mb: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb) = $ite(
        ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
        | ( Nb = 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,Mb),Nb)),Nb)) ) ).

% div_if
tff(fact_4218_div__geq,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb) = 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,Mb),Nb)),Nb)) ) ) ) ).

% div_geq
tff(fact_4219_add__eq__if,axiom,
    ! [Mb: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb) = $ite(Mb = zero_zero_nat,Nb,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,Mb),one_one_nat)),Nb))) ).

% add_eq_if
tff(fact_4220_nat__less__add__iff2,axiom,
    ! [I3: nat,J: nat,U: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I3),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,I3),U)),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Nb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),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),I3)),U)),Nb)) ) ) ).

% nat_less_add_iff2
tff(fact_4221_nat__less__add__iff1,axiom,
    ! [J: nat,I3: nat,U: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),I3)
     => ( 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,I3),U)),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Nb))
      <=> 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,I3),J)),U)),Mb)),Nb) ) ) ).

% nat_less_add_iff1
tff(fact_4222_mult__eq__if,axiom,
    ! [Mb: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb) = $ite(Mb = zero_zero_nat,zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)),Nb))) ).

% mult_eq_if
tff(fact_4223_Collect__subset,axiom,
    ! [A3: set_complex,P: fun_complex_o] : ord_le211207098394363844omplex(collect_complex(aa_fun2514317545224092796plex_o(aTP_Lamm_at(A3),P)),A3) ).

% Collect_subset
tff(fact_4224_Collect__subset,axiom,
    ! [A3: set_real,P: fun_real_o] : ord_less_eq_set_real(collect_real(aa_fun5868358341814070780real_o(aTP_Lamm_au(A3),P)),A3) ).

% Collect_subset
tff(fact_4225_Collect__subset,axiom,
    ! [A3: set_list_nat,P: fun_list_nat_o] : ord_le6045566169113846134st_nat(collect_list_nat(aa_fun6155608546934989052_nat_o(aTP_Lamm_av(A3),P)),A3) ).

% Collect_subset
tff(fact_4226_Collect__subset,axiom,
    ! [A3: set_set_nat,P: fun_set_nat_o] : ord_le6893508408891458716et_nat(collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_aw(A3),P)),A3) ).

% Collect_subset
tff(fact_4227_Collect__subset,axiom,
    ! [A3: set_int,P: fun_int_o] : ord_less_eq_set_int(collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_ax(A3),P)),A3) ).

% Collect_subset
tff(fact_4228_Collect__subset,axiom,
    ! [A3: set_nat,P: fun_nat_o] : aa_set_nat_o(ord_less_eq_set_nat(collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_ay(A3),P))),A3) ).

% Collect_subset
tff(fact_4229_less__eq__set__def,axiom,
    ! [A3: set_complex,B4: set_complex] :
      ( ord_le211207098394363844omplex(A3,B4)
    <=> ord_le7644322562001619171plex_o(aTP_Lamm_a(A3),aTP_Lamm_a(B4)) ) ).

% less_eq_set_def
tff(fact_4230_less__eq__set__def,axiom,
    ! [A3: set_real,B4: set_real] :
      ( ord_less_eq_set_real(A3,B4)
    <=> ord_le3605727316730466277real_o(aTP_Lamm_aa(A3),aTP_Lamm_aa(B4)) ) ).

% less_eq_set_def
tff(fact_4231_less__eq__set__def,axiom,
    ! [A3: set_set_nat,B4: set_set_nat] :
      ( ord_le6893508408891458716et_nat(A3,B4)
    <=> ord_le6468474429079055883_nat_o(aTP_Lamm_ac(A3),aTP_Lamm_ac(B4)) ) ).

% less_eq_set_def
tff(fact_4232_less__eq__set__def,axiom,
    ! [A3: set_int,B4: set_int] :
      ( ord_less_eq_set_int(A3,B4)
    <=> ord_le5418975059635765733_int_o(aTP_Lamm_ae(A3),aTP_Lamm_ae(B4)) ) ).

% less_eq_set_def
tff(fact_4233_less__eq__set__def,axiom,
    ! [A3: set_nat,B4: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A3),B4)
    <=> ord_le7240017405068453697_nat_o(aTP_Lamm_ad(A3),aTP_Lamm_ad(B4)) ) ).

% less_eq_set_def
tff(fact_4234_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [Nb: nat,Mb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) != 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,Nb),Mb)) != zero_zero_nat ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
tff(fact_4235_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [Nb: nat,Mb: nat] :
      ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb) != 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,Nb),Mb)) != zero_zero_int ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
tff(fact_4236_power__diff__power__eq,axiom,
    ! [A: nat,Mb: nat,Nb: 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),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),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,Nb),Mb)))) ) ) ).

% power_diff_power_eq
tff(fact_4237_power__diff__power__eq,axiom,
    ! [A: int,Mb: nat,Nb: nat] :
      ( ( A != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(power_power_int(A),Mb)),aa_nat_int(power_power_int(A),Nb)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb),aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),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,Nb),Mb)))) ) ) ).

% power_diff_power_eq
tff(fact_4238_power__eq__if,axiom,
    ! [P2: complex,Mb: nat] :
      aa_nat_complex(power_power_complex(P2),Mb) = $ite(Mb = zero_zero_nat,one_one_complex,aa_complex_complex(times_times_complex(P2),aa_nat_complex(power_power_complex(P2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)))) ).

% power_eq_if
tff(fact_4239_power__eq__if,axiom,
    ! [P2: real,Mb: nat] :
      aa_nat_real(aa_real_fun_nat_real(power_power_real,P2),Mb) = $ite(Mb = zero_zero_nat,one_one_real,aa_real_real(times_times_real(P2),aa_nat_real(aa_real_fun_nat_real(power_power_real,P2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)))) ).

% power_eq_if
tff(fact_4240_power__eq__if,axiom,
    ! [P2: rat,Mb: nat] :
      aa_nat_rat(power_power_rat(P2),Mb) = $ite(Mb = zero_zero_nat,one_one_rat,aa_rat_rat(times_times_rat(P2),aa_nat_rat(power_power_rat(P2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)))) ).

% power_eq_if
tff(fact_4241_power__eq__if,axiom,
    ! [P2: nat,Mb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,P2),Mb) = $ite(Mb = zero_zero_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,P2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,P2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)))) ).

% power_eq_if
tff(fact_4242_power__eq__if,axiom,
    ! [P2: int,Mb: nat] :
      aa_nat_int(power_power_int(P2),Mb) = $ite(Mb = zero_zero_nat,one_one_int,aa_int_int(aa_int_fun_int_int(times_times_int,P2),aa_nat_int(power_power_int(P2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)))) ).

% power_eq_if
tff(fact_4243_power__minus__mult,axiom,
    ! [Nb: nat,A: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))),A) = aa_nat_complex(power_power_complex(A),Nb) ) ) ).

% power_minus_mult
tff(fact_4244_power__minus__mult,axiom,
    ! [Nb: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))),A) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb) ) ) ).

% power_minus_mult
tff(fact_4245_power__minus__mult,axiom,
    ! [Nb: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))),A) = aa_nat_rat(power_power_rat(A),Nb) ) ) ).

% power_minus_mult
tff(fact_4246_power__minus__mult,axiom,
    ! [Nb: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( 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,Nb),one_one_nat))),A) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb) ) ) ).

% power_minus_mult
tff(fact_4247_power__minus__mult,axiom,
    ! [Nb: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( 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,Nb),one_one_nat))),A) = aa_nat_int(power_power_int(A),Nb) ) ) ).

% power_minus_mult
tff(fact_4248_diff__le__diff__pow,axiom,
    ! [K: nat,Mb: nat,Nb: 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,Mb),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K),Nb))) ) ).

% diff_le_diff_pow
tff(fact_4249_le__div__geq,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb) = 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,Mb),Nb)),Nb)) ) ) ) ).

% le_div_geq
tff(fact_4250_order__le__imp__less__or__eq,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Yb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Yb)
        | ( Xb = Yb ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_4251_order__le__imp__less__or__eq,axiom,
    ! [Xb: set_nat,Yb: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(Xb),Yb)
     => ( aa_set_nat_o(ord_less_set_nat(Xb),Yb)
        | ( Xb = Yb ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_4252_order__le__imp__less__or__eq,axiom,
    ! [Xb: rat,Yb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Yb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Yb)
        | ( Xb = Yb ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_4253_order__le__imp__less__or__eq,axiom,
    ! [Xb: num,Yb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Yb)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Yb)
        | ( Xb = Yb ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_4254_order__le__imp__less__or__eq,axiom,
    ! [Xb: nat,Yb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Yb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Yb)
        | ( Xb = Yb ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_4255_order__le__imp__less__or__eq,axiom,
    ! [Xb: int,Yb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Yb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Yb)
        | ( Xb = Yb ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_4256_linorder__le__less__linear,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Yb)
      | aa_real_o(aa_real_fun_real_o(ord_less_real,Yb),Xb) ) ).

% linorder_le_less_linear
tff(fact_4257_linorder__le__less__linear,axiom,
    ! [Xb: rat,Yb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Yb)
      | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Yb),Xb) ) ).

% linorder_le_less_linear
tff(fact_4258_linorder__le__less__linear,axiom,
    ! [Xb: num,Yb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Yb)
      | aa_num_o(aa_num_fun_num_o(ord_less_num,Yb),Xb) ) ).

% linorder_le_less_linear
tff(fact_4259_linorder__le__less__linear,axiom,
    ! [Xb: nat,Yb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Yb)
      | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Yb),Xb) ) ).

% linorder_le_less_linear
tff(fact_4260_linorder__le__less__linear,axiom,
    ! [Xb: int,Yb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Yb)
      | aa_int_o(aa_int_fun_int_o(ord_less_int,Yb),Xb) ) ).

% linorder_le_less_linear
tff(fact_4261_order__less__le__subst2,axiom,
    ! [A: real,B: real,F: fun_real_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_eq_real,aa_real_real(F,B)),C)
       => ( ! [X3: real,Y: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,X3)),aa_real_real(F,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,A)),C) ) ) ) ).

% order_less_le_subst2
tff(fact_4262_order__less__le__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_real,C: real] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_rat_real(F,B)),C)
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,X3)),aa_rat_real(F,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,A)),C) ) ) ) ).

% order_less_le_subst2
tff(fact_4263_order__less__le__subst2,axiom,
    ! [A: num,B: num,F: fun_num_real,C: real] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(F,B)),C)
       => ( ! [X3: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(F,X3)),aa_num_real(F,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(F,A)),C) ) ) ) ).

% order_less_le_subst2
tff(fact_4264_order__less__le__subst2,axiom,
    ! [A: nat,B: nat,F: fun_nat_real,C: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,B)),C)
       => ( ! [X3: nat,Y: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,X3)),aa_nat_real(F,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,A)),C) ) ) ) ).

% order_less_le_subst2
tff(fact_4265_order__less__le__subst2,axiom,
    ! [A: int,B: int,F: fun_int_real,C: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(F,B)),C)
       => ( ! [X3: int,Y: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(F,X3)),aa_int_real(F,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(F,A)),C) ) ) ) ).

% order_less_le_subst2
tff(fact_4266_order__less__le__subst2,axiom,
    ! [A: real,B: real,F: fun_real_rat,C: rat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_real_rat(F,B)),C)
       => ( ! [X3: real,Y: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,X3)),aa_real_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,A)),C) ) ) ) ).

% order_less_le_subst2
tff(fact_4267_order__less__le__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_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_eq_rat,aa_rat_rat(F,B)),C)
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,X3)),aa_rat_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,A)),C) ) ) ) ).

% order_less_le_subst2
tff(fact_4268_order__less__le__subst2,axiom,
    ! [A: num,B: num,F: fun_num_rat,C: rat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,B)),C)
       => ( ! [X3: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(F,X3)),aa_num_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(F,A)),C) ) ) ) ).

% order_less_le_subst2
tff(fact_4269_order__less__le__subst2,axiom,
    ! [A: nat,B: nat,F: fun_nat_rat,C: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,B)),C)
       => ( ! [X3: nat,Y: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,X3)),aa_nat_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,A)),C) ) ) ) ).

% order_less_le_subst2
tff(fact_4270_order__less__le__subst2,axiom,
    ! [A: int,B: int,F: fun_int_rat,C: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,B)),C)
       => ( ! [X3: int,Y: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(F,X3)),aa_int_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(F,A)),C) ) ) ) ).

% order_less_le_subst2
tff(fact_4271_order__less__le__subst1,axiom,
    ! [A: real,F: fun_rat_real,B: rat,C: rat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_rat_real(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_rat_real(F,X3)),aa_rat_real(F,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_rat_real(F,C)) ) ) ) ).

% order_less_le_subst1
tff(fact_4272_order__less__le__subst1,axiom,
    ! [A: rat,F: fun_rat_rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,X3)),aa_rat_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(F,C)) ) ) ) ).

% order_less_le_subst1
tff(fact_4273_order__less__le__subst1,axiom,
    ! [A: num,F: fun_rat_num,B: rat,C: rat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),aa_rat_num(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,X3)),aa_rat_num(F,Y)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,A),aa_rat_num(F,C)) ) ) ) ).

% order_less_le_subst1
tff(fact_4274_order__less__le__subst1,axiom,
    ! [A: nat,F: fun_rat_nat,B: rat,C: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_rat_nat(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,X3)),aa_rat_nat(F,Y)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_rat_nat(F,C)) ) ) ) ).

% order_less_le_subst1
tff(fact_4275_order__less__le__subst1,axiom,
    ! [A: int,F: fun_rat_int,B: rat,C: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_rat_int(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),C)
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,X3)),aa_rat_int(F,Y)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_rat_int(F,C)) ) ) ) ).

% order_less_le_subst1
tff(fact_4276_order__less__le__subst1,axiom,
    ! [A: real,F: fun_num_real,B: num,C: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_num_real(F,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C)
       => ( ! [X3: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(F,X3)),aa_num_real(F,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_num_real(F,C)) ) ) ) ).

% order_less_le_subst1
tff(fact_4277_order__less__le__subst1,axiom,
    ! [A: rat,F: fun_num_rat,B: num,C: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_num_rat(F,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C)
       => ( ! [X3: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,X3)),aa_num_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_num_rat(F,C)) ) ) ) ).

% order_less_le_subst1
tff(fact_4278_order__less__le__subst1,axiom,
    ! [A: num,F: fun_num_num,B: num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),aa_num_num(F,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C)
       => ( ! [X3: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,X3)),aa_num_num(F,Y)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,A),aa_num_num(F,C)) ) ) ) ).

% order_less_le_subst1
tff(fact_4279_order__less__le__subst1,axiom,
    ! [A: nat,F: fun_num_nat,B: num,C: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_num_nat(F,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C)
       => ( ! [X3: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,X3)),aa_num_nat(F,Y)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_num_nat(F,C)) ) ) ) ).

% order_less_le_subst1
tff(fact_4280_order__less__le__subst1,axiom,
    ! [A: int,F: fun_num_int,B: num,C: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_num_int(F,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C)
       => ( ! [X3: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(F,X3)),aa_num_int(F,Y)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_num_int(F,C)) ) ) ) ).

% order_less_le_subst1
tff(fact_4281_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_real,C: real] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,B)),C)
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_rat_real(F,X3)),aa_rat_real(F,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,A)),C) ) ) ) ).

% order_le_less_subst2
tff(fact_4282_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_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_rat,aa_rat_rat(F,B)),C)
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,X3)),aa_rat_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,A)),C) ) ) ) ).

% order_le_less_subst2
tff(fact_4283_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_num,C: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F,B)),C)
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,X3)),aa_rat_num(F,Y)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F,A)),C) ) ) ) ).

% order_le_less_subst2
tff(fact_4284_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_nat,C: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,B)),C)
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,X3)),aa_rat_nat(F,Y)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,A)),C) ) ) ) ).

% order_le_less_subst2
tff(fact_4285_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: fun_rat_int,C: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(F,B)),C)
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,X3)),aa_rat_int(F,Y)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(F,A)),C) ) ) ) ).

% order_le_less_subst2
tff(fact_4286_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: fun_num_real,C: real] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(F,B)),C)
       => ( ! [X3: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(F,X3)),aa_num_real(F,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(F,A)),C) ) ) ) ).

% order_le_less_subst2
tff(fact_4287_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: fun_num_rat,C: rat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(F,B)),C)
       => ( ! [X3: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,X3)),aa_num_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(F,A)),C) ) ) ) ).

% order_le_less_subst2
tff(fact_4288_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: fun_num_num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(F,B)),C)
       => ( ! [X3: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,X3)),aa_num_num(F,Y)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(F,A)),C) ) ) ) ).

% order_le_less_subst2
tff(fact_4289_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: fun_num_nat,C: nat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(F,B)),C)
       => ( ! [X3: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,X3)),aa_num_nat(F,Y)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(F,A)),C) ) ) ) ).

% order_le_less_subst2
tff(fact_4290_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: fun_num_int,C: int] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(F,B)),C)
       => ( ! [X3: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(F,X3)),aa_num_int(F,Y)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(F,A)),C) ) ) ) ).

% order_le_less_subst2
tff(fact_4291_order__le__less__subst1,axiom,
    ! [A: real,F: fun_real_real,B: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(F,B))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
       => ( ! [X3: real,Y: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,X3)),aa_real_real(F,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(F,C)) ) ) ) ).

% order_le_less_subst1
tff(fact_4292_order__le__less__subst1,axiom,
    ! [A: real,F: fun_rat_real,B: rat,C: rat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_rat_real(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,X3)),aa_rat_real(F,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_rat_real(F,C)) ) ) ) ).

% order_le_less_subst1
tff(fact_4293_order__le__less__subst1,axiom,
    ! [A: real,F: fun_num_real,B: num,C: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_num_real(F,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),C)
       => ( ! [X3: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(F,X3)),aa_num_real(F,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_num_real(F,C)) ) ) ) ).

% order_le_less_subst1
tff(fact_4294_order__le__less__subst1,axiom,
    ! [A: real,F: fun_nat_real,B: nat,C: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_nat_real(F,B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
       => ( ! [X3: nat,Y: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,X3)),aa_nat_real(F,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_nat_real(F,C)) ) ) ) ).

% order_le_less_subst1
tff(fact_4295_order__le__less__subst1,axiom,
    ! [A: real,F: fun_int_real,B: int,C: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_int_real(F,B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C)
       => ( ! [X3: int,Y: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(F,X3)),aa_int_real(F,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_int_real(F,C)) ) ) ) ).

% order_le_less_subst1
tff(fact_4296_order__le__less__subst1,axiom,
    ! [A: rat,F: fun_real_rat,B: real,C: real] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_real_rat(F,B))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),C)
       => ( ! [X3: real,Y: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,X3)),aa_real_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_real_rat(F,C)) ) ) ) ).

% order_le_less_subst1
tff(fact_4297_order__le__less__subst1,axiom,
    ! [A: rat,F: fun_rat_rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(F,B))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),C)
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,X3)),aa_rat_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(F,C)) ) ) ) ).

% order_le_less_subst1
tff(fact_4298_order__le__less__subst1,axiom,
    ! [A: rat,F: fun_num_rat,B: num,C: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_num_rat(F,B))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),C)
       => ( ! [X3: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(F,X3)),aa_num_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_num_rat(F,C)) ) ) ) ).

% order_le_less_subst1
tff(fact_4299_order__le__less__subst1,axiom,
    ! [A: rat,F: fun_nat_rat,B: nat,C: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_nat_rat(F,B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
       => ( ! [X3: nat,Y: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,X3)),aa_nat_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_nat_rat(F,C)) ) ) ) ).

% order_le_less_subst1
tff(fact_4300_order__le__less__subst1,axiom,
    ! [A: rat,F: fun_int_rat,B: int,C: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_int_rat(F,B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C)
       => ( ! [X3: int,Y: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(F,X3)),aa_int_rat(F,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_int_rat(F,C)) ) ) ) ).

% order_le_less_subst1
tff(fact_4301_order__less__le__trans,axiom,
    ! [Xb: real,Yb: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Yb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Yb),Z)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Z) ) ) ).

% order_less_le_trans
tff(fact_4302_order__less__le__trans,axiom,
    ! [Xb: set_nat,Yb: set_nat,Z: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(Xb),Yb)
     => ( aa_set_nat_o(ord_less_eq_set_nat(Yb),Z)
       => aa_set_nat_o(ord_less_set_nat(Xb),Z) ) ) ).

% order_less_le_trans
tff(fact_4303_order__less__le__trans,axiom,
    ! [Xb: rat,Yb: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Yb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Yb),Z)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Z) ) ) ).

% order_less_le_trans
tff(fact_4304_order__less__le__trans,axiom,
    ! [Xb: num,Yb: num,Z: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Yb)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Yb),Z)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Z) ) ) ).

% order_less_le_trans
tff(fact_4305_order__less__le__trans,axiom,
    ! [Xb: nat,Yb: nat,Z: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Yb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Yb),Z)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Z) ) ) ).

% order_less_le_trans
tff(fact_4306_order__less__le__trans,axiom,
    ! [Xb: int,Yb: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Yb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Yb),Z)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Z) ) ) ).

% order_less_le_trans
tff(fact_4307_order__le__less__trans,axiom,
    ! [Xb: real,Yb: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Yb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Yb),Z)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Z) ) ) ).

% order_le_less_trans
tff(fact_4308_order__le__less__trans,axiom,
    ! [Xb: set_nat,Yb: set_nat,Z: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(Xb),Yb)
     => ( aa_set_nat_o(ord_less_set_nat(Yb),Z)
       => aa_set_nat_o(ord_less_set_nat(Xb),Z) ) ) ).

% order_le_less_trans
tff(fact_4309_order__le__less__trans,axiom,
    ! [Xb: rat,Yb: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Yb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Yb),Z)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Z) ) ) ).

% order_le_less_trans
tff(fact_4310_order__le__less__trans,axiom,
    ! [Xb: num,Yb: num,Z: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Yb)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,Yb),Z)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Z) ) ) ).

% order_le_less_trans
tff(fact_4311_order__le__less__trans,axiom,
    ! [Xb: nat,Yb: nat,Z: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Yb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Yb),Z)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Z) ) ) ).

% order_le_less_trans
tff(fact_4312_order__le__less__trans,axiom,
    ! [Xb: int,Yb: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Yb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Yb),Z)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Z) ) ) ).

% order_le_less_trans
tff(fact_4313_order__neq__le__trans,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_real,A),B) ) ) ).

% order_neq_le_trans
tff(fact_4314_order__neq__le__trans,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( A != B )
     => ( aa_set_nat_o(ord_less_eq_set_nat(A),B)
       => aa_set_nat_o(ord_less_set_nat(A),B) ) ) ).

% order_neq_le_trans
tff(fact_4315_order__neq__le__trans,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_rat,A),B) ) ) ).

% order_neq_le_trans
tff(fact_4316_order__neq__le__trans,axiom,
    ! [A: num,B: num] :
      ( ( A != B )
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,A),B) ) ) ).

% order_neq_le_trans
tff(fact_4317_order__neq__le__trans,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_nat,A),B) ) ) ).

% order_neq_le_trans
tff(fact_4318_order__neq__le__trans,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_int,A),B) ) ) ).

% order_neq_le_trans
tff(fact_4319_order__le__neq__trans,axiom,
    ! [A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B)
     => ( ( A != B )
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ) ).

% order_le_neq_trans
tff(fact_4320_order__le__neq__trans,axiom,
    ! [A: set_nat,B: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A),B)
     => ( ( A != B )
       => aa_set_nat_o(ord_less_set_nat(A),B) ) ) ).

% order_le_neq_trans
tff(fact_4321_order__le__neq__trans,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B)
     => ( ( A != B )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ) ).

% order_le_neq_trans
tff(fact_4322_order__le__neq__trans,axiom,
    ! [A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( ( A != B )
       => aa_num_o(aa_num_fun_num_o(ord_less_num,A),B) ) ) ).

% order_le_neq_trans
tff(fact_4323_order__le__neq__trans,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( ( A != B )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ) ).

% order_le_neq_trans
tff(fact_4324_order__le__neq__trans,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( ( A != B )
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ) ).

% order_le_neq_trans
tff(fact_4325_order__less__imp__le,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Yb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Yb) ) ).

% order_less_imp_le
tff(fact_4326_order__less__imp__le,axiom,
    ! [Xb: set_nat,Yb: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(Xb),Yb)
     => aa_set_nat_o(ord_less_eq_set_nat(Xb),Yb) ) ).

% order_less_imp_le
tff(fact_4327_order__less__imp__le,axiom,
    ! [Xb: rat,Yb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Yb)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Yb) ) ).

% order_less_imp_le
tff(fact_4328_order__less__imp__le,axiom,
    ! [Xb: num,Yb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Yb)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Yb) ) ).

% order_less_imp_le
tff(fact_4329_order__less__imp__le,axiom,
    ! [Xb: nat,Yb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Yb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Yb) ) ).

% order_less_imp_le
tff(fact_4330_order__less__imp__le,axiom,
    ! [Xb: int,Yb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Yb)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Yb) ) ).

% order_less_imp_le
tff(fact_4331_linorder__not__less,axiom,
    ! [Xb: real,Yb: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Yb)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Yb),Xb) ) ).

% linorder_not_less
tff(fact_4332_linorder__not__less,axiom,
    ! [Xb: rat,Yb: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Yb)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Yb),Xb) ) ).

% linorder_not_less
tff(fact_4333_linorder__not__less,axiom,
    ! [Xb: num,Yb: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Yb)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Yb),Xb) ) ).

% linorder_not_less
tff(fact_4334_linorder__not__less,axiom,
    ! [Xb: nat,Yb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Yb)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Yb),Xb) ) ).

% linorder_not_less
tff(fact_4335_linorder__not__less,axiom,
    ! [Xb: int,Yb: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Yb)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Yb),Xb) ) ).

% linorder_not_less
tff(fact_4336_linorder__not__le,axiom,
    ! [Xb: real,Yb: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Yb)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Yb),Xb) ) ).

% linorder_not_le
tff(fact_4337_linorder__not__le,axiom,
    ! [Xb: rat,Yb: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Yb)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Yb),Xb) ) ).

% linorder_not_le
tff(fact_4338_linorder__not__le,axiom,
    ! [Xb: num,Yb: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Yb)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Yb),Xb) ) ).

% linorder_not_le
tff(fact_4339_linorder__not__le,axiom,
    ! [Xb: nat,Yb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Yb)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Yb),Xb) ) ).

% linorder_not_le
tff(fact_4340_linorder__not__le,axiom,
    ! [Xb: int,Yb: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Yb)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Yb),Xb) ) ).

% linorder_not_le
tff(fact_4341_order__less__le,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Yb)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Yb)
        & ( Xb != Yb ) ) ) ).

% order_less_le
tff(fact_4342_order__less__le,axiom,
    ! [Xb: set_nat,Yb: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(Xb),Yb)
    <=> ( aa_set_nat_o(ord_less_eq_set_nat(Xb),Yb)
        & ( Xb != Yb ) ) ) ).

% order_less_le
tff(fact_4343_order__less__le,axiom,
    ! [Xb: rat,Yb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Yb)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Yb)
        & ( Xb != Yb ) ) ) ).

% order_less_le
tff(fact_4344_order__less__le,axiom,
    ! [Xb: num,Yb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Yb)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Yb)
        & ( Xb != Yb ) ) ) ).

% order_less_le
tff(fact_4345_order__less__le,axiom,
    ! [Xb: nat,Yb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Yb)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Yb)
        & ( Xb != Yb ) ) ) ).

% order_less_le
tff(fact_4346_order__less__le,axiom,
    ! [Xb: int,Yb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Yb)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Yb)
        & ( Xb != Yb ) ) ) ).

% order_less_le
tff(fact_4347_order__le__less,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Yb)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Yb)
        | ( Xb = Yb ) ) ) ).

% order_le_less
tff(fact_4348_order__le__less,axiom,
    ! [Xb: set_nat,Yb: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(Xb),Yb)
    <=> ( aa_set_nat_o(ord_less_set_nat(Xb),Yb)
        | ( Xb = Yb ) ) ) ).

% order_le_less
tff(fact_4349_order__le__less,axiom,
    ! [Xb: rat,Yb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Yb)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Yb)
        | ( Xb = Yb ) ) ) ).

% order_le_less
tff(fact_4350_order__le__less,axiom,
    ! [Xb: num,Yb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Yb)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Yb)
        | ( Xb = Yb ) ) ) ).

% order_le_less
tff(fact_4351_order__le__less,axiom,
    ! [Xb: nat,Yb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Yb)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Yb)
        | ( Xb = Yb ) ) ) ).

% order_le_less
tff(fact_4352_order__le__less,axiom,
    ! [Xb: int,Yb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Yb)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Yb)
        | ( Xb = Yb ) ) ) ).

% order_le_less
tff(fact_4353_dual__order_Ostrict__implies__order,axiom,
    ! [B: real,A: real] :
      ( 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,B),A) ) ).

% dual_order.strict_implies_order
tff(fact_4354_dual__order_Ostrict__implies__order,axiom,
    ! [B: set_nat,A: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(B),A)
     => aa_set_nat_o(ord_less_eq_set_nat(B),A) ) ).

% dual_order.strict_implies_order
tff(fact_4355_dual__order_Ostrict__implies__order,axiom,
    ! [B: rat,A: rat] :
      ( 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,B),A) ) ).

% dual_order.strict_implies_order
tff(fact_4356_dual__order_Ostrict__implies__order,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A) ) ).

% dual_order.strict_implies_order
tff(fact_4357_dual__order_Ostrict__implies__order,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A) ) ).

% dual_order.strict_implies_order
tff(fact_4358_dual__order_Ostrict__implies__order,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ).

% dual_order.strict_implies_order
tff(fact_4359_order_Ostrict__implies__order,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_eq_real,A),B) ) ).

% order.strict_implies_order
tff(fact_4360_order_Ostrict__implies__order,axiom,
    ! [A: set_nat,B: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(A),B)
     => aa_set_nat_o(ord_less_eq_set_nat(A),B) ) ).

% order.strict_implies_order
tff(fact_4361_order_Ostrict__implies__order,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_eq_rat,A),B) ) ).

% order.strict_implies_order
tff(fact_4362_order_Ostrict__implies__order,axiom,
    ! [A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B) ) ).

% order.strict_implies_order
tff(fact_4363_order_Ostrict__implies__order,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,A),B) ) ).

% order.strict_implies_order
tff(fact_4364_order_Ostrict__implies__order,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,A),B) ) ).

% order.strict_implies_order
tff(fact_4365_dual__order_Ostrict__iff__not,axiom,
    ! [B: real,A: real] :
      ( 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,B),A)
        & ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ).

% dual_order.strict_iff_not
tff(fact_4366_dual__order_Ostrict__iff__not,axiom,
    ! [B: set_nat,A: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(B),A)
    <=> ( aa_set_nat_o(ord_less_eq_set_nat(B),A)
        & ~ aa_set_nat_o(ord_less_eq_set_nat(A),B) ) ) ).

% dual_order.strict_iff_not
tff(fact_4367_dual__order_Ostrict__iff__not,axiom,
    ! [B: rat,A: rat] :
      ( 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,B),A)
        & ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ) ).

% dual_order.strict_iff_not
tff(fact_4368_dual__order_Ostrict__iff__not,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
        & ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B) ) ) ).

% dual_order.strict_iff_not
tff(fact_4369_dual__order_Ostrict__iff__not,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
    <=> ( 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,A),B) ) ) ).

% dual_order.strict_iff_not
tff(fact_4370_dual__order_Ostrict__iff__not,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
    <=> ( 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,A),B) ) ) ).

% dual_order.strict_iff_not
tff(fact_4371_dual__order_Ostrict__trans2,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_eq_real,C),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,C),A) ) ) ).

% dual_order.strict_trans2
tff(fact_4372_dual__order_Ostrict__trans2,axiom,
    ! [B: set_nat,A: set_nat,C: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(B),A)
     => ( aa_set_nat_o(ord_less_eq_set_nat(C),B)
       => aa_set_nat_o(ord_less_set_nat(C),A) ) ) ).

% dual_order.strict_trans2
tff(fact_4373_dual__order_Ostrict__trans2,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_eq_rat,C),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),A) ) ) ).

% dual_order.strict_trans2
tff(fact_4374_dual__order_Ostrict__trans2,axiom,
    ! [B: num,A: num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),B)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,C),A) ) ) ).

% dual_order.strict_trans2
tff(fact_4375_dual__order_Ostrict__trans2,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),A) ) ) ).

% dual_order.strict_trans2
tff(fact_4376_dual__order_Ostrict__trans2,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_eq_int,C),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,C),A) ) ) ).

% dual_order.strict_trans2
tff(fact_4377_dual__order_Ostrict__trans1,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_real,C),B)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,C),A) ) ) ).

% dual_order.strict_trans1
tff(fact_4378_dual__order_Ostrict__trans1,axiom,
    ! [B: set_nat,A: set_nat,C: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(B),A)
     => ( aa_set_nat_o(ord_less_set_nat(C),B)
       => aa_set_nat_o(ord_less_set_nat(C),A) ) ) ).

% dual_order.strict_trans1
tff(fact_4379_dual__order_Ostrict__trans1,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_rat,C),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),A) ) ) ).

% dual_order.strict_trans1
tff(fact_4380_dual__order_Ostrict__trans1,axiom,
    ! [B: num,A: num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,C),B)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,C),A) ) ) ).

% dual_order.strict_trans1
tff(fact_4381_dual__order_Ostrict__trans1,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( 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,C),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),A) ) ) ).

% dual_order.strict_trans1
tff(fact_4382_dual__order_Ostrict__trans1,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_int,C),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,C),A) ) ) ).

% dual_order.strict_trans1
tff(fact_4383_dual__order_Ostrict__iff__order,axiom,
    ! [B: real,A: real] :
      ( 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,B),A)
        & ( A != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_4384_dual__order_Ostrict__iff__order,axiom,
    ! [B: set_nat,A: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(B),A)
    <=> ( aa_set_nat_o(ord_less_eq_set_nat(B),A)
        & ( A != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_4385_dual__order_Ostrict__iff__order,axiom,
    ! [B: rat,A: rat] :
      ( 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,B),A)
        & ( A != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_4386_dual__order_Ostrict__iff__order,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
        & ( A != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_4387_dual__order_Ostrict__iff__order,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
        & ( A != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_4388_dual__order_Ostrict__iff__order,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
        & ( A != B ) ) ) ).

% dual_order.strict_iff_order
tff(fact_4389_dual__order_Oorder__iff__strict,axiom,
    ! [B: real,A: 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_real,B),A)
        | ( A = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_4390_dual__order_Oorder__iff__strict,axiom,
    ! [B: set_nat,A: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(B),A)
    <=> ( aa_set_nat_o(ord_less_set_nat(B),A)
        | ( A = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_4391_dual__order_Oorder__iff__strict,axiom,
    ! [B: rat,A: 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_rat,B),A)
        | ( A = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_4392_dual__order_Oorder__iff__strict,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
        | ( A = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_4393_dual__order_Oorder__iff__strict,axiom,
    ! [B: nat,A: nat] :
      ( 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,B),A)
        | ( A = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_4394_dual__order_Oorder__iff__strict,axiom,
    ! [B: int,A: 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_int,B),A)
        | ( A = B ) ) ) ).

% dual_order.order_iff_strict
tff(fact_4395_dense__le__bounded,axiom,
    ! [Xb: real,Yb: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Yb)
     => ( ! [W2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),W2)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,W2),Yb)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,W2),Z) ) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Yb),Z) ) ) ).

% dense_le_bounded
tff(fact_4396_dense__le__bounded,axiom,
    ! [Xb: rat,Yb: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Yb)
     => ( ! [W2: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),W2)
           => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,W2),Yb)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,W2),Z) ) )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Yb),Z) ) ) ).

% dense_le_bounded
tff(fact_4397_dense__ge__bounded,axiom,
    ! [Z: real,Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z),Xb)
     => ( ! [W2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z),W2)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,W2),Xb)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Yb),W2) ) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Yb),Z) ) ) ).

% dense_ge_bounded
tff(fact_4398_dense__ge__bounded,axiom,
    ! [Z: rat,Xb: rat,Yb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),Xb)
     => ( ! [W2: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),W2)
           => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,W2),Xb)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Yb),W2) ) )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Yb),Z) ) ) ).

% dense_ge_bounded
tff(fact_4399_order_Ostrict__iff__not,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_eq_real,A),B)
        & ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A) ) ) ).

% order.strict_iff_not
tff(fact_4400_order_Ostrict__iff__not,axiom,
    ! [A: set_nat,B: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(A),B)
    <=> ( aa_set_nat_o(ord_less_eq_set_nat(A),B)
        & ~ aa_set_nat_o(ord_less_eq_set_nat(B),A) ) ) ).

% order.strict_iff_not
tff(fact_4401_order_Ostrict__iff__not,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_eq_rat,A),B)
        & ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A) ) ) ).

% order.strict_iff_not
tff(fact_4402_order_Ostrict__iff__not,axiom,
    ! [A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
        & ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A) ) ) ).

% order.strict_iff_not
tff(fact_4403_order_Ostrict__iff__not,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,A),B)
        & ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A) ) ) ).

% order.strict_iff_not
tff(fact_4404_order_Ostrict__iff__not,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,A),B)
        & ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ) ).

% order.strict_iff_not
tff(fact_4405_order_Ostrict__trans2,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_eq_real,B),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A),C) ) ) ).

% order.strict_trans2
tff(fact_4406_order_Ostrict__trans2,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(A),B)
     => ( aa_set_nat_o(ord_less_eq_set_nat(B),C)
       => aa_set_nat_o(ord_less_set_nat(A),C) ) ) ).

% order.strict_trans2
tff(fact_4407_order_Ostrict__trans2,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_eq_rat,B),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),C) ) ) ).

% order.strict_trans2
tff(fact_4408_order_Ostrict__trans2,axiom,
    ! [A: num,B: num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),C)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,A),C) ) ) ).

% order.strict_trans2
tff(fact_4409_order_Ostrict__trans2,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,B),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),C) ) ) ).

% order.strict_trans2
tff(fact_4410_order_Ostrict__trans2,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_eq_int,B),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),C) ) ) ).

% order.strict_trans2
tff(fact_4411_order_Ostrict__trans1,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_real,B),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A),C) ) ) ).

% order.strict_trans1
tff(fact_4412_order_Ostrict__trans1,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A),B)
     => ( aa_set_nat_o(ord_less_set_nat(B),C)
       => aa_set_nat_o(ord_less_set_nat(A),C) ) ) ).

% order.strict_trans1
tff(fact_4413_order_Ostrict__trans1,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_rat,B),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),C) ) ) ).

% order.strict_trans1
tff(fact_4414_order_Ostrict__trans1,axiom,
    ! [A: num,B: num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),C)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,A),C) ) ) ).

% order.strict_trans1
tff(fact_4415_order_Ostrict__trans1,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_nat,B),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),C) ) ) ).

% order.strict_trans1
tff(fact_4416_order_Ostrict__trans1,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_int,B),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),C) ) ) ).

% order.strict_trans1
tff(fact_4417_order_Ostrict__iff__order,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_eq_real,A),B)
        & ( A != B ) ) ) ).

% order.strict_iff_order
tff(fact_4418_order_Ostrict__iff__order,axiom,
    ! [A: set_nat,B: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(A),B)
    <=> ( aa_set_nat_o(ord_less_eq_set_nat(A),B)
        & ( A != B ) ) ) ).

% order.strict_iff_order
tff(fact_4419_order_Ostrict__iff__order,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_eq_rat,A),B)
        & ( A != B ) ) ) ).

% order.strict_iff_order
tff(fact_4420_order_Ostrict__iff__order,axiom,
    ! [A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
        & ( A != B ) ) ) ).

% order.strict_iff_order
tff(fact_4421_order_Ostrict__iff__order,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,A),B)
        & ( A != B ) ) ) ).

% order.strict_iff_order
tff(fact_4422_order_Ostrict__iff__order,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,A),B)
        & ( A != B ) ) ) ).

% order.strict_iff_order
tff(fact_4423_order_Oorder__iff__strict,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,A),B)
        | ( A = B ) ) ) ).

% order.order_iff_strict
tff(fact_4424_order_Oorder__iff__strict,axiom,
    ! [A: set_nat,B: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A),B)
    <=> ( aa_set_nat_o(ord_less_set_nat(A),B)
        | ( A = B ) ) ) ).

% order.order_iff_strict
tff(fact_4425_order_Oorder__iff__strict,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,A),B)
        | ( A = B ) ) ) ).

% order.order_iff_strict
tff(fact_4426_order_Oorder__iff__strict,axiom,
    ! [A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
        | ( A = B ) ) ) ).

% order.order_iff_strict
tff(fact_4427_order_Oorder__iff__strict,axiom,
    ! [A: nat,B: 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,A),B)
        | ( A = B ) ) ) ).

% order.order_iff_strict
tff(fact_4428_order_Oorder__iff__strict,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_int,A),B)
        | ( A = B ) ) ) ).

% order.order_iff_strict
tff(fact_4429_not__le__imp__less,axiom,
    ! [Yb: real,Xb: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Yb),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Yb) ) ).

% not_le_imp_less
tff(fact_4430_not__le__imp__less,axiom,
    ! [Yb: rat,Xb: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Yb),Xb)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Yb) ) ).

% not_le_imp_less
tff(fact_4431_not__le__imp__less,axiom,
    ! [Yb: num,Xb: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Yb),Xb)
     => aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Yb) ) ).

% not_le_imp_less
tff(fact_4432_not__le__imp__less,axiom,
    ! [Yb: nat,Xb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Yb),Xb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Yb) ) ).

% not_le_imp_less
tff(fact_4433_not__le__imp__less,axiom,
    ! [Yb: int,Xb: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Yb),Xb)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Yb) ) ).

% not_le_imp_less
tff(fact_4434_less__le__not__le,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Yb)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Yb)
        & ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Yb),Xb) ) ) ).

% less_le_not_le
tff(fact_4435_less__le__not__le,axiom,
    ! [Xb: set_nat,Yb: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(Xb),Yb)
    <=> ( aa_set_nat_o(ord_less_eq_set_nat(Xb),Yb)
        & ~ aa_set_nat_o(ord_less_eq_set_nat(Yb),Xb) ) ) ).

% less_le_not_le
tff(fact_4436_less__le__not__le,axiom,
    ! [Xb: rat,Yb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Yb)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Yb)
        & ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Yb),Xb) ) ) ).

% less_le_not_le
tff(fact_4437_less__le__not__le,axiom,
    ! [Xb: num,Yb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Yb)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Yb)
        & ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Yb),Xb) ) ) ).

% less_le_not_le
tff(fact_4438_less__le__not__le,axiom,
    ! [Xb: nat,Yb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Yb)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Yb)
        & ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Yb),Xb) ) ) ).

% less_le_not_le
tff(fact_4439_less__le__not__le,axiom,
    ! [Xb: int,Yb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Yb)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Yb)
        & ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Yb),Xb) ) ) ).

% less_le_not_le
tff(fact_4440_dense__le,axiom,
    ! [Yb: real,Z: real] :
      ( ! [X3: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Yb)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),Z) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Yb),Z) ) ).

% dense_le
tff(fact_4441_dense__le,axiom,
    ! [Yb: rat,Z: rat] :
      ( ! [X3: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Yb)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Z) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Yb),Z) ) ).

% dense_le
tff(fact_4442_dense__ge,axiom,
    ! [Z: real,Yb: real] :
      ( ! [X3: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z),X3)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Yb),X3) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Yb),Z) ) ).

% dense_ge
tff(fact_4443_dense__ge,axiom,
    ! [Z: rat,Yb: rat] :
      ( ! [X3: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),X3)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Yb),X3) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Yb),Z) ) ).

% dense_ge
tff(fact_4444_antisym__conv2,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Yb)
     => ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Yb)
      <=> ( Xb = Yb ) ) ) ).

% antisym_conv2
tff(fact_4445_antisym__conv2,axiom,
    ! [Xb: set_nat,Yb: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(Xb),Yb)
     => ( ~ aa_set_nat_o(ord_less_set_nat(Xb),Yb)
      <=> ( Xb = Yb ) ) ) ).

% antisym_conv2
tff(fact_4446_antisym__conv2,axiom,
    ! [Xb: rat,Yb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Yb)
     => ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Yb)
      <=> ( Xb = Yb ) ) ) ).

% antisym_conv2
tff(fact_4447_antisym__conv2,axiom,
    ! [Xb: num,Yb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Yb)
     => ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Yb)
      <=> ( Xb = Yb ) ) ) ).

% antisym_conv2
tff(fact_4448_antisym__conv2,axiom,
    ! [Xb: nat,Yb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Yb)
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Yb)
      <=> ( Xb = Yb ) ) ) ).

% antisym_conv2
tff(fact_4449_antisym__conv2,axiom,
    ! [Xb: int,Yb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Yb)
     => ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Yb)
      <=> ( Xb = Yb ) ) ) ).

% antisym_conv2
tff(fact_4450_antisym__conv1,axiom,
    ! [Xb: real,Yb: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Yb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Yb)
      <=> ( Xb = Yb ) ) ) ).

% antisym_conv1
tff(fact_4451_antisym__conv1,axiom,
    ! [Xb: set_nat,Yb: set_nat] :
      ( ~ aa_set_nat_o(ord_less_set_nat(Xb),Yb)
     => ( aa_set_nat_o(ord_less_eq_set_nat(Xb),Yb)
      <=> ( Xb = Yb ) ) ) ).

% antisym_conv1
tff(fact_4452_antisym__conv1,axiom,
    ! [Xb: rat,Yb: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Yb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Yb)
      <=> ( Xb = Yb ) ) ) ).

% antisym_conv1
tff(fact_4453_antisym__conv1,axiom,
    ! [Xb: num,Yb: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Yb)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Yb)
      <=> ( Xb = Yb ) ) ) ).

% antisym_conv1
tff(fact_4454_antisym__conv1,axiom,
    ! [Xb: nat,Yb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Yb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Yb)
      <=> ( Xb = Yb ) ) ) ).

% antisym_conv1
tff(fact_4455_antisym__conv1,axiom,
    ! [Xb: int,Yb: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Yb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Yb)
      <=> ( Xb = Yb ) ) ) ).

% antisym_conv1
tff(fact_4456_nless__le,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_eq_real,A),B)
        | ( A = B ) ) ) ).

% nless_le
tff(fact_4457_nless__le,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ~ aa_set_nat_o(ord_less_set_nat(A),B)
    <=> ( ~ aa_set_nat_o(ord_less_eq_set_nat(A),B)
        | ( A = B ) ) ) ).

% nless_le
tff(fact_4458_nless__le,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_eq_rat,A),B)
        | ( A = B ) ) ) ).

% nless_le
tff(fact_4459_nless__le,axiom,
    ! [A: num,B: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,A),B)
    <=> ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B)
        | ( A = B ) ) ) ).

% nless_le
tff(fact_4460_nless__le,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,A),B)
        | ( A = B ) ) ) ).

% nless_le
tff(fact_4461_nless__le,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,A),B)
        | ( A = B ) ) ) ).

% nless_le
tff(fact_4462_leI,axiom,
    ! [Xb: real,Yb: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Yb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Yb),Xb) ) ).

% leI
tff(fact_4463_leI,axiom,
    ! [Xb: rat,Yb: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Yb)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Yb),Xb) ) ).

% leI
tff(fact_4464_leI,axiom,
    ! [Xb: num,Yb: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Yb)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Yb),Xb) ) ).

% leI
tff(fact_4465_leI,axiom,
    ! [Xb: nat,Yb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Yb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Yb),Xb) ) ).

% leI
tff(fact_4466_leI,axiom,
    ! [Xb: int,Yb: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Yb)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Yb),Xb) ) ).

% leI
tff(fact_4467_leD,axiom,
    ! [Yb: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Yb),Xb)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Yb) ) ).

% leD
tff(fact_4468_leD,axiom,
    ! [Yb: set_nat,Xb: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(Yb),Xb)
     => ~ aa_set_nat_o(ord_less_set_nat(Xb),Yb) ) ).

% leD
tff(fact_4469_leD,axiom,
    ! [Yb: rat,Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Yb),Xb)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Yb) ) ).

% leD
tff(fact_4470_leD,axiom,
    ! [Yb: num,Xb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Yb),Xb)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Yb) ) ).

% leD
tff(fact_4471_leD,axiom,
    ! [Yb: nat,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Yb),Xb)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Yb) ) ).

% leD
tff(fact_4472_leD,axiom,
    ! [Yb: int,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Yb),Xb)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Yb) ) ).

% leD
tff(fact_4473_bot_Oextremum,axiom,
    ! [A: set_int] : ord_less_eq_set_int(bot_bot_set_int,A) ).

% bot.extremum
tff(fact_4474_bot_Oextremum,axiom,
    ! [A: set_real] : ord_less_eq_set_real(bot_bot_set_real,A) ).

% bot.extremum
tff(fact_4475_bot_Oextremum,axiom,
    ! [A: set_nat] : aa_set_nat_o(ord_less_eq_set_nat(bot_bot_set_nat),A) ).

% bot.extremum
tff(fact_4476_bot_Oextremum,axiom,
    ! [A: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,bot_bot_nat),A) ).

% bot.extremum
tff(fact_4477_bot_Oextremum__unique,axiom,
    ! [A: set_int] :
      ( ord_less_eq_set_int(A,bot_bot_set_int)
    <=> ( A = bot_bot_set_int ) ) ).

% bot.extremum_unique
tff(fact_4478_bot_Oextremum__unique,axiom,
    ! [A: set_real] :
      ( ord_less_eq_set_real(A,bot_bot_set_real)
    <=> ( A = bot_bot_set_real ) ) ).

% bot.extremum_unique
tff(fact_4479_bot_Oextremum__unique,axiom,
    ! [A: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A),bot_bot_set_nat)
    <=> ( A = bot_bot_set_nat ) ) ).

% bot.extremum_unique
tff(fact_4480_bot_Oextremum__unique,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),bot_bot_nat)
    <=> ( A = bot_bot_nat ) ) ).

% bot.extremum_unique
tff(fact_4481_bot_Oextremum__uniqueI,axiom,
    ! [A: set_int] :
      ( ord_less_eq_set_int(A,bot_bot_set_int)
     => ( A = bot_bot_set_int ) ) ).

% bot.extremum_uniqueI
tff(fact_4482_bot_Oextremum__uniqueI,axiom,
    ! [A: set_real] :
      ( ord_less_eq_set_real(A,bot_bot_set_real)
     => ( A = bot_bot_set_real ) ) ).

% bot.extremum_uniqueI
tff(fact_4483_bot_Oextremum__uniqueI,axiom,
    ! [A: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A),bot_bot_set_nat)
     => ( A = bot_bot_set_nat ) ) ).

% bot.extremum_uniqueI
tff(fact_4484_bot_Oextremum__uniqueI,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),bot_bot_nat)
     => ( A = bot_bot_nat ) ) ).

% bot.extremum_uniqueI
tff(fact_4485_bot_Onot__eq__extremum,axiom,
    ! [A: set_nat] :
      ( ( A != bot_bot_set_nat )
    <=> aa_set_nat_o(ord_less_set_nat(bot_bot_set_nat),A) ) ).

% bot.not_eq_extremum
tff(fact_4486_bot_Onot__eq__extremum,axiom,
    ! [A: set_int] :
      ( ( A != bot_bot_set_int )
    <=> ord_less_set_int(bot_bot_set_int,A) ) ).

% bot.not_eq_extremum
tff(fact_4487_bot_Onot__eq__extremum,axiom,
    ! [A: set_real] :
      ( ( A != bot_bot_set_real )
    <=> ord_less_set_real(bot_bot_set_real,A) ) ).

% bot.not_eq_extremum
tff(fact_4488_bot_Onot__eq__extremum,axiom,
    ! [A: nat] :
      ( ( A != bot_bot_nat )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,bot_bot_nat),A) ) ).

% bot.not_eq_extremum
tff(fact_4489_bot_Oextremum__strict,axiom,
    ! [A: set_nat] : ~ aa_set_nat_o(ord_less_set_nat(A),bot_bot_set_nat) ).

% bot.extremum_strict
tff(fact_4490_bot_Oextremum__strict,axiom,
    ! [A: set_int] : ~ ord_less_set_int(A,bot_bot_set_int) ).

% bot.extremum_strict
tff(fact_4491_bot_Oextremum__strict,axiom,
    ! [A: set_real] : ~ ord_less_set_real(A,bot_bot_set_real) ).

% bot.extremum_strict
tff(fact_4492_bot_Oextremum__strict,axiom,
    ! [A: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),bot_bot_nat) ).

% bot.extremum_strict
tff(fact_4493_list__decode_Ocases,axiom,
    ! [Xb: nat] :
      ( ( Xb != zero_zero_nat )
     => ~ ! [N: nat] : Xb != aa_nat_nat(suc,N) ) ).

% list_decode.cases
tff(fact_4494_max__absorb2,axiom,
    ! [Xb: set_nat,Yb: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(Xb),Yb)
     => ( ord_max_set_nat(Xb,Yb) = Yb ) ) ).

% max_absorb2
tff(fact_4495_max__absorb2,axiom,
    ! [Xb: rat,Yb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Yb)
     => ( aa_rat_rat(ord_max_rat(Xb),Yb) = Yb ) ) ).

% max_absorb2
tff(fact_4496_max__absorb2,axiom,
    ! [Xb: num,Yb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Yb)
     => ( aa_num_num(ord_max_num(Xb),Yb) = Yb ) ) ).

% max_absorb2
tff(fact_4497_max__absorb2,axiom,
    ! [Xb: nat,Yb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Yb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xb),Yb) = Yb ) ) ).

% max_absorb2
tff(fact_4498_max__absorb2,axiom,
    ! [Xb: int,Yb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Yb)
     => ( aa_int_int(aa_int_fun_int_int(ord_max_int,Xb),Yb) = Yb ) ) ).

% max_absorb2
tff(fact_4499_max__absorb1,axiom,
    ! [Yb: set_nat,Xb: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(Yb),Xb)
     => ( ord_max_set_nat(Xb,Yb) = Xb ) ) ).

% max_absorb1
tff(fact_4500_max__absorb1,axiom,
    ! [Yb: rat,Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Yb),Xb)
     => ( aa_rat_rat(ord_max_rat(Xb),Yb) = Xb ) ) ).

% max_absorb1
tff(fact_4501_max__absorb1,axiom,
    ! [Yb: num,Xb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Yb),Xb)
     => ( aa_num_num(ord_max_num(Xb),Yb) = Xb ) ) ).

% max_absorb1
tff(fact_4502_max__absorb1,axiom,
    ! [Yb: nat,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Yb),Xb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xb),Yb) = Xb ) ) ).

% max_absorb1
tff(fact_4503_max__absorb1,axiom,
    ! [Yb: int,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Yb),Xb)
     => ( aa_int_int(aa_int_fun_int_int(ord_max_int,Xb),Yb) = Xb ) ) ).

% max_absorb1
tff(fact_4504_max__def,axiom,
    ! [A: set_nat,B: set_nat] :
      ord_max_set_nat(A,B) = $ite(aa_set_nat_o(ord_less_eq_set_nat(A),B),B,A) ).

% max_def
tff(fact_4505_max__def,axiom,
    ! [A: rat,B: rat] :
      aa_rat_rat(ord_max_rat(A),B) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B),B,A) ).

% max_def
tff(fact_4506_max__def,axiom,
    ! [A: num,B: num] :
      aa_num_num(ord_max_num(A),B) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B),B,A) ).

% max_def
tff(fact_4507_max__def,axiom,
    ! [A: nat,B: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B),B,A) ).

% max_def
tff(fact_4508_max__def,axiom,
    ! [A: int,B: int] :
      aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B),B,A) ).

% max_def
tff(fact_4509_mult__exp__mod__exp__eq,axiom,
    ! [Mb: nat,Nb: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( 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))),Mb))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = 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,Nb),Mb)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)) ) ) ).

% mult_exp_mod_exp_eq
tff(fact_4510_mult__exp__mod__exp__eq,axiom,
    ! [Mb: nat,Nb: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_int_int(aa_int_fun_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))),Mb))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_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,Nb),Mb)))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Mb)) ) ) ).

% mult_exp_mod_exp_eq
tff(fact_4511_even__mod__4__div__2,axiom,
    ! [Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),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,Nb),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_4512_int__power__div__base,axiom,
    ! [Mb: nat,K: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
     => ( 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),Mb)),K) = aa_nat_int(power_power_int(K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),aa_nat_nat(suc,zero_zero_nat))) ) ) ) ).

% int_power_div_base
tff(fact_4513_even__mult__exp__div__exp__iff,axiom,
    ! [A: code_integer,Mb: nat,Nb: 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))),Mb)),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Nb)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb)
        | ( aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Nb) = zero_z3403309356797280102nteger )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
          & 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,Nb),Mb)))) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_4514_even__mult__exp__div__exp__iff,axiom,
    ! [A: nat,Mb: nat,Nb: 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))),Mb))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb)
        | ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
          & 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,Nb),Mb)))) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_4515_even__mult__exp__div__exp__iff,axiom,
    ! [A: int,Mb: nat,Nb: 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))),Mb))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb)
        | ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb) = zero_zero_int )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
          & 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,Nb),Mb)))) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_4516_real__average__minus__first,axiom,
    ! [A: real,B: real] : aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(A),B)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),A) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(B),A)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% real_average_minus_first
tff(fact_4517_real__average__minus__second,axiom,
    ! [B: real,A: real] : aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(B),A)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),A) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(B),A)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% real_average_minus_second
tff(fact_4518_vebt__member_Opelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Yb: $o] :
      ( ( aa_nat_o(vEBT_vebt_member(Xb),Xaa)
      <=> (Yb) )
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(Xb,Xaa))
       => ( ! [A4: $o,B3: $o] :
              ( ( Xb = vEBT_Leaf((A4),(B3)) )
             => ( ( (Yb)
                <=> $ite(
                      Xaa = zero_zero_nat,
                      (A4),
                      $ite(Xaa = one_one_nat,(B3),$false) ) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Leaf((A4),(B3)),Xaa)) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2) )
               => ( ~ (Yb)
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2),Xaa)) ) )
           => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2) )
                 => ( ~ (Yb)
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2),Xaa)) ) )
             => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2) )
                   => ( ~ (Yb)
                     => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2),Xaa)) ) )
               => ~ ! [Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary) )
                     => ( ( (Yb)
                        <=> $ite(
                              Xaa = Mi,
                              $true,
                              $ite(
                                Xaa = Ma,
                                $true,
                                $ite(
                                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),
                                  $false,
                                  $ite(
                                    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xaa),
                                    $false,
                                    $let(
                                      h: nat,
                                      h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_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)))),
                                      $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(TreeList,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary),Xaa)) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(1)
tff(fact_4519_vebt__member_Opelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(vEBT_vebt_member(Xb),Xaa)
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(Xb,Xaa))
       => ( ! [A4: $o,B3: $o] :
              ( ( Xb = vEBT_Leaf((A4),(B3)) )
             => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Leaf((A4),(B3)),Xaa))
               => $ite(
                    Xaa = zero_zero_nat,
                    (A4),
                    $ite(Xaa = one_one_nat,(B3),$false) ) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( Xb = 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),Xaa)) )
           => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( Xb = 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),Xaa)) )
             => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( Xb = 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),Xaa)) )
               => ~ ! [Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),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),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary),Xaa))
                       => $ite(
                            Xaa = Mi,
                            $true,
                            $ite(
                              Xaa = Ma,
                              $true,
                              $ite(
                                aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),
                                $false,
                                $ite(
                                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xaa),
                                  $false,
                                  $let(
                                    h: nat,
                                    h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_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)))),
                                    $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(TreeList,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(3)
tff(fact_4520_VEBT__internal_Onaive__member_Opelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Yb: $o] :
      ( ( aa_nat_o(vEBT_V5719532721284313246member(Xb),Xaa)
      <=> (Yb) )
     => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(Xb,Xaa))
       => ( ! [A4: $o,B3: $o] :
              ( ( Xb = vEBT_Leaf((A4),(B3)) )
             => ( ( (Yb)
                <=> $ite(
                      Xaa = zero_zero_nat,
                      (A4),
                      $ite(Xaa = one_one_nat,(B3),$false) ) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Leaf((A4),(B3)),Xaa)) ) )
         => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2) )
               => ( ~ (Yb)
                 => ~ accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2),Xaa)) ) )
           => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList: list_VEBT_VEBT,S: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList,S) )
                 => ( ( (Yb)
                    <=> $let(
                          pos: nat,
                          pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),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)))),
                          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList,S),Xaa)) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(1)
tff(fact_4521_VEBT__internal_Onaive__member_Opelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(vEBT_V5719532721284313246member(Xb),Xaa)
     => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(Xb,Xaa))
       => ( ! [A4: $o,B3: $o] :
              ( ( Xb = vEBT_Leaf((A4),(B3)) )
             => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Leaf((A4),(B3)),Xaa))
               => ~ $ite(
                      Xaa = zero_zero_nat,
                      (A4),
                      $ite(Xaa = one_one_nat,(B3),$false) ) ) )
         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList: list_VEBT_VEBT,S: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList,S) )
               => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList,S),Xaa))
                 => ~ $let(
                        pos: nat,
                        pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),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)))),
                        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(2)
tff(fact_4522_VEBT__internal_Onaive__member_Opelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(vEBT_V5719532721284313246member(Xb),Xaa)
     => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(Xb,Xaa))
       => ( ! [A4: $o,B3: $o] :
              ( ( Xb = vEBT_Leaf((A4),(B3)) )
             => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Leaf((A4),(B3)),Xaa))
               => $ite(
                    Xaa = zero_zero_nat,
                    (A4),
                    $ite(Xaa = one_one_nat,(B3),$false) ) ) )
         => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2),Xaa)) )
           => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList: list_VEBT_VEBT,S: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList,S) )
                 => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList,S),Xaa))
                   => $let(
                        pos: nat,
                        pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),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)))),
                        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(3)
tff(fact_4523_vebt__member_Opelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(vEBT_vebt_member(Xb),Xaa)
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(Xb,Xaa))
       => ( ! [A4: $o,B3: $o] :
              ( ( Xb = vEBT_Leaf((A4),(B3)) )
             => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Leaf((A4),(B3)),Xaa))
               => ~ $ite(
                      Xaa = zero_zero_nat,
                      (A4),
                      $ite(Xaa = one_one_nat,(B3),$false) ) ) )
         => ~ ! [Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),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),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary),Xaa))
                 => ~ $ite(
                        Xaa = Mi,
                        $true,
                        $ite(
                          Xaa = Ma,
                          $true,
                          $ite(
                            aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),
                            $false,
                            $ite(
                              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xaa),
                              $false,
                              $let(
                                h: nat,
                                h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_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)))),
                                $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(TreeList,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(2)
tff(fact_4524_diff__commute,axiom,
    ! [I3: 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,I3),J)),K) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I3),K)),J) ).

% diff_commute
tff(fact_4525_mult__commute__abs,axiom,
    ! [C: real] : aTP_Lamm_az(C) = times_times_real(C) ).

% mult_commute_abs
tff(fact_4526_mult__commute__abs,axiom,
    ! [C: rat] : aTP_Lamm_ba(C) = times_times_rat(C) ).

% mult_commute_abs
tff(fact_4527_mult__commute__abs,axiom,
    ! [C: nat] : aa_nat_fun_nat_nat(aTP_Lamm_bb,C) = aa_nat_fun_nat_nat(times_times_nat,C) ).

% mult_commute_abs
tff(fact_4528_mult__commute__abs,axiom,
    ! [C: int] : aa_int_fun_int_int(aTP_Lamm_bc,C) = aa_int_fun_int_int(times_times_int,C) ).

% mult_commute_abs
tff(fact_4529_VEBT__internal_Omembermima_Opelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(vEBT_VEBT_membermima(Xb),Xaa)
     => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(Xb,Xaa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( Xb = vEBT_Leaf((Uu2),(Uv2)) )
             => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(Uv2)),Xaa)) )
         => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( ( Xb = 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),Xaa)) )
           => ( ! [Mi: nat,Ma: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),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),Ma)),zero_zero_nat,Va3,Vb2),Xaa))
                   => ( ( Xaa = Mi )
                      | ( Xaa = Ma ) ) ) )
             => ( ! [Mi: nat,Ma: nat,V2: nat,TreeList: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),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),Ma)),aa_nat_nat(suc,V2),TreeList,Vc2),Xaa))
                     => ( ( Xaa = Mi )
                        | ( Xaa = Ma )
                        | $let(
                            pos: nat,
                            pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),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)))),
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) )
               => ~ ! [V2: nat,TreeList: list_VEBT_VEBT,Vd: vEBT_VEBT] :
                      ( ( Xb = 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),Xaa))
                       => $let(
                            pos: nat,
                            pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),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)))),
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(3)
tff(fact_4530_VEBT__internal_Omembermima_Opelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Yb: $o] :
      ( ( aa_nat_o(vEBT_VEBT_membermima(Xb),Xaa)
      <=> (Yb) )
     => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(Xb,Xaa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( Xb = vEBT_Leaf((Uu2),(Uv2)) )
             => ( ~ (Yb)
               => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(Uv2)),Xaa)) ) )
         => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2) )
               => ( ~ (Yb)
                 => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2),Xaa)) ) )
           => ( ! [Mi: nat,Ma: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,Va3,Vb2) )
                 => ( ( (Yb)
                    <=> ( ( Xaa = Mi )
                        | ( Xaa = Ma ) ) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,Va3,Vb2),Xaa)) ) )
             => ( ! [Mi: nat,Ma: nat,V2: nat,TreeList: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,V2),TreeList,Vc2) )
                   => ( ( (Yb)
                      <=> ( ( Xaa = Mi )
                          | ( Xaa = Ma )
                          | $let(
                              pos: nat,
                              pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),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)))),
                              $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,V2),TreeList,Vc2),Xaa)) ) )
               => ~ ! [V2: nat,TreeList: list_VEBT_VEBT,Vd: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList,Vd) )
                     => ( ( (Yb)
                        <=> $let(
                              pos: nat,
                              pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),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)))),
                              $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList,Vd),Xaa)) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(1)
tff(fact_4531_VEBT__internal_Omembermima_Opelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(vEBT_VEBT_membermima(Xb),Xaa)
     => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(Xb,Xaa))
       => ( ! [Mi: nat,Ma: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
              ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),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),Ma)),zero_zero_nat,Va3,Vb2),Xaa))
               => ~ ( ( Xaa = Mi )
                    | ( Xaa = Ma ) ) ) )
         => ( ! [Mi: nat,Ma: nat,V2: nat,TreeList: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),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),Ma)),aa_nat_nat(suc,V2),TreeList,Vc2),Xaa))
                 => ~ ( ( Xaa = Mi )
                      | ( Xaa = Ma )
                      | $let(
                          pos: nat,
                          pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),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)))),
                          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) )
           => ~ ! [V2: nat,TreeList: list_VEBT_VEBT,Vd: vEBT_VEBT] :
                  ( ( Xb = 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),Xaa))
                   => ~ $let(
                          pos: nat,
                          pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),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)))),
                          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(2)
tff(fact_4532_inrange,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => aa_set_nat_o(ord_less_eq_set_nat(vEBT_VEBT_set_vebt(Ta)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),one_one_nat))) ) ).

% inrange
tff(fact_4533_vebt__buildup_Oelims,axiom,
    ! [Xb: nat,Yb: vEBT_VEBT] :
      ( ( vEBT_vebt_buildup(Xb) = Yb )
     => ( ( ( Xb = zero_zero_nat )
         => ( Yb != vEBT_Leaf($false,$false) ) )
       => ( ( ( Xb = aa_nat_nat(suc,zero_zero_nat) )
           => ( Yb != vEBT_Leaf($false,$false) ) )
         => ~ ! [Va: nat] :
                ( ( Xb = aa_nat_nat(suc,aa_nat_nat(suc,Va)) )
               => ( Yb != $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_4534_triangle__def,axiom,
    ! [Nb: nat] : aa_nat_nat(nat_triangle,Nb) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(suc,Nb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% triangle_def
tff(fact_4535_intind,axiom,
    ! [I3: nat,Nb: nat,P: fun_nat_o,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),Nb)
     => ( aa_nat_o(P,Xb)
       => aa_nat_o(P,aa_nat_nat(nth_nat(replicate_nat(Nb,Xb)),I3)) ) ) ).

% intind
tff(fact_4536_intind,axiom,
    ! [I3: nat,Nb: nat,P: fun_int_o,Xb: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),Nb)
     => ( aa_int_o(P,Xb)
       => aa_int_o(P,aa_nat_int(nth_int(replicate_int(Nb,Xb)),I3)) ) ) ).

% intind
tff(fact_4537_intind,axiom,
    ! [I3: nat,Nb: nat,P: fun_VEBT_VEBT_o,Xb: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),Nb)
     => ( aa_VEBT_VEBT_o(P,Xb)
       => aa_VEBT_VEBT_o(P,nth_VEBT_VEBT(replicate_VEBT_VEBT(Nb,Xb),I3)) ) ) ).

% intind
tff(fact_4538_atLeastAtMost__iff,axiom,
    ! [I3: set_nat,L: set_nat,U: set_nat] :
      ( member_set_nat(I3,set_or4548717258645045905et_nat(L,U))
    <=> ( aa_set_nat_o(ord_less_eq_set_nat(L),I3)
        & aa_set_nat_o(ord_less_eq_set_nat(I3),U) ) ) ).

% atLeastAtMost_iff
tff(fact_4539_atLeastAtMost__iff,axiom,
    ! [I3: rat,L: rat,U: rat] :
      ( member_rat(I3,set_or633870826150836451st_rat(L,U))
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,L),I3)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,I3),U) ) ) ).

% atLeastAtMost_iff
tff(fact_4540_atLeastAtMost__iff,axiom,
    ! [I3: num,L: num,U: num] :
      ( member_num(I3,set_or7049704709247886629st_num(L,U))
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,L),I3)
        & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,I3),U) ) ) ).

% atLeastAtMost_iff
tff(fact_4541_atLeastAtMost__iff,axiom,
    ! [I3: nat,L: nat,U: nat] :
      ( aa_set_nat_o(member_nat(I3),aa_nat_set_nat(set_or1269000886237332187st_nat(L),U))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,L),I3)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I3),U) ) ) ).

% atLeastAtMost_iff
tff(fact_4542_atLeastAtMost__iff,axiom,
    ! [I3: int,L: int,U: int] :
      ( member_int(I3,set_or1266510415728281911st_int(L,U))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L),I3)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I3),U) ) ) ).

% atLeastAtMost_iff
tff(fact_4543_atLeastAtMost__iff,axiom,
    ! [I3: real,L: real,U: real] :
      ( member_real(I3,set_or1222579329274155063t_real(L,U))
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L),I3)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,I3),U) ) ) ).

% atLeastAtMost_iff
tff(fact_4544_Icc__eq__Icc,axiom,
    ! [L: set_nat,H: set_nat,L2: set_nat,H2: set_nat] :
      ( ( set_or4548717258645045905et_nat(L,H) = set_or4548717258645045905et_nat(L2,H2) )
    <=> ( ( ( L = L2 )
          & ( H = H2 ) )
        | ( ~ aa_set_nat_o(ord_less_eq_set_nat(L),H)
          & ~ aa_set_nat_o(ord_less_eq_set_nat(L2),H2) ) ) ) ).

% Icc_eq_Icc
tff(fact_4545_Icc__eq__Icc,axiom,
    ! [L: rat,H: rat,L2: rat,H2: rat] :
      ( ( set_or633870826150836451st_rat(L,H) = set_or633870826150836451st_rat(L2,H2) )
    <=> ( ( ( L = L2 )
          & ( H = H2 ) )
        | ( ~ 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,L2),H2) ) ) ) ).

% Icc_eq_Icc
tff(fact_4546_Icc__eq__Icc,axiom,
    ! [L: num,H: num,L2: num,H2: num] :
      ( ( set_or7049704709247886629st_num(L,H) = set_or7049704709247886629st_num(L2,H2) )
    <=> ( ( ( L = L2 )
          & ( H = H2 ) )
        | ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,L),H)
          & ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,L2),H2) ) ) ) ).

% Icc_eq_Icc
tff(fact_4547_Icc__eq__Icc,axiom,
    ! [L: nat,H: nat,L2: nat,H2: nat] :
      ( ( aa_nat_set_nat(set_or1269000886237332187st_nat(L),H) = aa_nat_set_nat(set_or1269000886237332187st_nat(L2),H2) )
    <=> ( ( ( L = L2 )
          & ( H = H2 ) )
        | ( ~ 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,L2),H2) ) ) ) ).

% Icc_eq_Icc
tff(fact_4548_Icc__eq__Icc,axiom,
    ! [L: int,H: int,L2: int,H2: int] :
      ( ( set_or1266510415728281911st_int(L,H) = set_or1266510415728281911st_int(L2,H2) )
    <=> ( ( ( L = L2 )
          & ( H = H2 ) )
        | ( ~ 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,L2),H2) ) ) ) ).

% Icc_eq_Icc
tff(fact_4549_Icc__eq__Icc,axiom,
    ! [L: real,H: real,L2: real,H2: real] :
      ( ( set_or1222579329274155063t_real(L,H) = set_or1222579329274155063t_real(L2,H2) )
    <=> ( ( ( L = L2 )
          & ( H = H2 ) )
        | ( ~ 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,L2),H2) ) ) ) ).

% Icc_eq_Icc
tff(fact_4550_replicate__eq__replicate,axiom,
    ! [Mb: nat,Xb: vEBT_VEBT,Nb: nat,Yb: vEBT_VEBT] :
      ( ( replicate_VEBT_VEBT(Mb,Xb) = replicate_VEBT_VEBT(Nb,Yb) )
    <=> ( ( Mb = Nb )
        & ( ( Mb != zero_zero_nat )
         => ( Xb = Yb ) ) ) ) ).

% replicate_eq_replicate
tff(fact_4551_length__replicate,axiom,
    ! [Nb: nat,Xb: vEBT_VEBT] : size_s6755466524823107622T_VEBT(replicate_VEBT_VEBT(Nb,Xb)) = Nb ).

% length_replicate
tff(fact_4552_length__replicate,axiom,
    ! [Nb: nat,Xb: $o] : size_size_list_o(replicate_o(Nb,(Xb))) = Nb ).

% length_replicate
tff(fact_4553_length__replicate,axiom,
    ! [Nb: nat,Xb: nat] : size_size_list_nat(replicate_nat(Nb,Xb)) = Nb ).

% length_replicate
tff(fact_4554_length__replicate,axiom,
    ! [Nb: nat,Xb: int] : size_size_list_int(replicate_int(Nb,Xb)) = Nb ).

% length_replicate
tff(fact_4555_triangle__0,axiom,
    aa_nat_nat(nat_triangle,zero_zero_nat) = zero_zero_nat ).

% triangle_0
tff(fact_4556_atLeastatMost__empty__iff2,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( bot_bot_set_set_nat = set_or4548717258645045905et_nat(A,B) )
    <=> ~ aa_set_nat_o(ord_less_eq_set_nat(A),B) ) ).

% atLeastatMost_empty_iff2
tff(fact_4557_atLeastatMost__empty__iff2,axiom,
    ! [A: rat,B: rat] :
      ( ( bot_bot_set_rat = set_or633870826150836451st_rat(A,B) )
    <=> ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).

% atLeastatMost_empty_iff2
tff(fact_4558_atLeastatMost__empty__iff2,axiom,
    ! [A: num,B: num] :
      ( ( bot_bot_set_num = set_or7049704709247886629st_num(A,B) )
    <=> ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B) ) ).

% atLeastatMost_empty_iff2
tff(fact_4559_atLeastatMost__empty__iff2,axiom,
    ! [A: nat,B: nat] :
      ( ( bot_bot_set_nat = aa_nat_set_nat(set_or1269000886237332187st_nat(A),B) )
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).

% atLeastatMost_empty_iff2
tff(fact_4560_atLeastatMost__empty__iff2,axiom,
    ! [A: int,B: int] :
      ( ( bot_bot_set_int = set_or1266510415728281911st_int(A,B) )
    <=> ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% atLeastatMost_empty_iff2
tff(fact_4561_atLeastatMost__empty__iff2,axiom,
    ! [A: real,B: real] :
      ( ( bot_bot_set_real = set_or1222579329274155063t_real(A,B) )
    <=> ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ).

% atLeastatMost_empty_iff2
tff(fact_4562_atLeastatMost__empty__iff,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( set_or4548717258645045905et_nat(A,B) = bot_bot_set_set_nat )
    <=> ~ aa_set_nat_o(ord_less_eq_set_nat(A),B) ) ).

% atLeastatMost_empty_iff
tff(fact_4563_atLeastatMost__empty__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( set_or633870826150836451st_rat(A,B) = bot_bot_set_rat )
    <=> ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).

% atLeastatMost_empty_iff
tff(fact_4564_atLeastatMost__empty__iff,axiom,
    ! [A: num,B: num] :
      ( ( set_or7049704709247886629st_num(A,B) = bot_bot_set_num )
    <=> ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B) ) ).

% atLeastatMost_empty_iff
tff(fact_4565_atLeastatMost__empty__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_set_nat(set_or1269000886237332187st_nat(A),B) = bot_bot_set_nat )
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).

% atLeastatMost_empty_iff
tff(fact_4566_atLeastatMost__empty__iff,axiom,
    ! [A: int,B: int] :
      ( ( set_or1266510415728281911st_int(A,B) = bot_bot_set_int )
    <=> ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% atLeastatMost_empty_iff
tff(fact_4567_atLeastatMost__empty__iff,axiom,
    ! [A: real,B: real] :
      ( ( set_or1222579329274155063t_real(A,B) = bot_bot_set_real )
    <=> ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ).

% atLeastatMost_empty_iff
tff(fact_4568_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_4569_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_4570_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(aa_num_fun_num_o(ord_less_eq_num,A),B)
        | ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),A)
          & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),D) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_4571_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_4572_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_4573_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_4574_atLeastatMost__empty,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),A)
     => ( set_or633870826150836451st_rat(A,B) = bot_bot_set_rat ) ) ).

% atLeastatMost_empty
tff(fact_4575_atLeastatMost__empty,axiom,
    ! [B: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B),A)
     => ( set_or7049704709247886629st_num(A,B) = bot_bot_set_num ) ) ).

% atLeastatMost_empty
tff(fact_4576_atLeastatMost__empty,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
     => ( aa_nat_set_nat(set_or1269000886237332187st_nat(A),B) = bot_bot_set_nat ) ) ).

% atLeastatMost_empty
tff(fact_4577_atLeastatMost__empty,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => ( set_or1266510415728281911st_int(A,B) = bot_bot_set_int ) ) ).

% atLeastatMost_empty
tff(fact_4578_atLeastatMost__empty,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B),A)
     => ( set_or1222579329274155063t_real(A,B) = bot_bot_set_real ) ) ).

% atLeastatMost_empty
tff(fact_4579_Ball__set__replicate,axiom,
    ! [Nb: nat,A: int,P: fun_int_o] :
      ( ! [X4: int] :
          ( member_int(X4,set_int2(replicate_int(Nb,A)))
         => aa_int_o(P,X4) )
    <=> ( aa_int_o(P,A)
        | ( Nb = zero_zero_nat ) ) ) ).

% Ball_set_replicate
tff(fact_4580_Ball__set__replicate,axiom,
    ! [Nb: nat,A: nat,P: fun_nat_o] :
      ( ! [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),set_nat2(replicate_nat(Nb,A)))
         => aa_nat_o(P,X4) )
    <=> ( aa_nat_o(P,A)
        | ( Nb = zero_zero_nat ) ) ) ).

% Ball_set_replicate
tff(fact_4581_Ball__set__replicate,axiom,
    ! [Nb: nat,A: vEBT_VEBT,P: fun_VEBT_VEBT_o] :
      ( ! [X4: vEBT_VEBT] :
          ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(replicate_VEBT_VEBT(Nb,A)))
         => aa_VEBT_VEBT_o(P,X4) )
    <=> ( aa_VEBT_VEBT_o(P,A)
        | ( Nb = zero_zero_nat ) ) ) ).

% Ball_set_replicate
tff(fact_4582_Bex__set__replicate,axiom,
    ! [Nb: nat,A: int,P: fun_int_o] :
      ( ? [X4: int] :
          ( member_int(X4,set_int2(replicate_int(Nb,A)))
          & aa_int_o(P,X4) )
    <=> ( aa_int_o(P,A)
        & ( Nb != zero_zero_nat ) ) ) ).

% Bex_set_replicate
tff(fact_4583_Bex__set__replicate,axiom,
    ! [Nb: nat,A: nat,P: fun_nat_o] :
      ( ? [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),set_nat2(replicate_nat(Nb,A)))
          & aa_nat_o(P,X4) )
    <=> ( aa_nat_o(P,A)
        & ( Nb != zero_zero_nat ) ) ) ).

% Bex_set_replicate
tff(fact_4584_Bex__set__replicate,axiom,
    ! [Nb: nat,A: vEBT_VEBT,P: fun_VEBT_VEBT_o] :
      ( ? [X4: vEBT_VEBT] :
          ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(replicate_VEBT_VEBT(Nb,A)))
          & aa_VEBT_VEBT_o(P,X4) )
    <=> ( aa_VEBT_VEBT_o(P,A)
        & ( Nb != zero_zero_nat ) ) ) ).

% Bex_set_replicate
tff(fact_4585_in__set__replicate,axiom,
    ! [Xb: complex,Nb: nat,Yb: complex] :
      ( member_complex(Xb,set_complex2(replicate_complex(Nb,Yb)))
    <=> ( ( Xb = Yb )
        & ( Nb != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_4586_in__set__replicate,axiom,
    ! [Xb: real,Nb: nat,Yb: real] :
      ( member_real(Xb,set_real2(replicate_real(Nb,Yb)))
    <=> ( ( Xb = Yb )
        & ( Nb != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_4587_in__set__replicate,axiom,
    ! [Xb: set_nat,Nb: nat,Yb: set_nat] :
      ( member_set_nat(Xb,set_set_nat2(replicate_set_nat(Nb,Yb)))
    <=> ( ( Xb = Yb )
        & ( Nb != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_4588_in__set__replicate,axiom,
    ! [Xb: int,Nb: nat,Yb: int] :
      ( member_int(Xb,set_int2(replicate_int(Nb,Yb)))
    <=> ( ( Xb = Yb )
        & ( Nb != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_4589_in__set__replicate,axiom,
    ! [Xb: nat,Nb: nat,Yb: nat] :
      ( aa_set_nat_o(member_nat(Xb),set_nat2(replicate_nat(Nb,Yb)))
    <=> ( ( Xb = Yb )
        & ( Nb != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_4590_in__set__replicate,axiom,
    ! [Xb: vEBT_VEBT,Nb: nat,Yb: vEBT_VEBT] :
      ( member_VEBT_VEBT(Xb,set_VEBT_VEBT2(replicate_VEBT_VEBT(Nb,Yb)))
    <=> ( ( Xb = Yb )
        & ( Nb != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_4591_nth__replicate,axiom,
    ! [I3: nat,Nb: nat,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),Nb)
     => ( aa_nat_nat(nth_nat(replicate_nat(Nb,Xb)),I3) = Xb ) ) ).

% nth_replicate
tff(fact_4592_nth__replicate,axiom,
    ! [I3: nat,Nb: nat,Xb: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),Nb)
     => ( aa_nat_int(nth_int(replicate_int(Nb,Xb)),I3) = Xb ) ) ).

% nth_replicate
tff(fact_4593_nth__replicate,axiom,
    ! [I3: nat,Nb: nat,Xb: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),Nb)
     => ( nth_VEBT_VEBT(replicate_VEBT_VEBT(Nb,Xb),I3) = Xb ) ) ).

% nth_replicate
tff(fact_4594_triangle__Suc,axiom,
    ! [Nb: nat] : aa_nat_nat(nat_triangle,aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(nat_triangle,Nb)),aa_nat_nat(suc,Nb)) ).

% triangle_Suc
tff(fact_4595_all__nat__less,axiom,
    ! [Nb: nat,P: fun_nat_o] :
      ( ! [M5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M5),Nb)
         => aa_nat_o(P,M5) )
    <=> ! [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb))
         => aa_nat_o(P,X4) ) ) ).

% all_nat_less
tff(fact_4596_ex__nat__less,axiom,
    ! [Nb: nat,P: fun_nat_o] :
      ( ? [M5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M5),Nb)
          & aa_nat_o(P,M5) )
    <=> ? [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb))
          & aa_nat_o(P,X4) ) ) ).

% ex_nat_less
tff(fact_4597_replicate__length__same,axiom,
    ! [Xs: list_VEBT_VEBT,Xb: vEBT_VEBT] :
      ( ! [X3: vEBT_VEBT] :
          ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(Xs))
         => ( X3 = Xb ) )
     => ( replicate_VEBT_VEBT(size_s6755466524823107622T_VEBT(Xs),Xb) = Xs ) ) ).

% replicate_length_same
tff(fact_4598_replicate__length__same,axiom,
    ! [Xs: list_o,Xb: $o] :
      ( ! [X3: $o] :
          ( member_o((X3),set_o2(Xs))
         => ( (X3)
          <=> (Xb) ) )
     => ( replicate_o(size_size_list_o(Xs),(Xb)) = Xs ) ) ).

% replicate_length_same
tff(fact_4599_replicate__length__same,axiom,
    ! [Xs: list_nat,Xb: nat] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),set_nat2(Xs))
         => ( X3 = Xb ) )
     => ( replicate_nat(size_size_list_nat(Xs),Xb) = Xs ) ) ).

% replicate_length_same
tff(fact_4600_replicate__length__same,axiom,
    ! [Xs: list_int,Xb: int] :
      ( ! [X3: int] :
          ( member_int(X3,set_int2(Xs))
         => ( X3 = Xb ) )
     => ( replicate_int(size_size_list_int(Xs),Xb) = Xs ) ) ).

% replicate_length_same
tff(fact_4601_replicate__eqI,axiom,
    ! [Xs: list_complex,Nb: nat,Xb: complex] :
      ( ( size_s3451745648224563538omplex(Xs) = Nb )
     => ( ! [Y: complex] :
            ( member_complex(Y,set_complex2(Xs))
           => ( Y = Xb ) )
       => ( Xs = replicate_complex(Nb,Xb) ) ) ) ).

% replicate_eqI
tff(fact_4602_replicate__eqI,axiom,
    ! [Xs: list_real,Nb: nat,Xb: real] :
      ( ( size_size_list_real(Xs) = Nb )
     => ( ! [Y: real] :
            ( member_real(Y,set_real2(Xs))
           => ( Y = Xb ) )
       => ( Xs = replicate_real(Nb,Xb) ) ) ) ).

% replicate_eqI
tff(fact_4603_replicate__eqI,axiom,
    ! [Xs: list_set_nat,Nb: nat,Xb: set_nat] :
      ( ( size_s3254054031482475050et_nat(Xs) = Nb )
     => ( ! [Y: set_nat] :
            ( member_set_nat(Y,set_set_nat2(Xs))
           => ( Y = Xb ) )
       => ( Xs = replicate_set_nat(Nb,Xb) ) ) ) ).

% replicate_eqI
tff(fact_4604_replicate__eqI,axiom,
    ! [Xs: list_VEBT_VEBT,Nb: nat,Xb: vEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT(Xs) = Nb )
     => ( ! [Y: vEBT_VEBT] :
            ( member_VEBT_VEBT(Y,set_VEBT_VEBT2(Xs))
           => ( Y = Xb ) )
       => ( Xs = replicate_VEBT_VEBT(Nb,Xb) ) ) ) ).

% replicate_eqI
tff(fact_4605_replicate__eqI,axiom,
    ! [Xs: list_o,Nb: nat,Xb: $o] :
      ( ( size_size_list_o(Xs) = Nb )
     => ( ! [Y: $o] :
            ( member_o((Y),set_o2(Xs))
           => ( (Y)
            <=> (Xb) ) )
       => ( Xs = replicate_o(Nb,(Xb)) ) ) ) ).

% replicate_eqI
tff(fact_4606_replicate__eqI,axiom,
    ! [Xs: list_nat,Nb: nat,Xb: nat] :
      ( ( size_size_list_nat(Xs) = Nb )
     => ( ! [Y: nat] :
            ( aa_set_nat_o(member_nat(Y),set_nat2(Xs))
           => ( Y = Xb ) )
       => ( Xs = replicate_nat(Nb,Xb) ) ) ) ).

% replicate_eqI
tff(fact_4607_replicate__eqI,axiom,
    ! [Xs: list_int,Nb: nat,Xb: int] :
      ( ( size_size_list_int(Xs) = Nb )
     => ( ! [Y: int] :
            ( member_int(Y,set_int2(Xs))
           => ( Y = Xb ) )
       => ( Xs = replicate_int(Nb,Xb) ) ) ) ).

% replicate_eqI
tff(fact_4608_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_4609_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_4610_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(aa_num_fun_num_o(ord_less_eq_num,A),B)
          | ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),A)
            & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),D)
            & ( aa_num_o(aa_num_fun_num_o(ord_less_num,C),A)
              | aa_num_o(aa_num_fun_num_o(ord_less_num,B),D) ) ) )
        & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),D) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_4611_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_4612_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_4613_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_4614_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_4615_signed__take__bit__rec,axiom,
    ! [Nb: nat,A: code_integer] :
      bit_ri6519982836138164636nteger(Nb,A) = $ite(Nb = 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,Nb),one_one_nat),divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))))))) ).

% signed_take_bit_rec
tff(fact_4616_signed__take__bit__rec,axiom,
    ! [Nb: nat,A: int] :
      aa_int_int(bit_ri631733984087533419it_int(Nb),A) = $ite(Nb = zero_zero_nat,aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_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(aa_int_fun_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,Nb),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_4617_concat__bit__Suc,axiom,
    ! [Nb: nat,K: int,L: int] : aa_int_int(aa_int_fun_int_int(bit_concat_bit(aa_nat_nat(suc,Nb)),K),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_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(aa_int_fun_int_int(bit_concat_bit(Nb),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_4618_vebt__buildup_Opelims,axiom,
    ! [Xb: nat,Yb: vEBT_VEBT] :
      ( ( vEBT_vebt_buildup(Xb) = Yb )
     => ( aa_nat_o(accp_nat(vEBT_v4011308405150292612up_rel),Xb)
       => ( ( ( Xb = zero_zero_nat )
           => ( ( Yb = vEBT_Leaf($false,$false) )
             => ~ aa_nat_o(accp_nat(vEBT_v4011308405150292612up_rel),zero_zero_nat) ) )
         => ( ( ( Xb = aa_nat_nat(suc,zero_zero_nat) )
             => ( ( Yb = vEBT_Leaf($false,$false) )
               => ~ aa_nat_o(accp_nat(vEBT_v4011308405150292612up_rel),aa_nat_nat(suc,zero_zero_nat)) ) )
           => ~ ! [Va: nat] :
                  ( ( Xb = aa_nat_nat(suc,aa_nat_nat(suc,Va)) )
                 => ( ( Yb = $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_4619_flip__bit__0,axiom,
    ! [A: code_integer] : bit_se1345352211410354436nteger(zero_zero_nat,A) = plus_p5714425477246183910nteger(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_4620_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_4621_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,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_4622_set__decode__0,axiom,
    ! [Xb: nat] :
      ( aa_set_nat_o(member_nat(zero_zero_nat),aa_nat_set_nat(nat_set_decode,Xb))
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Xb) ) ).

% set_decode_0
tff(fact_4623_set__decode__Suc,axiom,
    ! [Nb: nat,Xb: nat] :
      ( aa_set_nat_o(member_nat(aa_nat_nat(suc,Nb)),aa_nat_set_nat(nat_set_decode,Xb))
    <=> aa_set_nat_o(member_nat(Nb),aa_nat_set_nat(nat_set_decode,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% set_decode_Suc
tff(fact_4624_diff__shunt__var,axiom,
    ! [Xb: set_int,Yb: set_int] :
      ( ( minus_minus_set_int(Xb,Yb) = bot_bot_set_int )
    <=> ord_less_eq_set_int(Xb,Yb) ) ).

% diff_shunt_var
tff(fact_4625_diff__shunt__var,axiom,
    ! [Xb: set_real,Yb: set_real] :
      ( ( minus_minus_set_real(Xb,Yb) = bot_bot_set_real )
    <=> ord_less_eq_set_real(Xb,Yb) ) ).

% diff_shunt_var
tff(fact_4626_diff__shunt__var,axiom,
    ! [Xb: set_nat,Yb: set_nat] :
      ( ( minus_minus_set_nat(Xb,Yb) = bot_bot_set_nat )
    <=> aa_set_nat_o(ord_less_eq_set_nat(Xb),Yb) ) ).

% diff_shunt_var
tff(fact_4627_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_4628_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_4629_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_4630_neg__equal__iff__equal,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,A) = aa_rat_rat(uminus_uminus_rat,B) )
    <=> ( A = B ) ) ).

% neg_equal_iff_equal
tff(fact_4631_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_4632_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_4633_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_4634_add_Oinverse__inverse,axiom,
    ! [A: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(uminus1482373934393186551omplex,A)) = A ).

% add.inverse_inverse
tff(fact_4635_add_Oinverse__inverse,axiom,
    ! [A: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(uminus_uminus_rat,A)) = A ).

% add.inverse_inverse
tff(fact_4636_add_Oinverse__inverse,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) = A ).

% add.inverse_inverse
tff(fact_4637_Compl__subset__Compl__iff,axiom,
    ! [A3: set_nat,B4: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(uminus5710092332889474511et_nat(A3)),uminus5710092332889474511et_nat(B4))
    <=> aa_set_nat_o(ord_less_eq_set_nat(B4),A3) ) ).

% Compl_subset_Compl_iff
tff(fact_4638_Compl__anti__mono,axiom,
    ! [A3: set_nat,B4: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A3),B4)
     => aa_set_nat_o(ord_less_eq_set_nat(uminus5710092332889474511et_nat(B4)),uminus5710092332889474511et_nat(A3)) ) ).

% Compl_anti_mono
tff(fact_4639_compl__le__compl__iff,axiom,
    ! [Xb: set_nat,Yb: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(uminus5710092332889474511et_nat(Xb)),uminus5710092332889474511et_nat(Yb))
    <=> aa_set_nat_o(ord_less_eq_set_nat(Yb),Xb) ) ).

% compl_le_compl_iff
tff(fact_4640_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_4641_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_4642_neg__le__iff__le,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(uminus_uminus_rat,A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B) ) ).

% neg_le_iff_le
tff(fact_4643_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_4644_add_Oinverse__neutral,axiom,
    aa_real_real(uminus_uminus_real,zero_zero_real) = zero_zero_real ).

% add.inverse_neutral
tff(fact_4645_add_Oinverse__neutral,axiom,
    aa_int_int(uminus_uminus_int,zero_zero_int) = zero_zero_int ).

% add.inverse_neutral
tff(fact_4646_add_Oinverse__neutral,axiom,
    aa_complex_complex(uminus1482373934393186551omplex,zero_zero_complex) = zero_zero_complex ).

% add.inverse_neutral
tff(fact_4647_add_Oinverse__neutral,axiom,
    aa_rat_rat(uminus_uminus_rat,zero_zero_rat) = zero_zero_rat ).

% add.inverse_neutral
tff(fact_4648_add_Oinverse__neutral,axiom,
    aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% add.inverse_neutral
tff(fact_4649_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_4650_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_4651_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_4652_neg__0__equal__iff__equal,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat = aa_rat_rat(uminus_uminus_rat,A) )
    <=> ( zero_zero_rat = A ) ) ).

% neg_0_equal_iff_equal
tff(fact_4653_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_4654_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_4655_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_4656_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_4657_neg__equal__0__iff__equal,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% neg_equal_0_iff_equal
tff(fact_4658_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_4659_equal__neg__zero,axiom,
    ! [A: real] :
      ( ( A = aa_real_real(uminus_uminus_real,A) )
    <=> ( A = zero_zero_real ) ) ).

% equal_neg_zero
tff(fact_4660_equal__neg__zero,axiom,
    ! [A: int] :
      ( ( A = aa_int_int(uminus_uminus_int,A) )
    <=> ( A = zero_zero_int ) ) ).

% equal_neg_zero
tff(fact_4661_equal__neg__zero,axiom,
    ! [A: rat] :
      ( ( A = aa_rat_rat(uminus_uminus_rat,A) )
    <=> ( A = zero_zero_rat ) ) ).

% equal_neg_zero
tff(fact_4662_equal__neg__zero,axiom,
    ! [A: code_integer] :
      ( ( A = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% equal_neg_zero
tff(fact_4663_neg__equal__zero,axiom,
    ! [A: real] :
      ( ( aa_real_real(uminus_uminus_real,A) = A )
    <=> ( A = zero_zero_real ) ) ).

% neg_equal_zero
tff(fact_4664_neg__equal__zero,axiom,
    ! [A: int] :
      ( ( aa_int_int(uminus_uminus_int,A) = A )
    <=> ( A = zero_zero_int ) ) ).

% neg_equal_zero
tff(fact_4665_neg__equal__zero,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,A) = A )
    <=> ( A = zero_zero_rat ) ) ).

% neg_equal_zero
tff(fact_4666_neg__equal__zero,axiom,
    ! [A: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = A )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% neg_equal_zero
tff(fact_4667_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_4668_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_4669_neg__less__iff__less,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(uminus_uminus_rat,A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B) ) ).

% neg_less_iff_less
tff(fact_4670_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_4671_neg__numeral__eq__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( ( aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb)) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb)) )
    <=> ( Mb = Nb ) ) ).

% neg_numeral_eq_iff
tff(fact_4672_neg__numeral__eq__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( ( aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)) )
    <=> ( Mb = Nb ) ) ).

% neg_numeral_eq_iff
tff(fact_4673_neg__numeral__eq__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Mb)) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb)) )
    <=> ( Mb = Nb ) ) ).

% neg_numeral_eq_iff
tff(fact_4674_neg__numeral__eq__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( ( aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Mb)) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb)) )
    <=> ( Mb = Nb ) ) ).

% neg_numeral_eq_iff
tff(fact_4675_neg__numeral__eq__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Mb)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb)) )
    <=> ( Mb = Nb ) ) ).

% neg_numeral_eq_iff
tff(fact_4676_mult__minus__right,axiom,
    ! [A: real,B: real] : aa_real_real(times_times_real(A),aa_real_real(uminus_uminus_real,B)) = aa_real_real(uminus_uminus_real,aa_real_real(times_times_real(A),B)) ).

% mult_minus_right
tff(fact_4677_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_4678_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_4679_mult__minus__right,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(times_times_rat(A),aa_rat_rat(uminus_uminus_rat,B)) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(times_times_rat(A),B)) ).

% mult_minus_right
tff(fact_4680_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_4681_minus__mult__minus,axiom,
    ! [A: real,B: real] : aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,A)),aa_real_real(uminus_uminus_real,B)) = aa_real_real(times_times_real(A),B) ).

% minus_mult_minus
tff(fact_4682_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_4683_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_4684_minus__mult__minus,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(uminus_uminus_rat,B)) = aa_rat_rat(times_times_rat(A),B) ).

% minus_mult_minus
tff(fact_4685_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_4686_mult__minus__left,axiom,
    ! [A: real,B: real] : aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,A)),B) = aa_real_real(uminus_uminus_real,aa_real_real(times_times_real(A),B)) ).

% mult_minus_left
tff(fact_4687_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_4688_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_4689_mult__minus__left,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,A)),B) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(times_times_rat(A),B)) ).

% mult_minus_left
tff(fact_4690_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_4691_add__minus__cancel,axiom,
    ! [A: real,B: real] : aa_real_real(plus_plus_real(A),aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,A)),B)) = B ).

% add_minus_cancel
tff(fact_4692_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_4693_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_4694_add__minus__cancel,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(plus_plus_rat(A),aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,A)),B)) = B ).

% add_minus_cancel
tff(fact_4695_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_4696_minus__add__cancel,axiom,
    ! [A: real,B: real] : aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,A)),aa_real_real(plus_plus_real(A),B)) = B ).

% minus_add_cancel
tff(fact_4697_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_4698_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_4699_minus__add__cancel,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(plus_plus_rat(A),B)) = B ).

% minus_add_cancel
tff(fact_4700_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_4701_minus__add__distrib,axiom,
    ! [A: real,B: real] : aa_real_real(uminus_uminus_real,aa_real_real(plus_plus_real(A),B)) = aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,A)),aa_real_real(uminus_uminus_real,B)) ).

% minus_add_distrib
tff(fact_4702_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_4703_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_4704_minus__add__distrib,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(plus_plus_rat(A),B)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(uminus_uminus_rat,B)) ).

% minus_add_distrib
tff(fact_4705_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_4706_minus__diff__eq,axiom,
    ! [A: real,B: real] : aa_real_real(uminus_uminus_real,aa_real_real(minus_minus_real(A),B)) = aa_real_real(minus_minus_real(B),A) ).

% minus_diff_eq
tff(fact_4707_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_4708_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_4709_minus__diff__eq,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(minus_minus_rat(A),B)) = aa_rat_rat(minus_minus_rat(B),A) ).

% minus_diff_eq
tff(fact_4710_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_4711_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_4712_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_4713_minus__dvd__iff,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_real_real(uminus_uminus_real,Xb)),Yb)
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Xb),Yb) ) ).

% minus_dvd_iff
tff(fact_4714_minus__dvd__iff,axiom,
    ! [Xb: int,Yb: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(uminus_uminus_int,Xb)),Yb)
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xb),Yb) ) ).

% minus_dvd_iff
tff(fact_4715_minus__dvd__iff,axiom,
    ! [Xb: complex,Yb: complex] :
      ( aa_complex_o(dvd_dvd_complex(aa_complex_complex(uminus1482373934393186551omplex,Xb)),Yb)
    <=> aa_complex_o(dvd_dvd_complex(Xb),Yb) ) ).

% minus_dvd_iff
tff(fact_4716_minus__dvd__iff,axiom,
    ! [Xb: rat,Yb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(uminus_uminus_rat,Xb)),Yb)
    <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Xb),Yb) ) ).

% minus_dvd_iff
tff(fact_4717_minus__dvd__iff,axiom,
    ! [Xb: code_integer,Yb: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Xb)),Yb)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Xb),Yb) ) ).

% minus_dvd_iff
tff(fact_4718_dvd__minus__iff,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Xb),aa_real_real(uminus_uminus_real,Yb))
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Xb),Yb) ) ).

% dvd_minus_iff
tff(fact_4719_dvd__minus__iff,axiom,
    ! [Xb: int,Yb: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xb),aa_int_int(uminus_uminus_int,Yb))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xb),Yb) ) ).

% dvd_minus_iff
tff(fact_4720_dvd__minus__iff,axiom,
    ! [Xb: complex,Yb: complex] :
      ( aa_complex_o(dvd_dvd_complex(Xb),aa_complex_complex(uminus1482373934393186551omplex,Yb))
    <=> aa_complex_o(dvd_dvd_complex(Xb),Yb) ) ).

% dvd_minus_iff
tff(fact_4721_dvd__minus__iff,axiom,
    ! [Xb: rat,Yb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Xb),aa_rat_rat(uminus_uminus_rat,Yb))
    <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Xb),Yb) ) ).

% dvd_minus_iff
tff(fact_4722_dvd__minus__iff,axiom,
    ! [Xb: code_integer,Yb: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Xb),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Yb))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Xb),Yb) ) ).

% dvd_minus_iff
tff(fact_4723_mod__minus__minus,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_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(aa_int_fun_int_int(modulo_modulo_int,A),B)) ).

% mod_minus_minus
tff(fact_4724_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_4725_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_4726_of__bool__less__eq__iff,axiom,
    ! [P: $o,Q: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_n2687167440665602831ol_nat((P))),zero_n2687167440665602831ol_nat((Q)))
    <=> ( (P)
       => (Q) ) ) ).

% of_bool_less_eq_iff
tff(fact_4727_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_4728_of__bool__less__eq__iff,axiom,
    ! [P: $o,Q: $o] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_n356916108424825756nteger((P))),zero_n356916108424825756nteger((Q)))
    <=> ( (P)
       => (Q) ) ) ).

% of_bool_less_eq_iff
tff(fact_4729_of__bool__eq_I1_J,axiom,
    zero_n1201886186963655149omplex($false) = zero_zero_complex ).

% of_bool_eq(1)
tff(fact_4730_of__bool__eq_I1_J,axiom,
    zero_n3304061248610475627l_real($false) = zero_zero_real ).

% of_bool_eq(1)
tff(fact_4731_of__bool__eq_I1_J,axiom,
    zero_n2052037380579107095ol_rat($false) = zero_zero_rat ).

% of_bool_eq(1)
tff(fact_4732_of__bool__eq_I1_J,axiom,
    zero_n2687167440665602831ol_nat($false) = zero_zero_nat ).

% of_bool_eq(1)
tff(fact_4733_of__bool__eq_I1_J,axiom,
    aa_o_int(zero_n2684676970156552555ol_int,$false) = zero_zero_int ).

% of_bool_eq(1)
tff(fact_4734_of__bool__eq_I1_J,axiom,
    zero_n356916108424825756nteger($false) = zero_z3403309356797280102nteger ).

% of_bool_eq(1)
tff(fact_4735_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( zero_n1201886186963655149omplex((P)) = zero_zero_complex )
    <=> ~ (P) ) ).

% of_bool_eq_0_iff
tff(fact_4736_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( zero_n3304061248610475627l_real((P)) = zero_zero_real )
    <=> ~ (P) ) ).

% of_bool_eq_0_iff
tff(fact_4737_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( zero_n2052037380579107095ol_rat((P)) = zero_zero_rat )
    <=> ~ (P) ) ).

% of_bool_eq_0_iff
tff(fact_4738_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( zero_n2687167440665602831ol_nat((P)) = zero_zero_nat )
    <=> ~ (P) ) ).

% of_bool_eq_0_iff
tff(fact_4739_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_4740_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( zero_n356916108424825756nteger((P)) = zero_z3403309356797280102nteger )
    <=> ~ (P) ) ).

% of_bool_eq_0_iff
tff(fact_4741_real__add__minus__iff,axiom,
    ! [Xb: real,A: real] :
      ( ( aa_real_real(plus_plus_real(Xb),aa_real_real(uminus_uminus_real,A)) = zero_zero_real )
    <=> ( Xb = A ) ) ).

% real_add_minus_iff
tff(fact_4742_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_4743_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_4744_of__bool__less__iff,axiom,
    ! [P: $o,Q: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_n2687167440665602831ol_nat((P))),zero_n2687167440665602831ol_nat((Q)))
    <=> ( ~ (P)
        & (Q) ) ) ).

% of_bool_less_iff
tff(fact_4745_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_4746_of__bool__less__iff,axiom,
    ! [P: $o,Q: $o] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_n356916108424825756nteger((P))),zero_n356916108424825756nteger((Q)))
    <=> ( ~ (P)
        & (Q) ) ) ).

% of_bool_less_iff
tff(fact_4747_of__bool__eq_I2_J,axiom,
    zero_n1201886186963655149omplex($true) = one_one_complex ).

% of_bool_eq(2)
tff(fact_4748_of__bool__eq_I2_J,axiom,
    zero_n3304061248610475627l_real($true) = one_one_real ).

% of_bool_eq(2)
tff(fact_4749_of__bool__eq_I2_J,axiom,
    zero_n2052037380579107095ol_rat($true) = one_one_rat ).

% of_bool_eq(2)
tff(fact_4750_of__bool__eq_I2_J,axiom,
    zero_n2687167440665602831ol_nat($true) = one_one_nat ).

% of_bool_eq(2)
tff(fact_4751_of__bool__eq_I2_J,axiom,
    aa_o_int(zero_n2684676970156552555ol_int,$true) = one_one_int ).

% of_bool_eq(2)
tff(fact_4752_of__bool__eq_I2_J,axiom,
    zero_n356916108424825756nteger($true) = one_one_Code_integer ).

% of_bool_eq(2)
tff(fact_4753_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( zero_n1201886186963655149omplex((P)) = one_one_complex )
    <=> (P) ) ).

% of_bool_eq_1_iff
tff(fact_4754_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( zero_n3304061248610475627l_real((P)) = one_one_real )
    <=> (P) ) ).

% of_bool_eq_1_iff
tff(fact_4755_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( zero_n2052037380579107095ol_rat((P)) = one_one_rat )
    <=> (P) ) ).

% of_bool_eq_1_iff
tff(fact_4756_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( zero_n2687167440665602831ol_nat((P)) = one_one_nat )
    <=> (P) ) ).

% of_bool_eq_1_iff
tff(fact_4757_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_4758_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( zero_n356916108424825756nteger((P)) = one_one_Code_integer )
    <=> (P) ) ).

% of_bool_eq_1_iff
tff(fact_4759_of__bool__or__iff,axiom,
    ! [P: $o,Q: $o] :
      zero_n2687167440665602831ol_nat(( (P)
        | (Q) )) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,zero_n2687167440665602831ol_nat((P))),zero_n2687167440665602831ol_nat((Q))) ).

% of_bool_or_iff
tff(fact_4760_of__bool__or__iff,axiom,
    ! [P: $o,Q: $o] :
      aa_o_int(zero_n2684676970156552555ol_int,
        ( (P)
        | (Q) )) = aa_int_int(aa_int_fun_int_int(ord_max_int,aa_o_int(zero_n2684676970156552555ol_int,(P))),aa_o_int(zero_n2684676970156552555ol_int,(Q))) ).

% of_bool_or_iff
tff(fact_4761_of__bool__or__iff,axiom,
    ! [P: $o,Q: $o] :
      zero_n356916108424825756nteger(( (P)
        | (Q) )) = ord_max_Code_integer(zero_n356916108424825756nteger((P)),zero_n356916108424825756nteger((Q))) ).

% of_bool_or_iff
tff(fact_4762_concat__bit__0,axiom,
    ! [K: int,L: int] : aa_int_int(aa_int_fun_int_int(bit_concat_bit(zero_zero_nat),K),L) = L ).

% concat_bit_0
tff(fact_4763_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_4764_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_4765_neg__0__le__iff__le,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_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_4766_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_4767_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_4768_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_4769_neg__le__0__iff__le,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_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_4770_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_4771_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_4772_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_4773_less__eq__neg__nonpos,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(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_4774_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_4775_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_4776_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_4777_neg__less__eq__nonneg,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_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_4778_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_4779_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_4780_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_4781_less__neg__neg,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(uminus_uminus_rat,A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ).

% less_neg_neg
tff(fact_4782_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_4783_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_4784_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_4785_neg__less__pos,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_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_4786_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_4787_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_4788_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_4789_neg__0__less__iff__less,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_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_4790_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_4791_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_4792_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_4793_neg__less__0__iff__less,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_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_4794_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_4795_add_Oright__inverse,axiom,
    ! [A: real] : aa_real_real(plus_plus_real(A),aa_real_real(uminus_uminus_real,A)) = zero_zero_real ).

% add.right_inverse
tff(fact_4796_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_4797_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_4798_add_Oright__inverse,axiom,
    ! [A: rat] : aa_rat_rat(plus_plus_rat(A),aa_rat_rat(uminus_uminus_rat,A)) = zero_zero_rat ).

% add.right_inverse
tff(fact_4799_add_Oright__inverse,axiom,
    ! [A: code_integer] : plus_p5714425477246183910nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) = zero_z3403309356797280102nteger ).

% add.right_inverse
tff(fact_4800_ab__left__minus,axiom,
    ! [A: real] : aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,A)),A) = zero_zero_real ).

% ab_left_minus
tff(fact_4801_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_4802_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_4803_ab__left__minus,axiom,
    ! [A: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,A)),A) = zero_zero_rat ).

% ab_left_minus
tff(fact_4804_ab__left__minus,axiom,
    ! [A: code_integer] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),A) = zero_z3403309356797280102nteger ).

% ab_left_minus
tff(fact_4805_verit__minus__simplify_I3_J,axiom,
    ! [B: real] : aa_real_real(minus_minus_real(zero_zero_real),B) = aa_real_real(uminus_uminus_real,B) ).

% verit_minus_simplify(3)
tff(fact_4806_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_4807_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_4808_verit__minus__simplify_I3_J,axiom,
    ! [B: rat] : aa_rat_rat(minus_minus_rat(zero_zero_rat),B) = aa_rat_rat(uminus_uminus_rat,B) ).

% verit_minus_simplify(3)
tff(fact_4809_verit__minus__simplify_I3_J,axiom,
    ! [B: code_integer] : minus_8373710615458151222nteger(zero_z3403309356797280102nteger,B) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) ).

% verit_minus_simplify(3)
tff(fact_4810_diff__0,axiom,
    ! [A: real] : aa_real_real(minus_minus_real(zero_zero_real),A) = aa_real_real(uminus_uminus_real,A) ).

% diff_0
tff(fact_4811_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_4812_diff__0,axiom,
    ! [A: complex] : aa_complex_complex(minus_minus_complex(zero_zero_complex),A) = aa_complex_complex(uminus1482373934393186551omplex,A) ).

% diff_0
tff(fact_4813_diff__0,axiom,
    ! [A: rat] : aa_rat_rat(minus_minus_rat(zero_zero_rat),A) = aa_rat_rat(uminus_uminus_rat,A) ).

% diff_0
tff(fact_4814_diff__0,axiom,
    ! [A: code_integer] : minus_8373710615458151222nteger(zero_z3403309356797280102nteger,A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ).

% diff_0
tff(fact_4815_add__neg__numeral__simps_I3_J,axiom,
    ! [Mb: num,Nb: num] : aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb))),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))) = aa_real_real(uminus_uminus_real,aa_real_real(plus_plus_real(aa_num_real(numeral_numeral_real,Mb)),aa_num_real(numeral_numeral_real,Nb))) ).

% add_neg_numeral_simps(3)
tff(fact_4816_add__neg__numeral__simps_I3_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,Nb))) ).

% add_neg_numeral_simps(3)
tff(fact_4817_add__neg__numeral__simps_I3_J,axiom,
    ! [Mb: num,Nb: num] : aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Mb))),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb))) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(Mb)),numera6690914467698888265omplex(Nb))) ).

% add_neg_numeral_simps(3)
tff(fact_4818_add__neg__numeral__simps_I3_J,axiom,
    ! [Mb: num,Nb: num] : aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Mb))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,Mb)),aa_num_rat(numeral_numeral_rat,Nb))) ).

% add_neg_numeral_simps(3)
tff(fact_4819_add__neg__numeral__simps_I3_J,axiom,
    ! [Mb: num,Nb: num] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Mb)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,plus_p5714425477246183910nteger(numera6620942414471956472nteger(Mb),numera6620942414471956472nteger(Nb))) ).

% add_neg_numeral_simps(3)
tff(fact_4820_mult__minus1,axiom,
    ! [Z: real] : aa_real_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_4821_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_4822_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_4823_mult__minus1,axiom,
    ! [Z: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Z) = aa_rat_rat(uminus_uminus_rat,Z) ).

% mult_minus1
tff(fact_4824_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_4825_mult__minus1__right,axiom,
    ! [Z: real] : aa_real_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_4826_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_4827_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_4828_mult__minus1__right,axiom,
    ! [Z: rat] : aa_rat_rat(times_times_rat(Z),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_rat_rat(uminus_uminus_rat,Z) ).

% mult_minus1_right
tff(fact_4829_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_4830_diff__minus__eq__add,axiom,
    ! [A: real,B: real] : aa_real_real(minus_minus_real(A),aa_real_real(uminus_uminus_real,B)) = aa_real_real(plus_plus_real(A),B) ).

% diff_minus_eq_add
tff(fact_4831_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_4832_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_4833_diff__minus__eq__add,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(minus_minus_rat(A),aa_rat_rat(uminus_uminus_rat,B)) = aa_rat_rat(plus_plus_rat(A),B) ).

% diff_minus_eq_add
tff(fact_4834_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_4835_uminus__add__conv__diff,axiom,
    ! [A: real,B: real] : aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,A)),B) = aa_real_real(minus_minus_real(B),A) ).

% uminus_add_conv_diff
tff(fact_4836_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_4837_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_4838_uminus__add__conv__diff,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,A)),B) = aa_rat_rat(minus_minus_rat(B),A) ).

% uminus_add_conv_diff
tff(fact_4839_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_4840_divide__minus1,axiom,
    ! [Xb: real] : aa_real_real(divide_divide_real(Xb),aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,Xb) ).

% divide_minus1
tff(fact_4841_divide__minus1,axiom,
    ! [Xb: complex] : aa_complex_complex(divide1717551699836669952omplex(Xb),aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = aa_complex_complex(uminus1482373934393186551omplex,Xb) ).

% divide_minus1
tff(fact_4842_divide__minus1,axiom,
    ! [Xb: rat] : aa_rat_rat(divide_divide_rat(Xb),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_rat_rat(uminus_uminus_rat,Xb) ).

% divide_minus1
tff(fact_4843_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_4844_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_4845_minus__mod__self1,axiom,
    ! [B: int,A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A)),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,A)),B) ).

% minus_mod_self1
tff(fact_4846_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_4847_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_4848_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_4849_zero__less__of__bool__iff,axiom,
    ! [P: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),zero_n2687167440665602831ol_nat((P)))
    <=> (P) ) ).

% zero_less_of_bool_iff
tff(fact_4850_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_4851_zero__less__of__bool__iff,axiom,
    ! [P: $o] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),zero_n356916108424825756nteger((P)))
    <=> (P) ) ).

% zero_less_of_bool_iff
tff(fact_4852_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_4853_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_4854_of__bool__less__one__iff,axiom,
    ! [P: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_n2687167440665602831ol_nat((P))),one_one_nat)
    <=> ~ (P) ) ).

% of_bool_less_one_iff
tff(fact_4855_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_4856_of__bool__less__one__iff,axiom,
    ! [P: $o] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_n356916108424825756nteger((P))),one_one_Code_integer)
    <=> ~ (P) ) ).

% of_bool_less_one_iff
tff(fact_4857_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_4858_of__bool__not__iff,axiom,
    ! [P: $o] : zero_n3304061248610475627l_real(~ (P)) = aa_real_real(minus_minus_real(one_one_real),zero_n3304061248610475627l_real((P))) ).

% of_bool_not_iff
tff(fact_4859_of__bool__not__iff,axiom,
    ! [P: $o] : zero_n2052037380579107095ol_rat(~ (P)) = aa_rat_rat(minus_minus_rat(one_one_rat),zero_n2052037380579107095ol_rat((P))) ).

% of_bool_not_iff
tff(fact_4860_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_4861_of__bool__not__iff,axiom,
    ! [P: $o] : zero_n356916108424825756nteger(~ (P)) = minus_8373710615458151222nteger(one_one_Code_integer,zero_n356916108424825756nteger((P))) ).

% of_bool_not_iff
tff(fact_4862_Suc__0__mod__eq,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,zero_zero_nat)),Nb) = zero_n2687167440665602831ol_nat(Nb != aa_nat_nat(suc,zero_zero_nat)) ).

% Suc_0_mod_eq
tff(fact_4863_signed__take__bit__of__minus__1,axiom,
    ! [Nb: nat] : bit_ri6519982836138164636nteger(Nb,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% signed_take_bit_of_minus_1
tff(fact_4864_signed__take__bit__of__minus__1,axiom,
    ! [Nb: nat] : aa_int_int(bit_ri631733984087533419it_int(Nb),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_4865_concat__bit__nonnegative__iff,axiom,
    ! [Nb: nat,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(bit_concat_bit(Nb),K),L))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),L) ) ).

% concat_bit_nonnegative_iff
tff(fact_4866_concat__bit__negative__iff,axiom,
    ! [Nb: nat,K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_concat_bit(Nb),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_4867_dbl__simps_I1_J,axiom,
    ! [K: num] : aa_real_real(neg_numeral_dbl_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,K))) = aa_real_real(uminus_uminus_real,aa_real_real(neg_numeral_dbl_real,aa_num_real(numeral_numeral_real,K))) ).

% dbl_simps(1)
tff(fact_4868_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_4869_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_4870_dbl__simps_I1_J,axiom,
    ! [K: num] : aa_rat_rat(neg_numeral_dbl_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,K))) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(neg_numeral_dbl_rat,aa_num_rat(numeral_numeral_rat,K))) ).

% dbl_simps(1)
tff(fact_4871_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_4872_set__decode__zero,axiom,
    aa_nat_set_nat(nat_set_decode,zero_zero_nat) = bot_bot_set_nat ).

% set_decode_zero
tff(fact_4873_add__neg__numeral__special_I8_J,axiom,
    aa_real_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_4874_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_4875_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_4876_add__neg__numeral__special_I8_J,axiom,
    aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),one_one_rat) = zero_zero_rat ).

% add_neg_numeral_special(8)
tff(fact_4877_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_4878_add__neg__numeral__special_I7_J,axiom,
    aa_real_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_4879_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_4880_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_4881_add__neg__numeral__special_I7_J,axiom,
    aa_rat_rat(plus_plus_rat(one_one_rat),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = zero_zero_rat ).

% add_neg_numeral_special(7)
tff(fact_4882_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_4883_diff__numeral__special_I12_J,axiom,
    aa_real_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_4884_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_4885_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_4886_diff__numeral__special_I12_J,axiom,
    aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = zero_zero_rat ).

% diff_numeral_special(12)
tff(fact_4887_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_4888_neg__one__eq__numeral__iff,axiom,
    ! [Nb: num] :
      ( ( aa_real_real(uminus_uminus_real,one_one_real) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb)) )
    <=> ( Nb = one ) ) ).

% neg_one_eq_numeral_iff
tff(fact_4889_neg__one__eq__numeral__iff,axiom,
    ! [Nb: num] :
      ( ( aa_int_int(uminus_uminus_int,one_one_int) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)) )
    <=> ( Nb = one ) ) ).

% neg_one_eq_numeral_iff
tff(fact_4890_neg__one__eq__numeral__iff,axiom,
    ! [Nb: num] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb)) )
    <=> ( Nb = one ) ) ).

% neg_one_eq_numeral_iff
tff(fact_4891_neg__one__eq__numeral__iff,axiom,
    ! [Nb: num] :
      ( ( aa_rat_rat(uminus_uminus_rat,one_one_rat) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb)) )
    <=> ( Nb = one ) ) ).

% neg_one_eq_numeral_iff
tff(fact_4892_neg__one__eq__numeral__iff,axiom,
    ! [Nb: num] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb)) )
    <=> ( Nb = one ) ) ).

% neg_one_eq_numeral_iff
tff(fact_4893_numeral__eq__neg__one__iff,axiom,
    ! [Nb: num] :
      ( ( aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb)) = aa_real_real(uminus_uminus_real,one_one_real) )
    <=> ( Nb = one ) ) ).

% numeral_eq_neg_one_iff
tff(fact_4894_numeral__eq__neg__one__iff,axiom,
    ! [Nb: num] :
      ( ( aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)) = aa_int_int(uminus_uminus_int,one_one_int) )
    <=> ( Nb = one ) ) ).

% numeral_eq_neg_one_iff
tff(fact_4895_numeral__eq__neg__one__iff,axiom,
    ! [Nb: num] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb)) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) )
    <=> ( Nb = one ) ) ).

% numeral_eq_neg_one_iff
tff(fact_4896_numeral__eq__neg__one__iff,axiom,
    ! [Nb: num] :
      ( ( aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb)) = aa_rat_rat(uminus_uminus_rat,one_one_rat) )
    <=> ( Nb = one ) ) ).

% numeral_eq_neg_one_iff
tff(fact_4897_numeral__eq__neg__one__iff,axiom,
    ! [Nb: num] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) )
    <=> ( Nb = one ) ) ).

% numeral_eq_neg_one_iff
tff(fact_4898_left__minus__one__mult__self,axiom,
    ! [Nb: nat,A: real] : aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Nb)),aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Nb)),A)) = A ).

% left_minus_one_mult_self
tff(fact_4899_left__minus__one__mult__self,axiom,
    ! [Nb: 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)),Nb)),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)),Nb)),A)) = A ).

% left_minus_one_mult_self
tff(fact_4900_left__minus__one__mult__self,axiom,
    ! [Nb: nat,A: complex] : aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Nb)),aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Nb)),A)) = A ).

% left_minus_one_mult_self
tff(fact_4901_left__minus__one__mult__self,axiom,
    ! [Nb: nat,A: rat] : aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Nb)),aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Nb)),A)) = A ).

% left_minus_one_mult_self
tff(fact_4902_left__minus__one__mult__self,axiom,
    ! [Nb: nat,A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Nb)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Nb)),A)) = A ).

% left_minus_one_mult_self
tff(fact_4903_minus__one__mult__self,axiom,
    ! [Nb: nat] : aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Nb)) = one_one_real ).

% minus_one_mult_self
tff(fact_4904_minus__one__mult__self,axiom,
    ! [Nb: 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)),Nb)),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),Nb)) = one_one_int ).

% minus_one_mult_self
tff(fact_4905_minus__one__mult__self,axiom,
    ! [Nb: nat] : aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Nb)),aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Nb)) = one_one_complex ).

% minus_one_mult_self
tff(fact_4906_minus__one__mult__self,axiom,
    ! [Nb: nat] : aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Nb)),aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Nb)) = one_one_rat ).

% minus_one_mult_self
tff(fact_4907_minus__one__mult__self,axiom,
    ! [Nb: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Nb)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Nb)) = one_one_Code_integer ).

% minus_one_mult_self
tff(fact_4908_mod__minus1__right,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(uminus_uminus_int,one_one_int)) = zero_zero_int ).

% mod_minus1_right
tff(fact_4909_mod__minus1__right,axiom,
    ! [A: code_integer] : modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = zero_z3403309356797280102nteger ).

% mod_minus1_right
tff(fact_4910_max__number__of_I4_J,axiom,
    ! [U: num,V: num] :
      aa_real_real(ord_max_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,U))),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))) = $ite(aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,U))),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,U))) ).

% max_number_of(4)
tff(fact_4911_max__number__of_I4_J,axiom,
    ! [U: num,V: num] :
      ord_max_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(U)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V))) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(U))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(U))) ).

% max_number_of(4)
tff(fact_4912_max__number__of_I4_J,axiom,
    ! [U: num,V: num] :
      aa_rat_rat(ord_max_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,U))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,U))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,U))) ).

% max_number_of(4)
tff(fact_4913_max__number__of_I4_J,axiom,
    ! [U: num,V: num] :
      aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))) = $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,U))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U))) ).

% max_number_of(4)
tff(fact_4914_max__number__of_I3_J,axiom,
    ! [U: num,V: num] :
      aa_real_real(ord_max_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,U))),aa_num_real(numeral_numeral_real,V)) = $ite(aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,U))),aa_num_real(numeral_numeral_real,V)),aa_num_real(numeral_numeral_real,V),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,U))) ).

% max_number_of(3)
tff(fact_4915_max__number__of_I3_J,axiom,
    ! [U: num,V: num] :
      ord_max_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(U)),numera6620942414471956472nteger(V)) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(U))),numera6620942414471956472nteger(V)),numera6620942414471956472nteger(V),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(U))) ).

% max_number_of(3)
tff(fact_4916_max__number__of_I3_J,axiom,
    ! [U: num,V: num] :
      aa_rat_rat(ord_max_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,U))),aa_num_rat(numeral_numeral_rat,V)) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,U))),aa_num_rat(numeral_numeral_rat,V)),aa_num_rat(numeral_numeral_rat,V),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,U))) ).

% max_number_of(3)
tff(fact_4917_max__number__of_I3_J,axiom,
    ! [U: num,V: num] :
      aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U))),aa_num_int(numeral_numeral_int,V)) = $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,U))),aa_num_int(numeral_numeral_int,V)),aa_num_int(numeral_numeral_int,V),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U))) ).

% max_number_of(3)
tff(fact_4918_max__number__of_I2_J,axiom,
    ! [U: num,V: num] :
      aa_real_real(ord_max_real(aa_num_real(numeral_numeral_real,U)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))) = $ite(aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,U)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V)),aa_num_real(numeral_numeral_real,U)) ).

% max_number_of(2)
tff(fact_4919_max__number__of_I2_J,axiom,
    ! [U: num,V: num] :
      ord_max_Code_integer(numera6620942414471956472nteger(U),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V))) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,numera6620942414471956472nteger(U)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V)),numera6620942414471956472nteger(U)) ).

% max_number_of(2)
tff(fact_4920_max__number__of_I2_J,axiom,
    ! [U: num,V: num] :
      aa_rat_rat(ord_max_rat(aa_num_rat(numeral_numeral_rat,U)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,U)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V)),aa_num_rat(numeral_numeral_rat,U)) ).

% max_number_of(2)
tff(fact_4921_max__number__of_I2_J,axiom,
    ! [U: num,V: num] :
      aa_int_int(aa_int_fun_int_int(ord_max_int,aa_num_int(numeral_numeral_int,U)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,U)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)),aa_num_int(numeral_numeral_int,U)) ).

% max_number_of(2)
tff(fact_4922_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Yb: real] : aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))),aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),Yb)) = aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(plus_plus_num(V),W)))),Yb) ).

% semiring_norm(168)
tff(fact_4923_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Yb: 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))),Yb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(plus_plus_num(V),W)))),Yb) ).

% semiring_norm(168)
tff(fact_4924_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Yb: 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))),Yb)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(plus_plus_num(V),W)))),Yb) ).

% semiring_norm(168)
tff(fact_4925_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Yb: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),Yb)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(plus_plus_num(V),W)))),Yb) ).

% semiring_norm(168)
tff(fact_4926_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Yb: code_integer] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V)),plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(W)),Yb)) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(plus_plus_num(V),W))),Yb) ).

% semiring_norm(168)
tff(fact_4927_diff__numeral__simps_I2_J,axiom,
    ! [Mb: num,Nb: num] : aa_real_real(minus_minus_real(aa_num_real(numeral_numeral_real,Mb)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))) = aa_num_real(numeral_numeral_real,aa_num_num(plus_plus_num(Mb),Nb)) ).

% diff_numeral_simps(2)
tff(fact_4928_diff__numeral__simps_I2_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_num_int(numeral_numeral_int,aa_num_num(plus_plus_num(Mb),Nb)) ).

% diff_numeral_simps(2)
tff(fact_4929_diff__numeral__simps_I2_J,axiom,
    ! [Mb: num,Nb: num] : aa_complex_complex(minus_minus_complex(numera6690914467698888265omplex(Mb)),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb))) = numera6690914467698888265omplex(aa_num_num(plus_plus_num(Mb),Nb)) ).

% diff_numeral_simps(2)
tff(fact_4930_diff__numeral__simps_I2_J,axiom,
    ! [Mb: num,Nb: num] : aa_rat_rat(minus_minus_rat(aa_num_rat(numeral_numeral_rat,Mb)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))) = aa_num_rat(numeral_numeral_rat,aa_num_num(plus_plus_num(Mb),Nb)) ).

% diff_numeral_simps(2)
tff(fact_4931_diff__numeral__simps_I2_J,axiom,
    ! [Mb: num,Nb: num] : minus_8373710615458151222nteger(numera6620942414471956472nteger(Mb),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb))) = numera6620942414471956472nteger(aa_num_num(plus_plus_num(Mb),Nb)) ).

% diff_numeral_simps(2)
tff(fact_4932_diff__numeral__simps_I3_J,axiom,
    ! [Mb: num,Nb: num] : aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb))),aa_num_real(numeral_numeral_real,Nb)) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(plus_plus_num(Mb),Nb))) ).

% diff_numeral_simps(3)
tff(fact_4933_diff__numeral__simps_I3_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),aa_num_int(numeral_numeral_int,Nb)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(plus_plus_num(Mb),Nb))) ).

% diff_numeral_simps(3)
tff(fact_4934_diff__numeral__simps_I3_J,axiom,
    ! [Mb: num,Nb: num] : aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Mb))),numera6690914467698888265omplex(Nb)) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(plus_plus_num(Mb),Nb))) ).

% diff_numeral_simps(3)
tff(fact_4935_diff__numeral__simps_I3_J,axiom,
    ! [Mb: num,Nb: num] : aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Mb))),aa_num_rat(numeral_numeral_rat,Nb)) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(plus_plus_num(Mb),Nb))) ).

% diff_numeral_simps(3)
tff(fact_4936_diff__numeral__simps_I3_J,axiom,
    ! [Mb: num,Nb: num] : minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Mb)),numera6620942414471956472nteger(Nb)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(plus_plus_num(Mb),Nb))) ).

% diff_numeral_simps(3)
tff(fact_4937_mult__neg__numeral__simps_I3_J,axiom,
    ! [Mb: num,Nb: num] : aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,Mb)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(times_times_num(Mb),Nb))) ).

% mult_neg_numeral_simps(3)
tff(fact_4938_mult__neg__numeral__simps_I3_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(times_times_num(Mb),Nb))) ).

% mult_neg_numeral_simps(3)
tff(fact_4939_mult__neg__numeral__simps_I3_J,axiom,
    ! [Mb: num,Nb: num] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(Mb)),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb))) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(times_times_num(Mb),Nb))) ).

% mult_neg_numeral_simps(3)
tff(fact_4940_mult__neg__numeral__simps_I3_J,axiom,
    ! [Mb: num,Nb: num] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,Mb)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(times_times_num(Mb),Nb))) ).

% mult_neg_numeral_simps(3)
tff(fact_4941_mult__neg__numeral__simps_I3_J,axiom,
    ! [Mb: num,Nb: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(Mb)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(times_times_num(Mb),Nb))) ).

% mult_neg_numeral_simps(3)
tff(fact_4942_mult__neg__numeral__simps_I2_J,axiom,
    ! [Mb: num,Nb: num] : aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb))),aa_num_real(numeral_numeral_real,Nb)) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(times_times_num(Mb),Nb))) ).

% mult_neg_numeral_simps(2)
tff(fact_4943_mult__neg__numeral__simps_I2_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),aa_num_int(numeral_numeral_int,Nb)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(times_times_num(Mb),Nb))) ).

% mult_neg_numeral_simps(2)
tff(fact_4944_mult__neg__numeral__simps_I2_J,axiom,
    ! [Mb: num,Nb: num] : aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Mb))),numera6690914467698888265omplex(Nb)) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(times_times_num(Mb),Nb))) ).

% mult_neg_numeral_simps(2)
tff(fact_4945_mult__neg__numeral__simps_I2_J,axiom,
    ! [Mb: num,Nb: num] : aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Mb))),aa_num_rat(numeral_numeral_rat,Nb)) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(times_times_num(Mb),Nb))) ).

% mult_neg_numeral_simps(2)
tff(fact_4946_mult__neg__numeral__simps_I2_J,axiom,
    ! [Mb: num,Nb: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Mb))),numera6620942414471956472nteger(Nb)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(times_times_num(Mb),Nb))) ).

% mult_neg_numeral_simps(2)
tff(fact_4947_mult__neg__numeral__simps_I1_J,axiom,
    ! [Mb: num,Nb: num] : aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb))),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))) = aa_num_real(numeral_numeral_real,aa_num_num(times_times_num(Mb),Nb)) ).

% mult_neg_numeral_simps(1)
tff(fact_4948_mult__neg__numeral__simps_I1_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_num_int(numeral_numeral_int,aa_num_num(times_times_num(Mb),Nb)) ).

% mult_neg_numeral_simps(1)
tff(fact_4949_mult__neg__numeral__simps_I1_J,axiom,
    ! [Mb: num,Nb: num] : aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Mb))),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb))) = numera6690914467698888265omplex(aa_num_num(times_times_num(Mb),Nb)) ).

% mult_neg_numeral_simps(1)
tff(fact_4950_mult__neg__numeral__simps_I1_J,axiom,
    ! [Mb: num,Nb: num] : aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Mb))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))) = aa_num_rat(numeral_numeral_rat,aa_num_num(times_times_num(Mb),Nb)) ).

% mult_neg_numeral_simps(1)
tff(fact_4951_mult__neg__numeral__simps_I1_J,axiom,
    ! [Mb: num,Nb: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Mb))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb))) = numera6620942414471956472nteger(aa_num_num(times_times_num(Mb),Nb)) ).

% mult_neg_numeral_simps(1)
tff(fact_4952_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Yb: real] : aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,W)),Yb)) = aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(times_times_num(V),W)))),Yb) ).

% semiring_norm(170)
tff(fact_4953_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Yb: 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)),Yb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(times_times_num(V),W)))),Yb) ).

% semiring_norm(170)
tff(fact_4954_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Yb: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(V))),aa_complex_complex(times_times_complex(numera6690914467698888265omplex(W)),Yb)) = aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(times_times_num(V),W)))),Yb) ).

% semiring_norm(170)
tff(fact_4955_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Yb: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W)),Yb)) = aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(times_times_num(V),W)))),Yb) ).

% semiring_norm(170)
tff(fact_4956_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Yb: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(W)),Yb)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(times_times_num(V),W)))),Yb) ).

% semiring_norm(170)
tff(fact_4957_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Yb: real] : aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,V)),aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),Yb)) = aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(times_times_num(V),W)))),Yb) ).

% semiring_norm(171)
tff(fact_4958_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Yb: 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))),Yb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(times_times_num(V),W)))),Yb) ).

% semiring_norm(171)
tff(fact_4959_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Yb: complex] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(V)),aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))),Yb)) = aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(times_times_num(V),W)))),Yb) ).

% semiring_norm(171)
tff(fact_4960_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Yb: rat] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,V)),aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),Yb)) = aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(times_times_num(V),W)))),Yb) ).

% semiring_norm(171)
tff(fact_4961_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Yb: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(V)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(W))),Yb)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(times_times_num(V),W)))),Yb) ).

% semiring_norm(171)
tff(fact_4962_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Yb: real] : aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))),aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),Yb)) = aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(times_times_num(V),W))),Yb) ).

% semiring_norm(172)
tff(fact_4963_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Yb: 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))),Yb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(times_times_num(V),W))),Yb) ).

% semiring_norm(172)
tff(fact_4964_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Yb: 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))),Yb)) = aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(times_times_num(V),W))),Yb) ).

% semiring_norm(172)
tff(fact_4965_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Yb: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),Yb)) = aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(times_times_num(V),W))),Yb) ).

% semiring_norm(172)
tff(fact_4966_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Yb: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(W))),Yb)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(times_times_num(V),W))),Yb) ).

% semiring_norm(172)
tff(fact_4967_neg__numeral__le__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb))),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Nb),Mb) ) ).

% neg_numeral_le_iff
tff(fact_4968_neg__numeral__le__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Mb))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Nb),Mb) ) ).

% neg_numeral_le_iff
tff(fact_4969_neg__numeral__le__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Mb))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Nb),Mb) ) ).

% neg_numeral_le_iff
tff(fact_4970_neg__numeral__le__iff,axiom,
    ! [Mb: num,Nb: 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,Mb))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Nb),Mb) ) ).

% neg_numeral_le_iff
tff(fact_4971_neg__numeral__less__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb))),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Nb),Mb) ) ).

% neg_numeral_less_iff
tff(fact_4972_neg__numeral__less__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Nb),Mb) ) ).

% neg_numeral_less_iff
tff(fact_4973_neg__numeral__less__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Mb))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Nb),Mb) ) ).

% neg_numeral_less_iff
tff(fact_4974_neg__numeral__less__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Mb))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Nb),Mb) ) ).

% neg_numeral_less_iff
tff(fact_4975_not__neg__one__le__neg__numeral__iff,axiom,
    ! [Mb: 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,aa_num_real(numeral_numeral_real,Mb)))
    <=> ( Mb != one ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_4976_not__neg__one__le__neg__numeral__iff,axiom,
    ! [Mb: num] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Mb)))
    <=> ( Mb != one ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_4977_not__neg__one__le__neg__numeral__iff,axiom,
    ! [Mb: num] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Mb)))
    <=> ( Mb != one ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_4978_not__neg__one__le__neg__numeral__iff,axiom,
    ! [Mb: 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,Mb)))
    <=> ( Mb != one ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_4979_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(divide_divide_real(B),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)))),A)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)))),B) ) ).

% divide_le_eq_numeral1(2)
tff(fact_4980_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,aa_rat_rat(divide_divide_rat(B),aa_rat_rat(uminus_uminus_rat,aa_num_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),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)))),B) ) ).

% divide_le_eq_numeral1(2)
tff(fact_4981_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(divide_divide_real(B),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_real_real(times_times_real(A),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)))) ) ).

% le_divide_eq_numeral1(2)
tff(fact_4982_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),aa_rat_rat(divide_divide_rat(B),aa_rat_rat(uminus_uminus_rat,aa_num_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),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)))) ) ).

% le_divide_eq_numeral1(2)
tff(fact_4983_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B: real,W: num,A: real] :
      ( ( aa_real_real(divide_divide_real(B),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))) = A )
    <=> $ite(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)) != zero_zero_real,B = aa_real_real(times_times_real(A),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),A = zero_zero_real) ) ).

% divide_eq_eq_numeral1(2)
tff(fact_4984_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_4985_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( ( aa_rat_rat(divide_divide_rat(B),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))) = A )
    <=> $ite(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)) != zero_zero_rat,B = aa_rat_rat(times_times_rat(A),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),A = zero_zero_rat) ) ).

% divide_eq_eq_numeral1(2)
tff(fact_4986_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: real,B: real,W: num] :
      ( ( A = aa_real_real(divide_divide_real(B),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))) )
    <=> $ite(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)) != zero_zero_real,aa_real_real(times_times_real(A),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))) = B,A = zero_zero_real) ) ).

% eq_divide_eq_numeral1(2)
tff(fact_4987_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_4988_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( ( A = aa_rat_rat(divide_divide_rat(B),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))) )
    <=> $ite(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)) != zero_zero_rat,aa_rat_rat(times_times_rat(A),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))) = B,A = zero_zero_rat) ) ).

% eq_divide_eq_numeral1(2)
tff(fact_4989_neg__numeral__less__neg__one__iff,axiom,
    ! [Mb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb))),aa_real_real(uminus_uminus_real,one_one_real))
    <=> ( Mb != one ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_4990_neg__numeral__less__neg__one__iff,axiom,
    ! [Mb: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),aa_int_int(uminus_uminus_int,one_one_int))
    <=> ( Mb != one ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_4991_neg__numeral__less__neg__one__iff,axiom,
    ! [Mb: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Mb))),aa_rat_rat(uminus_uminus_rat,one_one_rat))
    <=> ( Mb != one ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_4992_neg__numeral__less__neg__one__iff,axiom,
    ! [Mb: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Mb))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer))
    <=> ( Mb != one ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_4993_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(divide_divide_real(B),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)))),A)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(A),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)))),B) ) ).

% divide_less_eq_numeral1(2)
tff(fact_4994_divide__less__eq__numeral1_I2_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(B),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)))),A)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)))),B) ) ).

% divide_less_eq_numeral1(2)
tff(fact_4995_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(divide_divide_real(B),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B),aa_real_real(times_times_real(A),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)))) ) ).

% less_divide_eq_numeral1(2)
tff(fact_4996_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),aa_rat_rat(divide_divide_rat(B),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(times_times_rat(A),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)))) ) ).

% less_divide_eq_numeral1(2)
tff(fact_4997_power2__minus,axiom,
    ! [A: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,A)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_minus
tff(fact_4998_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_4999_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_5000_power2__minus,axiom,
    ! [A: rat] : aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,A)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_minus
tff(fact_5001_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_5002_odd__of__bool__self,axiom,
    ! [P2: $o] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),zero_n2687167440665602831ol_nat((P2)))
    <=> (P2) ) ).

% odd_of_bool_self
tff(fact_5003_odd__of__bool__self,axiom,
    ! [P2: $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,(P2)))
    <=> (P2) ) ).

% odd_of_bool_self
tff(fact_5004_odd__of__bool__self,axiom,
    ! [P2: $o] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),zero_n356916108424825756nteger((P2)))
    <=> (P2) ) ).

% odd_of_bool_self
tff(fact_5005_add__neg__numeral__special_I9_J,axiom,
    aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% add_neg_numeral_special(9)
tff(fact_5006_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_5007_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_5008_add__neg__numeral__special_I9_J,axiom,
    aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))) ).

% add_neg_numeral_special(9)
tff(fact_5009_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_5010_diff__numeral__special_I11_J,axiom,
    aa_real_real(minus_minus_real(one_one_real),aa_real_real(uminus_uminus_real,one_one_real)) = aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)) ).

% diff_numeral_special(11)
tff(fact_5011_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_5012_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_5013_diff__numeral__special_I11_J,axiom,
    aa_rat_rat(minus_minus_rat(one_one_rat),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)) ).

% diff_numeral_special(11)
tff(fact_5014_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_5015_diff__numeral__special_I10_J,axiom,
    aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,one_one_real)),one_one_real) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% diff_numeral_special(10)
tff(fact_5016_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_5017_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_5018_diff__numeral__special_I10_J,axiom,
    aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),one_one_rat) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))) ).

% diff_numeral_special(10)
tff(fact_5019_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_5020_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_5021_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_5022_minus__1__mod__2__eq,axiom,
    aa_int_int(aa_int_fun_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_5023_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_5024_bits__minus__1__mod__2__eq,axiom,
    aa_int_int(aa_int_fun_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_5025_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_5026_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% Power.ring_1_class.power_minus_even
tff(fact_5027_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: int,Nb: 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))),Nb)) = 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))),Nb)) ).

% Power.ring_1_class.power_minus_even
tff(fact_5028_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: complex,Nb: 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))),Nb)) = 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))),Nb)) ).

% Power.ring_1_class.power_minus_even
tff(fact_5029_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: rat,Nb: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = 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))),Nb)) ).

% Power.ring_1_class.power_minus_even
tff(fact_5030_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: code_integer,Nb: 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))),Nb)) = 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))),Nb)) ).

% Power.ring_1_class.power_minus_even
tff(fact_5031_of__bool__half__eq__0,axiom,
    ! [B: $o] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_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_5032_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_5033_of__bool__half__eq__0,axiom,
    ! [B: $o] : divide6298287555418463151nteger(zero_n356916108424825756nteger((B)),numera6620942414471956472nteger(aa_num_num(bit0,one))) = zero_z3403309356797280102nteger ).

% of_bool_half_eq_0
tff(fact_5034_power__minus__odd,axiom,
    ! [Nb: 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))),Nb)
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,A)),Nb) = aa_real_real(uminus_uminus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ) ) ).

% power_minus_odd
tff(fact_5035_power__minus__odd,axiom,
    ! [Nb: 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))),Nb)
     => ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A)),Nb) = aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(A),Nb)) ) ) ).

% power_minus_odd
tff(fact_5036_power__minus__odd,axiom,
    ! [Nb: 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))),Nb)
     => ( aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),Nb) = aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(power_power_complex(A),Nb)) ) ) ).

% power_minus_odd
tff(fact_5037_power__minus__odd,axiom,
    ! [Nb: 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))),Nb)
     => ( aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,A)),Nb) = aa_rat_rat(uminus_uminus_rat,aa_nat_rat(power_power_rat(A),Nb)) ) ) ).

% power_minus_odd
tff(fact_5038_power__minus__odd,axiom,
    ! [Nb: 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))),Nb)
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),Nb) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb)) ) ) ).

% power_minus_odd
tff(fact_5039_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [Nb: 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))),Nb)
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,A)),Nb) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb) ) ) ).

% Parity.ring_1_class.power_minus_even
tff(fact_5040_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [Nb: 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))),Nb)
     => ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A)),Nb) = aa_nat_int(power_power_int(A),Nb) ) ) ).

% Parity.ring_1_class.power_minus_even
tff(fact_5041_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [Nb: 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))),Nb)
     => ( aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),Nb) = aa_nat_complex(power_power_complex(A),Nb) ) ) ).

% Parity.ring_1_class.power_minus_even
tff(fact_5042_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [Nb: 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))),Nb)
     => ( aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,A)),Nb) = aa_nat_rat(power_power_rat(A),Nb) ) ) ).

% Parity.ring_1_class.power_minus_even
tff(fact_5043_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [Nb: 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))),Nb)
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),Nb) = aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb) ) ) ).

% Parity.ring_1_class.power_minus_even
tff(fact_5044_diff__numeral__special_I4_J,axiom,
    ! [Mb: num] : aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb))),one_one_real) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(plus_plus_num(Mb),one))) ).

% diff_numeral_special(4)
tff(fact_5045_diff__numeral__special_I4_J,axiom,
    ! [Mb: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),one_one_int) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(plus_plus_num(Mb),one))) ).

% diff_numeral_special(4)
tff(fact_5046_diff__numeral__special_I4_J,axiom,
    ! [Mb: num] : aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Mb))),one_one_complex) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(plus_plus_num(Mb),one))) ).

% diff_numeral_special(4)
tff(fact_5047_diff__numeral__special_I4_J,axiom,
    ! [Mb: num] : aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Mb))),one_one_rat) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(plus_plus_num(Mb),one))) ).

% diff_numeral_special(4)
tff(fact_5048_diff__numeral__special_I4_J,axiom,
    ! [Mb: num] : minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Mb)),one_one_Code_integer) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(plus_plus_num(Mb),one))) ).

% diff_numeral_special(4)
tff(fact_5049_diff__numeral__special_I3_J,axiom,
    ! [Nb: num] : aa_real_real(minus_minus_real(one_one_real),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))) = aa_num_real(numeral_numeral_real,aa_num_num(plus_plus_num(one),Nb)) ).

% diff_numeral_special(3)
tff(fact_5050_diff__numeral__special_I3_J,axiom,
    ! [Nb: 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,Nb))) = aa_num_int(numeral_numeral_int,aa_num_num(plus_plus_num(one),Nb)) ).

% diff_numeral_special(3)
tff(fact_5051_diff__numeral__special_I3_J,axiom,
    ! [Nb: num] : aa_complex_complex(minus_minus_complex(one_one_complex),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb))) = numera6690914467698888265omplex(aa_num_num(plus_plus_num(one),Nb)) ).

% diff_numeral_special(3)
tff(fact_5052_diff__numeral__special_I3_J,axiom,
    ! [Nb: num] : aa_rat_rat(minus_minus_rat(one_one_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))) = aa_num_rat(numeral_numeral_rat,aa_num_num(plus_plus_num(one),Nb)) ).

% diff_numeral_special(3)
tff(fact_5053_diff__numeral__special_I3_J,axiom,
    ! [Nb: num] : minus_8373710615458151222nteger(one_one_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb))) = numera6620942414471956472nteger(aa_num_num(plus_plus_num(one),Nb)) ).

% diff_numeral_special(3)
tff(fact_5054_signed__take__bit__Suc__minus__bit0,axiom,
    ! [Nb: nat,K: num] : aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(suc,Nb)),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(Nb),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_5055_dbl__simps_I4_J,axiom,
    aa_real_real(neg_numeral_dbl_real,aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% dbl_simps(4)
tff(fact_5056_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_5057_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_5058_dbl__simps_I4_J,axiom,
    aa_rat_rat(neg_numeral_dbl_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))) ).

% dbl_simps(4)
tff(fact_5059_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_5060_power__minus1__even,axiom,
    ! [Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = one_one_real ).

% power_minus1_even
tff(fact_5061_power__minus1__even,axiom,
    ! [Nb: 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))),Nb)) = one_one_int ).

% power_minus1_even
tff(fact_5062_power__minus1__even,axiom,
    ! [Nb: 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))),Nb)) = one_one_complex ).

% power_minus1_even
tff(fact_5063_power__minus1__even,axiom,
    ! [Nb: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = one_one_rat ).

% power_minus1_even
tff(fact_5064_power__minus1__even,axiom,
    ! [Nb: 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))),Nb)) = one_one_Code_integer ).

% power_minus1_even
tff(fact_5065_neg__one__odd__power,axiom,
    ! [Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Nb) = aa_real_real(uminus_uminus_real,one_one_real) ) ) ).

% neg_one_odd_power
tff(fact_5066_neg__one__odd__power,axiom,
    ! [Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),Nb) = aa_int_int(uminus_uminus_int,one_one_int) ) ) ).

% neg_one_odd_power
tff(fact_5067_neg__one__odd__power,axiom,
    ! [Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Nb) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ) ) ).

% neg_one_odd_power
tff(fact_5068_neg__one__odd__power,axiom,
    ! [Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Nb) = aa_rat_rat(uminus_uminus_rat,one_one_rat) ) ) ).

% neg_one_odd_power
tff(fact_5069_neg__one__odd__power,axiom,
    ! [Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Nb) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ) ) ).

% neg_one_odd_power
tff(fact_5070_neg__one__even__power,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Nb) = one_one_real ) ) ).

% neg_one_even_power
tff(fact_5071_neg__one__even__power,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),Nb) = one_one_int ) ) ).

% neg_one_even_power
tff(fact_5072_neg__one__even__power,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Nb) = one_one_complex ) ) ).

% neg_one_even_power
tff(fact_5073_neg__one__even__power,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Nb) = one_one_rat ) ) ).

% neg_one_even_power
tff(fact_5074_neg__one__even__power,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Nb) = one_one_Code_integer ) ) ).

% neg_one_even_power
tff(fact_5075_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_5076_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(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ).

% signed_take_bit_0
tff(fact_5077_bits__1__div__exp,axiom,
    ! [Nb: 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))),Nb)) = zero_n2687167440665602831ol_nat(Nb = zero_zero_nat) ).

% bits_1_div_exp
tff(fact_5078_bits__1__div__exp,axiom,
    ! [Nb: 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))),Nb)) = aa_o_int(zero_n2684676970156552555ol_int,Nb = zero_zero_nat) ).

% bits_1_div_exp
tff(fact_5079_bits__1__div__exp,axiom,
    ! [Nb: nat] : divide6298287555418463151nteger(one_one_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Nb)) = zero_n356916108424825756nteger(Nb = zero_zero_nat) ).

% bits_1_div_exp
tff(fact_5080_one__div__2__pow__eq,axiom,
    ! [Nb: 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))),Nb)) = zero_n2687167440665602831ol_nat(Nb = zero_zero_nat) ).

% one_div_2_pow_eq
tff(fact_5081_one__div__2__pow__eq,axiom,
    ! [Nb: 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))),Nb)) = aa_o_int(zero_n2684676970156552555ol_int,Nb = zero_zero_nat) ).

% one_div_2_pow_eq
tff(fact_5082_one__div__2__pow__eq,axiom,
    ! [Nb: nat] : divide6298287555418463151nteger(one_one_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Nb)) = zero_n356916108424825756nteger(Nb = zero_zero_nat) ).

% one_div_2_pow_eq
tff(fact_5083_one__mod__2__pow__eq,axiom,
    ! [Nb: 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))),Nb)) = zero_n2687167440665602831ol_nat(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)) ).

% one_mod_2_pow_eq
tff(fact_5084_one__mod__2__pow__eq,axiom,
    ! [Nb: nat] : aa_int_int(aa_int_fun_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))),Nb)) = aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)) ).

% one_mod_2_pow_eq
tff(fact_5085_one__mod__2__pow__eq,axiom,
    ! [Nb: nat] : modulo364778990260209775nteger(one_one_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Nb)) = zero_n356916108424825756nteger(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)) ).

% one_mod_2_pow_eq
tff(fact_5086_compl__le__swap2,axiom,
    ! [Yb: set_nat,Xb: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(uminus5710092332889474511et_nat(Yb)),Xb)
     => aa_set_nat_o(ord_less_eq_set_nat(uminus5710092332889474511et_nat(Xb)),Yb) ) ).

% compl_le_swap2
tff(fact_5087_compl__le__swap1,axiom,
    ! [Yb: set_nat,Xb: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(Yb),uminus5710092332889474511et_nat(Xb))
     => aa_set_nat_o(ord_less_eq_set_nat(Xb),uminus5710092332889474511et_nat(Yb)) ) ).

% compl_le_swap1
tff(fact_5088_compl__mono,axiom,
    ! [Xb: set_nat,Yb: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(Xb),Yb)
     => aa_set_nat_o(ord_less_eq_set_nat(uminus5710092332889474511et_nat(Yb)),uminus5710092332889474511et_nat(Xb)) ) ).

% compl_mono
tff(fact_5089_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_5090_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_5091_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_5092_minus__equation__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,A) = B )
    <=> ( aa_rat_rat(uminus_uminus_rat,B) = A ) ) ).

% minus_equation_iff
tff(fact_5093_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_5094_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_5095_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_5096_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_5097_equation__minus__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( A = aa_rat_rat(uminus_uminus_rat,B) )
    <=> ( B = aa_rat_rat(uminus_uminus_rat,A) ) ) ).

% equation_minus_iff
tff(fact_5098_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_5099_of__bool__eq__iff,axiom,
    ! [P2: $o,Q2: $o] :
      ( ( zero_n2687167440665602831ol_nat((P2)) = zero_n2687167440665602831ol_nat((Q2)) )
    <=> ( (P2)
      <=> (Q2) ) ) ).

% of_bool_eq_iff
tff(fact_5100_of__bool__eq__iff,axiom,
    ! [P2: $o,Q2: $o] :
      ( ( aa_o_int(zero_n2684676970156552555ol_int,(P2)) = aa_o_int(zero_n2684676970156552555ol_int,(Q2)) )
    <=> ( (P2)
      <=> (Q2) ) ) ).

% of_bool_eq_iff
tff(fact_5101_of__bool__eq__iff,axiom,
    ! [P2: $o,Q2: $o] :
      ( ( zero_n356916108424825756nteger((P2)) = zero_n356916108424825756nteger((Q2)) )
    <=> ( (P2)
      <=> (Q2) ) ) ).

% of_bool_eq_iff
tff(fact_5102_of__bool__conj,axiom,
    ! [P: $o,Q: $o] :
      zero_n3304061248610475627l_real(( (P)
        & (Q) )) = aa_real_real(times_times_real(zero_n3304061248610475627l_real((P))),zero_n3304061248610475627l_real((Q))) ).

% of_bool_conj
tff(fact_5103_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_5104_of__bool__conj,axiom,
    ! [P: $o,Q: $o] :
      zero_n2687167440665602831ol_nat(( (P)
        & (Q) )) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,zero_n2687167440665602831ol_nat((P))),zero_n2687167440665602831ol_nat((Q))) ).

% of_bool_conj
tff(fact_5105_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_5106_of__bool__conj,axiom,
    ! [P: $o,Q: $o] :
      zero_n356916108424825756nteger(( (P)
        & (Q) )) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,zero_n356916108424825756nteger((P))),zero_n356916108424825756nteger((Q))) ).

% of_bool_conj
tff(fact_5107_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_5108_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_5109_le__minus__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(uminus_uminus_rat,B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),aa_rat_rat(uminus_uminus_rat,A)) ) ).

% le_minus_iff
tff(fact_5110_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_5111_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_5112_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_5113_minus__le__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A)),B)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),A) ) ).

% minus_le_iff
tff(fact_5114_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_5115_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_5116_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_5117_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,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(uminus_uminus_rat,A)) ) ).

% le_imp_neg_le
tff(fact_5118_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_5119_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_5120_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_5121_less__minus__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(uminus_uminus_rat,B))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B),aa_rat_rat(uminus_uminus_rat,A)) ) ).

% less_minus_iff
tff(fact_5122_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_5123_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_5124_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_5125_minus__less__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,A)),B)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B)),A) ) ).

% minus_less_iff
tff(fact_5126_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_5127_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_5128_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_5129_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,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(uminus_uminus_rat,A)) ) ).

% verit_negate_coefficient(2)
tff(fact_5130_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_5131_neg__numeral__neq__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb)) != aa_num_real(numeral_numeral_real,Nb) ).

% neg_numeral_neq_numeral
tff(fact_5132_neg__numeral__neq__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb)) != aa_num_int(numeral_numeral_int,Nb) ).

% neg_numeral_neq_numeral
tff(fact_5133_neg__numeral__neq__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Mb)) != numera6690914467698888265omplex(Nb) ).

% neg_numeral_neq_numeral
tff(fact_5134_neg__numeral__neq__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Mb)) != aa_num_rat(numeral_numeral_rat,Nb) ).

% neg_numeral_neq_numeral
tff(fact_5135_neg__numeral__neq__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Mb)) != numera6620942414471956472nteger(Nb) ).

% neg_numeral_neq_numeral
tff(fact_5136_numeral__neq__neg__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_num_real(numeral_numeral_real,Mb) != aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb)) ).

% numeral_neq_neg_numeral
tff(fact_5137_numeral__neq__neg__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_num_int(numeral_numeral_int,Mb) != aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)) ).

% numeral_neq_neg_numeral
tff(fact_5138_numeral__neq__neg__numeral,axiom,
    ! [Mb: num,Nb: num] : numera6690914467698888265omplex(Mb) != aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb)) ).

% numeral_neq_neg_numeral
tff(fact_5139_numeral__neq__neg__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_num_rat(numeral_numeral_rat,Mb) != aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb)) ).

% numeral_neq_neg_numeral
tff(fact_5140_numeral__neq__neg__numeral,axiom,
    ! [Mb: num,Nb: num] : numera6620942414471956472nteger(Mb) != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb)) ).

% numeral_neq_neg_numeral
tff(fact_5141_minus__mult__commute,axiom,
    ! [A: real,B: real] : aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,A)),B) = aa_real_real(times_times_real(A),aa_real_real(uminus_uminus_real,B)) ).

% minus_mult_commute
tff(fact_5142_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_5143_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_5144_minus__mult__commute,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,A)),B) = aa_rat_rat(times_times_rat(A),aa_rat_rat(uminus_uminus_rat,B)) ).

% minus_mult_commute
tff(fact_5145_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_5146_square__eq__iff,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(times_times_real(A),A) = aa_real_real(times_times_real(B),B) )
    <=> ( ( A = B )
        | ( A = aa_real_real(uminus_uminus_real,B) ) ) ) ).

% square_eq_iff
tff(fact_5147_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_5148_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_5149_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 = aa_rat_rat(uminus_uminus_rat,B) ) ) ) ).

% square_eq_iff
tff(fact_5150_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_5151_one__neq__neg__one,axiom,
    one_one_real != aa_real_real(uminus_uminus_real,one_one_real) ).

% one_neq_neg_one
tff(fact_5152_one__neq__neg__one,axiom,
    one_one_int != aa_int_int(uminus_uminus_int,one_one_int) ).

% one_neq_neg_one
tff(fact_5153_one__neq__neg__one,axiom,
    one_one_complex != aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% one_neq_neg_one
tff(fact_5154_one__neq__neg__one,axiom,
    one_one_rat != aa_rat_rat(uminus_uminus_rat,one_one_rat) ).

% one_neq_neg_one
tff(fact_5155_one__neq__neg__one,axiom,
    one_one_Code_integer != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% one_neq_neg_one
tff(fact_5156_is__num__normalize_I8_J,axiom,
    ! [A: real,B: real] : aa_real_real(uminus_uminus_real,aa_real_real(plus_plus_real(A),B)) = aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,B)),aa_real_real(uminus_uminus_real,A)) ).

% is_num_normalize(8)
tff(fact_5157_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_5158_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_5159_is__num__normalize_I8_J,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(plus_plus_rat(A),B)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(uminus_uminus_rat,A)) ).

% is_num_normalize(8)
tff(fact_5160_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_5161_group__cancel_Oneg1,axiom,
    ! [A3: real,K: real,A: real] :
      ( ( A3 = aa_real_real(plus_plus_real(K),A) )
     => ( aa_real_real(uminus_uminus_real,A3) = aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,K)),aa_real_real(uminus_uminus_real,A)) ) ) ).

% group_cancel.neg1
tff(fact_5162_group__cancel_Oneg1,axiom,
    ! [A3: int,K: int,A: int] :
      ( ( A3 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),A) )
     => ( aa_int_int(uminus_uminus_int,A3) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,K)),aa_int_int(uminus_uminus_int,A)) ) ) ).

% group_cancel.neg1
tff(fact_5163_group__cancel_Oneg1,axiom,
    ! [A3: complex,K: complex,A: complex] :
      ( ( A3 = aa_complex_complex(plus_plus_complex(K),A) )
     => ( aa_complex_complex(uminus1482373934393186551omplex,A3) = aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,K)),aa_complex_complex(uminus1482373934393186551omplex,A)) ) ) ).

% group_cancel.neg1
tff(fact_5164_group__cancel_Oneg1,axiom,
    ! [A3: rat,K: rat,A: rat] :
      ( ( A3 = aa_rat_rat(plus_plus_rat(K),A) )
     => ( aa_rat_rat(uminus_uminus_rat,A3) = aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,K)),aa_rat_rat(uminus_uminus_rat,A)) ) ) ).

% group_cancel.neg1
tff(fact_5165_group__cancel_Oneg1,axiom,
    ! [A3: code_integer,K: code_integer,A: code_integer] :
      ( ( A3 = plus_p5714425477246183910nteger(K,A) )
     => ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A3) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,K),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) ) ) ).

% group_cancel.neg1
tff(fact_5166_add_Oinverse__distrib__swap,axiom,
    ! [A: real,B: real] : aa_real_real(uminus_uminus_real,aa_real_real(plus_plus_real(A),B)) = aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,B)),aa_real_real(uminus_uminus_real,A)) ).

% add.inverse_distrib_swap
tff(fact_5167_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_5168_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_5169_add_Oinverse__distrib__swap,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(plus_plus_rat(A),B)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(uminus_uminus_rat,A)) ).

% add.inverse_distrib_swap
tff(fact_5170_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_5171_minus__diff__commute,axiom,
    ! [B: real,A: real] : aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,B)),A) = aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,A)),B) ).

% minus_diff_commute
tff(fact_5172_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_5173_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_5174_minus__diff__commute,axiom,
    ! [B: rat,A: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,B)),A) = aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,A)),B) ).

% minus_diff_commute
tff(fact_5175_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_5176_minus__diff__minus,axiom,
    ! [A: real,B: real] : aa_real_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(minus_minus_real(A),B)) ).

% minus_diff_minus
tff(fact_5177_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_5178_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_5179_minus__diff__minus,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(uminus_uminus_rat,B)) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(minus_minus_rat(A),B)) ).

% minus_diff_minus
tff(fact_5180_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_5181_minus__divide__right,axiom,
    ! [A: real,B: real] : aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(A),B)) = aa_real_real(divide_divide_real(A),aa_real_real(uminus_uminus_real,B)) ).

% minus_divide_right
tff(fact_5182_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_5183_minus__divide__right,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A),B)) = aa_rat_rat(divide_divide_rat(A),aa_rat_rat(uminus_uminus_rat,B)) ).

% minus_divide_right
tff(fact_5184_minus__divide__divide,axiom,
    ! [A: real,B: real] : aa_real_real(divide_divide_real(aa_real_real(uminus_uminus_real,A)),aa_real_real(uminus_uminus_real,B)) = aa_real_real(divide_divide_real(A),B) ).

% minus_divide_divide
tff(fact_5185_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_5186_minus__divide__divide,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(divide_divide_rat(aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(uminus_uminus_rat,B)) = aa_rat_rat(divide_divide_rat(A),B) ).

% minus_divide_divide
tff(fact_5187_minus__divide__left,axiom,
    ! [A: real,B: real] : aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(A),B)) = aa_real_real(divide_divide_real(aa_real_real(uminus_uminus_real,A)),B) ).

% minus_divide_left
tff(fact_5188_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_5189_minus__divide__left,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A),B)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(uminus_uminus_rat,A)),B) ).

% minus_divide_left
tff(fact_5190_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_5191_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_5192_mod__minus__right,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(uminus_uminus_int,B)) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,A)),B)) ).

% mod_minus_right
tff(fact_5193_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_5194_mod__minus__cong,axiom,
    ! [A: int,B: int,A2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B) )
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,A)),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,A2)),B) ) ) ).

% mod_minus_cong
tff(fact_5195_mod__minus__cong,axiom,
    ! [A: code_integer,B: code_integer,A2: code_integer] :
      ( ( modulo364778990260209775nteger(A,B) = modulo364778990260209775nteger(A2,B) )
     => ( modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),B) = modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2),B) ) ) ).

% mod_minus_cong
tff(fact_5196_mod__minus__eq,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B))),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,A)),B) ).

% mod_minus_eq
tff(fact_5197_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_5198_uminus__int__code_I1_J,axiom,
    aa_int_int(uminus_uminus_int,zero_zero_int) = zero_zero_int ).

% uminus_int_code(1)
tff(fact_5199_signed__take__bit__minus,axiom,
    ! [Nb: nat,K: int] : aa_int_int(bit_ri631733984087533419it_int(Nb),aa_int_int(uminus_uminus_int,aa_int_int(bit_ri631733984087533419it_int(Nb),K))) = aa_int_int(bit_ri631733984087533419it_int(Nb),aa_int_int(uminus_uminus_int,K)) ).

% signed_take_bit_minus
tff(fact_5200_concat__bit__assoc,axiom,
    ! [Nb: nat,K: int,Mb: nat,L: int,R2: int] : aa_int_int(aa_int_fun_int_int(bit_concat_bit(Nb),K),aa_int_int(aa_int_fun_int_int(bit_concat_bit(Mb),L),R2)) = aa_int_int(aa_int_fun_int_int(bit_concat_bit(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)),aa_int_int(aa_int_fun_int_int(bit_concat_bit(Nb),K),L)),R2) ).

% concat_bit_assoc
tff(fact_5201_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_5202_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_5203_zero__less__eq__of__bool,axiom,
    ! [P: $o] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),zero_n2687167440665602831ol_nat((P))) ).

% zero_less_eq_of_bool
tff(fact_5204_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_5205_zero__less__eq__of__bool,axiom,
    ! [P: $o] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),zero_n356916108424825756nteger((P))) ).

% zero_less_eq_of_bool
tff(fact_5206_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_5207_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_5208_of__bool__less__eq__one,axiom,
    ! [P: $o] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_n2687167440665602831ol_nat((P))),one_one_nat) ).

% of_bool_less_eq_one
tff(fact_5209_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_5210_of__bool__less__eq__one,axiom,
    ! [P: $o] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_n356916108424825756nteger((P))),one_one_Code_integer) ).

% of_bool_less_eq_one
tff(fact_5211_of__bool__def,axiom,
    ! [P2: $o] :
      zero_n1201886186963655149omplex((P2)) = $ite((P2),one_one_complex,zero_zero_complex) ).

% of_bool_def
tff(fact_5212_of__bool__def,axiom,
    ! [P2: $o] :
      zero_n3304061248610475627l_real((P2)) = $ite((P2),one_one_real,zero_zero_real) ).

% of_bool_def
tff(fact_5213_of__bool__def,axiom,
    ! [P2: $o] :
      zero_n2052037380579107095ol_rat((P2)) = $ite((P2),one_one_rat,zero_zero_rat) ).

% of_bool_def
tff(fact_5214_of__bool__def,axiom,
    ! [P2: $o] :
      zero_n2687167440665602831ol_nat((P2)) = $ite((P2),one_one_nat,zero_zero_nat) ).

% of_bool_def
tff(fact_5215_of__bool__def,axiom,
    ! [P2: $o] :
      aa_o_int(zero_n2684676970156552555ol_int,(P2)) = $ite((P2),one_one_int,zero_zero_int) ).

% of_bool_def
tff(fact_5216_of__bool__def,axiom,
    ! [P2: $o] :
      zero_n356916108424825756nteger((P2)) = $ite((P2),one_one_Code_integer,zero_z3403309356797280102nteger) ).

% of_bool_def
tff(fact_5217_split__of__bool,axiom,
    ! [P: fun_complex_o,P2: $o] :
      ( aa_complex_o(P,zero_n1201886186963655149omplex((P2)))
    <=> ( ( (P2)
         => aa_complex_o(P,one_one_complex) )
        & ( ~ (P2)
         => aa_complex_o(P,zero_zero_complex) ) ) ) ).

% split_of_bool
tff(fact_5218_split__of__bool,axiom,
    ! [P: fun_real_o,P2: $o] :
      ( aa_real_o(P,zero_n3304061248610475627l_real((P2)))
    <=> ( ( (P2)
         => aa_real_o(P,one_one_real) )
        & ( ~ (P2)
         => aa_real_o(P,zero_zero_real) ) ) ) ).

% split_of_bool
tff(fact_5219_split__of__bool,axiom,
    ! [P: fun_rat_o,P2: $o] :
      ( aa_rat_o(P,zero_n2052037380579107095ol_rat((P2)))
    <=> ( ( (P2)
         => aa_rat_o(P,one_one_rat) )
        & ( ~ (P2)
         => aa_rat_o(P,zero_zero_rat) ) ) ) ).

% split_of_bool
tff(fact_5220_split__of__bool,axiom,
    ! [P: fun_nat_o,P2: $o] :
      ( aa_nat_o(P,zero_n2687167440665602831ol_nat((P2)))
    <=> ( ( (P2)
         => aa_nat_o(P,one_one_nat) )
        & ( ~ (P2)
         => aa_nat_o(P,zero_zero_nat) ) ) ) ).

% split_of_bool
tff(fact_5221_split__of__bool,axiom,
    ! [P: fun_int_o,P2: $o] :
      ( aa_int_o(P,aa_o_int(zero_n2684676970156552555ol_int,(P2)))
    <=> ( ( (P2)
         => aa_int_o(P,one_one_int) )
        & ( ~ (P2)
         => aa_int_o(P,zero_zero_int) ) ) ) ).

% split_of_bool
tff(fact_5222_split__of__bool,axiom,
    ! [P: fun_Code_integer_o,P2: $o] :
      ( aa_Code_integer_o(P,zero_n356916108424825756nteger((P2)))
    <=> ( ( (P2)
         => aa_Code_integer_o(P,one_one_Code_integer) )
        & ( ~ (P2)
         => aa_Code_integer_o(P,zero_z3403309356797280102nteger) ) ) ) ).

% split_of_bool
tff(fact_5223_split__of__bool__asm,axiom,
    ! [P: fun_complex_o,P2: $o] :
      ( aa_complex_o(P,zero_n1201886186963655149omplex((P2)))
    <=> ~ ( ( (P2)
            & ~ aa_complex_o(P,one_one_complex) )
          | ( ~ (P2)
            & ~ aa_complex_o(P,zero_zero_complex) ) ) ) ).

% split_of_bool_asm
tff(fact_5224_split__of__bool__asm,axiom,
    ! [P: fun_real_o,P2: $o] :
      ( aa_real_o(P,zero_n3304061248610475627l_real((P2)))
    <=> ~ ( ( (P2)
            & ~ aa_real_o(P,one_one_real) )
          | ( ~ (P2)
            & ~ aa_real_o(P,zero_zero_real) ) ) ) ).

% split_of_bool_asm
tff(fact_5225_split__of__bool__asm,axiom,
    ! [P: fun_rat_o,P2: $o] :
      ( aa_rat_o(P,zero_n2052037380579107095ol_rat((P2)))
    <=> ~ ( ( (P2)
            & ~ aa_rat_o(P,one_one_rat) )
          | ( ~ (P2)
            & ~ aa_rat_o(P,zero_zero_rat) ) ) ) ).

% split_of_bool_asm
tff(fact_5226_split__of__bool__asm,axiom,
    ! [P: fun_nat_o,P2: $o] :
      ( aa_nat_o(P,zero_n2687167440665602831ol_nat((P2)))
    <=> ~ ( ( (P2)
            & ~ aa_nat_o(P,one_one_nat) )
          | ( ~ (P2)
            & ~ aa_nat_o(P,zero_zero_nat) ) ) ) ).

% split_of_bool_asm
tff(fact_5227_split__of__bool__asm,axiom,
    ! [P: fun_int_o,P2: $o] :
      ( aa_int_o(P,aa_o_int(zero_n2684676970156552555ol_int,(P2)))
    <=> ~ ( ( (P2)
            & ~ aa_int_o(P,one_one_int) )
          | ( ~ (P2)
            & ~ aa_int_o(P,zero_zero_int) ) ) ) ).

% split_of_bool_asm
tff(fact_5228_split__of__bool__asm,axiom,
    ! [P: fun_Code_integer_o,P2: $o] :
      ( aa_Code_integer_o(P,zero_n356916108424825756nteger((P2)))
    <=> ~ ( ( (P2)
            & ~ aa_Code_integer_o(P,one_one_Code_integer) )
          | ( ~ (P2)
            & ~ aa_Code_integer_o(P,zero_z3403309356797280102nteger) ) ) ) ).

% split_of_bool_asm
tff(fact_5229_neg__numeral__le__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb))),aa_num_real(numeral_numeral_real,Nb)) ).

% neg_numeral_le_numeral
tff(fact_5230_neg__numeral__le__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Mb))),numera6620942414471956472nteger(Nb)) ).

% neg_numeral_le_numeral
tff(fact_5231_neg__numeral__le__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Mb))),aa_num_rat(numeral_numeral_rat,Nb)) ).

% neg_numeral_le_numeral
tff(fact_5232_neg__numeral__le__numeral,axiom,
    ! [Mb: num,Nb: 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,Mb))),aa_num_int(numeral_numeral_int,Nb)) ).

% neg_numeral_le_numeral
tff(fact_5233_not__numeral__le__neg__numeral,axiom,
    ! [Mb: num,Nb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,Mb)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))) ).

% not_numeral_le_neg_numeral
tff(fact_5234_not__numeral__le__neg__numeral,axiom,
    ! [Mb: num,Nb: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,numera6620942414471956472nteger(Mb)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb))) ).

% not_numeral_le_neg_numeral
tff(fact_5235_not__numeral__le__neg__numeral,axiom,
    ! [Mb: num,Nb: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,Mb)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))) ).

% not_numeral_le_neg_numeral
tff(fact_5236_not__numeral__le__neg__numeral,axiom,
    ! [Mb: num,Nb: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) ).

% not_numeral_le_neg_numeral
tff(fact_5237_zero__neq__neg__numeral,axiom,
    ! [Nb: num] : zero_zero_real != aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb)) ).

% zero_neq_neg_numeral
tff(fact_5238_zero__neq__neg__numeral,axiom,
    ! [Nb: num] : zero_zero_int != aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)) ).

% zero_neq_neg_numeral
tff(fact_5239_zero__neq__neg__numeral,axiom,
    ! [Nb: num] : zero_zero_complex != aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb)) ).

% zero_neq_neg_numeral
tff(fact_5240_zero__neq__neg__numeral,axiom,
    ! [Nb: num] : zero_zero_rat != aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb)) ).

% zero_neq_neg_numeral
tff(fact_5241_zero__neq__neg__numeral,axiom,
    ! [Nb: num] : zero_z3403309356797280102nteger != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb)) ).

% zero_neq_neg_numeral
tff(fact_5242_neg__numeral__less__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb))),aa_num_real(numeral_numeral_real,Nb)) ).

% neg_numeral_less_numeral
tff(fact_5243_neg__numeral__less__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),aa_num_int(numeral_numeral_int,Nb)) ).

% neg_numeral_less_numeral
tff(fact_5244_neg__numeral__less__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Mb))),aa_num_rat(numeral_numeral_rat,Nb)) ).

% neg_numeral_less_numeral
tff(fact_5245_neg__numeral__less__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Mb))),numera6620942414471956472nteger(Nb)) ).

% neg_numeral_less_numeral
tff(fact_5246_not__numeral__less__neg__numeral,axiom,
    ! [Mb: num,Nb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,Mb)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))) ).

% not_numeral_less_neg_numeral
tff(fact_5247_not__numeral__less__neg__numeral,axiom,
    ! [Mb: num,Nb: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) ).

% not_numeral_less_neg_numeral
tff(fact_5248_not__numeral__less__neg__numeral,axiom,
    ! [Mb: num,Nb: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,Mb)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))) ).

% not_numeral_less_neg_numeral
tff(fact_5249_not__numeral__less__neg__numeral,axiom,
    ! [Mb: num,Nb: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,numera6620942414471956472nteger(Mb)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb))) ).

% not_numeral_less_neg_numeral
tff(fact_5250_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_5251_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_5252_le__minus__one__simps_I4_J,axiom,
    ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_rat_rat(uminus_uminus_rat,one_one_rat)) ).

% le_minus_one_simps(4)
tff(fact_5253_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_5254_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_5255_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_5256_le__minus__one__simps_I2_J,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),one_one_rat) ).

% le_minus_one_simps(2)
tff(fact_5257_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_5258_zero__neq__neg__one,axiom,
    zero_zero_real != aa_real_real(uminus_uminus_real,one_one_real) ).

% zero_neq_neg_one
tff(fact_5259_zero__neq__neg__one,axiom,
    zero_zero_int != aa_int_int(uminus_uminus_int,one_one_int) ).

% zero_neq_neg_one
tff(fact_5260_zero__neq__neg__one,axiom,
    zero_zero_complex != aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% zero_neq_neg_one
tff(fact_5261_zero__neq__neg__one,axiom,
    zero_zero_rat != aa_rat_rat(uminus_uminus_rat,one_one_rat) ).

% zero_neq_neg_one
tff(fact_5262_zero__neq__neg__one,axiom,
    zero_z3403309356797280102nteger != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% zero_neq_neg_one
tff(fact_5263_add__eq__0__iff,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(plus_plus_real(A),B) = zero_zero_real )
    <=> ( B = aa_real_real(uminus_uminus_real,A) ) ) ).

% add_eq_0_iff
tff(fact_5264_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_5265_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_5266_add__eq__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(plus_plus_rat(A),B) = zero_zero_rat )
    <=> ( B = aa_rat_rat(uminus_uminus_rat,A) ) ) ).

% add_eq_0_iff
tff(fact_5267_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_5268_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: real] : aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,A)),A) = zero_zero_real ).

% ab_group_add_class.ab_left_minus
tff(fact_5269_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_5270_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_5271_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,A)),A) = zero_zero_rat ).

% ab_group_add_class.ab_left_minus
tff(fact_5272_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_5273_add_Oinverse__unique,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(plus_plus_real(A),B) = zero_zero_real )
     => ( aa_real_real(uminus_uminus_real,A) = B ) ) ).

% add.inverse_unique
tff(fact_5274_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_5275_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_5276_add_Oinverse__unique,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(plus_plus_rat(A),B) = zero_zero_rat )
     => ( aa_rat_rat(uminus_uminus_rat,A) = B ) ) ).

% add.inverse_unique
tff(fact_5277_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_5278_eq__neg__iff__add__eq__0,axiom,
    ! [A: real,B: real] :
      ( ( A = aa_real_real(uminus_uminus_real,B) )
    <=> ( aa_real_real(plus_plus_real(A),B) = zero_zero_real ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_5279_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_5280_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_5281_eq__neg__iff__add__eq__0,axiom,
    ! [A: rat,B: rat] :
      ( ( A = aa_rat_rat(uminus_uminus_rat,B) )
    <=> ( aa_rat_rat(plus_plus_rat(A),B) = zero_zero_rat ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_5282_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_5283_neg__eq__iff__add__eq__0,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(uminus_uminus_real,A) = B )
    <=> ( aa_real_real(plus_plus_real(A),B) = zero_zero_real ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_5284_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_5285_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_5286_neg__eq__iff__add__eq__0,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,A) = B )
    <=> ( aa_rat_rat(plus_plus_rat(A),B) = zero_zero_rat ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_5287_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_5288_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_5289_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_5290_less__minus__one__simps_I4_J,axiom,
    ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_rat_rat(uminus_uminus_rat,one_one_rat)) ).

% less_minus_one_simps(4)
tff(fact_5291_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_5292_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_5293_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_5294_less__minus__one__simps_I2_J,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),one_one_rat) ).

% less_minus_one_simps(2)
tff(fact_5295_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_5296_numeral__times__minus__swap,axiom,
    ! [W: num,Xb: real] : aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,W)),aa_real_real(uminus_uminus_real,Xb)) = aa_real_real(times_times_real(Xb),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))) ).

% numeral_times_minus_swap
tff(fact_5297_numeral__times__minus__swap,axiom,
    ! [W: num,Xb: 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,Xb)) = aa_int_int(aa_int_fun_int_int(times_times_int,Xb),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))) ).

% numeral_times_minus_swap
tff(fact_5298_numeral__times__minus__swap,axiom,
    ! [W: num,Xb: complex] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(W)),aa_complex_complex(uminus1482373934393186551omplex,Xb)) = aa_complex_complex(times_times_complex(Xb),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))) ).

% numeral_times_minus_swap
tff(fact_5299_numeral__times__minus__swap,axiom,
    ! [W: num,Xb: rat] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W)),aa_rat_rat(uminus_uminus_rat,Xb)) = aa_rat_rat(times_times_rat(Xb),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))) ).

% numeral_times_minus_swap
tff(fact_5300_numeral__times__minus__swap,axiom,
    ! [W: num,Xb: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(W)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Xb)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Xb),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(W))) ).

% numeral_times_minus_swap
tff(fact_5301_nonzero__minus__divide__divide,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( aa_real_real(divide_divide_real(aa_real_real(uminus_uminus_real,A)),aa_real_real(uminus_uminus_real,B)) = aa_real_real(divide_divide_real(A),B) ) ) ).

% nonzero_minus_divide_divide
tff(fact_5302_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_5303_nonzero__minus__divide__divide,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(uminus_uminus_rat,B)) = aa_rat_rat(divide_divide_rat(A),B) ) ) ).

% nonzero_minus_divide_divide
tff(fact_5304_nonzero__minus__divide__right,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(A),B)) = aa_real_real(divide_divide_real(A),aa_real_real(uminus_uminus_real,B)) ) ) ).

% nonzero_minus_divide_right
tff(fact_5305_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_5306_nonzero__minus__divide__right,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A),B)) = aa_rat_rat(divide_divide_rat(A),aa_rat_rat(uminus_uminus_rat,B)) ) ) ).

% nonzero_minus_divide_right
tff(fact_5307_numeral__neq__neg__one,axiom,
    ! [Nb: num] : aa_num_real(numeral_numeral_real,Nb) != aa_real_real(uminus_uminus_real,one_one_real) ).

% numeral_neq_neg_one
tff(fact_5308_numeral__neq__neg__one,axiom,
    ! [Nb: num] : aa_num_int(numeral_numeral_int,Nb) != aa_int_int(uminus_uminus_int,one_one_int) ).

% numeral_neq_neg_one
tff(fact_5309_numeral__neq__neg__one,axiom,
    ! [Nb: num] : numera6690914467698888265omplex(Nb) != aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% numeral_neq_neg_one
tff(fact_5310_numeral__neq__neg__one,axiom,
    ! [Nb: num] : aa_num_rat(numeral_numeral_rat,Nb) != aa_rat_rat(uminus_uminus_rat,one_one_rat) ).

% numeral_neq_neg_one
tff(fact_5311_numeral__neq__neg__one,axiom,
    ! [Nb: num] : numera6620942414471956472nteger(Nb) != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% numeral_neq_neg_one
tff(fact_5312_one__neq__neg__numeral,axiom,
    ! [Nb: num] : one_one_real != aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb)) ).

% one_neq_neg_numeral
tff(fact_5313_one__neq__neg__numeral,axiom,
    ! [Nb: num] : one_one_int != aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)) ).

% one_neq_neg_numeral
tff(fact_5314_one__neq__neg__numeral,axiom,
    ! [Nb: num] : one_one_complex != aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb)) ).

% one_neq_neg_numeral
tff(fact_5315_one__neq__neg__numeral,axiom,
    ! [Nb: num] : one_one_rat != aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb)) ).

% one_neq_neg_numeral
tff(fact_5316_one__neq__neg__numeral,axiom,
    ! [Nb: num] : one_one_Code_integer != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb)) ).

% one_neq_neg_numeral
tff(fact_5317_square__eq__1__iff,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(times_times_real(Xb),Xb) = one_one_real )
    <=> ( ( Xb = one_one_real )
        | ( Xb = aa_real_real(uminus_uminus_real,one_one_real) ) ) ) ).

% square_eq_1_iff
tff(fact_5318_square__eq__1__iff,axiom,
    ! [Xb: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,Xb),Xb) = one_one_int )
    <=> ( ( Xb = one_one_int )
        | ( Xb = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% square_eq_1_iff
tff(fact_5319_square__eq__1__iff,axiom,
    ! [Xb: complex] :
      ( ( aa_complex_complex(times_times_complex(Xb),Xb) = one_one_complex )
    <=> ( ( Xb = one_one_complex )
        | ( Xb = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ) ) ) ).

% square_eq_1_iff
tff(fact_5320_square__eq__1__iff,axiom,
    ! [Xb: rat] :
      ( ( aa_rat_rat(times_times_rat(Xb),Xb) = one_one_rat )
    <=> ( ( Xb = one_one_rat )
        | ( Xb = aa_rat_rat(uminus_uminus_rat,one_one_rat) ) ) ) ).

% square_eq_1_iff
tff(fact_5321_square__eq__1__iff,axiom,
    ! [Xb: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Xb),Xb) = one_one_Code_integer )
    <=> ( ( Xb = one_one_Code_integer )
        | ( Xb = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ) ) ) ).

% square_eq_1_iff
tff(fact_5322_group__cancel_Osub2,axiom,
    ! [B4: real,K: real,B: real,A: real] :
      ( ( B4 = aa_real_real(plus_plus_real(K),B) )
     => ( aa_real_real(minus_minus_real(A),B4) = aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,K)),aa_real_real(minus_minus_real(A),B)) ) ) ).

% group_cancel.sub2
tff(fact_5323_group__cancel_Osub2,axiom,
    ! [B4: int,K: int,B: int,A: int] :
      ( ( B4 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),B) )
     => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B4) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,K)),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)) ) ) ).

% group_cancel.sub2
tff(fact_5324_group__cancel_Osub2,axiom,
    ! [B4: complex,K: complex,B: complex,A: complex] :
      ( ( B4 = aa_complex_complex(plus_plus_complex(K),B) )
     => ( aa_complex_complex(minus_minus_complex(A),B4) = aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,K)),aa_complex_complex(minus_minus_complex(A),B)) ) ) ).

% group_cancel.sub2
tff(fact_5325_group__cancel_Osub2,axiom,
    ! [B4: rat,K: rat,B: rat,A: rat] :
      ( ( B4 = aa_rat_rat(plus_plus_rat(K),B) )
     => ( aa_rat_rat(minus_minus_rat(A),B4) = aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,K)),aa_rat_rat(minus_minus_rat(A),B)) ) ) ).

% group_cancel.sub2
tff(fact_5326_group__cancel_Osub2,axiom,
    ! [B4: code_integer,K: code_integer,B: code_integer,A: code_integer] :
      ( ( B4 = plus_p5714425477246183910nteger(K,B) )
     => ( minus_8373710615458151222nteger(A,B4) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,K),minus_8373710615458151222nteger(A,B)) ) ) ).

% group_cancel.sub2
tff(fact_5327_diff__conv__add__uminus,axiom,
    ! [A: real,B: real] : aa_real_real(minus_minus_real(A),B) = aa_real_real(plus_plus_real(A),aa_real_real(uminus_uminus_real,B)) ).

% diff_conv_add_uminus
tff(fact_5328_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_5329_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_5330_diff__conv__add__uminus,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(minus_minus_rat(A),B) = aa_rat_rat(plus_plus_rat(A),aa_rat_rat(uminus_uminus_rat,B)) ).

% diff_conv_add_uminus
tff(fact_5331_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_5332_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A: real,B: real] : aa_real_real(minus_minus_real(A),B) = aa_real_real(plus_plus_real(A),aa_real_real(uminus_uminus_real,B)) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_5333_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_5334_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_5335_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(minus_minus_rat(A),B) = aa_rat_rat(plus_plus_rat(A),aa_rat_rat(uminus_uminus_rat,B)) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_5336_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_5337_dvd__div__neg,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,B),A)
     => ( aa_real_real(divide_divide_real(A),aa_real_real(uminus_uminus_real,B)) = aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(A),B)) ) ) ).

% dvd_div_neg
tff(fact_5338_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_5339_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_5340_dvd__div__neg,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,B),A)
     => ( aa_rat_rat(divide_divide_rat(A),aa_rat_rat(uminus_uminus_rat,B)) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A),B)) ) ) ).

% dvd_div_neg
tff(fact_5341_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_5342_dvd__neg__div,axiom,
    ! [B: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,B),A)
     => ( aa_real_real(divide_divide_real(aa_real_real(uminus_uminus_real,A)),B) = aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(A),B)) ) ) ).

% dvd_neg_div
tff(fact_5343_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_5344_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_5345_dvd__neg__div,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,B),A)
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(uminus_uminus_rat,A)),B) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A),B)) ) ) ).

% dvd_neg_div
tff(fact_5346_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_5347_subset__Compl__self__eq,axiom,
    ! [A3: set_int] :
      ( ord_less_eq_set_int(A3,uminus1532241313380277803et_int(A3))
    <=> ( A3 = bot_bot_set_int ) ) ).

% subset_Compl_self_eq
tff(fact_5348_subset__Compl__self__eq,axiom,
    ! [A3: set_real] :
      ( ord_less_eq_set_real(A3,uminus612125837232591019t_real(A3))
    <=> ( A3 = bot_bot_set_real ) ) ).

% subset_Compl_self_eq
tff(fact_5349_subset__Compl__self__eq,axiom,
    ! [A3: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A3),uminus5710092332889474511et_nat(A3))
    <=> ( A3 = bot_bot_set_nat ) ) ).

% subset_Compl_self_eq
tff(fact_5350_real__minus__mult__self__le,axiom,
    ! [U: real,Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(times_times_real(U),U))),aa_real_real(times_times_real(Xb),Xb)) ).

% real_minus_mult_self_le
tff(fact_5351_zmult__eq__1__iff,axiom,
    ! [Mb: int,Nb: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,Mb),Nb) = one_one_int )
    <=> ( ( ( Mb = one_one_int )
          & ( Nb = one_one_int ) )
        | ( ( Mb = aa_int_int(uminus_uminus_int,one_one_int) )
          & ( Nb = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ) ).

% zmult_eq_1_iff
tff(fact_5352_pos__zmult__eq__1__iff__lemma,axiom,
    ! [Mb: int,Nb: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,Mb),Nb) = one_one_int )
     => ( ( Mb = one_one_int )
        | ( Mb = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% pos_zmult_eq_1_iff_lemma
tff(fact_5353_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_5354_zmod__zminus2__not__zero,axiom,
    ! [K: int,L: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),aa_int_int(uminus_uminus_int,L)) != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L) != zero_zero_int ) ) ).

% zmod_zminus2_not_zero
tff(fact_5355_zmod__zminus1__not__zero,axiom,
    ! [K: int,L: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,K)),L) != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L) != zero_zero_int ) ) ).

% zmod_zminus1_not_zero
tff(fact_5356_minus__real__def,axiom,
    ! [Xb: real,Yb: real] : aa_real_real(minus_minus_real(Xb),Yb) = aa_real_real(plus_plus_real(Xb),aa_real_real(uminus_uminus_real,Yb)) ).

% minus_real_def
tff(fact_5357_neg__numeral__le__zero,axiom,
    ! [Nb: num] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))),zero_zero_real) ).

% neg_numeral_le_zero
tff(fact_5358_neg__numeral__le__zero,axiom,
    ! [Nb: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb))),zero_z3403309356797280102nteger) ).

% neg_numeral_le_zero
tff(fact_5359_neg__numeral__le__zero,axiom,
    ! [Nb: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))),zero_zero_rat) ).

% neg_numeral_le_zero
tff(fact_5360_neg__numeral__le__zero,axiom,
    ! [Nb: 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,Nb))),zero_zero_int) ).

% neg_numeral_le_zero
tff(fact_5361_not__zero__le__neg__numeral,axiom,
    ! [Nb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))) ).

% not_zero_le_neg_numeral
tff(fact_5362_not__zero__le__neg__numeral,axiom,
    ! [Nb: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb))) ).

% not_zero_le_neg_numeral
tff(fact_5363_not__zero__le__neg__numeral,axiom,
    ! [Nb: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))) ).

% not_zero_le_neg_numeral
tff(fact_5364_not__zero__le__neg__numeral,axiom,
    ! [Nb: 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,Nb))) ).

% not_zero_le_neg_numeral
tff(fact_5365_neg__numeral__less__zero,axiom,
    ! [Nb: num] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))),zero_zero_real) ).

% neg_numeral_less_zero
tff(fact_5366_neg__numeral__less__zero,axiom,
    ! [Nb: num] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))),zero_zero_int) ).

% neg_numeral_less_zero
tff(fact_5367_neg__numeral__less__zero,axiom,
    ! [Nb: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))),zero_zero_rat) ).

% neg_numeral_less_zero
tff(fact_5368_neg__numeral__less__zero,axiom,
    ! [Nb: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb))),zero_z3403309356797280102nteger) ).

% neg_numeral_less_zero
tff(fact_5369_not__zero__less__neg__numeral,axiom,
    ! [Nb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))) ).

% not_zero_less_neg_numeral
tff(fact_5370_not__zero__less__neg__numeral,axiom,
    ! [Nb: 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,Nb))) ).

% not_zero_less_neg_numeral
tff(fact_5371_not__zero__less__neg__numeral,axiom,
    ! [Nb: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))) ).

% not_zero_less_neg_numeral
tff(fact_5372_not__zero__less__neg__numeral,axiom,
    ! [Nb: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb))) ).

% not_zero_less_neg_numeral
tff(fact_5373_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_5374_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_5375_le__minus__one__simps_I3_J,axiom,
    ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,one_one_rat)) ).

% le_minus_one_simps(3)
tff(fact_5376_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_5377_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_5378_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_5379_le__minus__one__simps_I1_J,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),zero_zero_rat) ).

% le_minus_one_simps(1)
tff(fact_5380_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_5381_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_5382_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_5383_less__minus__one__simps_I3_J,axiom,
    ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,one_one_rat)) ).

% less_minus_one_simps(3)
tff(fact_5384_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_5385_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_5386_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_5387_less__minus__one__simps_I1_J,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),zero_zero_rat) ).

% less_minus_one_simps(1)
tff(fact_5388_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_5389_neg__numeral__le__one,axiom,
    ! [Mb: num] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb))),one_one_real) ).

% neg_numeral_le_one
tff(fact_5390_neg__numeral__le__one,axiom,
    ! [Mb: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Mb))),one_one_Code_integer) ).

% neg_numeral_le_one
tff(fact_5391_neg__numeral__le__one,axiom,
    ! [Mb: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Mb))),one_one_rat) ).

% neg_numeral_le_one
tff(fact_5392_neg__numeral__le__one,axiom,
    ! [Mb: 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,Mb))),one_one_int) ).

% neg_numeral_le_one
tff(fact_5393_neg__one__le__numeral,axiom,
    ! [Mb: num] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_num_real(numeral_numeral_real,Mb)) ).

% neg_one_le_numeral
tff(fact_5394_neg__one__le__numeral,axiom,
    ! [Mb: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),numera6620942414471956472nteger(Mb)) ).

% neg_one_le_numeral
tff(fact_5395_neg__one__le__numeral,axiom,
    ! [Mb: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_num_rat(numeral_numeral_rat,Mb)) ).

% neg_one_le_numeral
tff(fact_5396_neg__one__le__numeral,axiom,
    ! [Mb: 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,Mb)) ).

% neg_one_le_numeral
tff(fact_5397_neg__numeral__le__neg__one,axiom,
    ! [Mb: num] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb))),aa_real_real(uminus_uminus_real,one_one_real)) ).

% neg_numeral_le_neg_one
tff(fact_5398_neg__numeral__le__neg__one,axiom,
    ! [Mb: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Mb))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).

% neg_numeral_le_neg_one
tff(fact_5399_neg__numeral__le__neg__one,axiom,
    ! [Mb: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Mb))),aa_rat_rat(uminus_uminus_rat,one_one_rat)) ).

% neg_numeral_le_neg_one
tff(fact_5400_neg__numeral__le__neg__one,axiom,
    ! [Mb: 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,Mb))),aa_int_int(uminus_uminus_int,one_one_int)) ).

% neg_numeral_le_neg_one
tff(fact_5401_not__numeral__le__neg__one,axiom,
    ! [Mb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,Mb)),aa_real_real(uminus_uminus_real,one_one_real)) ).

% not_numeral_le_neg_one
tff(fact_5402_not__numeral__le__neg__one,axiom,
    ! [Mb: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,numera6620942414471956472nteger(Mb)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).

% not_numeral_le_neg_one
tff(fact_5403_not__numeral__le__neg__one,axiom,
    ! [Mb: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,Mb)),aa_rat_rat(uminus_uminus_rat,one_one_rat)) ).

% not_numeral_le_neg_one
tff(fact_5404_not__numeral__le__neg__one,axiom,
    ! [Mb: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(uminus_uminus_int,one_one_int)) ).

% not_numeral_le_neg_one
tff(fact_5405_not__one__le__neg__numeral,axiom,
    ! [Mb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb))) ).

% not_one_le_neg_numeral
tff(fact_5406_not__one__le__neg__numeral,axiom,
    ! [Mb: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Mb))) ).

% not_one_le_neg_numeral
tff(fact_5407_not__one__le__neg__numeral,axiom,
    ! [Mb: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Mb))) ).

% not_one_le_neg_numeral
tff(fact_5408_not__one__le__neg__numeral,axiom,
    ! [Mb: 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,Mb))) ).

% not_one_le_neg_numeral
tff(fact_5409_neg__numeral__less__one,axiom,
    ! [Mb: num] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb))),one_one_real) ).

% neg_numeral_less_one
tff(fact_5410_neg__numeral__less__one,axiom,
    ! [Mb: num] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),one_one_int) ).

% neg_numeral_less_one
tff(fact_5411_neg__numeral__less__one,axiom,
    ! [Mb: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Mb))),one_one_rat) ).

% neg_numeral_less_one
tff(fact_5412_neg__numeral__less__one,axiom,
    ! [Mb: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Mb))),one_one_Code_integer) ).

% neg_numeral_less_one
tff(fact_5413_neg__one__less__numeral,axiom,
    ! [Mb: num] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_num_real(numeral_numeral_real,Mb)) ).

% neg_one_less_numeral
tff(fact_5414_neg__one__less__numeral,axiom,
    ! [Mb: 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,Mb)) ).

% neg_one_less_numeral
tff(fact_5415_neg__one__less__numeral,axiom,
    ! [Mb: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_num_rat(numeral_numeral_rat,Mb)) ).

% neg_one_less_numeral
tff(fact_5416_neg__one__less__numeral,axiom,
    ! [Mb: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),numera6620942414471956472nteger(Mb)) ).

% neg_one_less_numeral
tff(fact_5417_not__numeral__less__neg__one,axiom,
    ! [Mb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,Mb)),aa_real_real(uminus_uminus_real,one_one_real)) ).

% not_numeral_less_neg_one
tff(fact_5418_not__numeral__less__neg__one,axiom,
    ! [Mb: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(uminus_uminus_int,one_one_int)) ).

% not_numeral_less_neg_one
tff(fact_5419_not__numeral__less__neg__one,axiom,
    ! [Mb: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,Mb)),aa_rat_rat(uminus_uminus_rat,one_one_rat)) ).

% not_numeral_less_neg_one
tff(fact_5420_not__numeral__less__neg__one,axiom,
    ! [Mb: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,numera6620942414471956472nteger(Mb)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).

% not_numeral_less_neg_one
tff(fact_5421_not__one__less__neg__numeral,axiom,
    ! [Mb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb))) ).

% not_one_less_neg_numeral
tff(fact_5422_not__one__less__neg__numeral,axiom,
    ! [Mb: 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,Mb))) ).

% not_one_less_neg_numeral
tff(fact_5423_not__one__less__neg__numeral,axiom,
    ! [Mb: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Mb))) ).

% not_one_less_neg_numeral
tff(fact_5424_not__one__less__neg__numeral,axiom,
    ! [Mb: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Mb))) ).

% not_one_less_neg_numeral
tff(fact_5425_not__neg__one__less__neg__numeral,axiom,
    ! [Mb: 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,aa_num_real(numeral_numeral_real,Mb))) ).

% not_neg_one_less_neg_numeral
tff(fact_5426_not__neg__one__less__neg__numeral,axiom,
    ! [Mb: 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,Mb))) ).

% not_neg_one_less_neg_numeral
tff(fact_5427_not__neg__one__less__neg__numeral,axiom,
    ! [Mb: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Mb))) ).

% not_neg_one_less_neg_numeral
tff(fact_5428_not__neg__one__less__neg__numeral,axiom,
    ! [Mb: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Mb))) ).

% not_neg_one_less_neg_numeral
tff(fact_5429_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(divide_divide_real(A),B)) )
      <=> ( aa_real_real(times_times_real(C),B) = aa_real_real(uminus_uminus_real,A) ) ) ) ).

% nonzero_neg_divide_eq_eq2
tff(fact_5430_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_5431_nonzero__neg__divide__eq__eq2,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( ( C = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A),B)) )
      <=> ( aa_rat_rat(times_times_rat(C),B) = aa_rat_rat(uminus_uminus_rat,A) ) ) ) ).

% nonzero_neg_divide_eq_eq2
tff(fact_5432_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(divide_divide_real(A),B)) = C )
      <=> ( aa_real_real(uminus_uminus_real,A) = aa_real_real(times_times_real(C),B) ) ) ) ).

% nonzero_neg_divide_eq_eq
tff(fact_5433_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_5434_nonzero__neg__divide__eq__eq,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( B != zero_zero_rat )
     => ( ( aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A),B)) = C )
      <=> ( aa_rat_rat(uminus_uminus_rat,A) = aa_rat_rat(times_times_rat(C),B) ) ) ) ).

% nonzero_neg_divide_eq_eq
tff(fact_5435_minus__divide__eq__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( ( aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(B),C)) = A )
    <=> $ite(C != zero_zero_real,aa_real_real(uminus_uminus_real,B) = aa_real_real(times_times_real(A),C),A = zero_zero_real) ) ).

% minus_divide_eq_eq
tff(fact_5436_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_5437_minus__divide__eq__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C)) = A )
    <=> $ite(C != zero_zero_rat,aa_rat_rat(uminus_uminus_rat,B) = aa_rat_rat(times_times_rat(A),C),A = zero_zero_rat) ) ).

% minus_divide_eq_eq
tff(fact_5438_eq__minus__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( A = aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(B),C)) )
    <=> $ite(C != zero_zero_real,aa_real_real(times_times_real(A),C) = aa_real_real(uminus_uminus_real,B),A = zero_zero_real) ) ).

% eq_minus_divide_eq
tff(fact_5439_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_5440_eq__minus__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( A = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C)) )
    <=> $ite(C != zero_zero_rat,aa_rat_rat(times_times_rat(A),C) = aa_rat_rat(uminus_uminus_rat,B),A = zero_zero_rat) ) ).

% eq_minus_divide_eq
tff(fact_5441_mult__1s__ring__1_I2_J,axiom,
    ! [B: real] : aa_real_real(times_times_real(B),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,one))) = aa_real_real(uminus_uminus_real,B) ).

% mult_1s_ring_1(2)
tff(fact_5442_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_5443_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_5444_mult__1s__ring__1_I2_J,axiom,
    ! [B: rat] : aa_rat_rat(times_times_rat(B),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,one))) = aa_rat_rat(uminus_uminus_rat,B) ).

% mult_1s_ring_1(2)
tff(fact_5445_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_5446_mult__1s__ring__1_I1_J,axiom,
    ! [B: real] : aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,one))),B) = aa_real_real(uminus_uminus_real,B) ).

% mult_1s_ring_1(1)
tff(fact_5447_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_5448_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_5449_mult__1s__ring__1_I1_J,axiom,
    ! [B: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,one))),B) = aa_rat_rat(uminus_uminus_rat,B) ).

% mult_1s_ring_1(1)
tff(fact_5450_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_5451_divide__eq__minus__1__iff,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_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_5452_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_5453_divide__eq__minus__1__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(divide_divide_rat(A),B) = aa_rat_rat(uminus_uminus_rat,one_one_rat) )
    <=> ( ( B != zero_zero_rat )
        & ( A = aa_rat_rat(uminus_uminus_rat,B) ) ) ) ).

% divide_eq_minus_1_iff
tff(fact_5454_uminus__numeral__One,axiom,
    aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,one)) = aa_real_real(uminus_uminus_real,one_one_real) ).

% uminus_numeral_One
tff(fact_5455_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_5456_uminus__numeral__One,axiom,
    aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(one)) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% uminus_numeral_One
tff(fact_5457_uminus__numeral__One,axiom,
    aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,one)) = aa_rat_rat(uminus_uminus_rat,one_one_rat) ).

% uminus_numeral_One
tff(fact_5458_uminus__numeral__One,axiom,
    aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(one)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% uminus_numeral_One
tff(fact_5459_power__minus,axiom,
    ! [A: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,A)),Nb) = aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ).

% power_minus
tff(fact_5460_power__minus,axiom,
    ! [A: int,Nb: nat] : aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A)),Nb) = 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)),Nb)),aa_nat_int(power_power_int(A),Nb)) ).

% power_minus
tff(fact_5461_power__minus,axiom,
    ! [A: complex,Nb: nat] : aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),Nb) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Nb)),aa_nat_complex(power_power_complex(A),Nb)) ).

% power_minus
tff(fact_5462_power__minus,axiom,
    ! [A: rat,Nb: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,A)),Nb) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Nb)),aa_nat_rat(power_power_rat(A),Nb)) ).

% power_minus
tff(fact_5463_power__minus,axiom,
    ! [A: code_integer,Nb: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),Nb) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Nb)),aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb)) ).

% power_minus
tff(fact_5464_bset_I1_J,axiom,
    ! [D4: int,B4: set_int,P: fun_int_o,Q: fun_int_o] :
      ( ! [X3: int] :
          ( ! [Xa: int] :
              ( member_int(Xa,set_or1266510415728281911st_int(one_one_int,D4))
             => ! [Xb2: int] :
                  ( member_int(Xb2,B4)
                 => ( X3 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa) ) ) )
         => ( aa_int_o(P,X3)
           => aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D4)) ) )
     => ( ! [X3: int] :
            ( ! [Xa: int] :
                ( member_int(Xa,set_or1266510415728281911st_int(one_one_int,D4))
               => ! [Xb2: int] :
                    ( member_int(Xb2,B4)
                   => ( X3 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa) ) ) )
           => ( aa_int_o(Q,X3)
             => aa_int_o(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D4)) ) )
       => ! [X: int] :
            ( ! [Xa2: int] :
                ( member_int(Xa2,set_or1266510415728281911st_int(one_one_int,D4))
               => ! [Xb3: int] :
                    ( member_int(Xb3,B4)
                   => ( X != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa2) ) ) )
           => ( ( aa_int_o(P,X)
                & aa_int_o(Q,X) )
             => ( aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),D4))
                & aa_int_o(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),D4)) ) ) ) ) ) ).

% bset(1)
tff(fact_5465_bset_I2_J,axiom,
    ! [D4: int,B4: set_int,P: fun_int_o,Q: fun_int_o] :
      ( ! [X3: int] :
          ( ! [Xa: int] :
              ( member_int(Xa,set_or1266510415728281911st_int(one_one_int,D4))
             => ! [Xb2: int] :
                  ( member_int(Xb2,B4)
                 => ( X3 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa) ) ) )
         => ( aa_int_o(P,X3)
           => aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D4)) ) )
     => ( ! [X3: int] :
            ( ! [Xa: int] :
                ( member_int(Xa,set_or1266510415728281911st_int(one_one_int,D4))
               => ! [Xb2: int] :
                    ( member_int(Xb2,B4)
                   => ( X3 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa) ) ) )
           => ( aa_int_o(Q,X3)
             => aa_int_o(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D4)) ) )
       => ! [X: int] :
            ( ! [Xa2: int] :
                ( member_int(Xa2,set_or1266510415728281911st_int(one_one_int,D4))
               => ! [Xb3: int] :
                    ( member_int(Xb3,B4)
                   => ( X != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa2) ) ) )
           => ( ( aa_int_o(P,X)
                | aa_int_o(Q,X) )
             => ( aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),D4))
                | aa_int_o(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),D4)) ) ) ) ) ) ).

% bset(2)
tff(fact_5466_aset_I1_J,axiom,
    ! [D4: int,A3: set_int,P: fun_int_o,Q: fun_int_o] :
      ( ! [X3: int] :
          ( ! [Xa: int] :
              ( member_int(Xa,set_or1266510415728281911st_int(one_one_int,D4))
             => ! [Xb2: int] :
                  ( member_int(Xb2,A3)
                 => ( X3 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa) ) ) )
         => ( aa_int_o(P,X3)
           => aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D4)) ) )
     => ( ! [X3: int] :
            ( ! [Xa: int] :
                ( member_int(Xa,set_or1266510415728281911st_int(one_one_int,D4))
               => ! [Xb2: int] :
                    ( member_int(Xb2,A3)
                   => ( X3 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa) ) ) )
           => ( aa_int_o(Q,X3)
             => aa_int_o(Q,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D4)) ) )
       => ! [X: int] :
            ( ! [Xa2: int] :
                ( member_int(Xa2,set_or1266510415728281911st_int(one_one_int,D4))
               => ! [Xb3: int] :
                    ( member_int(Xb3,A3)
                   => ( X != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa2) ) ) )
           => ( ( aa_int_o(P,X)
                & aa_int_o(Q,X) )
             => ( aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),D4))
                & aa_int_o(Q,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),D4)) ) ) ) ) ) ).

% aset(1)
tff(fact_5467_aset_I2_J,axiom,
    ! [D4: int,A3: set_int,P: fun_int_o,Q: fun_int_o] :
      ( ! [X3: int] :
          ( ! [Xa: int] :
              ( member_int(Xa,set_or1266510415728281911st_int(one_one_int,D4))
             => ! [Xb2: int] :
                  ( member_int(Xb2,A3)
                 => ( X3 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa) ) ) )
         => ( aa_int_o(P,X3)
           => aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D4)) ) )
     => ( ! [X3: int] :
            ( ! [Xa: int] :
                ( member_int(Xa,set_or1266510415728281911st_int(one_one_int,D4))
               => ! [Xb2: int] :
                    ( member_int(Xb2,A3)
                   => ( X3 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa) ) ) )
           => ( aa_int_o(Q,X3)
             => aa_int_o(Q,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D4)) ) )
       => ! [X: int] :
            ( ! [Xa2: int] :
                ( member_int(Xa2,set_or1266510415728281911st_int(one_one_int,D4))
               => ! [Xb3: int] :
                    ( member_int(Xb3,A3)
                   => ( X != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa2) ) ) )
           => ( ( aa_int_o(P,X)
                | aa_int_o(Q,X) )
             => ( aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),D4))
                | aa_int_o(Q,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),D4)) ) ) ) ) ) ).

% aset(2)
tff(fact_5468_power__minus__Bit0,axiom,
    ! [Xb: real,K: num] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) ).

% power_minus_Bit0
tff(fact_5469_power__minus__Bit0,axiom,
    ! [Xb: int,K: num] : aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) = aa_nat_int(power_power_int(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) ).

% power_minus_Bit0
tff(fact_5470_power__minus__Bit0,axiom,
    ! [Xb: complex,K: num] : aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) = aa_nat_complex(power_power_complex(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) ).

% power_minus_Bit0
tff(fact_5471_power__minus__Bit0,axiom,
    ! [Xb: rat,K: num] : aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) = aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) ).

% power_minus_Bit0
tff(fact_5472_power__minus__Bit0,axiom,
    ! [Xb: code_integer,K: num] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) = aa_nat_Code_integer(power_8256067586552552935nteger(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) ).

% power_minus_Bit0
tff(fact_5473_real__0__less__add__iff,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(plus_plus_real(Xb),Yb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,Xb)),Yb) ) ).

% real_0_less_add_iff
tff(fact_5474_real__add__less__0__iff,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(Xb),Yb)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Yb),aa_real_real(uminus_uminus_real,Xb)) ) ).

% real_add_less_0_iff
tff(fact_5475_real__add__le__0__iff,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(Xb),Yb)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Yb),aa_real_real(uminus_uminus_real,Xb)) ) ).

% real_add_le_0_iff
tff(fact_5476_real__0__le__add__iff,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(plus_plus_real(Xb),Yb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,Xb)),Yb) ) ).

% real_0_le_add_iff
tff(fact_5477_zmod__zminus1__eq__if,axiom,
    ! [A: int,B: int] :
      aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,A)),B) = $ite(aa_int_int(aa_int_fun_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(aa_int_fun_int_int(modulo_modulo_int,A),B))) ).

% zmod_zminus1_eq_if
tff(fact_5478_zmod__zminus2__eq__if,axiom,
    ! [A: int,B: int] :
      aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(uminus_uminus_int,B)) = $ite(aa_int_int(aa_int_fun_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(aa_int_fun_int_int(modulo_modulo_int,A),B)),B)) ).

% zmod_zminus2_eq_if
tff(fact_5479_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(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(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(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_5480_less__minus__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(uminus_uminus_rat,aa_rat_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)),aa_rat_rat(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,aa_rat_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_5481_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(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(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(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_5482_minus__divide__less__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_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,aa_rat_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)),aa_rat_rat(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_5483_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(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(times_times_real(A),C)) ) ) ).

% neg_less_minus_divide_eq
tff(fact_5484_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),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C)))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(times_times_rat(A),C)) ) ) ).

% neg_less_minus_divide_eq
tff(fact_5485_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(divide_divide_real(B),C))),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(A),C)),aa_real_real(uminus_uminus_real,B)) ) ) ).

% neg_minus_divide_less_eq
tff(fact_5486_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,aa_rat_rat(uminus_uminus_rat,aa_rat_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)),aa_rat_rat(uminus_uminus_rat,B)) ) ) ).

% neg_minus_divide_less_eq
tff(fact_5487_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(divide_divide_real(B),C)))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(times_times_real(A),C)),aa_real_real(uminus_uminus_real,B)) ) ) ).

% pos_less_minus_divide_eq
tff(fact_5488_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),aa_rat_rat(uminus_uminus_rat,aa_rat_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)),aa_rat_rat(uminus_uminus_rat,B)) ) ) ).

% pos_less_minus_divide_eq
tff(fact_5489_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(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(times_times_real(A),C)) ) ) ).

% pos_minus_divide_less_eq
tff(fact_5490_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,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C))),A)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(times_times_rat(A),C)) ) ) ).

% pos_minus_divide_less_eq
tff(fact_5491_divide__eq__eq__numeral_I2_J,axiom,
    ! [B: real,C: real,W: num] :
      ( ( aa_real_real(divide_divide_real(B),C) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)) )
    <=> $ite(C != zero_zero_real,B = aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),C),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)) = zero_zero_real) ) ).

% divide_eq_eq_numeral(2)
tff(fact_5492_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_5493_divide__eq__eq__numeral_I2_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( ( aa_rat_rat(divide_divide_rat(B),C) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)) )
    <=> $ite(C != zero_zero_rat,B = aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),C),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)) = zero_zero_rat) ) ).

% divide_eq_eq_numeral(2)
tff(fact_5494_eq__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: real,C: real] :
      ( ( aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)) = aa_real_real(divide_divide_real(B),C) )
    <=> $ite(C != zero_zero_real,aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),C) = B,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)) = zero_zero_real) ) ).

% eq_divide_eq_numeral(2)
tff(fact_5495_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_5496_eq__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)) = aa_rat_rat(divide_divide_rat(B),C) )
    <=> $ite(C != zero_zero_rat,aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),C) = B,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)) = zero_zero_rat) ) ).

% eq_divide_eq_numeral(2)
tff(fact_5497_subset__decode__imp__le,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(nat_set_decode,Mb)),aa_nat_set_nat(nat_set_decode,Nb))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% subset_decode_imp_le
tff(fact_5498_add__divide__eq__if__simps_I3_J,axiom,
    ! [A: real,Z: real,B: real] :
      aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(A),Z))),B) = $ite(Z = zero_zero_real,B,aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,A)),aa_real_real(times_times_real(B),Z))),Z)) ).

% add_divide_eq_if_simps(3)
tff(fact_5499_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_5500_add__divide__eq__if__simps_I3_J,axiom,
    ! [A: rat,Z: rat,B: rat] :
      aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A),Z))),B) = $ite(Z = zero_zero_rat,B,aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(times_times_rat(B),Z))),Z)) ).

% add_divide_eq_if_simps(3)
tff(fact_5501_minus__divide__add__eq__iff,axiom,
    ! [Z: real,Xb: real,Yb: real] :
      ( ( Z != zero_zero_real )
     => ( aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(Xb),Z))),Yb) = aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,Xb)),aa_real_real(times_times_real(Yb),Z))),Z) ) ) ).

% minus_divide_add_eq_iff
tff(fact_5502_minus__divide__add__eq__iff,axiom,
    ! [Z: complex,Xb: complex,Yb: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(Xb),Z))),Yb) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,Xb)),aa_complex_complex(times_times_complex(Yb),Z))),Z) ) ) ).

% minus_divide_add_eq_iff
tff(fact_5503_minus__divide__add__eq__iff,axiom,
    ! [Z: rat,Xb: rat,Yb: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(Xb),Z))),Yb) = aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,Xb)),aa_rat_rat(times_times_rat(Yb),Z))),Z) ) ) ).

% minus_divide_add_eq_iff
tff(fact_5504_minus__divide__diff__eq__iff,axiom,
    ! [Z: real,Xb: real,Yb: real] :
      ( ( Z != zero_zero_real )
     => ( aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(Xb),Z))),Yb) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,Xb)),aa_real_real(times_times_real(Yb),Z))),Z) ) ) ).

% minus_divide_diff_eq_iff
tff(fact_5505_minus__divide__diff__eq__iff,axiom,
    ! [Z: complex,Xb: complex,Yb: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(Xb),Z))),Yb) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,Xb)),aa_complex_complex(times_times_complex(Yb),Z))),Z) ) ) ).

% minus_divide_diff_eq_iff
tff(fact_5506_minus__divide__diff__eq__iff,axiom,
    ! [Z: rat,Xb: rat,Yb: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(Xb),Z))),Yb) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,Xb)),aa_rat_rat(times_times_rat(Yb),Z))),Z) ) ) ).

% minus_divide_diff_eq_iff
tff(fact_5507_add__divide__eq__if__simps_I5_J,axiom,
    ! [A: real,Z: real,B: real] :
      aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(A),Z)),B) = $ite(Z = zero_zero_real,aa_real_real(uminus_uminus_real,B),aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(A),aa_real_real(times_times_real(B),Z))),Z)) ).

% add_divide_eq_if_simps(5)
tff(fact_5508_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_5509_add__divide__eq__if__simps_I5_J,axiom,
    ! [A: rat,Z: rat,B: rat] :
      aa_rat_rat(minus_minus_rat(aa_rat_rat(divide_divide_rat(A),Z)),B) = $ite(Z = zero_zero_rat,aa_rat_rat(uminus_uminus_rat,B),aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(A),aa_rat_rat(times_times_rat(B),Z))),Z)) ).

% add_divide_eq_if_simps(5)
tff(fact_5510_add__divide__eq__if__simps_I6_J,axiom,
    ! [A: real,Z: real,B: real] :
      aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(A),Z))),B) = $ite(Z = zero_zero_real,aa_real_real(uminus_uminus_real,B),aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,A)),aa_real_real(times_times_real(B),Z))),Z)) ).

% add_divide_eq_if_simps(6)
tff(fact_5511_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_5512_add__divide__eq__if__simps_I6_J,axiom,
    ! [A: rat,Z: rat,B: rat] :
      aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A),Z))),B) = $ite(Z = zero_zero_rat,aa_rat_rat(uminus_uminus_rat,B),aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(times_times_rat(B),Z))),Z)) ).

% add_divide_eq_if_simps(6)
tff(fact_5513_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_5514_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_5515_power2__eq__iff,axiom,
    ! [Xb: real,Yb: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
    <=> ( ( Xb = Yb )
        | ( Xb = aa_real_real(uminus_uminus_real,Yb) ) ) ) ).

% power2_eq_iff
tff(fact_5516_power2__eq__iff,axiom,
    ! [Xb: int,Yb: int] :
      ( ( aa_nat_int(power_power_int(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_int(power_power_int(Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
    <=> ( ( Xb = Yb )
        | ( Xb = aa_int_int(uminus_uminus_int,Yb) ) ) ) ).

% power2_eq_iff
tff(fact_5517_power2__eq__iff,axiom,
    ! [Xb: complex,Yb: complex] :
      ( ( aa_nat_complex(power_power_complex(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_complex(power_power_complex(Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
    <=> ( ( Xb = Yb )
        | ( Xb = aa_complex_complex(uminus1482373934393186551omplex,Yb) ) ) ) ).

% power2_eq_iff
tff(fact_5518_power2__eq__iff,axiom,
    ! [Xb: rat,Yb: rat] :
      ( ( aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_rat(power_power_rat(Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
    <=> ( ( Xb = Yb )
        | ( Xb = aa_rat_rat(uminus_uminus_rat,Yb) ) ) ) ).

% power2_eq_iff
tff(fact_5519_power2__eq__iff,axiom,
    ! [Xb: code_integer,Yb: code_integer] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_Code_integer(power_8256067586552552935nteger(Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
    <=> ( ( Xb = Yb )
        | ( Xb = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Yb) ) ) ) ).

% power2_eq_iff
tff(fact_5520_aset_I10_J,axiom,
    ! [D: int,D4: int,A3: set_int,Ta: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),D4)
     => ! [X: int] :
          ( ! [Xa2: int] :
              ( member_int(Xa2,set_or1266510415728281911st_int(one_one_int,D4))
             => ! [Xb3: int] :
                  ( member_int(Xb3,A3)
                 => ( X != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa2) ) ) )
         => ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Ta))
           => ~ 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,X),D4)),Ta)) ) ) ) ).

% aset(10)
tff(fact_5521_aset_I9_J,axiom,
    ! [D: int,D4: int,A3: set_int,Ta: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),D4)
     => ! [X: int] :
          ( ! [Xa2: int] :
              ( member_int(Xa2,set_or1266510415728281911st_int(one_one_int,D4))
             => ! [Xb3: int] :
                  ( member_int(Xb3,A3)
                 => ( X != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa2) ) ) )
         => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Ta))
           => 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,X),D4)),Ta)) ) ) ) ).

% aset(9)
tff(fact_5522_bset_I10_J,axiom,
    ! [D: int,D4: int,B4: set_int,Ta: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),D4)
     => ! [X: int] :
          ( ! [Xa2: int] :
              ( member_int(Xa2,set_or1266510415728281911st_int(one_one_int,D4))
             => ! [Xb3: int] :
                  ( member_int(Xb3,B4)
                 => ( X != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa2) ) ) )
         => ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Ta))
           => ~ 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,X),D4)),Ta)) ) ) ) ).

% bset(10)
tff(fact_5523_bset_I9_J,axiom,
    ! [D: int,D4: int,B4: set_int,Ta: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),D4)
     => ! [X: int] :
          ( ! [Xa2: int] :
              ( member_int(Xa2,set_or1266510415728281911st_int(one_one_int,D4))
             => ! [Xb3: int] :
                  ( member_int(Xb3,B4)
                 => ( X != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa2) ) ) )
         => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Ta))
           => 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,X),D4)),Ta)) ) ) ) ).

% bset(9)
tff(fact_5524_verit__less__mono__div__int2,axiom,
    ! [A3: int,B4: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A3),B4)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(uminus_uminus_int,Nb))
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,B4),Nb)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A3),Nb)) ) ) ).

% verit_less_mono_div_int2
tff(fact_5525_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_5526_of__bool__odd__eq__mod__2,axiom,
    ! [A: 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_5527_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(aa_int_fun_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_5528_of__bool__odd__eq__mod__2,axiom,
    ! [A: 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_5529_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(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(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(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_5530_le__minus__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(uminus_uminus_rat,aa_rat_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)),aa_rat_rat(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,aa_rat_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_5531_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(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(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(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_5532_minus__divide__le__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_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,aa_rat_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)),aa_rat_rat(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_5533_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(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(times_times_real(A),C)) ) ) ).

% neg_le_minus_divide_eq
tff(fact_5534_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),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C)))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(times_times_rat(A),C)) ) ) ).

% neg_le_minus_divide_eq
tff(fact_5535_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(divide_divide_real(B),C))),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A),C)),aa_real_real(uminus_uminus_real,B)) ) ) ).

% neg_minus_divide_le_eq
tff(fact_5536_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,aa_rat_rat(uminus_uminus_rat,aa_rat_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)),aa_rat_rat(uminus_uminus_rat,B)) ) ) ).

% neg_minus_divide_le_eq
tff(fact_5537_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(divide_divide_real(B),C)))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(A),C)),aa_real_real(uminus_uminus_real,B)) ) ) ).

% pos_le_minus_divide_eq
tff(fact_5538_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),aa_rat_rat(uminus_uminus_rat,aa_rat_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)),aa_rat_rat(uminus_uminus_rat,B)) ) ) ).

% pos_le_minus_divide_eq
tff(fact_5539_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(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(times_times_real(A),C)) ) ) ).

% pos_minus_divide_le_eq
tff(fact_5540_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,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B),C))),A)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B)),aa_rat_rat(times_times_rat(A),C)) ) ) ).

% pos_minus_divide_le_eq
tff(fact_5541_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(divide_divide_real(B),C)),aa_real_real(uminus_uminus_real,aa_num_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(times_times_real(aa_real_real(uminus_uminus_real,aa_num_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(times_times_real(aa_real_real(uminus_uminus_real,aa_num_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,aa_num_real(numeral_numeral_real,W)))) ) ) ).

% divide_less_eq_numeral(2)
tff(fact_5542_divide__less__eq__numeral_I2_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(B),C)),aa_rat_rat(uminus_uminus_rat,aa_num_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(aa_rat_rat(uminus_uminus_rat,aa_num_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(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),C)),B),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)))) ) ) ).

% divide_less_eq_numeral(2)
tff(fact_5543_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,aa_num_real(numeral_numeral_real,W))),aa_real_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(times_times_real(aa_real_real(uminus_uminus_real,aa_num_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(times_times_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),C)),aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),zero_zero_real)) ) ) ).

% less_divide_eq_numeral(2)
tff(fact_5544_less__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),aa_rat_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(aa_rat_rat(uminus_uminus_rat,aa_num_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(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),zero_zero_rat)) ) ) ).

% less_divide_eq_numeral(2)
tff(fact_5545_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)
     => ( ! [X3: int,K2: 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,K2),D))) )
       => ( ? [X_13: int] : aa_int_o(P,X_13)
        <=> ? [X4: int] :
              ( member_int(X4,set_or1266510415728281911st_int(one_one_int,D))
              & aa_int_o(P,X4) ) ) ) ) ).

% periodic_finite_ex
tff(fact_5546_bset_I3_J,axiom,
    ! [D4: int,Ta: int,B4: 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,Ta),one_one_int),B4)
       => ! [X: int] :
            ( ! [Xa2: int] :
                ( member_int(Xa2,set_or1266510415728281911st_int(one_one_int,D4))
               => ! [Xb3: int] :
                    ( member_int(Xb3,B4)
                   => ( X != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa2) ) ) )
           => ( ( X = Ta )
             => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,X),D4) = Ta ) ) ) ) ) ).

% bset(3)
tff(fact_5547_bset_I4_J,axiom,
    ! [D4: int,Ta: int,B4: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D4)
     => ( member_int(Ta,B4)
       => ! [X: int] :
            ( ! [Xa2: int] :
                ( member_int(Xa2,set_or1266510415728281911st_int(one_one_int,D4))
               => ! [Xb3: int] :
                    ( member_int(Xb3,B4)
                   => ( X != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa2) ) ) )
           => ( ( X != Ta )
             => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,X),D4) != Ta ) ) ) ) ) ).

% bset(4)
tff(fact_5548_bset_I5_J,axiom,
    ! [D4: int,B4: set_int,Ta: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D4)
     => ! [X: int] :
          ( ! [Xa2: int] :
              ( member_int(Xa2,set_or1266510415728281911st_int(one_one_int,D4))
             => ! [Xb3: int] :
                  ( member_int(Xb3,B4)
                 => ( X != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa2) ) ) )
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Ta)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),D4)),Ta) ) ) ) ).

% bset(5)
tff(fact_5549_bset_I7_J,axiom,
    ! [D4: int,Ta: int,B4: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D4)
     => ( member_int(Ta,B4)
       => ! [X: int] :
            ( ! [Xa2: int] :
                ( member_int(Xa2,set_or1266510415728281911st_int(one_one_int,D4))
               => ! [Xb3: int] :
                    ( member_int(Xb3,B4)
                   => ( X != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa2) ) ) )
           => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Ta),X)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,Ta),aa_int_int(aa_int_fun_int_int(minus_minus_int,X),D4)) ) ) ) ) ).

% bset(7)
tff(fact_5550_aset_I3_J,axiom,
    ! [D4: int,Ta: int,A3: 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,Ta),one_one_int),A3)
       => ! [X: int] :
            ( ! [Xa2: int] :
                ( member_int(Xa2,set_or1266510415728281911st_int(one_one_int,D4))
               => ! [Xb3: int] :
                    ( member_int(Xb3,A3)
                   => ( X != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa2) ) ) )
           => ( ( X = Ta )
             => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,X),D4) = Ta ) ) ) ) ) ).

% aset(3)
tff(fact_5551_aset_I4_J,axiom,
    ! [D4: int,Ta: int,A3: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D4)
     => ( member_int(Ta,A3)
       => ! [X: int] :
            ( ! [Xa2: int] :
                ( member_int(Xa2,set_or1266510415728281911st_int(one_one_int,D4))
               => ! [Xb3: int] :
                    ( member_int(Xb3,A3)
                   => ( X != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa2) ) ) )
           => ( ( X != Ta )
             => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,X),D4) != Ta ) ) ) ) ) ).

% aset(4)
tff(fact_5552_aset_I5_J,axiom,
    ! [D4: int,Ta: int,A3: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D4)
     => ( member_int(Ta,A3)
       => ! [X: int] :
            ( ! [Xa2: int] :
                ( member_int(Xa2,set_or1266510415728281911st_int(one_one_int,D4))
               => ! [Xb3: int] :
                    ( member_int(Xb3,A3)
                   => ( X != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa2) ) ) )
           => ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Ta)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),D4)),Ta) ) ) ) ) ).

% aset(5)
tff(fact_5553_aset_I7_J,axiom,
    ! [D4: int,A3: set_int,Ta: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D4)
     => ! [X: int] :
          ( ! [Xa2: int] :
              ( member_int(Xa2,set_or1266510415728281911st_int(one_one_int,D4))
             => ! [Xb3: int] :
                  ( member_int(Xb3,A3)
                 => ( X != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa2) ) ) )
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Ta),X)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,Ta),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),D4)) ) ) ) ).

% aset(7)
tff(fact_5554_power2__eq__1__iff,axiom,
    ! [A: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_real )
    <=> ( ( A = one_one_real )
        | ( A = aa_real_real(uminus_uminus_real,one_one_real) ) ) ) ).

% power2_eq_1_iff
tff(fact_5555_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_5556_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_5557_power2__eq__1__iff,axiom,
    ! [A: rat] :
      ( ( aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_rat )
    <=> ( ( A = one_one_rat )
        | ( A = aa_rat_rat(uminus_uminus_rat,one_one_rat) ) ) ) ).

% power2_eq_1_iff
tff(fact_5558_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_5559_uminus__power__if,axiom,
    ! [A: real,Nb: nat] :
      aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,A)),Nb) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb),aa_real_real(uminus_uminus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb))) ).

% uminus_power_if
tff(fact_5560_uminus__power__if,axiom,
    ! [A: int,Nb: nat] :
      aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A)),Nb) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb),aa_nat_int(power_power_int(A),Nb),aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(A),Nb))) ).

% uminus_power_if
tff(fact_5561_uminus__power__if,axiom,
    ! [A: complex,Nb: nat] :
      aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),Nb) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb),aa_nat_complex(power_power_complex(A),Nb),aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(power_power_complex(A),Nb))) ).

% uminus_power_if
tff(fact_5562_uminus__power__if,axiom,
    ! [A: rat,Nb: nat] :
      aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,A)),Nb) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb),aa_nat_rat(power_power_rat(A),Nb),aa_rat_rat(uminus_uminus_rat,aa_nat_rat(power_power_rat(A),Nb))) ).

% uminus_power_if
tff(fact_5563_uminus__power__if,axiom,
    ! [A: code_integer,Nb: nat] :
      aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),Nb) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb),aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb))) ).

% uminus_power_if
tff(fact_5564_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),K)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_5565_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => ( 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,Nb),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,Nb),K)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_5566_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => ( aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),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,Nb),K)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_5567_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => ( aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),K)) = aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_5568_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),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,Nb),K)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_5569_realpow__square__minus__le,axiom,
    ! [U: real,Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,U),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% realpow_square_minus_le
tff(fact_5570_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(aa_int_fun_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(aa_int_fun_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_5571_zmod__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(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_5572_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(aa_int_fun_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_5573_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(aa_int_fun_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_5574_zminus1__lemma,axiom,
    ! [A: int,B: int,Q2: int,R2: int] :
      ( aa_Pro4592068669816069595_int_o(eucl_rel_int(A,B),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q2),R2))
     => ( ( B != zero_zero_int )
       => aa_Pro4592068669816069595_int_o(eucl_rel_int(aa_int_int(uminus_uminus_int,A),B),
            aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,
                $ite(R2 = zero_zero_int,aa_int_int(uminus_uminus_int,Q2),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,Q2)),one_one_int))),
              $ite(R2 = zero_zero_int,zero_zero_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),R2)))) ) ) ).

% zminus1_lemma
tff(fact_5575_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,B3: $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,zero_n2687167440665602831ol_nat((B3))),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,zero_n2687167440665602831ol_nat((B3))),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_5576_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,B3: $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,(B3))),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,(B3))),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_5577_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,B3: $o] :
            ( aa_Code_integer_o(P,A4)
           => ( ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(zero_n356916108424825756nteger((B3)),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(zero_n356916108424825756nteger((B3)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),A4))) ) )
       => aa_Code_integer_o(P,A) ) ) ).

% bits_induct
tff(fact_5578_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(divide_divide_real(B),C)),aa_real_real(uminus_uminus_real,aa_num_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(times_times_real(aa_real_real(uminus_uminus_real,aa_num_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(times_times_real(aa_real_real(uminus_uminus_real,aa_num_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,aa_num_real(numeral_numeral_real,W)))) ) ) ).

% divide_le_eq_numeral(2)
tff(fact_5579_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,aa_rat_rat(divide_divide_rat(B),C)),aa_rat_rat(uminus_uminus_rat,aa_num_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(aa_rat_rat(uminus_uminus_rat,aa_num_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(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),C)),B),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)))) ) ) ).

% divide_le_eq_numeral(2)
tff(fact_5580_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,aa_num_real(numeral_numeral_real,W))),aa_real_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(times_times_real(aa_real_real(uminus_uminus_real,aa_num_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(times_times_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),C)),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),zero_zero_real)) ) ) ).

% le_divide_eq_numeral(2)
tff(fact_5581_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,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),aa_rat_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(aa_rat_rat(uminus_uminus_rat,aa_num_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(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),zero_zero_rat)) ) ) ).

% le_divide_eq_numeral(2)
tff(fact_5582_bset_I6_J,axiom,
    ! [D4: int,B4: set_int,Ta: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D4)
     => ! [X: int] :
          ( ! [Xa2: int] :
              ( member_int(Xa2,set_or1266510415728281911st_int(one_one_int,D4))
             => ! [Xb3: int] :
                  ( member_int(Xb3,B4)
                 => ( X != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa2) ) ) )
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Ta)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),D4)),Ta) ) ) ) ).

% bset(6)
tff(fact_5583_bset_I8_J,axiom,
    ! [D4: int,Ta: int,B4: 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,Ta),one_one_int),B4)
       => ! [X: int] :
            ( ! [Xa2: int] :
                ( member_int(Xa2,set_or1266510415728281911st_int(one_one_int,D4))
               => ! [Xb3: int] :
                    ( member_int(Xb3,B4)
                   => ( X != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa2) ) ) )
           => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ta),X)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ta),aa_int_int(aa_int_fun_int_int(minus_minus_int,X),D4)) ) ) ) ) ).

% bset(8)
tff(fact_5584_aset_I6_J,axiom,
    ! [D4: int,Ta: int,A3: 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,Ta),one_one_int),A3)
       => ! [X: int] :
            ( ! [Xa2: int] :
                ( member_int(Xa2,set_or1266510415728281911st_int(one_one_int,D4))
               => ! [Xb3: int] :
                    ( member_int(Xb3,A3)
                   => ( X != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa2) ) ) )
           => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Ta)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),D4)),Ta) ) ) ) ) ).

% aset(6)
tff(fact_5585_aset_I8_J,axiom,
    ! [D4: int,A3: set_int,Ta: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D4)
     => ! [X: int] :
          ( ! [Xa2: int] :
              ( member_int(Xa2,set_or1266510415728281911st_int(one_one_int,D4))
             => ! [Xb3: int] :
                  ( member_int(Xb3,A3)
                 => ( X != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa2) ) ) )
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ta),X)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ta),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),D4)) ) ) ) ).

% aset(8)
tff(fact_5586_cpmi,axiom,
    ! [D4: int,P: fun_int_o,P3: fun_int_o,B4: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D4)
     => ( ? [Z3: int] :
          ! [X3: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Z3)
           => ( aa_int_o(P,X3)
            <=> aa_int_o(P3,X3) ) )
       => ( ! [X3: int] :
              ( ! [Xa: int] :
                  ( member_int(Xa,set_or1266510415728281911st_int(one_one_int,D4))
                 => ! [Xb2: int] :
                      ( member_int(Xb2,B4)
                     => ( X3 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa) ) ) )
             => ( aa_int_o(P,X3)
               => aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D4)) ) )
         => ( ! [X3: int,K2: int] :
                ( aa_int_o(P3,X3)
              <=> aa_int_o(P3,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D4))) )
           => ( ? [X_13: int] : aa_int_o(P,X_13)
            <=> ( ? [X4: int] :
                    ( member_int(X4,set_or1266510415728281911st_int(one_one_int,D4))
                    & aa_int_o(P3,X4) )
                | ? [X4: int] :
                    ( member_int(X4,set_or1266510415728281911st_int(one_one_int,D4))
                    & ? [Xa3: int] :
                        ( member_int(Xa3,B4)
                        & aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xa3),X4)) ) ) ) ) ) ) ) ) ).

% cpmi
tff(fact_5587_cppi,axiom,
    ! [D4: int,P: fun_int_o,P3: fun_int_o,A3: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D4)
     => ( ? [Z3: int] :
          ! [X3: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),X3)
           => ( aa_int_o(P,X3)
            <=> aa_int_o(P3,X3) ) )
       => ( ! [X3: int] :
              ( ! [Xa: int] :
                  ( member_int(Xa,set_or1266510415728281911st_int(one_one_int,D4))
                 => ! [Xb2: int] :
                      ( member_int(Xb2,A3)
                     => ( X3 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa) ) ) )
             => ( aa_int_o(P,X3)
               => aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D4)) ) )
         => ( ! [X3: int,K2: int] :
                ( aa_int_o(P3,X3)
              <=> aa_int_o(P3,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D4))) )
           => ( ? [X_13: int] : aa_int_o(P,X_13)
            <=> ( ? [X4: int] :
                    ( member_int(X4,set_or1266510415728281911st_int(one_one_int,D4))
                    & aa_int_o(P3,X4) )
                | ? [X4: int] :
                    ( member_int(X4,set_or1266510415728281911st_int(one_one_int,D4))
                    & ? [Xa3: int] :
                        ( member_int(Xa3,A3)
                        & aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xa3),X4)) ) ) ) ) ) ) ) ) ).

% cppi
tff(fact_5588_square__le__1,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real) ) ) ).

% square_le_1
tff(fact_5589_square__le__1,axiom,
    ! [Xb: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Xb)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Xb),one_one_Code_integer)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_Code_integer) ) ) ).

% square_le_1
tff(fact_5590_square__le__1,axiom,
    ! [Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),Xb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),one_one_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_rat) ) ) ).

% square_le_1
tff(fact_5591_square__le__1,axiom,
    ! [Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),Xb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),one_one_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_int) ) ) ).

% square_le_1
tff(fact_5592_minus__power__mult__self,axiom,
    ! [A: real,Nb: nat] : aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,A)),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,A)),Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% minus_power_mult_self
tff(fact_5593_minus__power__mult__self,axiom,
    ! [A: int,Nb: 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)),Nb)),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A)),Nb)) = 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))),Nb)) ).

% minus_power_mult_self
tff(fact_5594_minus__power__mult__self,axiom,
    ! [A: complex,Nb: nat] : aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),Nb)),aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),Nb)) = 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))),Nb)) ).

% minus_power_mult_self
tff(fact_5595_minus__power__mult__self,axiom,
    ! [A: rat,Nb: nat] : aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,A)),Nb)),aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,A)),Nb)) = 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))),Nb)) ).

% minus_power_mult_self
tff(fact_5596_minus__power__mult__self,axiom,
    ! [A: code_integer,Nb: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),Nb)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),Nb)) = 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))),Nb)) ).

% minus_power_mult_self
tff(fact_5597_minus__one__power__iff,axiom,
    ! [Nb: nat] :
      aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Nb) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb),one_one_real,aa_real_real(uminus_uminus_real,one_one_real)) ).

% minus_one_power_iff
tff(fact_5598_minus__one__power__iff,axiom,
    ! [Nb: nat] :
      aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),Nb) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb),one_one_int,aa_int_int(uminus_uminus_int,one_one_int)) ).

% minus_one_power_iff
tff(fact_5599_minus__one__power__iff,axiom,
    ! [Nb: nat] :
      aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Nb) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb),one_one_complex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) ).

% minus_one_power_iff
tff(fact_5600_minus__one__power__iff,axiom,
    ! [Nb: nat] :
      aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Nb) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb),one_one_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)) ).

% minus_one_power_iff
tff(fact_5601_minus__one__power__iff,axiom,
    ! [Nb: nat] :
      aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Nb) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb),one_one_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).

% minus_one_power_iff
tff(fact_5602_minus__1__div__exp__eq__int,axiom,
    ! [Nb: 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))),Nb)) = aa_int_int(uminus_uminus_int,one_one_int) ).

% minus_1_div_exp_eq_int
tff(fact_5603_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_5604_signed__take__bit__int__greater__eq__minus__exp,axiom,
    ! [Nb: 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))),Nb))),aa_int_int(bit_ri631733984087533419it_int(Nb),K)) ).

% signed_take_bit_int_greater_eq_minus_exp
tff(fact_5605_signed__take__bit__int__less__eq__self__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(bit_ri631733984087533419it_int(Nb),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))),Nb))),K) ) ).

% signed_take_bit_int_less_eq_self_iff
tff(fact_5606_signed__take__bit__int__greater__self__iff,axiom,
    ! [K: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_int_int(bit_ri631733984087533419it_int(Nb),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))),Nb))) ) ).

% signed_take_bit_int_greater_self_iff
tff(fact_5607_exp__mod__exp,axiom,
    ! [Mb: nat,Nb: 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))),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,zero_n2687167440665602831ol_nat(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)) ).

% exp_mod_exp
tff(fact_5608_exp__mod__exp,axiom,
    ! [Mb: nat,Nb: nat] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Mb)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) = 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,Mb),Nb))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Mb)) ).

% exp_mod_exp
tff(fact_5609_exp__mod__exp,axiom,
    ! [Mb: nat,Nb: nat] : modulo364778990260209775nteger(aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Mb),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Nb)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,zero_n356916108424825756nteger(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb))),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Mb)) ).

% exp_mod_exp
tff(fact_5610_power__minus1__odd,axiom,
    ! [Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) = aa_real_real(uminus_uminus_real,one_one_real) ).

% power_minus1_odd
tff(fact_5611_power__minus1__odd,axiom,
    ! [Nb: 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))),Nb))) = aa_int_int(uminus_uminus_int,one_one_int) ).

% power_minus1_odd
tff(fact_5612_power__minus1__odd,axiom,
    ! [Nb: 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))),Nb))) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% power_minus1_odd
tff(fact_5613_power__minus1__odd,axiom,
    ! [Nb: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) = aa_rat_rat(uminus_uminus_rat,one_one_rat) ).

% power_minus1_odd
tff(fact_5614_power__minus1__odd,axiom,
    ! [Nb: 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))),Nb))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% power_minus1_odd
tff(fact_5615_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))
       => ( ! [K2: int] :
              ( aa_int_o(P,K2)
             => ( ( K2 != zero_zero_int )
               => aa_int_o(P,aa_int_int(aa_int_fun_int_int(times_times_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ) )
         => ( ! [K2: int] :
                ( aa_int_o(P,K2)
               => ( ( K2 != 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,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))) ) )
           => aa_int_o(P,K) ) ) ) ) ).

% int_bit_induct
tff(fact_5616_signed__take__bit__int__eq__self__iff,axiom,
    ! [Nb: nat,K: int] :
      ( ( aa_int_int(bit_ri631733984087533419it_int(Nb),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))),Nb))),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))),Nb)) ) ) ).

% signed_take_bit_int_eq_self_iff
tff(fact_5617_signed__take__bit__int__eq__self,axiom,
    ! [Nb: 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))),Nb))),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))),Nb))
       => ( aa_int_int(bit_ri631733984087533419it_int(Nb),K) = K ) ) ) ).

% signed_take_bit_int_eq_self
tff(fact_5618_exp__div__exp__eq,axiom,
    ! [Mb: nat,Nb: 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))),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_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))),Mb) != zero_zero_nat )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb) ))),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,Mb),Nb))) ).

% exp_div_exp_eq
tff(fact_5619_exp__div__exp__eq,axiom,
    ! [Mb: nat,Nb: 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))),Mb)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) = 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))),Mb) != zero_zero_int )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb) ))),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,Mb),Nb))) ).

% exp_div_exp_eq
tff(fact_5620_exp__div__exp__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      divide6298287555418463151nteger(aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Mb),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Nb)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,
          zero_n356916108424825756nteger(( ( aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Mb) != zero_z3403309356797280102nteger )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb) ))),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb))) ).

% exp_div_exp_eq
tff(fact_5621_signed__take__bit__int__greater__eq,axiom,
    ! [K: int,Nb: 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))),Nb)))
     => 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,Nb)))),aa_int_int(bit_ri631733984087533419it_int(Nb),K)) ) ).

% signed_take_bit_int_greater_eq
tff(fact_5622_set__decode__def,axiom,
    ! [Xb: nat] : aa_nat_set_nat(nat_set_decode,Xb) = collect_nat(aa_nat_fun_nat_o(aTP_Lamm_bd,Xb)) ).

% set_decode_def
tff(fact_5623_Divides_Oadjust__div__eq,axiom,
    ! [Q2: int,R2: int] : aa_Pro6799258434465214121nt_int(adjust_div,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q2),R2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Q2),aa_o_int(zero_n2684676970156552555ol_int,R2 != zero_zero_int)) ).

% Divides.adjust_div_eq
tff(fact_5624_signed__take__bit__Suc__minus__bit1,axiom,
    ! [Nb: nat,K: num] : aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(suc,Nb)),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(Nb),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_5625_divmod__step__def,axiom,
    ! [L: num,Qr: product_prod_nat_nat] : unique5026877609467782581ep_nat(L,Qr) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_be(L)),Qr) ).

% divmod_step_def
tff(fact_5626_divmod__step__def,axiom,
    ! [L: num,Qr: product_prod_int_int] : aa_Pro6970504127699319548nt_int(unique5024387138958732305ep_int(L),Qr) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_bf(L)),Qr) ).

% divmod_step_def
tff(fact_5627_divmod__step__def,axiom,
    ! [L: num,Qr: produc8923325533196201883nteger] : unique4921790084139445826nteger(L,Qr) = produc6916734918728496179nteger(aTP_Lamm_bg(L),Qr) ).

% divmod_step_def
tff(fact_5628_signed__take__bit__Suc__bit1,axiom,
    ! [Nb: nat,K: num] : aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(suc,Nb)),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(Nb),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_5629_take__bit__rec,axiom,
    ! [Nb: nat,A: int] :
      aa_int_int(bit_se2923211474154528505it_int(Nb),A) = $ite(Nb = 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,Nb),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(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))) ).

% take_bit_rec
tff(fact_5630_take__bit__rec,axiom,
    ! [Nb: nat,A: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),A) = $ite(Nb = 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,Nb),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_5631_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(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),
            $ite(aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int,l,aa_real_real(plus_plus_real(l),one_one_real)) ) ) ) ).

% of_int_code_if
tff(fact_5632_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(aa_int_fun_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_5633_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(aa_int_fun_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_5634_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),
          aa_rat_rat(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(aa_num_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(aa_int_fun_int_int(modulo_modulo_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int,l,aa_rat_rat(plus_plus_rat(l),one_one_rat)) ) ) ) ).

% of_int_code_if
tff(fact_5635_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(aa_int_fun_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_5636_sqrt__sum__squares__half__less,axiom,
    ! [Xb: real,U: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(divide_divide_real(U),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Yb),aa_real_real(divide_divide_real(U),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Yb)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),U) ) ) ) ) ).

% sqrt_sum_squares_half_less
tff(fact_5637_semiring__norm_I90_J,axiom,
    ! [Mb: num,Nb: num] :
      ( ( aa_num_num(bit1,Mb) = aa_num_num(bit1,Nb) )
    <=> ( Mb = Nb ) ) ).

% semiring_norm(90)
tff(fact_5638_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_5639_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_5640_semiring__norm_I88_J,axiom,
    ! [Mb: num,Nb: num] : aa_num_num(bit0,Mb) != aa_num_num(bit1,Nb) ).

% semiring_norm(88)
tff(fact_5641_semiring__norm_I89_J,axiom,
    ! [Mb: num,Nb: num] : aa_num_num(bit1,Mb) != aa_num_num(bit0,Nb) ).

% semiring_norm(89)
tff(fact_5642_semiring__norm_I84_J,axiom,
    ! [Nb: num] : one != aa_num_num(bit1,Nb) ).

% semiring_norm(84)
tff(fact_5643_semiring__norm_I86_J,axiom,
    ! [Mb: num] : aa_num_num(bit1,Mb) != one ).

% semiring_norm(86)
tff(fact_5644_take__bit__of__0,axiom,
    ! [Nb: nat] : aa_int_int(bit_se2923211474154528505it_int(Nb),zero_zero_int) = zero_zero_int ).

% take_bit_of_0
tff(fact_5645_take__bit__of__0,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),zero_zero_nat) = zero_zero_nat ).

% take_bit_of_0
tff(fact_5646_real__sqrt__one,axiom,
    aa_real_real(sqrt,one_one_real) = one_one_real ).

% real_sqrt_one
tff(fact_5647_real__sqrt__eq__1__iff,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(sqrt,Xb) = one_one_real )
    <=> ( Xb = one_one_real ) ) ).

% real_sqrt_eq_1_iff
tff(fact_5648_concat__bit__of__zero__2,axiom,
    ! [Nb: nat,K: int] : aa_int_int(aa_int_fun_int_int(bit_concat_bit(Nb),K),zero_zero_int) = aa_int_int(bit_se2923211474154528505it_int(Nb),K) ).

% concat_bit_of_zero_2
tff(fact_5649_semiring__norm_I73_J,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(bit1,Mb)),aa_num_num(bit1,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Mb),Nb) ) ).

% semiring_norm(73)
tff(fact_5650_semiring__norm_I80_J,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(bit1,Mb)),aa_num_num(bit1,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Nb) ) ).

% semiring_norm(80)
tff(fact_5651_case__prod__conv,axiom,
    ! [F: fun_na1668100129766719005at_nat,A: nat,B: nat] : aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(F),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B)) = aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(F,A),B) ).

% case_prod_conv
tff(fact_5652_case__prod__conv,axiom,
    ! [F: fun_na5914473681949881652_nat_o,A: nat,B: nat] : aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(F),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B)) = aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(F,A),B) ).

% case_prod_conv
tff(fact_5653_case__prod__conv,axiom,
    ! [F: fun_in1696471567231710611nt_int,A: int,B: int] : aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(F),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(F,A),B) ).

% case_prod_conv
tff(fact_5654_case__prod__conv,axiom,
    ! [F: fun_int_fun_int_o,A: int,B: int] :
      ( aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(F),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B))
    <=> aa_int_o(aa_int_fun_int_o(F,A),B) ) ).

% case_prod_conv
tff(fact_5655_case__prod__conv,axiom,
    ! [F: fun_int_fun_int_int,A: int,B: int] : aa_Pro6799258434465214121nt_int(produc8211389475949308722nt_int(F),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B)) = aa_int_int(aa_int_fun_int_int(F,A),B) ).

% case_prod_conv
tff(fact_5656_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_5657_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_5658_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_5659_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_5660_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_5661_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_5662_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_5663_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_5664_of__int__0,axiom,
    aa_int_complex(ring_17405671764205052669omplex,zero_zero_int) = zero_zero_complex ).

% of_int_0
tff(fact_5665_of__int__0,axiom,
    aa_int_int(ring_1_of_int_int,zero_zero_int) = zero_zero_int ).

% of_int_0
tff(fact_5666_of__int__0,axiom,
    aa_int_real(ring_1_of_int_real,zero_zero_int) = zero_zero_real ).

% of_int_0
tff(fact_5667_of__int__0,axiom,
    aa_int_rat(ring_1_of_int_rat,zero_zero_int) = zero_zero_rat ).

% of_int_0
tff(fact_5668_take__bit__0,axiom,
    ! [A: int] : aa_int_int(bit_se2923211474154528505it_int(zero_zero_nat),A) = zero_zero_int ).

% take_bit_0
tff(fact_5669_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_5670_take__bit__Suc__1,axiom,
    ! [Nb: nat] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Nb)),one_one_int) = one_one_int ).

% take_bit_Suc_1
tff(fact_5671_take__bit__Suc__1,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_nat_nat(suc,Nb)),one_one_nat) = one_one_nat ).

% take_bit_Suc_1
tff(fact_5672_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_5673_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_5674_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_5675_of__int__eq__numeral__iff,axiom,
    ! [Z: int,Nb: num] :
      ( ( aa_int_complex(ring_17405671764205052669omplex,Z) = numera6690914467698888265omplex(Nb) )
    <=> ( Z = aa_num_int(numeral_numeral_int,Nb) ) ) ).

% of_int_eq_numeral_iff
tff(fact_5676_of__int__eq__numeral__iff,axiom,
    ! [Z: int,Nb: num] :
      ( ( aa_int_real(ring_1_of_int_real,Z) = aa_num_real(numeral_numeral_real,Nb) )
    <=> ( Z = aa_num_int(numeral_numeral_int,Nb) ) ) ).

% of_int_eq_numeral_iff
tff(fact_5677_of__int__eq__numeral__iff,axiom,
    ! [Z: int,Nb: num] :
      ( ( aa_int_rat(ring_1_of_int_rat,Z) = aa_num_rat(numeral_numeral_rat,Nb) )
    <=> ( Z = aa_num_int(numeral_numeral_int,Nb) ) ) ).

% of_int_eq_numeral_iff
tff(fact_5678_of__int__eq__numeral__iff,axiom,
    ! [Z: int,Nb: num] :
      ( ( aa_int_int(ring_1_of_int_int,Z) = aa_num_int(numeral_numeral_int,Nb) )
    <=> ( Z = aa_num_int(numeral_numeral_int,Nb) ) ) ).

% of_int_eq_numeral_iff
tff(fact_5679_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_5680_of__int__numeral,axiom,
    ! [K: num] : aa_int_real(ring_1_of_int_real,aa_num_int(numeral_numeral_int,K)) = aa_num_real(numeral_numeral_real,K) ).

% of_int_numeral
tff(fact_5681_of__int__numeral,axiom,
    ! [K: num] : aa_int_rat(ring_1_of_int_rat,aa_num_int(numeral_numeral_int,K)) = aa_num_rat(numeral_numeral_rat,K) ).

% of_int_numeral
tff(fact_5682_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_5683_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_5684_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_5685_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_5686_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_5687_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_5688_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_5689_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_5690_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_5691_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_5692_of__int__1,axiom,
    aa_int_complex(ring_17405671764205052669omplex,one_one_int) = one_one_complex ).

% of_int_1
tff(fact_5693_of__int__1,axiom,
    aa_int_int(ring_1_of_int_int,one_one_int) = one_one_int ).

% of_int_1
tff(fact_5694_of__int__1,axiom,
    aa_int_real(ring_1_of_int_real,one_one_int) = one_one_real ).

% of_int_1
tff(fact_5695_of__int__1,axiom,
    aa_int_rat(ring_1_of_int_rat,one_one_int) = one_one_rat ).

% of_int_1
tff(fact_5696_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(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_5697_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_5698_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_5699_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_5700_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(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_5701_of__int__add,axiom,
    ! [W: int,Z: int] : aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,W),Z)) = aa_rat_rat(plus_plus_rat(aa_int_rat(ring_1_of_int_rat,W)),aa_int_rat(ring_1_of_int_rat,Z)) ).

% of_int_add
tff(fact_5702_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_5703_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_5704_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_5705_of__int__minus,axiom,
    ! [Z: int] : aa_int_rat(ring_1_of_int_rat,aa_int_int(uminus_uminus_int,Z)) = aa_rat_rat(uminus_uminus_rat,aa_int_rat(ring_1_of_int_rat,Z)) ).

% of_int_minus
tff(fact_5706_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_5707_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(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_5708_of__int__diff,axiom,
    ! [W: int,Z: int] : aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(minus_minus_int,W),Z)) = aa_rat_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,W)),aa_int_rat(ring_1_of_int_rat,Z)) ).

% of_int_diff
tff(fact_5709_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_5710_real__sqrt__gt__1__iff,axiom,
    ! [Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(sqrt,Yb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Yb) ) ).

% real_sqrt_gt_1_iff
tff(fact_5711_real__sqrt__lt__1__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,Xb)),one_one_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),one_one_real) ) ).

% real_sqrt_lt_1_iff
tff(fact_5712_real__sqrt__ge__1__iff,axiom,
    ! [Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(sqrt,Yb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Yb) ) ).

% real_sqrt_ge_1_iff
tff(fact_5713_real__sqrt__le__1__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,Xb)),one_one_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real) ) ).

% real_sqrt_le_1_iff
tff(fact_5714_semiring__norm_I9_J,axiom,
    ! [Mb: num,Nb: num] : aa_num_num(plus_plus_num(aa_num_num(bit1,Mb)),aa_num_num(bit0,Nb)) = aa_num_num(bit1,aa_num_num(plus_plus_num(Mb),Nb)) ).

% semiring_norm(9)
tff(fact_5715_semiring__norm_I7_J,axiom,
    ! [Mb: num,Nb: num] : aa_num_num(plus_plus_num(aa_num_num(bit0,Mb)),aa_num_num(bit1,Nb)) = aa_num_num(bit1,aa_num_num(plus_plus_num(Mb),Nb)) ).

% semiring_norm(7)
tff(fact_5716_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [Xb: int,B: int,W: nat] :
      ( ( aa_int_rat(ring_1_of_int_rat,Xb) = aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B)),W) )
    <=> ( Xb = aa_nat_int(power_power_int(B),W) ) ) ).

% of_int_power_eq_of_int_cancel_iff
tff(fact_5717_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [Xb: int,B: int,W: nat] :
      ( ( aa_int_int(ring_1_of_int_int,Xb) = aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,B)),W) )
    <=> ( Xb = aa_nat_int(power_power_int(B),W) ) ) ).

% of_int_power_eq_of_int_cancel_iff
tff(fact_5718_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [Xb: int,B: int,W: nat] :
      ( ( aa_int_real(ring_1_of_int_real,Xb) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,B)),W) )
    <=> ( Xb = aa_nat_int(power_power_int(B),W) ) ) ).

% of_int_power_eq_of_int_cancel_iff
tff(fact_5719_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [Xb: int,B: int,W: nat] :
      ( ( aa_int_complex(ring_17405671764205052669omplex,Xb) = aa_nat_complex(power_power_complex(aa_int_complex(ring_17405671764205052669omplex,B)),W) )
    <=> ( Xb = aa_nat_int(power_power_int(B),W) ) ) ).

% of_int_power_eq_of_int_cancel_iff
tff(fact_5720_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,Xb: 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,Xb) )
    <=> ( aa_nat_int(power_power_int(B),W) = Xb ) ) ).

% of_int_eq_of_int_power_cancel_iff
tff(fact_5721_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,Xb: 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,Xb) )
    <=> ( aa_nat_int(power_power_int(B),W) = Xb ) ) ).

% of_int_eq_of_int_power_cancel_iff
tff(fact_5722_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,Xb: int] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,B)),W) = aa_int_real(ring_1_of_int_real,Xb) )
    <=> ( aa_nat_int(power_power_int(B),W) = Xb ) ) ).

% of_int_eq_of_int_power_cancel_iff
tff(fact_5723_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,Xb: int] :
      ( ( aa_nat_complex(power_power_complex(aa_int_complex(ring_17405671764205052669omplex,B)),W) = aa_int_complex(ring_17405671764205052669omplex,Xb) )
    <=> ( aa_nat_int(power_power_int(B),W) = Xb ) ) ).

% of_int_eq_of_int_power_cancel_iff
tff(fact_5724_of__int__power,axiom,
    ! [Z: int,Nb: nat] : aa_int_rat(ring_1_of_int_rat,aa_nat_int(power_power_int(Z),Nb)) = aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,Z)),Nb) ).

% of_int_power
tff(fact_5725_of__int__power,axiom,
    ! [Z: int,Nb: nat] : aa_int_int(ring_1_of_int_int,aa_nat_int(power_power_int(Z),Nb)) = aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,Z)),Nb) ).

% of_int_power
tff(fact_5726_of__int__power,axiom,
    ! [Z: int,Nb: nat] : aa_int_real(ring_1_of_int_real,aa_nat_int(power_power_int(Z),Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,Z)),Nb) ).

% of_int_power
tff(fact_5727_of__int__power,axiom,
    ! [Z: int,Nb: nat] : aa_int_complex(ring_17405671764205052669omplex,aa_nat_int(power_power_int(Z),Nb)) = aa_nat_complex(power_power_complex(aa_int_complex(ring_17405671764205052669omplex,Z)),Nb) ).

% of_int_power
tff(fact_5728_semiring__norm_I15_J,axiom,
    ! [Mb: num,Nb: num] : aa_num_num(times_times_num(aa_num_num(bit1,Mb)),aa_num_num(bit0,Nb)) = aa_num_num(bit0,aa_num_num(times_times_num(aa_num_num(bit1,Mb)),Nb)) ).

% semiring_norm(15)
tff(fact_5729_semiring__norm_I14_J,axiom,
    ! [Mb: num,Nb: num] : aa_num_num(times_times_num(aa_num_num(bit0,Mb)),aa_num_num(bit1,Nb)) = aa_num_num(bit0,aa_num_num(times_times_num(Mb),aa_num_num(bit1,Nb))) ).

% semiring_norm(14)
tff(fact_5730_semiring__norm_I81_J,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(bit1,Mb)),aa_num_num(bit0,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Nb) ) ).

% semiring_norm(81)
tff(fact_5731_semiring__norm_I72_J,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(bit0,Mb)),aa_num_num(bit1,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Mb),Nb) ) ).

% semiring_norm(72)
tff(fact_5732_semiring__norm_I77_J,axiom,
    ! [Nb: num] : aa_num_o(aa_num_fun_num_o(ord_less_num,one),aa_num_num(bit1,Nb)) ).

% semiring_norm(77)
tff(fact_5733_semiring__norm_I70_J,axiom,
    ! [Mb: num] : ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(bit1,Mb)),one) ).

% semiring_norm(70)
tff(fact_5734_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_5735_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_5736_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_5737_of__int__of__bool,axiom,
    ! [P: $o] : aa_int_Code_integer(ring_18347121197199848620nteger,aa_o_int(zero_n2684676970156552555ol_int,(P))) = zero_n356916108424825756nteger((P)) ).

% of_int_of_bool
tff(fact_5738_take__bit__of__1__eq__0__iff,axiom,
    ! [Nb: nat] :
      ( ( aa_int_int(bit_se2923211474154528505it_int(Nb),one_one_int) = zero_zero_int )
    <=> ( Nb = zero_zero_nat ) ) ).

% take_bit_of_1_eq_0_iff
tff(fact_5739_take__bit__of__1__eq__0__iff,axiom,
    ! [Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),one_one_nat) = zero_zero_nat )
    <=> ( Nb = zero_zero_nat ) ) ).

% take_bit_of_1_eq_0_iff
tff(fact_5740_real__sqrt__four,axiom,
    aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)) ).

% real_sqrt_four
tff(fact_5741_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_5742_semiring__norm_I3_J,axiom,
    ! [Nb: num] : aa_num_num(plus_plus_num(one),aa_num_num(bit0,Nb)) = aa_num_num(bit1,Nb) ).

% semiring_norm(3)
tff(fact_5743_semiring__norm_I4_J,axiom,
    ! [Nb: num] : aa_num_num(plus_plus_num(one),aa_num_num(bit1,Nb)) = aa_num_num(bit0,aa_num_num(plus_plus_num(Nb),one)) ).

% semiring_norm(4)
tff(fact_5744_semiring__norm_I5_J,axiom,
    ! [Mb: num] : aa_num_num(plus_plus_num(aa_num_num(bit0,Mb)),one) = aa_num_num(bit1,Mb) ).

% semiring_norm(5)
tff(fact_5745_semiring__norm_I8_J,axiom,
    ! [Mb: num] : aa_num_num(plus_plus_num(aa_num_num(bit1,Mb)),one) = aa_num_num(bit0,aa_num_num(plus_plus_num(Mb),one)) ).

% semiring_norm(8)
tff(fact_5746_semiring__norm_I10_J,axiom,
    ! [Mb: num,Nb: num] : aa_num_num(plus_plus_num(aa_num_num(bit1,Mb)),aa_num_num(bit1,Nb)) = aa_num_num(bit0,aa_num_num(plus_plus_num(aa_num_num(plus_plus_num(Mb),Nb)),one)) ).

% semiring_norm(10)
tff(fact_5747_take__bit__of__Suc__0,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),aa_nat_nat(suc,zero_zero_nat)) = zero_n2687167440665602831ol_nat(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)) ).

% take_bit_of_Suc_0
tff(fact_5748_semiring__norm_I16_J,axiom,
    ! [Mb: num,Nb: num] : aa_num_num(times_times_num(aa_num_num(bit1,Mb)),aa_num_num(bit1,Nb)) = aa_num_num(bit1,aa_num_num(plus_plus_num(aa_num_num(plus_plus_num(Mb),Nb)),aa_num_num(bit0,aa_num_num(times_times_num(Mb),Nb)))) ).

% semiring_norm(16)
tff(fact_5749_semiring__norm_I79_J,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(bit0,Mb)),aa_num_num(bit1,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Mb),Nb) ) ).

% semiring_norm(79)
tff(fact_5750_semiring__norm_I74_J,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(bit1,Mb)),aa_num_num(bit0,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Nb) ) ).

% semiring_norm(74)
tff(fact_5751_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_5752_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_5753_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_5754_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_5755_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_5756_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_5757_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_5758_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_5759_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_5760_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_5761_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_5762_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_5763_of__int__numeral__le__iff,axiom,
    ! [Nb: num,Z: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,Nb)),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,Nb)),Z) ) ).

% of_int_numeral_le_iff
tff(fact_5764_of__int__numeral__le__iff,axiom,
    ! [Nb: num,Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,Nb)),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,Nb)),Z) ) ).

% of_int_numeral_le_iff
tff(fact_5765_of__int__numeral__le__iff,axiom,
    ! [Nb: num,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Nb)),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,Nb)),Z) ) ).

% of_int_numeral_le_iff
tff(fact_5766_of__int__le__numeral__iff,axiom,
    ! [Z: int,Nb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z)),aa_num_real(numeral_numeral_real,Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),aa_num_int(numeral_numeral_int,Nb)) ) ).

% of_int_le_numeral_iff
tff(fact_5767_of__int__le__numeral__iff,axiom,
    ! [Z: int,Nb: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z)),aa_num_rat(numeral_numeral_rat,Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),aa_num_int(numeral_numeral_int,Nb)) ) ).

% of_int_le_numeral_iff
tff(fact_5768_of__int__le__numeral__iff,axiom,
    ! [Z: int,Nb: 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,Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),aa_num_int(numeral_numeral_int,Nb)) ) ).

% of_int_le_numeral_iff
tff(fact_5769_of__int__numeral__less__iff,axiom,
    ! [Nb: num,Z: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,Nb)),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,Nb)),Z) ) ).

% of_int_numeral_less_iff
tff(fact_5770_of__int__numeral__less__iff,axiom,
    ! [Nb: num,Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,Nb)),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,Nb)),Z) ) ).

% of_int_numeral_less_iff
tff(fact_5771_of__int__numeral__less__iff,axiom,
    ! [Nb: num,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Nb)),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,Nb)),Z) ) ).

% of_int_numeral_less_iff
tff(fact_5772_of__int__less__numeral__iff,axiom,
    ! [Z: int,Nb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Z)),aa_num_real(numeral_numeral_real,Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),aa_num_int(numeral_numeral_int,Nb)) ) ).

% of_int_less_numeral_iff
tff(fact_5773_of__int__less__numeral__iff,axiom,
    ! [Z: int,Nb: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Z)),aa_num_rat(numeral_numeral_rat,Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),aa_num_int(numeral_numeral_int,Nb)) ) ).

% of_int_less_numeral_iff
tff(fact_5774_of__int__less__numeral__iff,axiom,
    ! [Z: int,Nb: 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,Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),aa_num_int(numeral_numeral_int,Nb)) ) ).

% of_int_less_numeral_iff
tff(fact_5775_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_5776_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_5777_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_5778_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_5779_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_5780_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_5781_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_5782_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_5783_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_5784_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_5785_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_5786_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_5787_take__bit__of__1,axiom,
    ! [Nb: nat] : bit_se1745604003318907178nteger(Nb,one_one_Code_integer) = zero_n356916108424825756nteger(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)) ).

% take_bit_of_1
tff(fact_5788_take__bit__of__1,axiom,
    ! [Nb: nat] : aa_int_int(bit_se2923211474154528505it_int(Nb),one_one_int) = aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)) ).

% take_bit_of_1
tff(fact_5789_take__bit__of__1,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),one_one_nat) = zero_n2687167440665602831ol_nat(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)) ).

% take_bit_of_1
tff(fact_5790_of__int__power__le__of__int__cancel__iff,axiom,
    ! [Xb: 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,Xb)),aa_nat_real(aa_real_fun_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,Xb),aa_nat_int(power_power_int(B),W)) ) ).

% of_int_power_le_of_int_cancel_iff
tff(fact_5791_of__int__power__le__of__int__cancel__iff,axiom,
    ! [Xb: 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,Xb)),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,Xb),aa_nat_int(power_power_int(B),W)) ) ).

% of_int_power_le_of_int_cancel_iff
tff(fact_5792_of__int__power__le__of__int__cancel__iff,axiom,
    ! [Xb: 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,Xb)),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,Xb),aa_nat_int(power_power_int(B),W)) ) ).

% of_int_power_le_of_int_cancel_iff
tff(fact_5793_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,Xb: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,B)),W)),aa_int_real(ring_1_of_int_real,Xb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(B),W)),Xb) ) ).

% of_int_le_of_int_power_cancel_iff
tff(fact_5794_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,Xb: 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,Xb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(B),W)),Xb) ) ).

% of_int_le_of_int_power_cancel_iff
tff(fact_5795_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,Xb: 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,Xb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(B),W)),Xb) ) ).

% of_int_le_of_int_power_cancel_iff
tff(fact_5796_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Yb: int,Xb: num,Nb: nat] :
      ( ( aa_int_complex(ring_17405671764205052669omplex,Yb) = aa_nat_complex(power_power_complex(numera6690914467698888265omplex(Xb)),Nb) )
    <=> ( Yb = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb) ) ) ).

% of_int_eq_numeral_power_cancel_iff
tff(fact_5797_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Yb: int,Xb: num,Nb: nat] :
      ( ( aa_int_real(ring_1_of_int_real,Yb) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,Xb)),Nb) )
    <=> ( Yb = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb) ) ) ).

% of_int_eq_numeral_power_cancel_iff
tff(fact_5798_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Yb: int,Xb: num,Nb: nat] :
      ( ( aa_int_rat(ring_1_of_int_rat,Yb) = aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,Xb)),Nb) )
    <=> ( Yb = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb) ) ) ).

% of_int_eq_numeral_power_cancel_iff
tff(fact_5799_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Yb: int,Xb: num,Nb: nat] :
      ( ( aa_int_int(ring_1_of_int_int,Yb) = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb) )
    <=> ( Yb = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb) ) ) ).

% of_int_eq_numeral_power_cancel_iff
tff(fact_5800_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,Yb: int] :
      ( ( aa_nat_complex(power_power_complex(numera6690914467698888265omplex(Xb)),Nb) = aa_int_complex(ring_17405671764205052669omplex,Yb) )
    <=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb) = Yb ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_5801_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,Yb: int] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,Xb)),Nb) = aa_int_real(ring_1_of_int_real,Yb) )
    <=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb) = Yb ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_5802_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,Yb: int] :
      ( ( aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,Xb)),Nb) = aa_int_rat(ring_1_of_int_rat,Yb) )
    <=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb) = Yb ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_5803_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,Yb: int] :
      ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb) = aa_int_int(ring_1_of_int_int,Yb) )
    <=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb) = Yb ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_5804_of__int__power__less__of__int__cancel__iff,axiom,
    ! [Xb: int,B: int,W: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Xb)),aa_nat_real(aa_real_fun_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,Xb),aa_nat_int(power_power_int(B),W)) ) ).

% of_int_power_less_of_int_cancel_iff
tff(fact_5805_of__int__power__less__of__int__cancel__iff,axiom,
    ! [Xb: int,B: int,W: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Xb)),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,Xb),aa_nat_int(power_power_int(B),W)) ) ).

% of_int_power_less_of_int_cancel_iff
tff(fact_5806_of__int__power__less__of__int__cancel__iff,axiom,
    ! [Xb: int,B: int,W: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(ring_1_of_int_int,Xb)),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,Xb),aa_nat_int(power_power_int(B),W)) ) ).

% of_int_power_less_of_int_cancel_iff
tff(fact_5807_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,Xb: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,B)),W)),aa_int_real(ring_1_of_int_real,Xb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(B),W)),Xb) ) ).

% of_int_less_of_int_power_cancel_iff
tff(fact_5808_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,Xb: 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,Xb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(B),W)),Xb) ) ).

% of_int_less_of_int_power_cancel_iff
tff(fact_5809_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,Xb: 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,Xb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(B),W)),Xb) ) ).

% of_int_less_of_int_power_cancel_iff
tff(fact_5810_even__take__bit__eq,axiom,
    ! [Nb: nat,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_se1745604003318907178nteger(Nb,A))
    <=> ( ( Nb = 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_5811_even__take__bit__eq,axiom,
    ! [Nb: 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(Nb),A))
    <=> ( ( Nb = 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_5812_even__take__bit__eq,axiom,
    ! [Nb: 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,Nb),A))
    <=> ( ( Nb = 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_5813_Suc__div__eq__add3__div__numeral,axiom,
    ! [Mb: 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,Mb)))),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))),Mb)),aa_num_nat(numeral_numeral_nat,V)) ).

% Suc_div_eq_add3_div_numeral
tff(fact_5814_div__Suc__eq__div__add3,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,Nb)))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),Nb)) ).

% div_Suc_eq_div_add3
tff(fact_5815_Suc__mod__eq__add3__mod__numeral,axiom,
    ! [Mb: 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,Mb)))),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))),Mb)),aa_num_nat(numeral_numeral_nat,V)) ).

% Suc_mod_eq_add3_mod_numeral
tff(fact_5816_mod__Suc__eq__mod__add3,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,Nb)))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),Nb)) ).

% mod_Suc_eq_mod_add3
tff(fact_5817_take__bit__Suc__0,axiom,
    ! [A: int] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,zero_zero_nat)),A) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% take_bit_Suc_0
tff(fact_5818_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_5819_real__sqrt__pow2__iff,axiom,
    ! [Xb: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sqrt,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = Xb )
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb) ) ).

% real_sqrt_pow2_iff
tff(fact_5820_real__sqrt__pow2,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sqrt,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = Xb ) ) ).

% real_sqrt_pow2
tff(fact_5821_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,Xb: num,Nb: 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(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,Xb)),Nb))
    <=> 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,Xb)),Nb)) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_5822_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,Xb: num,Nb: 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(aa_num_rat(numeral_numeral_rat,Xb)),Nb))
    <=> 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,Xb)),Nb)) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_5823_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,Xb: num,Nb: 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,Xb)),Nb))
    <=> 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,Xb)),Nb)) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_5824_numeral__power__le__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,A: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,Xb)),Nb)),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,Xb)),Nb)),A) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_5825_numeral__power__le__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,A: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,Xb)),Nb)),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,Xb)),Nb)),A) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_5826_numeral__power__le__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: 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,Xb)),Nb)),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,Xb)),Nb)),A) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_5827_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,Xb: num,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,A)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,Xb)),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb)) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_5828_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,Xb: num,Nb: 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(aa_num_rat(numeral_numeral_rat,Xb)),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb)) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_5829_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,Xb: num,Nb: 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,Xb)),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb)) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_5830_numeral__power__less__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,A: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,Xb)),Nb)),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,Xb)),Nb)),A) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_5831_numeral__power__less__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,A: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,Xb)),Nb)),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,Xb)),Nb)),A) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_5832_numeral__power__less__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: 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,Xb)),Nb)),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,Xb)),Nb)),A) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_5833_real__sqrt__sum__squares__mult__squared__eq,axiom,
    ! [Xb: real,Yb: real,Xaa: real,Yaa: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sqrt,aa_real_real(times_times_real(aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Xaa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Yaa),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(times_times_real(aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Xaa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Yaa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% real_sqrt_sum_squares_mult_squared_eq
tff(fact_5834_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Yb: int,Xb: num,Nb: nat] :
      ( ( aa_int_real(ring_1_of_int_real,Yb) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Xb))),Nb) )
    <=> ( Yb = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_5835_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Yb: int,Xb: num,Nb: nat] :
      ( ( aa_int_int(ring_1_of_int_int,Yb) = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb) )
    <=> ( Yb = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_5836_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Yb: int,Xb: num,Nb: nat] :
      ( ( aa_int_complex(ring_17405671764205052669omplex,Yb) = aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Xb))),Nb) )
    <=> ( Yb = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_5837_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Yb: int,Xb: num,Nb: nat] :
      ( ( aa_int_rat(ring_1_of_int_rat,Yb) = aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Xb))),Nb) )
    <=> ( Yb = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_5838_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Yb: int,Xb: num,Nb: nat] :
      ( ( aa_int_Code_integer(ring_18347121197199848620nteger,Yb) = aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Xb))),Nb) )
    <=> ( Yb = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_5839_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,Yb: int] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Xb))),Nb) = aa_int_real(ring_1_of_int_real,Yb) )
    <=> ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb) = Yb ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_5840_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,Yb: int] :
      ( ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb) = aa_int_int(ring_1_of_int_int,Yb) )
    <=> ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb) = Yb ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_5841_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,Yb: int] :
      ( ( aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Xb))),Nb) = aa_int_complex(ring_17405671764205052669omplex,Yb) )
    <=> ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb) = Yb ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_5842_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,Yb: int] :
      ( ( aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Xb))),Nb) = aa_int_rat(ring_1_of_int_rat,Yb) )
    <=> ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb) = Yb ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_5843_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,Yb: int] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Xb))),Nb) = aa_int_Code_integer(ring_18347121197199848620nteger,Yb) )
    <=> ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb) = Yb ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_5844_take__bit__of__exp,axiom,
    ! [Mb: nat,Nb: nat] : bit_se1745604003318907178nteger(Mb,aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Nb)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,zero_n356916108424825756nteger(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb))),aa_nat_Code_integer(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))),Nb)) ).

% take_bit_of_exp
tff(fact_5845_take__bit__of__exp,axiom,
    ! [Mb: nat,Nb: nat] : aa_int_int(bit_se2923211474154528505it_int(Mb),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) = 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,Nb),Mb))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ).

% take_bit_of_exp
tff(fact_5846_take__bit__of__exp,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,zero_n2687167440665602831ol_nat(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% take_bit_of_exp
tff(fact_5847_take__bit__of__2,axiom,
    ! [Nb: nat] : bit_se1745604003318907178nteger(Nb,numera6620942414471956472nteger(aa_num_num(bit0,one))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,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))),Nb))),numera6620942414471956472nteger(aa_num_num(bit0,one))) ).

% take_bit_of_2
tff(fact_5848_take__bit__of__2,axiom,
    ! [Nb: nat] : aa_int_int(bit_se2923211474154528505it_int(Nb),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))),Nb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% take_bit_of_2
tff(fact_5849_take__bit__of__2,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_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))),Nb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% take_bit_of_2
tff(fact_5850_zmod__numeral__Bit1,axiom,
    ! [V: num,W: num] : aa_int_int(aa_int_fun_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(aa_int_fun_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_5851_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,Xb: num,Nb: 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(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Xb))),Nb))
    <=> 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,Xb))),Nb)) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_5852_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,Xb: num,Nb: 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(Xb))),Nb))
    <=> 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,Xb))),Nb)) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_5853_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,Xb: num,Nb: 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(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Xb))),Nb))
    <=> 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,Xb))),Nb)) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_5854_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,Xb: num,Nb: 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,Xb))),Nb))
    <=> 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,Xb))),Nb)) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_5855_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,A: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Xb))),Nb)),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,Xb))),Nb)),A) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_5856_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,A: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Xb))),Nb)),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,Xb))),Nb)),A) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_5857_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,A: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Xb))),Nb)),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,Xb))),Nb)),A) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_5858_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: 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,Xb))),Nb)),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,Xb))),Nb)),A) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_5859_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,Xb: num,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,A)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Xb))),Nb))
    <=> 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,Xb))),Nb)) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_5860_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,Xb: num,Nb: 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,Xb))),Nb))
    <=> 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,Xb))),Nb)) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_5861_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,Xb: num,Nb: 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(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Xb))),Nb))
    <=> 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,Xb))),Nb)) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_5862_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,Xb: num,Nb: 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(Xb))),Nb))
    <=> 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,Xb))),Nb)) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_5863_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,A: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Xb))),Nb)),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,Xb))),Nb)),A) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_5864_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: 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,Xb))),Nb)),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,Xb))),Nb)),A) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_5865_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,A: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Xb))),Nb)),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,Xb))),Nb)),A) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_5866_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,A: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Xb))),Nb)),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,Xb))),Nb)),A) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_5867_take__bit__minus,axiom,
    ! [Nb: nat,K: int] : aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(uminus_uminus_int,aa_int_int(bit_se2923211474154528505it_int(Nb),K))) = aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(uminus_uminus_int,K)) ).

% take_bit_minus
tff(fact_5868_prod_Ocase__distrib,axiom,
    ! [H: fun_o_o,F: fun_int_fun_int_o,Prod: product_prod_int_int] :
      ( aa_o_o(H,aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(F),Prod))
    <=> aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(aa_fun6133342665515662012_int_o(aTP_Lamm_bh(H),F)),Prod) ) ).

% prod.case_distrib
tff(fact_5869_prod_Ocase__distrib,axiom,
    ! [H: fun_o_int,F: fun_int_fun_int_o,Prod: product_prod_int_int] : aa_o_int(H,aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(F),Prod)) = aa_Pro6799258434465214121nt_int(produc8211389475949308722nt_int(aa_fun7323606162024495668nt_int(aTP_Lamm_bi(H),F)),Prod) ).

% prod.case_distrib
tff(fact_5870_prod_Ocase__distrib,axiom,
    ! [H: fun_int_o,F: fun_int_fun_int_int,Prod: product_prod_int_int] :
      ( aa_int_o(H,aa_Pro6799258434465214121nt_int(produc8211389475949308722nt_int(F),Prod))
    <=> aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(aa_fun82482788810740534_int_o(aTP_Lamm_bj(H),F)),Prod) ) ).

% prod.case_distrib
tff(fact_5871_prod_Ocase__distrib,axiom,
    ! [H: fun_int_int,F: fun_int_fun_int_int,Prod: product_prod_int_int] : aa_int_int(H,aa_Pro6799258434465214121nt_int(produc8211389475949308722nt_int(F),Prod)) = aa_Pro6799258434465214121nt_int(produc8211389475949308722nt_int(aa_fun6119123525526402106nt_int(aTP_Lamm_bk(H),F)),Prod) ).

% prod.case_distrib
tff(fact_5872_prod_Ocase__distrib,axiom,
    ! [H: fun_Pr3238374608701176162_int_o,F: fun_in1696471567231710611nt_int,Prod: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(H,aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(F),Prod))
    <=> aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(aa_fun4545072666309863515_int_o(aTP_Lamm_bl(H),F)),Prod) ) ).

% prod.case_distrib
tff(fact_5873_prod_Ocase__distrib,axiom,
    ! [H: fun_Pr982841511892225912nt_int,F: fun_in1696471567231710611nt_int,Prod: product_prod_int_int] : aa_Pro6799258434465214121nt_int(H,aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(F),Prod)) = aa_Pro6799258434465214121nt_int(produc8211389475949308722nt_int(aa_fun7461304428735181397nt_int(aTP_Lamm_bm(H),F)),Prod) ).

% prod.case_distrib
tff(fact_5874_prod_Ocase__distrib,axiom,
    ! [H: fun_o_8988054265482332260nt_int,F: fun_int_fun_int_o,Prod: product_prod_int_int] : aa_o_P3134438092330336605nt_int(H,aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(F),Prod)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aa_fun4023131440832735197nt_int(aTP_Lamm_bn(H),F)),Prod) ).

% prod.case_distrib
tff(fact_5875_prod_Ocase__distrib,axiom,
    ! [H: fun_in4211310926729683942nt_int,F: fun_int_fun_int_int,Prod: product_prod_int_int] : aa_int7462565939943104719nt_int(H,aa_Pro6799258434465214121nt_int(produc8211389475949308722nt_int(F),Prod)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aa_fun1034830265760468567nt_int(aTP_Lamm_bo(H),F)),Prod) ).

% prod.case_distrib
tff(fact_5876_prod_Ocase__distrib,axiom,
    ! [H: fun_Pr2478283820680522243nt_int,F: fun_in1696471567231710611nt_int,Prod: product_prod_int_int] : aa_Pro6970504127699319548nt_int(H,aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(F),Prod)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aa_fun4152872853588681340nt_int(aTP_Lamm_bp(H),F)),Prod) ).

% prod.case_distrib
tff(fact_5877_prod_Ocase__distrib,axiom,
    ! [H: fun_fu3275390434952296369_nat_o,F: fun_na5914473681949881652_nat_o,Prod: product_prod_nat_nat] : aa_fun359341089595923442_nat_o(H,aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(F),Prod)) = aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(aa_fun824319262034445606_nat_o(aTP_Lamm_bq(H),F)),Prod) ).

% prod.case_distrib
tff(fact_5878_take__bit__of__int,axiom,
    ! [Nb: nat,K: int] : aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(ring_1_of_int_int,K)) = aa_int_int(ring_1_of_int_int,aa_int_int(bit_se2923211474154528505it_int(Nb),K)) ).

% take_bit_of_int
tff(fact_5879_take__bit__add,axiom,
    ! [Nb: nat,A: int,B: int] : aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(bit_se2923211474154528505it_int(Nb),A)),aa_int_int(bit_se2923211474154528505it_int(Nb),B))) = aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ).

% take_bit_add
tff(fact_5880_take__bit__add,axiom,
    ! [Nb: nat,A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),A)),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),B))) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ).

% take_bit_add
tff(fact_5881_take__bit__tightened,axiom,
    ! [Nb: nat,A: int,B: int,Mb: nat] :
      ( ( aa_int_int(bit_se2923211474154528505it_int(Nb),A) = aa_int_int(bit_se2923211474154528505it_int(Nb),B) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
       => ( aa_int_int(bit_se2923211474154528505it_int(Mb),A) = aa_int_int(bit_se2923211474154528505it_int(Mb),B) ) ) ) ).

% take_bit_tightened
tff(fact_5882_take__bit__tightened,axiom,
    ! [Nb: nat,A: nat,B: nat,Mb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),A) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),B) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Mb),A) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Mb),B) ) ) ) ).

% take_bit_tightened
tff(fact_5883_take__bit__tightened__less__eq__nat,axiom,
    ! [Mb: nat,Nb: nat,Q2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Mb),Q2)),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),Q2)) ) ).

% take_bit_tightened_less_eq_nat
tff(fact_5884_take__bit__nat__less__eq__self,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),Mb)),Mb) ).

% take_bit_nat_less_eq_self
tff(fact_5885_mult__of__int__commute,axiom,
    ! [Xb: int,Yb: real] : aa_real_real(times_times_real(aa_int_real(ring_1_of_int_real,Xb)),Yb) = aa_real_real(times_times_real(Yb),aa_int_real(ring_1_of_int_real,Xb)) ).

% mult_of_int_commute
tff(fact_5886_mult__of__int__commute,axiom,
    ! [Xb: int,Yb: rat] : aa_rat_rat(times_times_rat(aa_int_rat(ring_1_of_int_rat,Xb)),Yb) = aa_rat_rat(times_times_rat(Yb),aa_int_rat(ring_1_of_int_rat,Xb)) ).

% mult_of_int_commute
tff(fact_5887_mult__of__int__commute,axiom,
    ! [Xb: int,Yb: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(ring_1_of_int_int,Xb)),Yb) = aa_int_int(aa_int_fun_int_int(times_times_int,Yb),aa_int_int(ring_1_of_int_int,Xb)) ).

% mult_of_int_commute
tff(fact_5888_real__sqrt__mult,axiom,
    ! [Xb: real,Yb: real] : aa_real_real(sqrt,aa_real_real(times_times_real(Xb),Yb)) = aa_real_real(times_times_real(aa_real_real(sqrt,Xb)),aa_real_real(sqrt,Yb)) ).

% real_sqrt_mult
tff(fact_5889_take__bit__mult,axiom,
    ! [Nb: nat,K: int,L: int] : aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_se2923211474154528505it_int(Nb),K)),aa_int_int(bit_se2923211474154528505it_int(Nb),L))) = aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(aa_int_fun_int_int(times_times_int,K),L)) ).

% take_bit_mult
tff(fact_5890_verit__eq__simplify_I14_J,axiom,
    ! [X2: num,X32: num] : aa_num_num(bit0,X2) != aa_num_num(bit1,X32) ).

% verit_eq_simplify(14)
tff(fact_5891_verit__eq__simplify_I12_J,axiom,
    ! [X32: num] : one != aa_num_num(bit1,X32) ).

% verit_eq_simplify(12)
tff(fact_5892_take__bit__diff,axiom,
    ! [Nb: nat,K: int,L: int] : aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(bit_se2923211474154528505it_int(Nb),K)),aa_int_int(bit_se2923211474154528505it_int(Nb),L))) = aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(aa_int_fun_int_int(minus_minus_int,K),L)) ).

% take_bit_diff
tff(fact_5893_of__int__max,axiom,
    ! [Xb: int,Yb: int] : aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(ord_max_int,Xb),Yb)) = aa_real_real(ord_max_real(aa_int_real(ring_1_of_int_real,Xb)),aa_int_real(ring_1_of_int_real,Yb)) ).

% of_int_max
tff(fact_5894_of__int__max,axiom,
    ! [Xb: int,Yb: int] : aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(ord_max_int,Xb),Yb)) = aa_rat_rat(ord_max_rat(aa_int_rat(ring_1_of_int_rat,Xb)),aa_int_rat(ring_1_of_int_rat,Yb)) ).

% of_int_max
tff(fact_5895_of__int__max,axiom,
    ! [Xb: int,Yb: int] : aa_int_int(ring_1_of_int_int,aa_int_int(aa_int_fun_int_int(ord_max_int,Xb),Yb)) = aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(ring_1_of_int_int,Xb)),aa_int_int(ring_1_of_int_int,Yb)) ).

% of_int_max
tff(fact_5896_old_Oprod_Ocase,axiom,
    ! [F: fun_na1668100129766719005at_nat,X1: nat,X2: nat] : aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(F),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X1),X2)) = aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(F,X1),X2) ).

% old.prod.case
tff(fact_5897_old_Oprod_Ocase,axiom,
    ! [F: fun_na5914473681949881652_nat_o,X1: nat,X2: nat] : aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(F),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X1),X2)) = aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(F,X1),X2) ).

% old.prod.case
tff(fact_5898_old_Oprod_Ocase,axiom,
    ! [F: fun_in1696471567231710611nt_int,X1: int,X2: int] : aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(F),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X1),X2)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(F,X1),X2) ).

% old.prod.case
tff(fact_5899_old_Oprod_Ocase,axiom,
    ! [F: fun_int_fun_int_o,X1: int,X2: int] :
      ( aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(F),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X1),X2))
    <=> aa_int_o(aa_int_fun_int_o(F,X1),X2) ) ).

% old.prod.case
tff(fact_5900_old_Oprod_Ocase,axiom,
    ! [F: fun_int_fun_int_int,X1: int,X2: int] : aa_Pro6799258434465214121nt_int(produc8211389475949308722nt_int(F),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X1),X2)) = aa_int_int(aa_int_fun_int_int(F,X1),X2) ).

% old.prod.case
tff(fact_5901_concat__bit__eq__iff,axiom,
    ! [Nb: nat,K: int,L: int,R2: int,S2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(bit_concat_bit(Nb),K),L) = aa_int_int(aa_int_fun_int_int(bit_concat_bit(Nb),R2),S2) )
    <=> ( ( aa_int_int(bit_se2923211474154528505it_int(Nb),K) = aa_int_int(bit_se2923211474154528505it_int(Nb),R2) )
        & ( L = S2 ) ) ) ).

% concat_bit_eq_iff
tff(fact_5902_concat__bit__take__bit__eq,axiom,
    ! [Nb: nat,B: int] : aa_int_fun_int_int(bit_concat_bit(Nb),aa_int_int(bit_se2923211474154528505it_int(Nb),B)) = aa_int_fun_int_int(bit_concat_bit(Nb),B) ).

% concat_bit_take_bit_eq
tff(fact_5903_case__prodE2,axiom,
    ! [Q: fun_fu8391177546919232046_nat_o,P: fun_na1668100129766719005at_nat,Z: product_prod_nat_nat] :
      ( aa_fun3538214411680374423_nat_o(Q,aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(P),Z))
     => ~ ! [X3: nat,Y: nat] :
            ( ( Z = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),Y) )
           => ~ aa_fun3538214411680374423_nat_o(Q,aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(P,X3),Y)) ) ) ).

% case_prodE2
tff(fact_5904_case__prodE2,axiom,
    ! [Q: fun_fu6891447142836604759at_o_o,P: fun_na5914473681949881652_nat_o,Z: product_prod_nat_nat] :
      ( aa_fun8408151826566693824at_o_o(Q,aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(P),Z))
     => ~ ! [X3: nat,Y: nat] :
            ( ( Z = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),Y) )
           => ~ aa_fun8408151826566693824at_o_o(Q,aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(P,X3),Y)) ) ) ).

% case_prodE2
tff(fact_5905_case__prodE2,axiom,
    ! [Q: fun_Pr3238374608701176162_int_o,P: fun_in1696471567231710611nt_int,Z: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(Q,aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(P),Z))
     => ~ ! [X3: int,Y: int] :
            ( ( Z = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X3),Y) )
           => ~ aa_Pro4592068669816069595_int_o(Q,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(P,X3),Y)) ) ) ).

% case_prodE2
tff(fact_5906_case__prodE2,axiom,
    ! [Q: fun_o_o,P: fun_int_fun_int_o,Z: product_prod_int_int] :
      ( aa_o_o(Q,aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(P),Z))
     => ~ ! [X3: int,Y: int] :
            ( ( Z = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X3),Y) )
           => ~ aa_o_o(Q,aa_int_o(aa_int_fun_int_o(P,X3),Y)) ) ) ).

% case_prodE2
tff(fact_5907_case__prodE2,axiom,
    ! [Q: fun_int_o,P: fun_int_fun_int_int,Z: product_prod_int_int] :
      ( aa_int_o(Q,aa_Pro6799258434465214121nt_int(produc8211389475949308722nt_int(P),Z))
     => ~ ! [X3: int,Y: int] :
            ( ( Z = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X3),Y) )
           => ~ aa_int_o(Q,aa_int_int(aa_int_fun_int_int(P,X3),Y)) ) ) ).

% case_prodE2
tff(fact_5908_case__prod__eta,axiom,
    ! [F: fun_Pr2477239776880934807at_nat] : produc4528391901023682385at_nat(aTP_Lamm_br(F)) = F ).

% case_prod_eta
tff(fact_5909_case__prod__eta,axiom,
    ! [F: fun_Pr2028404973800837934_nat_o] : produc6395135316500328296_nat_o(aTP_Lamm_bs(F)) = F ).

% case_prod_eta
tff(fact_5910_case__prod__eta,axiom,
    ! [F: fun_Pr2478283820680522243nt_int] : produc4245557441103728435nt_int(aTP_Lamm_bt(F)) = F ).

% case_prod_eta
tff(fact_5911_case__prod__eta,axiom,
    ! [F: fun_Pr3238374608701176162_int_o] : produc4947309494688390418_int_o(aTP_Lamm_bu(F)) = F ).

% case_prod_eta
tff(fact_5912_case__prod__eta,axiom,
    ! [F: fun_Pr982841511892225912nt_int] : produc8211389475949308722nt_int(aTP_Lamm_bv(F)) = F ).

% case_prod_eta
tff(fact_5913_cond__case__prod__eta,axiom,
    ! [F: fun_na1668100129766719005at_nat,G: fun_Pr2477239776880934807at_nat] :
      ( ! [X3: nat,Y: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(F,X3),Y) = aa_Pro8023792847356852296at_nat(G,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),Y))
     => ( produc4528391901023682385at_nat(F) = G ) ) ).

% cond_case_prod_eta
tff(fact_5914_cond__case__prod__eta,axiom,
    ! [F: fun_na5914473681949881652_nat_o,G: fun_Pr2028404973800837934_nat_o] :
      ( ! [X3: nat,Y: nat] : aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(F,X3),Y) = aa_Pro989595756709598047_nat_o(G,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),Y))
     => ( produc6395135316500328296_nat_o(F) = G ) ) ).

% cond_case_prod_eta
tff(fact_5915_cond__case__prod__eta,axiom,
    ! [F: fun_in1696471567231710611nt_int,G: fun_Pr2478283820680522243nt_int] :
      ( ! [X3: int,Y: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(F,X3),Y) = aa_Pro6970504127699319548nt_int(G,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X3),Y))
     => ( produc4245557441103728435nt_int(F) = G ) ) ).

% cond_case_prod_eta
tff(fact_5916_cond__case__prod__eta,axiom,
    ! [F: fun_int_fun_int_o,G: fun_Pr3238374608701176162_int_o] :
      ( ! [X3: int,Y: int] :
          ( aa_int_o(aa_int_fun_int_o(F,X3),Y)
        <=> aa_Pro4592068669816069595_int_o(G,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X3),Y)) )
     => ( produc4947309494688390418_int_o(F) = G ) ) ).

% cond_case_prod_eta
tff(fact_5917_cond__case__prod__eta,axiom,
    ! [F: fun_int_fun_int_int,G: fun_Pr982841511892225912nt_int] :
      ( ! [X3: int,Y: int] : aa_int_int(aa_int_fun_int_int(F,X3),Y) = aa_Pro6799258434465214121nt_int(G,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X3),Y))
     => ( produc8211389475949308722nt_int(F) = G ) ) ).

% cond_case_prod_eta
tff(fact_5918_real__sqrt__ge__one,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(sqrt,Xb)) ) ).

% real_sqrt_ge_one
tff(fact_5919_take__bit__tightened__less__eq__int,axiom,
    ! [Mb: nat,Nb: nat,K: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(bit_se2923211474154528505it_int(Mb),K)),aa_int_int(bit_se2923211474154528505it_int(Nb),K)) ) ).

% take_bit_tightened_less_eq_int
tff(fact_5920_take__bit__nonnegative,axiom,
    ! [Nb: nat,K: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se2923211474154528505it_int(Nb),K)) ).

% take_bit_nonnegative
tff(fact_5921_take__bit__int__less__eq__self__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(bit_se2923211474154528505it_int(Nb),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_5922_signed__take__bit__eq__iff__take__bit__eq,axiom,
    ! [Nb: nat,A: int,B: int] :
      ( ( aa_int_int(bit_ri631733984087533419it_int(Nb),A) = aa_int_int(bit_ri631733984087533419it_int(Nb),B) )
    <=> ( aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Nb)),A) = aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Nb)),B) ) ) ).

% signed_take_bit_eq_iff_take_bit_eq
tff(fact_5923_not__take__bit__negative,axiom,
    ! [Nb: nat,K: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se2923211474154528505it_int(Nb),K)),zero_zero_int) ).

% not_take_bit_negative
tff(fact_5924_take__bit__int__greater__self__iff,axiom,
    ! [K: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_int_int(bit_se2923211474154528505it_int(Nb),K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int) ) ).

% take_bit_int_greater_self_iff
tff(fact_5925_signed__take__bit__take__bit,axiom,
    ! [Mb: nat,Nb: nat,A: int] :
      aa_int_int(bit_ri631733984087533419it_int(Mb),aa_int_int(bit_se2923211474154528505it_int(Nb),A)) = aa_int_int($ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb),bit_se2923211474154528505it_int(Nb),bit_ri631733984087533419it_int(Mb)),A) ).

% signed_take_bit_take_bit
tff(fact_5926_num_Oexhaust,axiom,
    ! [Yb: num] :
      ( ( Yb != one )
     => ( ! [X23: num] : Yb != aa_num_num(bit0,X23)
       => ~ ! [X33: num] : Yb != aa_num_num(bit1,X33) ) ) ).

% num.exhaust
tff(fact_5927_xor__num_Ocases,axiom,
    ! [Xb: product_prod_num_num] :
      ( ( Xb != product_Pair_num_num(one,one) )
     => ( ! [N: num] : Xb != product_Pair_num_num(one,aa_num_num(bit0,N))
       => ( ! [N: num] : Xb != product_Pair_num_num(one,aa_num_num(bit1,N))
         => ( ! [M2: num] : Xb != product_Pair_num_num(aa_num_num(bit0,M2),one)
           => ( ! [M2: num,N: num] : Xb != product_Pair_num_num(aa_num_num(bit0,M2),aa_num_num(bit0,N))
             => ( ! [M2: num,N: num] : Xb != product_Pair_num_num(aa_num_num(bit0,M2),aa_num_num(bit1,N))
               => ( ! [M2: num] : Xb != product_Pair_num_num(aa_num_num(bit1,M2),one)
                 => ( ! [M2: num,N: num] : Xb != product_Pair_num_num(aa_num_num(bit1,M2),aa_num_num(bit0,N))
                   => ~ ! [M2: num,N: num] : Xb != product_Pair_num_num(aa_num_num(bit1,M2),aa_num_num(bit1,N)) ) ) ) ) ) ) ) ) ).

% xor_num.cases
tff(fact_5928_take__bit__unset__bit__eq,axiom,
    ! [Nb: nat,Mb: nat,A: int] :
      aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,Mb),A)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb),aa_int_int(bit_se2923211474154528505it_int(Nb),A),aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,Mb),aa_int_int(bit_se2923211474154528505it_int(Nb),A))) ).

% take_bit_unset_bit_eq
tff(fact_5929_take__bit__unset__bit__eq,axiom,
    ! [Nb: nat,Mb: nat,A: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,Mb),A)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),A),aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),A))) ).

% take_bit_unset_bit_eq
tff(fact_5930_take__bit__set__bit__eq,axiom,
    ! [Nb: nat,Mb: nat,A: int] :
      aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,Mb),A)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb),aa_int_int(bit_se2923211474154528505it_int(Nb),A),aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,Mb),aa_int_int(bit_se2923211474154528505it_int(Nb),A))) ).

% take_bit_set_bit_eq
tff(fact_5931_take__bit__set__bit__eq,axiom,
    ! [Nb: nat,Mb: nat,A: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,Mb),A)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),A),aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),A))) ).

% take_bit_set_bit_eq
tff(fact_5932_take__bit__flip__bit__eq,axiom,
    ! [Nb: nat,Mb: nat,A: int] :
      aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(bit_se2159334234014336723it_int(Mb),A)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb),aa_int_int(bit_se2923211474154528505it_int(Nb),A),aa_int_int(bit_se2159334234014336723it_int(Mb),aa_int_int(bit_se2923211474154528505it_int(Nb),A))) ).

% take_bit_flip_bit_eq
tff(fact_5933_take__bit__flip__bit__eq,axiom,
    ! [Nb: nat,Mb: nat,A: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,Mb),A)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),A),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),A))) ).

% take_bit_flip_bit_eq
tff(fact_5934_sqrt__add__le__add__sqrt,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Yb)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,aa_real_real(plus_plus_real(Xb),Yb))),aa_real_real(plus_plus_real(aa_real_real(sqrt,Xb)),aa_real_real(sqrt,Yb))) ) ) ).

% sqrt_add_le_add_sqrt
tff(fact_5935_take__bit__signed__take__bit,axiom,
    ! [Mb: nat,Nb: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb))
     => ( aa_int_int(bit_se2923211474154528505it_int(Mb),aa_int_int(bit_ri631733984087533419it_int(Nb),A)) = aa_int_int(bit_se2923211474154528505it_int(Mb),A) ) ) ).

% take_bit_signed_take_bit
tff(fact_5936_of__int__neg__numeral,axiom,
    ! [K: num] : aa_int_real(ring_1_of_int_real,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,K)) ).

% of_int_neg_numeral
tff(fact_5937_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_5938_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_5939_of__int__neg__numeral,axiom,
    ! [K: num] : aa_int_rat(ring_1_of_int_rat,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,K)) ).

% of_int_neg_numeral
tff(fact_5940_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_5941_le__real__sqrt__sumsq,axiom,
    ! [Xb: real,Yb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(sqrt,aa_real_real(plus_plus_real(aa_real_real(times_times_real(Xb),Xb)),aa_real_real(times_times_real(Yb),Yb)))) ).

% le_real_sqrt_sumsq
tff(fact_5942_numeral__Bit1,axiom,
    ! [Nb: num] : numera6690914467698888265omplex(aa_num_num(bit1,Nb)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(Nb)),numera6690914467698888265omplex(Nb))),one_one_complex) ).

% numeral_Bit1
tff(fact_5943_numeral__Bit1,axiom,
    ! [Nb: num] : aa_num_real(numeral_numeral_real,aa_num_num(bit1,Nb)) = aa_real_real(plus_plus_real(aa_real_real(plus_plus_real(aa_num_real(numeral_numeral_real,Nb)),aa_num_real(numeral_numeral_real,Nb))),one_one_real) ).

% numeral_Bit1
tff(fact_5944_numeral__Bit1,axiom,
    ! [Nb: num] : aa_num_rat(numeral_numeral_rat,aa_num_num(bit1,Nb)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,Nb)),aa_num_rat(numeral_numeral_rat,Nb))),one_one_rat) ).

% numeral_Bit1
tff(fact_5945_numeral__Bit1,axiom,
    ! [Nb: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Nb)) = 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,Nb)),aa_num_nat(numeral_numeral_nat,Nb))),one_one_nat) ).

% numeral_Bit1
tff(fact_5946_numeral__Bit1,axiom,
    ! [Nb: num] : aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)) = 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,Nb)),aa_num_int(numeral_numeral_int,Nb))),one_one_int) ).

% numeral_Bit1
tff(fact_5947_take__bit__decr__eq,axiom,
    ! [Nb: nat,K: int] :
      ( ( aa_int_int(bit_se2923211474154528505it_int(Nb),K) != zero_zero_int )
     => ( aa_int_int(bit_se2923211474154528505it_int(Nb),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(Nb),K)),one_one_int) ) ) ).

% take_bit_decr_eq
tff(fact_5948_eval__nat__numeral_I3_J,axiom,
    ! [Nb: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Nb)) = aa_nat_nat(suc,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Nb))) ).

% eval_nat_numeral(3)
tff(fact_5949_cong__exp__iff__simps_I10_J,axiom,
    ! [Mb: num,Q2: num,Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Mb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) != aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Nb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) ).

% cong_exp_iff_simps(10)
tff(fact_5950_cong__exp__iff__simps_I10_J,axiom,
    ! [Mb: num,Q2: num,Nb: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Mb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) != aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) ).

% cong_exp_iff_simps(10)
tff(fact_5951_cong__exp__iff__simps_I12_J,axiom,
    ! [Mb: num,Q2: num,Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Mb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) != aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Nb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) ).

% cong_exp_iff_simps(12)
tff(fact_5952_cong__exp__iff__simps_I12_J,axiom,
    ! [Mb: num,Q2: num,Nb: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Mb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) != aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) ).

% cong_exp_iff_simps(12)
tff(fact_5953_cong__exp__iff__simps_I13_J,axiom,
    ! [Mb: num,Q2: num,Nb: num] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Mb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Nb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Q2)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Nb)),aa_num_nat(numeral_numeral_nat,Q2)) ) ) ).

% cong_exp_iff_simps(13)
tff(fact_5954_cong__exp__iff__simps_I13_J,axiom,
    ! [Mb: num,Q2: num,Nb: num] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Mb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) )
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,Q2)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,Nb)),aa_num_int(numeral_numeral_int,Q2)) ) ) ).

% cong_exp_iff_simps(13)
tff(fact_5955_power__minus__Bit1,axiom,
    ! [Xb: real,K: num] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K))) = aa_real_real(uminus_uminus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K)))) ).

% power_minus_Bit1
tff(fact_5956_power__minus__Bit1,axiom,
    ! [Xb: int,K: num] : aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K))) = aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K)))) ).

% power_minus_Bit1
tff(fact_5957_power__minus__Bit1,axiom,
    ! [Xb: complex,K: num] : aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K))) = aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(power_power_complex(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K)))) ).

% power_minus_Bit1
tff(fact_5958_power__minus__Bit1,axiom,
    ! [Xb: rat,K: num] : aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K))) = aa_rat_rat(uminus_uminus_rat,aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K)))) ).

% power_minus_Bit1
tff(fact_5959_power__minus__Bit1,axiom,
    ! [Xb: code_integer,K: num] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(power_8256067586552552935nteger(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K)))) ).

% power_minus_Bit1
tff(fact_5960_real__of__int__div4,axiom,
    ! [Nb: int,Xb: 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,Nb),Xb))),aa_real_real(divide_divide_real(aa_int_real(ring_1_of_int_real,Nb)),aa_int_real(ring_1_of_int_real,Xb))) ).

% real_of_int_div4
tff(fact_5961_take__bit__Suc__bit1,axiom,
    ! [Nb: nat,K: num] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Nb)),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(Nb),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_5962_take__bit__Suc__bit1,axiom,
    ! [Nb: nat,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_nat_nat(suc,Nb)),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,Nb),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_5963_real__of__int__div,axiom,
    ! [D: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),Nb)
     => ( aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(divide_divide_int,Nb),D)) = aa_real_real(divide_divide_real(aa_int_real(ring_1_of_int_real,Nb)),aa_int_real(ring_1_of_int_real,D)) ) ) ).

% real_of_int_div
tff(fact_5964_numeral__code_I3_J,axiom,
    ! [Nb: num] :
      numera6690914467698888265omplex(aa_num_num(bit1,Nb)) = $let(
        m2: complex,
        m2:= numera6690914467698888265omplex(Nb),
        aa_complex_complex(plus_plus_complex(aa_complex_complex(plus_plus_complex(m2),m2)),one_one_complex) ) ).

% numeral_code(3)
tff(fact_5965_numeral__code_I3_J,axiom,
    ! [Nb: num] :
      aa_num_real(numeral_numeral_real,aa_num_num(bit1,Nb)) = $let(
        m2: real,
        m2:= aa_num_real(numeral_numeral_real,Nb),
        aa_real_real(plus_plus_real(aa_real_real(plus_plus_real(m2),m2)),one_one_real) ) ).

% numeral_code(3)
tff(fact_5966_numeral__code_I3_J,axiom,
    ! [Nb: num] :
      aa_num_rat(numeral_numeral_rat,aa_num_num(bit1,Nb)) = $let(
        m2: rat,
        m2:= aa_num_rat(numeral_numeral_rat,Nb),
        aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(m2),m2)),one_one_rat) ) ).

% numeral_code(3)
tff(fact_5967_numeral__code_I3_J,axiom,
    ! [Nb: num] :
      aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Nb)) = $let(
        m2: nat,
        m2:= aa_num_nat(numeral_numeral_nat,Nb),
        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_5968_numeral__code_I3_J,axiom,
    ! [Nb: num] :
      aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)) = $let(
        m2: int,
        m2:= aa_num_int(numeral_numeral_int,Nb),
        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_5969_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_5970_power__numeral__odd,axiom,
    ! [Z: real,W: num] :
      aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,W))) = $let(
        w: real,
        w:= aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),aa_num_nat(numeral_numeral_nat,W)),
        aa_real_real(times_times_real(aa_real_real(times_times_real(Z),w)),w) ) ).

% power_numeral_odd
tff(fact_5971_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_5972_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_5973_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_5974_sqrt2__less__2,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% sqrt2_less_2
tff(fact_5975_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_5976_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_5977_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_5978_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_5979_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_5980_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_5981_numeral__Bit1__div__2,axiom,
    ! [Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Nb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_num_nat(numeral_numeral_nat,Nb) ).

% numeral_Bit1_div_2
tff(fact_5982_numeral__Bit1__div__2,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_num_int(numeral_numeral_int,Nb) ).

% numeral_Bit1_div_2
tff(fact_5983_odd__numeral,axiom,
    ! [Nb: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),numera6620942414471956472nteger(aa_num_num(bit1,Nb))) ).

% odd_numeral
tff(fact_5984_odd__numeral,axiom,
    ! [Nb: 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,Nb))) ).

% odd_numeral
tff(fact_5985_odd__numeral,axiom,
    ! [Nb: 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,Nb))) ).

% odd_numeral
tff(fact_5986_cong__exp__iff__simps_I3_J,axiom,
    ! [Nb: num,Q2: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Nb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) != zero_zero_nat ).

% cong_exp_iff_simps(3)
tff(fact_5987_cong__exp__iff__simps_I3_J,axiom,
    ! [Nb: num,Q2: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) != zero_zero_int ).

% cong_exp_iff_simps(3)
tff(fact_5988_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_5989_power3__eq__cube,axiom,
    ! [A: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))) = aa_real_real(times_times_real(aa_real_real(times_times_real(A),A)),A) ).

% power3_eq_cube
tff(fact_5990_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_5991_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_5992_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_5993_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_5994_int__le__real__less,axiom,
    ! [Nb: int,Mb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Nb),Mb)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Nb)),aa_real_real(plus_plus_real(aa_int_real(ring_1_of_int_real,Mb)),one_one_real)) ) ).

% int_le_real_less
tff(fact_5995_Suc3__eq__add__3,axiom,
    ! [Nb: nat] : aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),Nb) ).

% Suc3_eq_add_3
tff(fact_5996_int__less__real__le,axiom,
    ! [Nb: int,Mb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Nb),Mb)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(aa_int_real(ring_1_of_int_real,Nb)),one_one_real)),aa_int_real(ring_1_of_int_real,Mb)) ) ).

% int_less_real_le
tff(fact_5997_real__of__int__div__aux,axiom,
    ! [Xb: int,D: int] : aa_real_real(divide_divide_real(aa_int_real(ring_1_of_int_real,Xb)),aa_int_real(ring_1_of_int_real,D)) = aa_real_real(plus_plus_real(aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(divide_divide_int,Xb),D))),aa_real_real(divide_divide_real(aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Xb),D))),aa_int_real(ring_1_of_int_real,D))) ).

% real_of_int_div_aux
tff(fact_5998_take__bit__Suc__bit0,axiom,
    ! [Nb: nat,K: num] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Nb)),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(Nb),aa_num_int(numeral_numeral_int,K))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% take_bit_Suc_bit0
tff(fact_5999_take__bit__Suc__bit0,axiom,
    ! [Nb: nat,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_nat_nat(suc,Nb)),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,Nb),aa_num_nat(numeral_numeral_nat,K))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% take_bit_Suc_bit0
tff(fact_6000_take__bit__eq__mod,axiom,
    ! [Nb: nat,A: int] : aa_int_int(bit_se2923211474154528505it_int(Nb),A) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ).

% take_bit_eq_mod
tff(fact_6001_take__bit__eq__mod,axiom,
    ! [Nb: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),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))),Nb)) ).

% take_bit_eq_mod
tff(fact_6002_take__bit__nat__eq__self__iff,axiom,
    ! [Nb: nat,Mb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),Mb) = Mb )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ) ).

% take_bit_nat_eq_self_iff
tff(fact_6003_take__bit__nat__less__exp,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% take_bit_nat_less_exp
tff(fact_6004_take__bit__nat__eq__self,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),Mb) = Mb ) ) ).

% take_bit_nat_eq_self
tff(fact_6005_real__less__rsqrt,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Yb)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(sqrt,Yb)) ) ).

% real_less_rsqrt
tff(fact_6006_real__le__rsqrt,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Yb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(sqrt,Yb)) ) ).

% real_le_rsqrt
tff(fact_6007_sqrt__le__D,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,Xb)),Yb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_nat_real(aa_real_fun_nat_real(power_power_real,Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% sqrt_le_D
tff(fact_6008_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_6009_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_6010_take__bit__nat__def,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),Mb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% take_bit_nat_def
tff(fact_6011_take__bit__Suc__minus__bit0,axiom,
    ! [Nb: nat,K: num] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Nb)),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(Nb),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_6012_take__bit__int__less__exp,axiom,
    ! [Nb: nat,K: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se2923211474154528505it_int(Nb),K)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ).

% take_bit_int_less_exp
tff(fact_6013_take__bit__int__def,axiom,
    ! [Nb: nat,K: int] : aa_int_int(bit_se2923211474154528505it_int(Nb),K) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ).

% take_bit_int_def
tff(fact_6014_cong__exp__iff__simps_I7_J,axiom,
    ! [Q2: num,Nb: 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,Q2))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Nb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Nb)),aa_num_nat(numeral_numeral_nat,Q2)) = zero_zero_nat ) ) ).

% cong_exp_iff_simps(7)
tff(fact_6015_cong__exp__iff__simps_I7_J,axiom,
    ! [Q2: num,Nb: num] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,one)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) )
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,Nb)),aa_num_int(numeral_numeral_int,Q2)) = zero_zero_int ) ) ).

% cong_exp_iff_simps(7)
tff(fact_6016_cong__exp__iff__simps_I11_J,axiom,
    ! [Mb: num,Q2: num] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Mb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,one)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q2))) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Q2)) = zero_zero_nat ) ) ).

% cong_exp_iff_simps(11)
tff(fact_6017_cong__exp__iff__simps_I11_J,axiom,
    ! [Mb: num,Q2: num] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Mb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,one)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) )
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,Q2)) = zero_zero_int ) ) ).

% cong_exp_iff_simps(11)
tff(fact_6018_Suc__div__eq__add3__div,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,Mb)))),Nb) = 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))),Mb)),Nb) ).

% Suc_div_eq_add3_div
tff(fact_6019_real__of__int__div2,axiom,
    ! [Nb: int,Xb: int] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(aa_int_real(ring_1_of_int_real,Nb)),aa_int_real(ring_1_of_int_real,Xb))),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(divide_divide_int,Nb),Xb)))) ).

% real_of_int_div2
tff(fact_6020_real__of__int__div3,axiom,
    ! [Nb: int,Xb: int] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(aa_int_real(ring_1_of_int_real,Nb)),aa_int_real(ring_1_of_int_real,Xb))),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(divide_divide_int,Nb),Xb)))),one_one_real) ).

% real_of_int_div3
tff(fact_6021_Suc__mod__eq__add3__mod,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,Mb)))),Nb) = 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))),Mb)),Nb) ).

% Suc_mod_eq_add3_mod
tff(fact_6022_take__bit__eq__0__iff,axiom,
    ! [Nb: nat,A: code_integer] :
      ( ( bit_se1745604003318907178nteger(Nb,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))),Nb)),A) ) ).

% take_bit_eq_0_iff
tff(fact_6023_take__bit__eq__0__iff,axiom,
    ! [Nb: nat,A: int] :
      ( ( aa_int_int(bit_se2923211474154528505it_int(Nb),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))),Nb)),A) ) ).

% take_bit_eq_0_iff
tff(fact_6024_take__bit__eq__0__iff,axiom,
    ! [Nb: nat,A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),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))),Nb)),A) ) ).

% take_bit_eq_0_iff
tff(fact_6025_take__bit__nat__less__self__iff,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),Mb)),Mb)
    <=> 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))),Nb)),Mb) ) ).

% take_bit_nat_less_self_iff
tff(fact_6026_real__sqrt__unique,axiom,
    ! [Yb: real,Xb: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = Xb )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Yb)
       => ( aa_real_real(sqrt,Xb) = Yb ) ) ) ).

% real_sqrt_unique
tff(fact_6027_real__le__lsqrt,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Yb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_nat_real(aa_real_fun_nat_real(power_power_real,Yb),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,Xb)),Yb) ) ) ) ).

% real_le_lsqrt
tff(fact_6028_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(divide_divide_real(U),aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),U) ) ).

% lemma_real_divide_sqrt_less
tff(fact_6029_real__sqrt__sum__squares__eq__cancel2,axiom,
    ! [Xb: real,Yb: real] :
      ( ( aa_real_real(sqrt,aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) = Yb )
     => ( Xb = zero_zero_real ) ) ).

% real_sqrt_sum_squares_eq_cancel2
tff(fact_6030_real__sqrt__sum__squares__eq__cancel,axiom,
    ! [Xb: real,Yb: real] :
      ( ( aa_real_real(sqrt,aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) = Xb )
     => ( Yb = zero_zero_real ) ) ).

% real_sqrt_sum_squares_eq_cancel
tff(fact_6031_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(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(plus_plus_real(A),C)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(plus_plus_real(B),D)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(plus_plus_real(aa_real_real(sqrt,aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(sqrt,aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,C),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,D),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ).

% real_sqrt_sum_squares_triangle_ineq
tff(fact_6032_real__sqrt__sum__squares__ge2,axiom,
    ! [Yb: real,Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Yb),aa_real_real(sqrt,aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% real_sqrt_sum_squares_ge2
tff(fact_6033_real__sqrt__sum__squares__ge1,axiom,
    ! [Xb: real,Yb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(sqrt,aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% real_sqrt_sum_squares_ge1
tff(fact_6034_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_6035_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_6036_take__bit__int__less__self__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se2923211474154528505it_int(Nb),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))),Nb)),K) ) ).

% take_bit_int_less_self_iff
tff(fact_6037_take__bit__int__greater__eq__self__iff,axiom,
    ! [K: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),aa_int_int(bit_se2923211474154528505it_int(Nb),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))),Nb)) ) ).

% take_bit_int_greater_eq_self_iff
tff(fact_6038_real__less__lsqrt,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Yb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_nat_real(aa_real_fun_nat_real(power_power_real,Yb),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,Xb)),Yb) ) ) ) ).

% real_less_lsqrt
tff(fact_6039_sqrt__sum__squares__le__sum,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Yb)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(plus_plus_real(Xb),Yb)) ) ) ).

% sqrt_sum_squares_le_sum
tff(fact_6040_sqrt__even__pow2,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_real_real(sqrt,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ).

% sqrt_even_pow2
tff(fact_6041_take__bit__int__eq__self,axiom,
    ! [K: int,Nb: 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))),Nb))
       => ( aa_int_int(bit_se2923211474154528505it_int(Nb),K) = K ) ) ) ).

% take_bit_int_eq_self
tff(fact_6042_take__bit__int__eq__self__iff,axiom,
    ! [Nb: nat,K: int] :
      ( ( aa_int_int(bit_se2923211474154528505it_int(Nb),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))),Nb)) ) ) ).

% take_bit_int_eq_self_iff
tff(fact_6043_take__bit__incr__eq,axiom,
    ! [Nb: nat,K: int] :
      ( ( aa_int_int(bit_se2923211474154528505it_int(Nb),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))),Nb)),one_one_int) )
     => ( aa_int_int(bit_se2923211474154528505it_int(Nb),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(Nb),K)) ) ) ).

% take_bit_incr_eq
tff(fact_6044_take__bit__Suc__minus__1__eq,axiom,
    ! [Nb: nat] : bit_se1745604003318907178nteger(aa_nat_nat(suc,Nb),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,Nb)),one_one_Code_integer) ).

% take_bit_Suc_minus_1_eq
tff(fact_6045_take__bit__Suc__minus__1__eq,axiom,
    ! [Nb: nat] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Nb)),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,Nb))),one_one_int) ).

% take_bit_Suc_minus_1_eq
tff(fact_6046_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_6047_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_6048_take__bit__Suc,axiom,
    ! [Nb: nat,A: int] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Nb)),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(Nb),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(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ).

% take_bit_Suc
tff(fact_6049_take__bit__Suc,axiom,
    ! [Nb: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_nat_nat(suc,Nb)),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,Nb),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_6050_real__sqrt__sum__squares__mult__ge__zero,axiom,
    ! [Xb: real,Yb: real,Xaa: real,Yaa: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(sqrt,aa_real_real(times_times_real(aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Xaa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Yaa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ).

% real_sqrt_sum_squares_mult_ge_zero
tff(fact_6051_real__sqrt__power__even,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sqrt,Xb)),Nb) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ) ).

% real_sqrt_power_even
tff(fact_6052_arith__geo__mean__sqrt,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Yb)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,aa_real_real(times_times_real(Xb),Yb))),aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(Xb),Yb)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ).

% arith_geo_mean_sqrt
tff(fact_6053_take__bit__int__less__eq,axiom,
    ! [Nb: 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))),Nb)),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(bit_se2923211474154528505it_int(Nb),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))),Nb))) ) ) ).

% take_bit_int_less_eq
tff(fact_6054_take__bit__int__greater__eq,axiom,
    ! [K: int,Nb: 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))),Nb))),aa_int_int(bit_se2923211474154528505it_int(Nb),K)) ) ).

% take_bit_int_greater_eq
tff(fact_6055_take__bit__minus__small__eq,axiom,
    ! [K: int,Nb: 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))),Nb))
       => ( aa_int_int(bit_se2923211474154528505it_int(Nb),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))),Nb)),K) ) ) ) ).

% take_bit_minus_small_eq
tff(fact_6056_signed__take__bit__eq__take__bit__shift,axiom,
    ! [Nb: nat,K: int] : aa_int_int(bit_ri631733984087533419it_int(Nb),K) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Nb)),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))),Nb)))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ).

% signed_take_bit_eq_take_bit_shift
tff(fact_6057_stable__imp__take__bit__eq,axiom,
    ! [A: code_integer,Nb: nat] :
      ( ( divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) = A )
     => ( bit_se1745604003318907178nteger(Nb,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))),Nb),one_one_Code_integer)) ) ) ).

% stable_imp_take_bit_eq
tff(fact_6058_stable__imp__take__bit__eq,axiom,
    ! [A: int,Nb: 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(Nb),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))),Nb)),one_one_int)) ) ) ).

% stable_imp_take_bit_eq
tff(fact_6059_stable__imp__take__bit__eq,axiom,
    ! [A: nat,Nb: 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,Nb),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))),Nb)),one_one_nat)) ) ) ).

% stable_imp_take_bit_eq
tff(fact_6060_divmod__step__nat__def,axiom,
    ! [L: num,Qr: product_prod_nat_nat] : unique5026877609467782581ep_nat(L,Qr) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_be(L)),Qr) ).

% divmod_step_nat_def
tff(fact_6061_divmod__step__int__def,axiom,
    ! [L: num,Qr: product_prod_int_int] : aa_Pro6970504127699319548nt_int(unique5024387138958732305ep_int(L),Qr) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_bf(L)),Qr) ).

% divmod_step_int_def
tff(fact_6062_odd__mod__4__div__2,axiom,
    ! [Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),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,Nb),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_6063_arsinh__real__aux,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(plus_plus_real(Xb),aa_real_real(sqrt,aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)))) ).

% arsinh_real_aux
tff(fact_6064_mod__exhaust__less__4,axiom,
    ! [Mb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),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,Mb),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,Mb),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,Mb),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_6065_floor__exists1,axiom,
    ! [Xb: real] :
    ? [X3: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,X3)),Xb)
      & aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),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)),Xb)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,Y3),one_one_int))) )
         => ( Y3 = X3 ) ) ) ).

% floor_exists1
tff(fact_6066_floor__exists1,axiom,
    ! [Xb: rat] :
    ? [X3: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,X3)),Xb)
      & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),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)),Xb)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,Y3),one_one_int))) )
         => ( Y3 = X3 ) ) ) ).

% floor_exists1
tff(fact_6067_floor__exists,axiom,
    ! [Xb: real] :
    ? [Z2: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z2)),Xb)
      & aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),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_6068_floor__exists,axiom,
    ! [Xb: rat] :
    ? [Z2: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z2)),Xb)
      & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),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_6069_divmod__algorithm__code_I6_J,axiom,
    ! [Mb: num,Nb: num] : unique5052692396658037445od_int(aa_num_num(bit1,Mb),aa_num_num(bit0,Nb)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_bw),unique5052692396658037445od_int(Mb,Nb)) ).

% divmod_algorithm_code(6)
tff(fact_6070_divmod__algorithm__code_I6_J,axiom,
    ! [Mb: num,Nb: num] : unique5055182867167087721od_nat(aa_num_num(bit1,Mb),aa_num_num(bit0,Nb)) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_bx),unique5055182867167087721od_nat(Mb,Nb)) ).

% divmod_algorithm_code(6)
tff(fact_6071_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_6072_dbl__dec__simps_I4_J,axiom,
    aa_real_real(neg_nu6075765906172075777c_real,aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(bit1,one))) ).

% dbl_dec_simps(4)
tff(fact_6073_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_6074_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_6075_dbl__dec__simps_I4_J,axiom,
    aa_rat_rat(neg_nu3179335615603231917ec_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit1,one))) ).

% dbl_dec_simps(4)
tff(fact_6076_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_6077_case__prodI2,axiom,
    ! [P2: produc6271795597528267376eger_o,C: fun_Co480733329897759773un_o_o] :
      ( ! [A4: code_integer,B3: $o] :
          ( ( P2 = produc6677183202524767010eger_o(A4,(B3)) )
         => aa_o_o(aa_Cod6379977994291289310un_o_o(C,A4),(B3)) )
     => aa_Pro5286387174248033258er_o_o(produc7828578312038201481er_o_o(C),P2) ) ).

% case_prodI2
tff(fact_6078_case__prodI2,axiom,
    ! [P2: product_prod_num_num,C: fun_num_fun_num_o] :
      ( ! [A4: num,B3: num] :
          ( ( P2 = product_Pair_num_num(A4,B3) )
         => aa_num_o(aa_num_fun_num_o(C,A4),B3) )
     => aa_Pro7685007914727650815_num_o(produc5703948589228662326_num_o(C),P2) ) ).

% case_prodI2
tff(fact_6079_case__prodI2,axiom,
    ! [P2: product_prod_nat_num,C: fun_nat_fun_num_o] :
      ( ! [A4: nat,B3: num] :
          ( ( P2 = product_Pair_nat_num(A4,B3) )
         => aa_num_o(aa_nat_fun_num_o(C,A4),B3) )
     => aa_Pro6908818167415475913_num_o(produc4927758841916487424_num_o(C),P2) ) ).

% case_prodI2
tff(fact_6080_case__prodI2,axiom,
    ! [P2: product_prod_nat_nat,C: fun_nat_fun_nat_o] :
      ( ! [A4: nat,B3: nat] :
          ( ( P2 = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A4),B3) )
         => aa_nat_o(aa_nat_fun_nat_o(C,A4),B3) )
     => aa_Pro6150637604445964947_nat_o(produc6081775807080527818_nat_o(C),P2) ) ).

% case_prodI2
tff(fact_6081_case__prodI2,axiom,
    ! [P2: product_prod_int_int,C: fun_int_fun_int_o] :
      ( ! [A4: int,B3: int] :
          ( ( P2 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A4),B3) )
         => aa_int_o(aa_int_fun_int_o(C,A4),B3) )
     => aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(C),P2) ) ).

% case_prodI2
tff(fact_6082_case__prodI,axiom,
    ! [F: fun_Co480733329897759773un_o_o,A: code_integer,B: $o] :
      ( aa_o_o(aa_Cod6379977994291289310un_o_o(F,A),(B))
     => aa_Pro5286387174248033258er_o_o(produc7828578312038201481er_o_o(F),produc6677183202524767010eger_o(A,(B))) ) ).

% case_prodI
tff(fact_6083_case__prodI,axiom,
    ! [F: fun_num_fun_num_o,A: num,B: num] :
      ( aa_num_o(aa_num_fun_num_o(F,A),B)
     => aa_Pro7685007914727650815_num_o(produc5703948589228662326_num_o(F),product_Pair_num_num(A,B)) ) ).

% case_prodI
tff(fact_6084_case__prodI,axiom,
    ! [F: fun_nat_fun_num_o,A: nat,B: num] :
      ( aa_num_o(aa_nat_fun_num_o(F,A),B)
     => aa_Pro6908818167415475913_num_o(produc4927758841916487424_num_o(F),product_Pair_nat_num(A,B)) ) ).

% case_prodI
tff(fact_6085_case__prodI,axiom,
    ! [F: fun_nat_fun_nat_o,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(F,A),B)
     => aa_Pro6150637604445964947_nat_o(produc6081775807080527818_nat_o(F),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B)) ) ).

% case_prodI
tff(fact_6086_case__prodI,axiom,
    ! [F: fun_int_fun_int_o,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(F,A),B)
     => aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(F),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B)) ) ).

% case_prodI
tff(fact_6087_mem__case__prodI2,axiom,
    ! [P2: produc6271795597528267376eger_o,Z: complex,C: fun_Co5015990272074005067omplex] :
      ( ! [A4: code_integer,B3: $o] :
          ( ( P2 = produc6677183202524767010eger_o(A4,(B3)) )
         => member_complex(Z,aa_o_set_complex(aa_Cod2779864889637189172omplex(C,A4),(B3))) )
     => member_complex(Z,produc1043322548047392435omplex(C,P2)) ) ).

% mem_case_prodI2
tff(fact_6088_mem__case__prodI2,axiom,
    ! [P2: produc6271795597528267376eger_o,Z: real,C: fun_Co5368275265270938953t_real] :
      ( ! [A4: code_integer,B3: $o] :
          ( ( P2 = produc6677183202524767010eger_o(A4,(B3)) )
         => member_real(Z,aa_o_set_real(aa_Cod7471715548787160626t_real(C,A4),(B3))) )
     => member_real(Z,produc242741666403216561t_real(C,P2)) ) ).

% mem_case_prodI2
tff(fact_6089_mem__case__prodI2,axiom,
    ! [P2: produc6271795597528267376eger_o,Z: nat,C: fun_Co7401227239307276909et_nat] :
      ( ! [A4: code_integer,B3: $o] :
          ( ( P2 = produc6677183202524767010eger_o(A4,(B3)) )
         => aa_set_nat_o(member_nat(Z),aa_o_set_nat(aa_Cod3851761462451373910et_nat(C,A4),(B3))) )
     => aa_set_nat_o(member_nat(Z),produc5431169771168744661et_nat(C,P2)) ) ).

% mem_case_prodI2
tff(fact_6090_mem__case__prodI2,axiom,
    ! [P2: produc6271795597528267376eger_o,Z: int,C: fun_Co2587678820519358153et_int] :
      ( ! [A4: code_integer,B3: $o] :
          ( ( P2 = produc6677183202524767010eger_o(A4,(B3)) )
         => member_int(Z,aa_o_set_int(aa_Cod1361678317519647154et_int(C,A4),(B3))) )
     => member_int(Z,produc1253318751659547953et_int(C,P2)) ) ).

% mem_case_prodI2
tff(fact_6091_mem__case__prodI2,axiom,
    ! [P2: product_prod_num_num,Z: complex,C: fun_nu9041852840018610706omplex] :
      ( ! [A4: num,B3: num] :
          ( ( P2 = product_Pair_num_num(A4,B3) )
         => member_complex(Z,aa_num_set_complex(aa_num241437687621652691omplex(C,A4),B3)) )
     => member_complex(Z,produc2866383454006189126omplex(C,P2)) ) ).

% mem_case_prodI2
tff(fact_6092_mem__case__prodI2,axiom,
    ! [P2: product_prod_num_num,Z: real,C: fun_nu4698968662094489488t_real] :
      ( ! [A4: num,B3: num] :
          ( ( P2 = product_Pair_num_num(A4,B3) )
         => member_real(Z,aa_num_set_real(aa_num2469771427700027217t_real(C,A4),B3)) )
     => member_real(Z,produc8296048397933160132t_real(C,P2)) ) ).

% mem_case_prodI2
tff(fact_6093_mem__case__prodI2,axiom,
    ! [P2: product_prod_num_num,Z: nat,C: fun_nu530747996811530292et_nat] :
      ( ! [A4: num,B3: num] :
          ( ( P2 = product_Pair_num_num(A4,B3) )
         => aa_set_nat_o(member_nat(Z),aa_num_set_nat(aa_num5617506412915843061et_nat(C,A4),B3)) )
     => aa_set_nat_o(member_nat(Z),produc1361121860356118632et_nat(C,P2)) ) ).

% mem_case_prodI2
tff(fact_6094_mem__case__prodI2,axiom,
    ! [P2: product_prod_num_num,Z: int,C: fun_nu4940571614878387344et_int] :
      ( ! [A4: num,B3: num] :
          ( ( P2 = product_Pair_num_num(A4,B3) )
         => member_int(Z,aa_num_set_int(aa_num3127423267984116305et_int(C,A4),B3)) )
     => member_int(Z,produc6406642877701697732et_int(C,P2)) ) ).

% mem_case_prodI2
tff(fact_6095_mem__case__prodI2,axiom,
    ! [P2: product_prod_nat_num,Z: complex,C: fun_na3925014923085203400omplex] :
      ( ! [A4: nat,B3: num] :
          ( ( P2 = product_Pair_nat_num(A4,B3) )
         => member_complex(Z,aa_num_set_complex(aa_nat4347971807543021193omplex(C,A4),B3)) )
     => member_complex(Z,produc6231982587499038204omplex(C,P2)) ) ).

% mem_case_prodI2
tff(fact_6096_mem__case__prodI2,axiom,
    ! [P2: product_prod_nat_num,Z: real,C: fun_na3666575854997362246t_real] :
      ( ! [A4: nat,B3: num] :
          ( ( P2 = product_Pair_nat_num(A4,B3) )
         => member_real(Z,aa_num_set_real(aa_nat1437378620602899975t_real(C,A4),B3)) )
     => member_real(Z,produc1435849484188172666t_real(C,P2)) ) ).

% mem_case_prodI2
tff(fact_6097_mem__case__prodI,axiom,
    ! [Z: complex,C: fun_Co5015990272074005067omplex,A: code_integer,B: $o] :
      ( member_complex(Z,aa_o_set_complex(aa_Cod2779864889637189172omplex(C,A),(B)))
     => member_complex(Z,produc1043322548047392435omplex(C,produc6677183202524767010eger_o(A,(B)))) ) ).

% mem_case_prodI
tff(fact_6098_mem__case__prodI,axiom,
    ! [Z: real,C: fun_Co5368275265270938953t_real,A: code_integer,B: $o] :
      ( member_real(Z,aa_o_set_real(aa_Cod7471715548787160626t_real(C,A),(B)))
     => member_real(Z,produc242741666403216561t_real(C,produc6677183202524767010eger_o(A,(B)))) ) ).

% mem_case_prodI
tff(fact_6099_mem__case__prodI,axiom,
    ! [Z: nat,C: fun_Co7401227239307276909et_nat,A: code_integer,B: $o] :
      ( aa_set_nat_o(member_nat(Z),aa_o_set_nat(aa_Cod3851761462451373910et_nat(C,A),(B)))
     => aa_set_nat_o(member_nat(Z),produc5431169771168744661et_nat(C,produc6677183202524767010eger_o(A,(B)))) ) ).

% mem_case_prodI
tff(fact_6100_mem__case__prodI,axiom,
    ! [Z: int,C: fun_Co2587678820519358153et_int,A: code_integer,B: $o] :
      ( member_int(Z,aa_o_set_int(aa_Cod1361678317519647154et_int(C,A),(B)))
     => member_int(Z,produc1253318751659547953et_int(C,produc6677183202524767010eger_o(A,(B)))) ) ).

% mem_case_prodI
tff(fact_6101_mem__case__prodI,axiom,
    ! [Z: complex,C: fun_nu9041852840018610706omplex,A: num,B: num] :
      ( member_complex(Z,aa_num_set_complex(aa_num241437687621652691omplex(C,A),B))
     => member_complex(Z,produc2866383454006189126omplex(C,product_Pair_num_num(A,B))) ) ).

% mem_case_prodI
tff(fact_6102_mem__case__prodI,axiom,
    ! [Z: real,C: fun_nu4698968662094489488t_real,A: num,B: num] :
      ( member_real(Z,aa_num_set_real(aa_num2469771427700027217t_real(C,A),B))
     => member_real(Z,produc8296048397933160132t_real(C,product_Pair_num_num(A,B))) ) ).

% mem_case_prodI
tff(fact_6103_mem__case__prodI,axiom,
    ! [Z: nat,C: fun_nu530747996811530292et_nat,A: num,B: num] :
      ( aa_set_nat_o(member_nat(Z),aa_num_set_nat(aa_num5617506412915843061et_nat(C,A),B))
     => aa_set_nat_o(member_nat(Z),produc1361121860356118632et_nat(C,product_Pair_num_num(A,B))) ) ).

% mem_case_prodI
tff(fact_6104_mem__case__prodI,axiom,
    ! [Z: int,C: fun_nu4940571614878387344et_int,A: num,B: num] :
      ( member_int(Z,aa_num_set_int(aa_num3127423267984116305et_int(C,A),B))
     => member_int(Z,produc6406642877701697732et_int(C,product_Pair_num_num(A,B))) ) ).

% mem_case_prodI
tff(fact_6105_mem__case__prodI,axiom,
    ! [Z: complex,C: fun_na3925014923085203400omplex,A: nat,B: num] :
      ( member_complex(Z,aa_num_set_complex(aa_nat4347971807543021193omplex(C,A),B))
     => member_complex(Z,produc6231982587499038204omplex(C,product_Pair_nat_num(A,B))) ) ).

% mem_case_prodI
tff(fact_6106_mem__case__prodI,axiom,
    ! [Z: real,C: fun_na3666575854997362246t_real,A: nat,B: num] :
      ( member_real(Z,aa_num_set_real(aa_nat1437378620602899975t_real(C,A),B))
     => member_real(Z,produc1435849484188172666t_real(C,product_Pair_nat_num(A,B))) ) ).

% mem_case_prodI
tff(fact_6107_case__prodI2_H,axiom,
    ! [P2: product_prod_nat_nat,C: fun_na5914473681949881652_nat_o,Xb: product_prod_nat_nat] :
      ( ! [A4: nat,B3: nat] :
          ( ( aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A4),B3) = P2 )
         => aa_Pro6150637604445964947_nat_o(aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(C,A4),B3),Xb) )
     => aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(C),P2),Xb) ) ).

% case_prodI2'
tff(fact_6108_dbl__dec__simps_I3_J,axiom,
    aa_complex_complex(neg_nu6511756317524482435omplex,one_one_complex) = one_one_complex ).

% dbl_dec_simps(3)
tff(fact_6109_dbl__dec__simps_I3_J,axiom,
    aa_real_real(neg_nu6075765906172075777c_real,one_one_real) = one_one_real ).

% dbl_dec_simps(3)
tff(fact_6110_dbl__dec__simps_I3_J,axiom,
    aa_rat_rat(neg_nu3179335615603231917ec_rat,one_one_rat) = one_one_rat ).

% dbl_dec_simps(3)
tff(fact_6111_dbl__dec__simps_I3_J,axiom,
    aa_int_int(neg_nu3811975205180677377ec_int,one_one_int) = one_one_int ).

% dbl_dec_simps(3)
tff(fact_6112_pred__numeral__simps_I1_J,axiom,
    aa_num_nat(pred_numeral,one) = zero_zero_nat ).

% pred_numeral_simps(1)
tff(fact_6113_Suc__eq__numeral,axiom,
    ! [Nb: nat,K: num] :
      ( ( aa_nat_nat(suc,Nb) = aa_num_nat(numeral_numeral_nat,K) )
    <=> ( Nb = aa_num_nat(pred_numeral,K) ) ) ).

% Suc_eq_numeral
tff(fact_6114_eq__numeral__Suc,axiom,
    ! [K: num,Nb: nat] :
      ( ( aa_num_nat(numeral_numeral_nat,K) = aa_nat_nat(suc,Nb) )
    <=> ( aa_num_nat(pred_numeral,K) = Nb ) ) ).

% eq_numeral_Suc
tff(fact_6115_less__Suc__numeral,axiom,
    ! [Nb: nat,K: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),aa_num_nat(numeral_numeral_nat,K))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_num_nat(pred_numeral,K)) ) ).

% less_Suc_numeral
tff(fact_6116_less__numeral__Suc,axiom,
    ! [K: num,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(suc,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(pred_numeral,K)),Nb) ) ).

% less_numeral_Suc
tff(fact_6117_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_6118_le__Suc__numeral,axiom,
    ! [Nb: nat,K: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Nb)),aa_num_nat(numeral_numeral_nat,K))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_num_nat(pred_numeral,K)) ) ).

% le_Suc_numeral
tff(fact_6119_le__numeral__Suc,axiom,
    ! [K: num,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(suc,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(pred_numeral,K)),Nb) ) ).

% le_numeral_Suc
tff(fact_6120_diff__Suc__numeral,axiom,
    ! [Nb: nat,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,Nb)),aa_num_nat(numeral_numeral_nat,K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_num_nat(pred_numeral,K)) ).

% diff_Suc_numeral
tff(fact_6121_diff__numeral__Suc,axiom,
    ! [K: num,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_num_nat(pred_numeral,K)),Nb) ).

% diff_numeral_Suc
tff(fact_6122_max__Suc__numeral,axiom,
    ! [Nb: nat,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(suc,Nb)),aa_num_nat(numeral_numeral_nat,K)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Nb),aa_num_nat(pred_numeral,K))) ).

% max_Suc_numeral
tff(fact_6123_max__numeral__Suc,axiom,
    ! [K: num,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(suc,Nb)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(pred_numeral,K)),Nb)) ).

% max_numeral_Suc
tff(fact_6124_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_6125_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_6126_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_6127_dbl__dec__simps_I2_J,axiom,
    aa_rat_rat(neg_nu3179335615603231917ec_rat,zero_zero_rat) = aa_rat_rat(uminus_uminus_rat,one_one_rat) ).

% dbl_dec_simps(2)
tff(fact_6128_dbl__dec__simps_I2_J,axiom,
    neg_nu7757733837767384882nteger(zero_z3403309356797280102nteger) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% dbl_dec_simps(2)
tff(fact_6129_numeral__div__minus__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_int_int(uminus_uminus_int,aa_Pro6799258434465214121nt_int(adjust_div,unique5052692396658037445od_int(Mb,Nb))) ).

% numeral_div_minus_numeral
tff(fact_6130_minus__numeral__div__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),aa_num_int(numeral_numeral_int,Nb)) = aa_int_int(uminus_uminus_int,aa_Pro6799258434465214121nt_int(adjust_div,unique5052692396658037445od_int(Mb,Nb))) ).

% minus_numeral_div_numeral
tff(fact_6131_dvd__numeral__simp,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(Mb)),numera6620942414471956472nteger(Nb))
    <=> unique5706413561485394159nteger(unique3479559517661332726nteger(Nb,Mb)) ) ).

% dvd_numeral_simp
tff(fact_6132_dvd__numeral__simp,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,Nb))
    <=> aa_Pro4592068669816069595_int_o(unique6319869463603278526ux_int,unique5052692396658037445od_int(Nb,Mb)) ) ).

% dvd_numeral_simp
tff(fact_6133_dvd__numeral__simp,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Nb))
    <=> aa_Pro6150637604445964947_nat_o(unique6322359934112328802ux_nat,unique5055182867167087721od_nat(Nb,Mb)) ) ).

% dvd_numeral_simp
tff(fact_6134_divmod__algorithm__code_I2_J,axiom,
    ! [Mb: num] : unique5052692396658037445od_int(Mb,one) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_num_int(numeral_numeral_int,Mb)),zero_zero_int) ).

% divmod_algorithm_code(2)
tff(fact_6135_divmod__algorithm__code_I2_J,axiom,
    ! [Mb: num] : unique5055182867167087721od_nat(Mb,one) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_num_nat(numeral_numeral_nat,Mb)),zero_zero_nat) ).

% divmod_algorithm_code(2)
tff(fact_6136_divmod__algorithm__code_I3_J,axiom,
    ! [Nb: num] : unique5052692396658037445od_int(one,aa_num_num(bit0,Nb)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),aa_num_int(numeral_numeral_int,one)) ).

% divmod_algorithm_code(3)
tff(fact_6137_divmod__algorithm__code_I3_J,axiom,
    ! [Nb: num] : unique5055182867167087721od_nat(one,aa_num_num(bit0,Nb)) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,one)) ).

% divmod_algorithm_code(3)
tff(fact_6138_divmod__algorithm__code_I4_J,axiom,
    ! [Nb: num] : unique5052692396658037445od_int(one,aa_num_num(bit1,Nb)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),aa_num_int(numeral_numeral_int,one)) ).

% divmod_algorithm_code(4)
tff(fact_6139_divmod__algorithm__code_I4_J,axiom,
    ! [Nb: num] : unique5055182867167087721od_nat(one,aa_num_num(bit1,Nb)) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,one)) ).

% divmod_algorithm_code(4)
tff(fact_6140_one__div__minus__numeral,axiom,
    ! [Nb: 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,Nb))) = aa_int_int(uminus_uminus_int,aa_Pro6799258434465214121nt_int(adjust_div,unique5052692396658037445od_int(one,Nb))) ).

% one_div_minus_numeral
tff(fact_6141_minus__one__div__numeral,axiom,
    ! [Nb: 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,Nb)) = aa_int_int(uminus_uminus_int,aa_Pro6799258434465214121nt_int(adjust_div,unique5052692396658037445od_int(one,Nb))) ).

% minus_one_div_numeral
tff(fact_6142_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_6143_divmod__algorithm__code_I5_J,axiom,
    ! [Mb: num,Nb: num] : unique5052692396658037445od_int(aa_num_num(bit0,Mb),aa_num_num(bit0,Nb)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_by),unique5052692396658037445od_int(Mb,Nb)) ).

% divmod_algorithm_code(5)
tff(fact_6144_divmod__algorithm__code_I5_J,axiom,
    ! [Mb: num,Nb: num] : unique5055182867167087721od_nat(aa_num_num(bit0,Mb),aa_num_num(bit0,Nb)) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_bz),unique5055182867167087721od_nat(Mb,Nb)) ).

% divmod_algorithm_code(5)
tff(fact_6145_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_6146_divmod__algorithm__code_I7_J,axiom,
    ! [Mb: num,Nb: num] :
      unique5055182867167087721od_nat(aa_num_num(bit0,Mb),aa_num_num(bit1,Nb)) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Mb),Nb),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Mb))),unique5026877609467782581ep_nat(aa_num_num(bit1,Nb),unique5055182867167087721od_nat(aa_num_num(bit0,Mb),aa_num_num(bit0,aa_num_num(bit1,Nb))))) ).

% divmod_algorithm_code(7)
tff(fact_6147_divmod__algorithm__code_I7_J,axiom,
    ! [Mb: num,Nb: num] :
      unique5052692396658037445od_int(aa_num_num(bit0,Mb),aa_num_num(bit1,Nb)) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Mb),Nb),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Mb))),aa_Pro6970504127699319548nt_int(unique5024387138958732305ep_int(aa_num_num(bit1,Nb)),unique5052692396658037445od_int(aa_num_num(bit0,Mb),aa_num_num(bit0,aa_num_num(bit1,Nb))))) ).

% divmod_algorithm_code(7)
tff(fact_6148_divmod__algorithm__code_I7_J,axiom,
    ! [Mb: num,Nb: num] :
      unique3479559517661332726nteger(aa_num_num(bit0,Mb),aa_num_num(bit1,Nb)) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Mb),Nb),produc1086072967326762835nteger(zero_z3403309356797280102nteger,numera6620942414471956472nteger(aa_num_num(bit0,Mb))),unique4921790084139445826nteger(aa_num_num(bit1,Nb),unique3479559517661332726nteger(aa_num_num(bit0,Mb),aa_num_num(bit0,aa_num_num(bit1,Nb))))) ).

% divmod_algorithm_code(7)
tff(fact_6149_divmod__algorithm__code_I8_J,axiom,
    ! [Mb: num,Nb: num] :
      unique5055182867167087721od_nat(aa_num_num(bit1,Mb),aa_num_num(bit1,Nb)) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Nb),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Mb))),unique5026877609467782581ep_nat(aa_num_num(bit1,Nb),unique5055182867167087721od_nat(aa_num_num(bit1,Mb),aa_num_num(bit0,aa_num_num(bit1,Nb))))) ).

% divmod_algorithm_code(8)
tff(fact_6150_divmod__algorithm__code_I8_J,axiom,
    ! [Mb: num,Nb: num] :
      unique5052692396658037445od_int(aa_num_num(bit1,Mb),aa_num_num(bit1,Nb)) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Nb),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),aa_num_int(numeral_numeral_int,aa_num_num(bit1,Mb))),aa_Pro6970504127699319548nt_int(unique5024387138958732305ep_int(aa_num_num(bit1,Nb)),unique5052692396658037445od_int(aa_num_num(bit1,Mb),aa_num_num(bit0,aa_num_num(bit1,Nb))))) ).

% divmod_algorithm_code(8)
tff(fact_6151_divmod__algorithm__code_I8_J,axiom,
    ! [Mb: num,Nb: num] :
      unique3479559517661332726nteger(aa_num_num(bit1,Mb),aa_num_num(bit1,Nb)) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Nb),produc1086072967326762835nteger(zero_z3403309356797280102nteger,numera6620942414471956472nteger(aa_num_num(bit1,Mb))),unique4921790084139445826nteger(aa_num_num(bit1,Nb),unique3479559517661332726nteger(aa_num_num(bit1,Mb),aa_num_num(bit0,aa_num_num(bit1,Nb))))) ).

% divmod_algorithm_code(8)
tff(fact_6152_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_6153_mem__case__prodE,axiom,
    ! [Z: complex,C: fun_Co5015990272074005067omplex,P2: produc6271795597528267376eger_o] :
      ( member_complex(Z,produc1043322548047392435omplex(C,P2))
     => ~ ! [X3: code_integer,Y: $o] :
            ( ( P2 = produc6677183202524767010eger_o(X3,(Y)) )
           => ~ member_complex(Z,aa_o_set_complex(aa_Cod2779864889637189172omplex(C,X3),(Y))) ) ) ).

% mem_case_prodE
tff(fact_6154_mem__case__prodE,axiom,
    ! [Z: real,C: fun_Co5368275265270938953t_real,P2: produc6271795597528267376eger_o] :
      ( member_real(Z,produc242741666403216561t_real(C,P2))
     => ~ ! [X3: code_integer,Y: $o] :
            ( ( P2 = produc6677183202524767010eger_o(X3,(Y)) )
           => ~ member_real(Z,aa_o_set_real(aa_Cod7471715548787160626t_real(C,X3),(Y))) ) ) ).

% mem_case_prodE
tff(fact_6155_mem__case__prodE,axiom,
    ! [Z: nat,C: fun_Co7401227239307276909et_nat,P2: produc6271795597528267376eger_o] :
      ( aa_set_nat_o(member_nat(Z),produc5431169771168744661et_nat(C,P2))
     => ~ ! [X3: code_integer,Y: $o] :
            ( ( P2 = produc6677183202524767010eger_o(X3,(Y)) )
           => ~ aa_set_nat_o(member_nat(Z),aa_o_set_nat(aa_Cod3851761462451373910et_nat(C,X3),(Y))) ) ) ).

% mem_case_prodE
tff(fact_6156_mem__case__prodE,axiom,
    ! [Z: int,C: fun_Co2587678820519358153et_int,P2: produc6271795597528267376eger_o] :
      ( member_int(Z,produc1253318751659547953et_int(C,P2))
     => ~ ! [X3: code_integer,Y: $o] :
            ( ( P2 = produc6677183202524767010eger_o(X3,(Y)) )
           => ~ member_int(Z,aa_o_set_int(aa_Cod1361678317519647154et_int(C,X3),(Y))) ) ) ).

% mem_case_prodE
tff(fact_6157_mem__case__prodE,axiom,
    ! [Z: complex,C: fun_nu9041852840018610706omplex,P2: product_prod_num_num] :
      ( member_complex(Z,produc2866383454006189126omplex(C,P2))
     => ~ ! [X3: num,Y: num] :
            ( ( P2 = product_Pair_num_num(X3,Y) )
           => ~ member_complex(Z,aa_num_set_complex(aa_num241437687621652691omplex(C,X3),Y)) ) ) ).

% mem_case_prodE
tff(fact_6158_mem__case__prodE,axiom,
    ! [Z: real,C: fun_nu4698968662094489488t_real,P2: product_prod_num_num] :
      ( member_real(Z,produc8296048397933160132t_real(C,P2))
     => ~ ! [X3: num,Y: num] :
            ( ( P2 = product_Pair_num_num(X3,Y) )
           => ~ member_real(Z,aa_num_set_real(aa_num2469771427700027217t_real(C,X3),Y)) ) ) ).

% mem_case_prodE
tff(fact_6159_mem__case__prodE,axiom,
    ! [Z: nat,C: fun_nu530747996811530292et_nat,P2: product_prod_num_num] :
      ( aa_set_nat_o(member_nat(Z),produc1361121860356118632et_nat(C,P2))
     => ~ ! [X3: num,Y: num] :
            ( ( P2 = product_Pair_num_num(X3,Y) )
           => ~ aa_set_nat_o(member_nat(Z),aa_num_set_nat(aa_num5617506412915843061et_nat(C,X3),Y)) ) ) ).

% mem_case_prodE
tff(fact_6160_mem__case__prodE,axiom,
    ! [Z: int,C: fun_nu4940571614878387344et_int,P2: product_prod_num_num] :
      ( member_int(Z,produc6406642877701697732et_int(C,P2))
     => ~ ! [X3: num,Y: num] :
            ( ( P2 = product_Pair_num_num(X3,Y) )
           => ~ member_int(Z,aa_num_set_int(aa_num3127423267984116305et_int(C,X3),Y)) ) ) ).

% mem_case_prodE
tff(fact_6161_mem__case__prodE,axiom,
    ! [Z: complex,C: fun_na3925014923085203400omplex,P2: product_prod_nat_num] :
      ( member_complex(Z,produc6231982587499038204omplex(C,P2))
     => ~ ! [X3: nat,Y: num] :
            ( ( P2 = product_Pair_nat_num(X3,Y) )
           => ~ member_complex(Z,aa_num_set_complex(aa_nat4347971807543021193omplex(C,X3),Y)) ) ) ).

% mem_case_prodE
tff(fact_6162_mem__case__prodE,axiom,
    ! [Z: real,C: fun_na3666575854997362246t_real,P2: product_prod_nat_num] :
      ( member_real(Z,produc1435849484188172666t_real(C,P2))
     => ~ ! [X3: nat,Y: num] :
            ( ( P2 = product_Pair_nat_num(X3,Y) )
           => ~ member_real(Z,aa_num_set_real(aa_nat1437378620602899975t_real(C,X3),Y)) ) ) ).

% mem_case_prodE
tff(fact_6163_case__prodE,axiom,
    ! [C: fun_Co480733329897759773un_o_o,P2: produc6271795597528267376eger_o] :
      ( aa_Pro5286387174248033258er_o_o(produc7828578312038201481er_o_o(C),P2)
     => ~ ! [X3: code_integer,Y: $o] :
            ( ( P2 = produc6677183202524767010eger_o(X3,(Y)) )
           => ~ aa_o_o(aa_Cod6379977994291289310un_o_o(C,X3),(Y)) ) ) ).

% case_prodE
tff(fact_6164_case__prodE,axiom,
    ! [C: fun_num_fun_num_o,P2: product_prod_num_num] :
      ( aa_Pro7685007914727650815_num_o(produc5703948589228662326_num_o(C),P2)
     => ~ ! [X3: num,Y: num] :
            ( ( P2 = product_Pair_num_num(X3,Y) )
           => ~ aa_num_o(aa_num_fun_num_o(C,X3),Y) ) ) ).

% case_prodE
tff(fact_6165_case__prodE,axiom,
    ! [C: fun_nat_fun_num_o,P2: product_prod_nat_num] :
      ( aa_Pro6908818167415475913_num_o(produc4927758841916487424_num_o(C),P2)
     => ~ ! [X3: nat,Y: num] :
            ( ( P2 = product_Pair_nat_num(X3,Y) )
           => ~ aa_num_o(aa_nat_fun_num_o(C,X3),Y) ) ) ).

% case_prodE
tff(fact_6166_case__prodE,axiom,
    ! [C: fun_nat_fun_nat_o,P2: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(produc6081775807080527818_nat_o(C),P2)
     => ~ ! [X3: nat,Y: nat] :
            ( ( P2 = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),Y) )
           => ~ aa_nat_o(aa_nat_fun_nat_o(C,X3),Y) ) ) ).

% case_prodE
tff(fact_6167_case__prodE,axiom,
    ! [C: fun_int_fun_int_o,P2: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(C),P2)
     => ~ ! [X3: int,Y: int] :
            ( ( P2 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X3),Y) )
           => ~ aa_int_o(aa_int_fun_int_o(C,X3),Y) ) ) ).

% case_prodE
tff(fact_6168_case__prodD,axiom,
    ! [F: fun_Co480733329897759773un_o_o,A: code_integer,B: $o] :
      ( aa_Pro5286387174248033258er_o_o(produc7828578312038201481er_o_o(F),produc6677183202524767010eger_o(A,(B)))
     => aa_o_o(aa_Cod6379977994291289310un_o_o(F,A),(B)) ) ).

% case_prodD
tff(fact_6169_case__prodD,axiom,
    ! [F: fun_num_fun_num_o,A: num,B: num] :
      ( aa_Pro7685007914727650815_num_o(produc5703948589228662326_num_o(F),product_Pair_num_num(A,B))
     => aa_num_o(aa_num_fun_num_o(F,A),B) ) ).

% case_prodD
tff(fact_6170_case__prodD,axiom,
    ! [F: fun_nat_fun_num_o,A: nat,B: num] :
      ( aa_Pro6908818167415475913_num_o(produc4927758841916487424_num_o(F),product_Pair_nat_num(A,B))
     => aa_num_o(aa_nat_fun_num_o(F,A),B) ) ).

% case_prodD
tff(fact_6171_case__prodD,axiom,
    ! [F: fun_nat_fun_nat_o,A: nat,B: nat] :
      ( aa_Pro6150637604445964947_nat_o(produc6081775807080527818_nat_o(F),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B))
     => aa_nat_o(aa_nat_fun_nat_o(F,A),B) ) ).

% case_prodD
tff(fact_6172_case__prodD,axiom,
    ! [F: fun_int_fun_int_o,A: int,B: int] :
      ( aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(F),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),B))
     => aa_int_o(aa_int_fun_int_o(F,A),B) ) ).

% case_prodD
tff(fact_6173_case__prodE_H,axiom,
    ! [C: fun_na5914473681949881652_nat_o,P2: product_prod_nat_nat,Z: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(C),P2),Z)
     => ~ ! [X3: nat,Y: nat] :
            ( ( P2 = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),Y) )
           => ~ aa_Pro6150637604445964947_nat_o(aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(C,X3),Y),Z) ) ) ).

% case_prodE'
tff(fact_6174_case__prodD_H,axiom,
    ! [R: fun_na5914473681949881652_nat_o,A: nat,B: nat,C: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(R),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B)),C)
     => aa_Pro6150637604445964947_nat_o(aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(R,A),B),C) ) ).

% case_prodD'
tff(fact_6175_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_6176_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_6177_divmod__int__def,axiom,
    ! [Mb: num,Nb: num] : unique5052692396658037445od_int(Mb,Nb) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,Nb))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,Nb))) ).

% divmod_int_def
tff(fact_6178_Divides_Oadjust__div__def,axiom,
    ! [Qr: product_prod_int_int] : aa_Pro6799258434465214121nt_int(adjust_div,Qr) = aa_Pro6799258434465214121nt_int(produc8211389475949308722nt_int(aTP_Lamm_ca),Qr) ).

% Divides.adjust_div_def
tff(fact_6179_divmod__def,axiom,
    ! [Mb: num,Nb: num] : unique5052692396658037445od_int(Mb,Nb) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,Nb))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,Nb))) ).

% divmod_def
tff(fact_6180_divmod__def,axiom,
    ! [Mb: num,Nb: num] : unique5055182867167087721od_nat(Mb,Nb) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Nb))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Nb))) ).

% divmod_def
tff(fact_6181_divmod_H__nat__def,axiom,
    ! [Mb: num,Nb: num] : unique5055182867167087721od_nat(Mb,Nb) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Nb))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Nb))) ).

% divmod'_nat_def
tff(fact_6182_dbl__dec__def,axiom,
    ! [Xb: complex] : aa_complex_complex(neg_nu6511756317524482435omplex,Xb) = aa_complex_complex(minus_minus_complex(aa_complex_complex(plus_plus_complex(Xb),Xb)),one_one_complex) ).

% dbl_dec_def
tff(fact_6183_dbl__dec__def,axiom,
    ! [Xb: real] : aa_real_real(neg_nu6075765906172075777c_real,Xb) = aa_real_real(minus_minus_real(aa_real_real(plus_plus_real(Xb),Xb)),one_one_real) ).

% dbl_dec_def
tff(fact_6184_dbl__dec__def,axiom,
    ! [Xb: rat] : aa_rat_rat(neg_nu3179335615603231917ec_rat,Xb) = aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(Xb),Xb)),one_one_rat) ).

% dbl_dec_def
tff(fact_6185_dbl__dec__def,axiom,
    ! [Xb: int] : aa_int_int(neg_nu3811975205180677377ec_int,Xb) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Xb)),one_one_int) ).

% dbl_dec_def
tff(fact_6186_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_6187_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_6188_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_6189_exists__least__lemma,axiom,
    ! [P: fun_nat_o] :
      ( ~ aa_nat_o(P,zero_zero_nat)
     => ( ? [X_1: nat] : aa_nat_o(P,X_1)
       => ? [N: nat] :
            ( ~ aa_nat_o(P,N)
            & aa_nat_o(P,aa_nat_nat(suc,N)) ) ) ) ).

% exists_least_lemma
tff(fact_6190_ex__le__of__int,axiom,
    ! [Xb: real] :
    ? [Z2: int] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_int_real(ring_1_of_int_real,Z2)) ).

% ex_le_of_int
tff(fact_6191_ex__le__of__int,axiom,
    ! [Xb: rat] :
    ? [Z2: int] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),aa_int_rat(ring_1_of_int_rat,Z2)) ).

% ex_le_of_int
tff(fact_6192_ex__of__int__less,axiom,
    ! [Xb: real] :
    ? [Z2: int] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Z2)),Xb) ).

% ex_of_int_less
tff(fact_6193_ex__of__int__less,axiom,
    ! [Xb: rat] :
    ? [Z2: int] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Z2)),Xb) ).

% ex_of_int_less
tff(fact_6194_ex__less__of__int,axiom,
    ! [Xb: real] :
    ? [Z2: int] : aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_int_real(ring_1_of_int_real,Z2)) ).

% ex_less_of_int
tff(fact_6195_ex__less__of__int,axiom,
    ! [Xb: rat] :
    ? [Z2: int] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),aa_int_rat(ring_1_of_int_rat,Z2)) ).

% ex_less_of_int
tff(fact_6196_eq__diff__eq_H,axiom,
    ! [Xb: real,Yb: real,Z: real] :
      ( ( Xb = aa_real_real(minus_minus_real(Yb),Z) )
    <=> ( Yb = aa_real_real(plus_plus_real(Xb),Z) ) ) ).

% eq_diff_eq'
tff(fact_6197_divmod__divmod__step,axiom,
    ! [Mb: num,Nb: num] :
      unique5055182867167087721od_nat(Mb,Nb) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Nb),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,Mb)),unique5026877609467782581ep_nat(Nb,unique5055182867167087721od_nat(Mb,aa_num_num(bit0,Nb)))) ).

% divmod_divmod_step
tff(fact_6198_divmod__divmod__step,axiom,
    ! [Mb: num,Nb: num] :
      unique5052692396658037445od_int(Mb,Nb) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Nb),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),aa_num_int(numeral_numeral_int,Mb)),aa_Pro6970504127699319548nt_int(unique5024387138958732305ep_int(Nb),unique5052692396658037445od_int(Mb,aa_num_num(bit0,Nb)))) ).

% divmod_divmod_step
tff(fact_6199_divmod__divmod__step,axiom,
    ! [Mb: num,Nb: num] :
      unique3479559517661332726nteger(Mb,Nb) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Nb),produc1086072967326762835nteger(zero_z3403309356797280102nteger,numera6620942414471956472nteger(Mb)),unique4921790084139445826nteger(Nb,unique3479559517661332726nteger(Mb,aa_num_num(bit0,Nb)))) ).

% divmod_divmod_step
tff(fact_6200_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_6201_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_6202_arcosh__1,axiom,
    aa_real_real(arcosh_real,one_one_real) = zero_zero_real ).

% arcosh_1
tff(fact_6203_take__bit__numeral__minus__bit1,axiom,
    ! [L: num,K: num] : aa_int_int(bit_se2923211474154528505it_int(aa_num_nat(numeral_numeral_nat,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_se2923211474154528505it_int(aa_num_nat(pred_numeral,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(inc,K))))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).

% take_bit_numeral_minus_bit1
tff(fact_6204_divmod__nat__if,axiom,
    ! [Mb: nat,Nb: nat] :
      divmod_nat(Mb,Nb) = $ite(
        ( ( Nb = zero_zero_nat )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ),
        aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),Mb),
        aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_cb),divmod_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb),Nb)) ) ).

% divmod_nat_if
tff(fact_6205_round__unique,axiom,
    ! [Xb: real,Yb: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(minus_minus_real(Xb),aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_int_real(ring_1_of_int_real,Yb))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Yb)),aa_real_real(plus_plus_real(Xb),aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => ( aa_real_int(archim8280529875227126926d_real,Xb) = Yb ) ) ) ).

% round_unique
tff(fact_6206_round__unique,axiom,
    ! [Xb: rat,Yb: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(Xb),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))),aa_int_rat(ring_1_of_int_rat,Yb))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Yb)),aa_rat_rat(plus_plus_rat(Xb),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))))
       => ( aa_rat_int(archim7778729529865785530nd_rat,Xb) = Yb ) ) ) ).

% round_unique
tff(fact_6207_arsinh__0,axiom,
    aa_real_real(arsinh_real,zero_zero_real) = zero_zero_real ).

% arsinh_0
tff(fact_6208_artanh__0,axiom,
    aa_real_real(artanh_real,zero_zero_real) = zero_zero_real ).

% artanh_0
tff(fact_6209_split__part,axiom,
    ! [P: $o,Q: fun_int_fun_int_o,X: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(aa_fun6133342665515662012_int_o(aTP_Lamm_cc((P)),Q)),X)
    <=> ( (P)
        & aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(Q),X) ) ) ).

% split_part
tff(fact_6210_round__0,axiom,
    aa_real_int(archim8280529875227126926d_real,zero_zero_real) = zero_zero_int ).

% round_0
tff(fact_6211_round__0,axiom,
    aa_rat_int(archim7778729529865785530nd_rat,zero_zero_rat) = zero_zero_int ).

% round_0
tff(fact_6212_round__numeral,axiom,
    ! [Nb: num] : aa_real_int(archim8280529875227126926d_real,aa_num_real(numeral_numeral_real,Nb)) = aa_num_int(numeral_numeral_int,Nb) ).

% round_numeral
tff(fact_6213_round__numeral,axiom,
    ! [Nb: num] : aa_rat_int(archim7778729529865785530nd_rat,aa_num_rat(numeral_numeral_rat,Nb)) = aa_num_int(numeral_numeral_int,Nb) ).

% round_numeral
tff(fact_6214_round__1,axiom,
    aa_real_int(archim8280529875227126926d_real,one_one_real) = one_one_int ).

% round_1
tff(fact_6215_round__1,axiom,
    aa_rat_int(archim7778729529865785530nd_rat,one_one_rat) = one_one_int ).

% round_1
tff(fact_6216_pred__numeral__inc,axiom,
    ! [K: num] : aa_num_nat(pred_numeral,aa_num_num(inc,K)) = aa_num_nat(numeral_numeral_nat,K) ).

% pred_numeral_inc
tff(fact_6217_add__neg__numeral__special_I5_J,axiom,
    ! [Nb: num] : aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(inc,Nb))) ).

% add_neg_numeral_special(5)
tff(fact_6218_add__neg__numeral__special_I5_J,axiom,
    ! [Nb: 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,Nb))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(inc,Nb))) ).

% add_neg_numeral_special(5)
tff(fact_6219_add__neg__numeral__special_I5_J,axiom,
    ! [Nb: num] : aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb))) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(inc,Nb))) ).

% add_neg_numeral_special(5)
tff(fact_6220_add__neg__numeral__special_I5_J,axiom,
    ! [Nb: num] : aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(inc,Nb))) ).

% add_neg_numeral_special(5)
tff(fact_6221_add__neg__numeral__special_I5_J,axiom,
    ! [Nb: num] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(inc,Nb))) ).

% add_neg_numeral_special(5)
tff(fact_6222_add__neg__numeral__special_I6_J,axiom,
    ! [Mb: num] : aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb))),aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(inc,Mb))) ).

% add_neg_numeral_special(6)
tff(fact_6223_add__neg__numeral__special_I6_J,axiom,
    ! [Mb: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(inc,Mb))) ).

% add_neg_numeral_special(6)
tff(fact_6224_add__neg__numeral__special_I6_J,axiom,
    ! [Mb: num] : aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Mb))),aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(inc,Mb))) ).

% add_neg_numeral_special(6)
tff(fact_6225_add__neg__numeral__special_I6_J,axiom,
    ! [Mb: num] : aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Mb))),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(inc,Mb))) ).

% add_neg_numeral_special(6)
tff(fact_6226_add__neg__numeral__special_I6_J,axiom,
    ! [Mb: num] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Mb)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(inc,Mb))) ).

% add_neg_numeral_special(6)
tff(fact_6227_diff__numeral__special_I6_J,axiom,
    ! [Mb: num] : aa_real_real(minus_minus_real(aa_num_real(numeral_numeral_real,Mb)),aa_real_real(uminus_uminus_real,one_one_real)) = aa_num_real(numeral_numeral_real,aa_num_num(inc,Mb)) ).

% diff_numeral_special(6)
tff(fact_6228_diff__numeral__special_I6_J,axiom,
    ! [Mb: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(uminus_uminus_int,one_one_int)) = aa_num_int(numeral_numeral_int,aa_num_num(inc,Mb)) ).

% diff_numeral_special(6)
tff(fact_6229_diff__numeral__special_I6_J,axiom,
    ! [Mb: num] : aa_complex_complex(minus_minus_complex(numera6690914467698888265omplex(Mb)),aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = numera6690914467698888265omplex(aa_num_num(inc,Mb)) ).

% diff_numeral_special(6)
tff(fact_6230_diff__numeral__special_I6_J,axiom,
    ! [Mb: num] : aa_rat_rat(minus_minus_rat(aa_num_rat(numeral_numeral_rat,Mb)),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_num_rat(numeral_numeral_rat,aa_num_num(inc,Mb)) ).

% diff_numeral_special(6)
tff(fact_6231_diff__numeral__special_I6_J,axiom,
    ! [Mb: num] : minus_8373710615458151222nteger(numera6620942414471956472nteger(Mb),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = numera6620942414471956472nteger(aa_num_num(inc,Mb)) ).

% diff_numeral_special(6)
tff(fact_6232_diff__numeral__special_I5_J,axiom,
    ! [Nb: num] : aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,one_one_real)),aa_num_real(numeral_numeral_real,Nb)) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(inc,Nb))) ).

% diff_numeral_special(5)
tff(fact_6233_diff__numeral__special_I5_J,axiom,
    ! [Nb: 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,Nb)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(inc,Nb))) ).

% diff_numeral_special(5)
tff(fact_6234_diff__numeral__special_I5_J,axiom,
    ! [Nb: num] : aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),numera6690914467698888265omplex(Nb)) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(inc,Nb))) ).

% diff_numeral_special(5)
tff(fact_6235_diff__numeral__special_I5_J,axiom,
    ! [Nb: num] : aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_num_rat(numeral_numeral_rat,Nb)) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(inc,Nb))) ).

% diff_numeral_special(5)
tff(fact_6236_diff__numeral__special_I5_J,axiom,
    ! [Nb: num] : minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),numera6620942414471956472nteger(Nb)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(inc,Nb))) ).

% diff_numeral_special(5)
tff(fact_6237_round__neg__numeral,axiom,
    ! [Nb: num] : aa_real_int(archim8280529875227126926d_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)) ).

% round_neg_numeral
tff(fact_6238_round__neg__numeral,axiom,
    ! [Nb: num] : aa_rat_int(archim7778729529865785530nd_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)) ).

% round_neg_numeral
tff(fact_6239_prod_Odisc__eq__case,axiom,
    ! [Prod: product_prod_int_int] : aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(aTP_Lamm_cd),Prod) ).

% prod.disc_eq_case
tff(fact_6240_Collect__case__prod__mono,axiom,
    ! [A3: fun_int_fun_int_o,B4: fun_int_fun_int_o] :
      ( ord_le77980682512674322_int_o(A3,B4)
     => ord_le2843351958646193337nt_int(collec213857154873943460nt_int(produc4947309494688390418_int_o(A3)),collec213857154873943460nt_int(produc4947309494688390418_int_o(B4))) ) ).

% Collect_case_prod_mono
tff(fact_6241_num__induct,axiom,
    ! [P: fun_num_o,Xb: num] :
      ( aa_num_o(P,one)
     => ( ! [X3: num] :
            ( aa_num_o(P,X3)
           => aa_num_o(P,aa_num_num(inc,X3)) )
       => aa_num_o(P,Xb) ) ) ).

% num_induct
tff(fact_6242_add__inc,axiom,
    ! [Xb: num,Yb: num] : aa_num_num(plus_plus_num(Xb),aa_num_num(inc,Yb)) = aa_num_num(inc,aa_num_num(plus_plus_num(Xb),Yb)) ).

% add_inc
tff(fact_6243_inc_Osimps_I1_J,axiom,
    aa_num_num(inc,one) = aa_num_num(bit0,one) ).

% inc.simps(1)
tff(fact_6244_inc_Osimps_I2_J,axiom,
    ! [Xb: num] : aa_num_num(inc,aa_num_num(bit0,Xb)) = aa_num_num(bit1,Xb) ).

% inc.simps(2)
tff(fact_6245_inc_Osimps_I3_J,axiom,
    ! [Xb: num] : aa_num_num(inc,aa_num_num(bit1,Xb)) = aa_num_num(bit0,aa_num_num(inc,Xb)) ).

% inc.simps(3)
tff(fact_6246_add__One,axiom,
    ! [Xb: num] : aa_num_num(plus_plus_num(Xb),one) = aa_num_num(inc,Xb) ).

% add_One
tff(fact_6247_round__mono,axiom,
    ! [Xb: rat,Yb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Yb)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim7778729529865785530nd_rat,Xb)),aa_rat_int(archim7778729529865785530nd_rat,Yb)) ) ).

% round_mono
tff(fact_6248_mult__inc,axiom,
    ! [Xb: num,Yb: num] : aa_num_num(times_times_num(Xb),aa_num_num(inc,Yb)) = aa_num_num(plus_plus_num(aa_num_num(times_times_num(Xb),Yb)),Xb) ).

% mult_inc
tff(fact_6249_numeral__inc,axiom,
    ! [Xb: num] : numera6690914467698888265omplex(aa_num_num(inc,Xb)) = aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(Xb)),one_one_complex) ).

% numeral_inc
tff(fact_6250_numeral__inc,axiom,
    ! [Xb: num] : aa_num_real(numeral_numeral_real,aa_num_num(inc,Xb)) = aa_real_real(plus_plus_real(aa_num_real(numeral_numeral_real,Xb)),one_one_real) ).

% numeral_inc
tff(fact_6251_numeral__inc,axiom,
    ! [Xb: num] : aa_num_rat(numeral_numeral_rat,aa_num_num(inc,Xb)) = aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,Xb)),one_one_rat) ).

% numeral_inc
tff(fact_6252_numeral__inc,axiom,
    ! [Xb: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(inc,Xb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,Xb)),one_one_nat) ).

% numeral_inc
tff(fact_6253_numeral__inc,axiom,
    ! [Xb: num] : aa_num_int(numeral_numeral_int,aa_num_num(inc,Xb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,Xb)),one_one_int) ).

% numeral_inc
tff(fact_6254_divmod__nat__def,axiom,
    ! [Mb: nat,Nb: nat] : divmod_nat(Mb,Nb) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb)) ).

% divmod_nat_def
tff(fact_6255_of__int__round__le,axiom,
    ! [Xb: 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,Xb))),aa_real_real(plus_plus_real(Xb),aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% of_int_round_le
tff(fact_6256_of__int__round__le,axiom,
    ! [Xb: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim7778729529865785530nd_rat,Xb))),aa_rat_rat(plus_plus_rat(Xb),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))) ).

% of_int_round_le
tff(fact_6257_of__int__round__ge,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(Xb),aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_int_real(ring_1_of_int_real,aa_real_int(archim8280529875227126926d_real,Xb))) ).

% of_int_round_ge
tff(fact_6258_of__int__round__ge,axiom,
    ! [Xb: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(Xb),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))),aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim7778729529865785530nd_rat,Xb))) ).

% of_int_round_ge
tff(fact_6259_of__int__round__gt,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(minus_minus_real(Xb),aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_int_real(ring_1_of_int_real,aa_real_int(archim8280529875227126926d_real,Xb))) ).

% of_int_round_gt
tff(fact_6260_of__int__round__gt,axiom,
    ! [Xb: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(Xb),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))),aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim7778729529865785530nd_rat,Xb))) ).

% of_int_round_gt
tff(fact_6261_take__bit__Suc__minus__bit1,axiom,
    ! [Nb: nat,K: num] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Nb)),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(Nb),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(inc,K))))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).

% take_bit_Suc_minus_bit1
tff(fact_6262_artanh__def,axiom,
    ! [Xb: real] : aa_real_real(artanh_real,Xb) = aa_real_real(divide_divide_real(aa_real_real(ln_ln_real,aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(one_one_real),Xb)),aa_real_real(minus_minus_real(one_one_real),Xb)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% artanh_def
tff(fact_6263_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_6264_dbl__inc__simps_I3_J,axiom,
    aa_real_real(neg_nu8295874005876285629c_real,one_one_real) = aa_num_real(numeral_numeral_real,aa_num_num(bit1,one)) ).

% dbl_inc_simps(3)
tff(fact_6265_dbl__inc__simps_I3_J,axiom,
    aa_rat_rat(neg_nu5219082963157363817nc_rat,one_one_rat) = aa_num_rat(numeral_numeral_rat,aa_num_num(bit1,one)) ).

% dbl_inc_simps(3)
tff(fact_6266_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_6267_or__int__unfold,axiom,
    ! [K: int,L: int] :
      aa_int_int(aa_int_fun_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(aa_int_fun_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(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(aa_int_fun_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_6268_ln__one__minus__pos__lower__bound,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,Xb)),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(ln_ln_real,aa_real_real(minus_minus_real(one_one_real),Xb))) ) ) ).

% ln_one_minus_pos_lower_bound
tff(fact_6269_divmod__BitM__2__eq,axiom,
    ! [Mb: num] : unique5052692396658037445od_int(aa_num_num(bitM,Mb),aa_num_num(bit0,one)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,Mb)),one_one_int)),one_one_int) ).

% divmod_BitM_2_eq
tff(fact_6270_or_Oright__idem,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,A),B)),B) = aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,A),B) ).

% or.right_idem
tff(fact_6271_or_Oright__idem,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,A),B)),B) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,A),B) ).

% or.right_idem
tff(fact_6272_or_Oleft__idem,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,A),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,A),B)) = aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,A),B) ).

% or.left_idem
tff(fact_6273_or_Oleft__idem,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,A),B) ).

% or.left_idem
tff(fact_6274_or_Oidem,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,A),A) = A ).

% or.idem
tff(fact_6275_or_Oidem,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,A),A) = A ).

% or.idem
tff(fact_6276_or_Oleft__neutral,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,zero_zero_int),A) = A ).

% or.left_neutral
tff(fact_6277_or_Oleft__neutral,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,zero_zero_nat),A) = A ).

% or.left_neutral
tff(fact_6278_or_Oright__neutral,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,A),zero_zero_int) = A ).

% or.right_neutral
tff(fact_6279_or_Oright__neutral,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,A),zero_zero_nat) = A ).

% or.right_neutral
tff(fact_6280_take__bit__or,axiom,
    ! [Nb: nat,A: int,B: int] : aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,A),B)) = aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(bit_se2923211474154528505it_int(Nb),A)),aa_int_int(bit_se2923211474154528505it_int(Nb),B)) ).

% take_bit_or
tff(fact_6281_take__bit__or,axiom,
    ! [Nb: nat,A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),A)),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),B)) ).

% take_bit_or
tff(fact_6282_bit_Odisj__one__right,axiom,
    ! [Xb: code_integer] : bit_se1080825931792720795nteger(Xb,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% bit.disj_one_right
tff(fact_6283_bit_Odisj__one__right,axiom,
    ! [Xb: int] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,Xb),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,one_one_int) ).

% bit.disj_one_right
tff(fact_6284_bit_Odisj__one__left,axiom,
    ! [Xb: code_integer] : bit_se1080825931792720795nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),Xb) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% bit.disj_one_left
tff(fact_6285_bit_Odisj__one__left,axiom,
    ! [Xb: int] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(uminus_uminus_int,one_one_int)),Xb) = aa_int_int(uminus_uminus_int,one_one_int) ).

% bit.disj_one_left
tff(fact_6286_ln__one,axiom,
    aa_real_real(ln_ln_real,one_one_real) = zero_zero_real ).

% ln_one
tff(fact_6287_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(aa_int_fun_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_6288_or__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_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_6289_dbl__inc__simps_I2_J,axiom,
    aa_complex_complex(neg_nu8557863876264182079omplex,zero_zero_complex) = one_one_complex ).

% dbl_inc_simps(2)
tff(fact_6290_dbl__inc__simps_I2_J,axiom,
    aa_real_real(neg_nu8295874005876285629c_real,zero_zero_real) = one_one_real ).

% dbl_inc_simps(2)
tff(fact_6291_dbl__inc__simps_I2_J,axiom,
    aa_rat_rat(neg_nu5219082963157363817nc_rat,zero_zero_rat) = one_one_rat ).

% dbl_inc_simps(2)
tff(fact_6292_dbl__inc__simps_I2_J,axiom,
    aa_int_int(neg_nu5851722552734809277nc_int,zero_zero_int) = one_one_int ).

% dbl_inc_simps(2)
tff(fact_6293_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_6294_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_6295_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_6296_dbl__inc__simps_I4_J,axiom,
    aa_rat_rat(neg_nu5219082963157363817nc_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_rat_rat(uminus_uminus_rat,one_one_rat) ).

% dbl_inc_simps(4)
tff(fact_6297_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_6298_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_6299_dbl__inc__simps_I5_J,axiom,
    ! [K: num] : aa_real_real(neg_nu8295874005876285629c_real,aa_num_real(numeral_numeral_real,K)) = aa_num_real(numeral_numeral_real,aa_num_num(bit1,K)) ).

% dbl_inc_simps(5)
tff(fact_6300_dbl__inc__simps_I5_J,axiom,
    ! [K: num] : aa_rat_rat(neg_nu5219082963157363817nc_rat,aa_num_rat(numeral_numeral_rat,K)) = aa_num_rat(numeral_numeral_rat,aa_num_num(bit1,K)) ).

% dbl_inc_simps(5)
tff(fact_6301_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_6302_dbl__dec__simps_I5_J,axiom,
    ! [K: num] : aa_complex_complex(neg_nu6511756317524482435omplex,numera6690914467698888265omplex(K)) = numera6690914467698888265omplex(aa_num_num(bitM,K)) ).

% dbl_dec_simps(5)
tff(fact_6303_dbl__dec__simps_I5_J,axiom,
    ! [K: num] : aa_real_real(neg_nu6075765906172075777c_real,aa_num_real(numeral_numeral_real,K)) = aa_num_real(numeral_numeral_real,aa_num_num(bitM,K)) ).

% dbl_dec_simps(5)
tff(fact_6304_dbl__dec__simps_I5_J,axiom,
    ! [K: num] : aa_rat_rat(neg_nu3179335615603231917ec_rat,aa_num_rat(numeral_numeral_rat,K)) = aa_num_rat(numeral_numeral_rat,aa_num_num(bitM,K)) ).

% dbl_dec_simps(5)
tff(fact_6305_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,aa_num_num(bitM,K)) ).

% dbl_dec_simps(5)
tff(fact_6306_or__numerals_I2_J,axiom,
    ! [Yb: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,one_one_int),aa_num_int(numeral_numeral_int,aa_num_num(bit1,Yb))) = aa_num_int(numeral_numeral_int,aa_num_num(bit1,Yb)) ).

% or_numerals(2)
tff(fact_6307_or__numerals_I2_J,axiom,
    ! [Yb: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Yb))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Yb)) ).

% or_numerals(2)
tff(fact_6308_or__numerals_I8_J,axiom,
    ! [Xb: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Xb))),one_one_int) = aa_num_int(numeral_numeral_int,aa_num_num(bit1,Xb)) ).

% or_numerals(8)
tff(fact_6309_or__numerals_I8_J,axiom,
    ! [Xb: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xb))),one_one_nat) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xb)) ).

% or_numerals(8)
tff(fact_6310_ln__less__zero__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(ln_ln_real,Xb)),zero_zero_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),one_one_real) ) ) ).

% ln_less_zero_iff
tff(fact_6311_ln__gt__zero__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(ln_ln_real,Xb))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb) ) ) ).

% ln_gt_zero_iff
tff(fact_6312_ln__eq__zero__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( ( aa_real_real(ln_ln_real,Xb) = zero_zero_real )
      <=> ( Xb = one_one_real ) ) ) ).

% ln_eq_zero_iff
tff(fact_6313_pred__numeral__simps_I2_J,axiom,
    ! [K: num] : aa_num_nat(pred_numeral,aa_num_num(bit0,K)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bitM,K)) ).

% pred_numeral_simps(2)
tff(fact_6314_dbl__dec__simps_I1_J,axiom,
    ! [K: num] : aa_real_real(neg_nu6075765906172075777c_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,K))) = aa_real_real(uminus_uminus_real,aa_real_real(neg_nu8295874005876285629c_real,aa_num_real(numeral_numeral_real,K))) ).

% dbl_dec_simps(1)
tff(fact_6315_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_6316_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_6317_dbl__dec__simps_I1_J,axiom,
    ! [K: num] : aa_rat_rat(neg_nu3179335615603231917ec_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,K))) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(neg_nu5219082963157363817nc_rat,aa_num_rat(numeral_numeral_rat,K))) ).

% dbl_dec_simps(1)
tff(fact_6318_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_6319_dbl__inc__simps_I1_J,axiom,
    ! [K: num] : aa_real_real(neg_nu8295874005876285629c_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,K))) = aa_real_real(uminus_uminus_real,aa_real_real(neg_nu6075765906172075777c_real,aa_num_real(numeral_numeral_real,K))) ).

% dbl_inc_simps(1)
tff(fact_6320_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_6321_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_6322_dbl__inc__simps_I1_J,axiom,
    ! [K: num] : aa_rat_rat(neg_nu5219082963157363817nc_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,K))) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(neg_nu3179335615603231917ec_rat,aa_num_rat(numeral_numeral_rat,K))) ).

% dbl_inc_simps(1)
tff(fact_6323_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_6324_or__numerals_I3_J,axiom,
    ! [Xb: num,Yb: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Xb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Yb))) = 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(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,Xb)),aa_num_int(numeral_numeral_int,Yb))) ).

% or_numerals(3)
tff(fact_6325_or__numerals_I3_J,axiom,
    ! [Xb: num,Yb: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Xb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Yb))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,Xb)),aa_num_nat(numeral_numeral_nat,Yb))) ).

% or_numerals(3)
tff(fact_6326_or__numerals_I5_J,axiom,
    ! [Xb: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Xb))),one_one_int) = aa_num_int(numeral_numeral_int,aa_num_num(bit1,Xb)) ).

% or_numerals(5)
tff(fact_6327_or__numerals_I5_J,axiom,
    ! [Xb: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Xb))),one_one_nat) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xb)) ).

% or_numerals(5)
tff(fact_6328_or__numerals_I1_J,axiom,
    ! [Yb: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,one_one_int),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Yb))) = aa_num_int(numeral_numeral_int,aa_num_num(bit1,Yb)) ).

% or_numerals(1)
tff(fact_6329_or__numerals_I1_J,axiom,
    ! [Yb: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Yb))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Yb)) ).

% or_numerals(1)
tff(fact_6330_ln__le__zero__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,Xb)),zero_zero_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real) ) ) ).

% ln_le_zero_iff
tff(fact_6331_ln__ge__zero__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(ln_ln_real,Xb))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xb) ) ) ).

% ln_ge_zero_iff
tff(fact_6332_or__minus__numerals_I2_J,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb))) ).

% or_minus_numerals(2)
tff(fact_6333_or__minus__numerals_I6_J,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))),one_one_int) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb))) ).

% or_minus_numerals(6)
tff(fact_6334_or__numerals_I7_J,axiom,
    ! [Xb: num,Yb: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Xb))),aa_num_int(numeral_numeral_int,aa_num_num(bit1,Yb))) = 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(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,Xb)),aa_num_int(numeral_numeral_int,Yb)))) ).

% or_numerals(7)
tff(fact_6335_or__numerals_I7_J,axiom,
    ! [Xb: num,Yb: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Yb))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,Xb)),aa_num_nat(numeral_numeral_nat,Yb)))) ).

% or_numerals(7)
tff(fact_6336_or__numerals_I6_J,axiom,
    ! [Xb: num,Yb: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Xb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Yb))) = 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(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,Xb)),aa_num_int(numeral_numeral_int,Yb)))) ).

% or_numerals(6)
tff(fact_6337_or__numerals_I6_J,axiom,
    ! [Xb: num,Yb: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Yb))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,Xb)),aa_num_nat(numeral_numeral_nat,Yb)))) ).

% or_numerals(6)
tff(fact_6338_or__numerals_I4_J,axiom,
    ! [Xb: num,Yb: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Xb))),aa_num_int(numeral_numeral_int,aa_num_num(bit1,Yb))) = 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(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,Xb)),aa_num_int(numeral_numeral_int,Yb)))) ).

% or_numerals(4)
tff(fact_6339_or__numerals_I4_J,axiom,
    ! [Xb: num,Yb: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Xb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Yb))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,Xb)),aa_num_nat(numeral_numeral_nat,Yb)))) ).

% or_numerals(4)
tff(fact_6340_or_Oleft__commute,axiom,
    ! [B: int,A: int,C: int] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,B),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,A),C)) = aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,A),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,B),C)) ).

% or.left_commute
tff(fact_6341_or_Oleft__commute,axiom,
    ! [B: nat,A: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,A),C)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,B),C)) ).

% or.left_commute
tff(fact_6342_or_Ocommute,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,A),B) = aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,B),A) ).

% or.commute
tff(fact_6343_or_Ocommute,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,B),A) ).

% or.commute
tff(fact_6344_or_Oassoc,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,A),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,B),C)) ).

% or.assoc
tff(fact_6345_or_Oassoc,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,B),C)) ).

% or.assoc
tff(fact_6346_of__int__or__eq,axiom,
    ! [K: int,L: int] : aa_int_int(ring_1_of_int_int,aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,K),L)) = aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(ring_1_of_int_int,K)),aa_int_int(ring_1_of_int_int,L)) ).

% of_int_or_eq
tff(fact_6347_or__eq__0__iff,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,A),B) = zero_zero_int )
    <=> ( ( A = zero_zero_int )
        & ( B = zero_zero_int ) ) ) ).

% or_eq_0_iff
tff(fact_6348_or__eq__0__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,A),B) = zero_zero_nat )
    <=> ( ( A = zero_zero_nat )
        & ( B = zero_zero_nat ) ) ) ).

% or_eq_0_iff
tff(fact_6349_bit_Odisj__zero__right,axiom,
    ! [Xb: int] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,Xb),zero_zero_int) = Xb ).

% bit.disj_zero_right
tff(fact_6350_semiring__norm_I26_J,axiom,
    aa_num_num(bitM,one) = one ).

% semiring_norm(26)
tff(fact_6351_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(aa_int_fun_int_int(bit_se1409905431419307370or_int,K),L)) ) ).

% or_greater_eq
tff(fact_6352_OR__lower,axiom,
    ! [Xb: int,Yb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Yb)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,Xb),Yb)) ) ) ).

% OR_lower
tff(fact_6353_semiring__norm_I28_J,axiom,
    ! [Nb: num] : aa_num_num(bitM,aa_num_num(bit1,Nb)) = aa_num_num(bit1,aa_num_num(bit0,Nb)) ).

% semiring_norm(28)
tff(fact_6354_semiring__norm_I27_J,axiom,
    ! [Nb: num] : aa_num_num(bitM,aa_num_num(bit0,Nb)) = aa_num_num(bit1,aa_num_num(bitM,Nb)) ).

% semiring_norm(27)
tff(fact_6355_inc__BitM__eq,axiom,
    ! [Nb: num] : aa_num_num(inc,aa_num_num(bitM,Nb)) = aa_num_num(bit0,Nb) ).

% inc_BitM_eq
tff(fact_6356_BitM__inc__eq,axiom,
    ! [Nb: num] : aa_num_num(bitM,aa_num_num(inc,Nb)) = aa_num_num(bit1,Nb) ).

% BitM_inc_eq
tff(fact_6357_ln__gt__zero,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(ln_ln_real,Xb)) ) ).

% ln_gt_zero
tff(fact_6358_ln__less__zero,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(ln_ln_real,Xb)),zero_zero_real) ) ) ).

% ln_less_zero
tff(fact_6359_ln__gt__zero__imp__gt__one,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(ln_ln_real,Xb))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb) ) ) ).

% ln_gt_zero_imp_gt_one
tff(fact_6360_ln__ge__zero,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(ln_ln_real,Xb)) ) ).

% ln_ge_zero
tff(fact_6361_eval__nat__numeral_I2_J,axiom,
    ! [Nb: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Nb)) = aa_nat_nat(suc,aa_num_nat(numeral_numeral_nat,aa_num_num(bitM,Nb))) ).

% eval_nat_numeral(2)
tff(fact_6362_ln__ge__zero__imp__ge__one,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(ln_ln_real,Xb))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xb) ) ) ).

% ln_ge_zero_imp_ge_one
tff(fact_6363_BitM__plus__one,axiom,
    ! [Nb: num] : aa_num_num(plus_plus_num(aa_num_num(bitM,Nb)),one) = aa_num_num(bit0,Nb) ).

% BitM_plus_one
tff(fact_6364_one__plus__BitM,axiom,
    ! [Nb: num] : aa_num_num(plus_plus_num(one),aa_num_num(bitM,Nb)) = aa_num_num(bit0,Nb) ).

% one_plus_BitM
tff(fact_6365_even__or__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_se1080825931792720795nteger(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_or_iff
tff(fact_6366_even__or__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(bit_se1409905431419307370or_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_or_iff
tff(fact_6367_even__or__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_se1412395901928357646or_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_or_iff
tff(fact_6368_ln__add__one__self__le__self,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,aa_real_real(plus_plus_real(one_one_real),Xb))),Xb) ) ).

% ln_add_one_self_le_self
tff(fact_6369_ln__mult,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Yb)
       => ( aa_real_real(ln_ln_real,aa_real_real(times_times_real(Xb),Yb)) = aa_real_real(plus_plus_real(aa_real_real(ln_ln_real,Xb)),aa_real_real(ln_ln_real,Yb)) ) ) ) ).

% ln_mult
tff(fact_6370_ln__eq__minus__one,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( ( aa_real_real(ln_ln_real,Xb) = aa_real_real(minus_minus_real(Xb),one_one_real) )
       => ( Xb = one_one_real ) ) ) ).

% ln_eq_minus_one
tff(fact_6371_dbl__inc__def,axiom,
    ! [Xb: complex] : aa_complex_complex(neg_nu8557863876264182079omplex,Xb) = aa_complex_complex(plus_plus_complex(aa_complex_complex(plus_plus_complex(Xb),Xb)),one_one_complex) ).

% dbl_inc_def
tff(fact_6372_dbl__inc__def,axiom,
    ! [Xb: real] : aa_real_real(neg_nu8295874005876285629c_real,Xb) = aa_real_real(plus_plus_real(aa_real_real(plus_plus_real(Xb),Xb)),one_one_real) ).

% dbl_inc_def
tff(fact_6373_dbl__inc__def,axiom,
    ! [Xb: rat] : aa_rat_rat(neg_nu5219082963157363817nc_rat,Xb) = aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(Xb),Xb)),one_one_rat) ).

% dbl_inc_def
tff(fact_6374_dbl__inc__def,axiom,
    ! [Xb: int] : aa_int_int(neg_nu5851722552734809277nc_int,Xb) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Xb)),one_one_int) ).

% dbl_inc_def
tff(fact_6375_ln__2__less__1,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(ln_ln_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),one_one_real) ).

% ln_2_less_1
tff(fact_6376_numeral__BitM,axiom,
    ! [Nb: num] : numera6690914467698888265omplex(aa_num_num(bitM,Nb)) = aa_complex_complex(minus_minus_complex(numera6690914467698888265omplex(aa_num_num(bit0,Nb))),one_one_complex) ).

% numeral_BitM
tff(fact_6377_numeral__BitM,axiom,
    ! [Nb: num] : aa_num_real(numeral_numeral_real,aa_num_num(bitM,Nb)) = aa_real_real(minus_minus_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,Nb))),one_one_real) ).

% numeral_BitM
tff(fact_6378_numeral__BitM,axiom,
    ! [Nb: num] : aa_num_rat(numeral_numeral_rat,aa_num_num(bitM,Nb)) = aa_rat_rat(minus_minus_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,Nb))),one_one_rat) ).

% numeral_BitM
tff(fact_6379_numeral__BitM,axiom,
    ! [Nb: num] : aa_num_int(numeral_numeral_int,aa_num_num(bitM,Nb)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb))),one_one_int) ).

% numeral_BitM
tff(fact_6380_odd__numeral__BitM,axiom,
    ! [W: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),numera6620942414471956472nteger(aa_num_num(bitM,W))) ).

% odd_numeral_BitM
tff(fact_6381_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,aa_num_num(bitM,W))) ).

% odd_numeral_BitM
tff(fact_6382_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,aa_num_num(bitM,W))) ).

% odd_numeral_BitM
tff(fact_6383_ln__le__minus__one,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,Xb)),aa_real_real(minus_minus_real(Xb),one_one_real)) ) ).

% ln_le_minus_one
tff(fact_6384_ln__add__one__self__le__self2,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,aa_real_real(plus_plus_real(one_one_real),Xb))),Xb) ) ).

% ln_add_one_self_le_self2
tff(fact_6385_ln__one__minus__pos__upper__bound,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,aa_real_real(minus_minus_real(one_one_real),Xb))),aa_real_real(uminus_uminus_real,Xb)) ) ) ).

% ln_one_minus_pos_upper_bound
tff(fact_6386_ln__sqrt,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_real(ln_ln_real,aa_real_real(sqrt,Xb)) = aa_real_real(divide_divide_real(aa_real_real(ln_ln_real,Xb)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) ) ).

% ln_sqrt
tff(fact_6387_arsinh__real__def,axiom,
    ! [Xb: real] : aa_real_real(arsinh_real,Xb) = aa_real_real(ln_ln_real,aa_real_real(plus_plus_real(Xb),aa_real_real(sqrt,aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)))) ).

% arsinh_real_def
tff(fact_6388_one__or__eq,axiom,
    ! [A: code_integer] : bit_se1080825931792720795nteger(one_one_Code_integer,A) = plus_p5714425477246183910nteger(A,zero_n356916108424825756nteger(aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))) ).

% one_or_eq
tff(fact_6389_one__or__eq,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,one_one_int),A) = 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))) ).

% one_or_eq
tff(fact_6390_one__or__eq,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,one_one_nat),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),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_or_eq
tff(fact_6391_or__one__eq,axiom,
    ! [A: code_integer] : bit_se1080825931792720795nteger(A,one_one_Code_integer) = plus_p5714425477246183910nteger(A,zero_n356916108424825756nteger(aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))) ).

% or_one_eq
tff(fact_6392_or__one__eq,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,A),one_one_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))) ).

% or_one_eq
tff(fact_6393_or__one__eq,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,A),one_one_nat) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),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))) ).

% or_one_eq
tff(fact_6394_OR__upper,axiom,
    ! [Xb: int,Nb: nat,Yb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Yb),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,Xb),Yb)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ) ) ) ).

% OR_upper
tff(fact_6395_or__int__rec,axiom,
    ! [K: int,L: int] :
      aa_int_int(aa_int_fun_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(aa_int_fun_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_6396_ln__one__plus__pos__lower__bound,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(Xb),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(ln_ln_real,aa_real_real(plus_plus_real(one_one_real),Xb))) ) ) ).

% ln_one_plus_pos_lower_bound
tff(fact_6397_arcosh__real__def,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xb)
     => ( aa_real_real(arcosh_real,Xb) = aa_real_real(ln_ln_real,aa_real_real(plus_plus_real(Xb),aa_real_real(sqrt,aa_real_real(minus_minus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)))) ) ) ).

% arcosh_real_def
tff(fact_6398_abs__ln__one__plus__x__minus__x__bound__nonpos,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(aa_real_real(ln_ln_real,aa_real_real(plus_plus_real(one_one_real),Xb))),Xb))),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ).

% abs_ln_one_plus_x_minus_x_bound_nonpos
tff(fact_6399_tanh__ln__real,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_real(tanh_real,aa_real_real(ln_ln_real,Xb)) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)),aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)) ) ) ).

% tanh_ln_real
tff(fact_6400_pred__subset__eq2,axiom,
    ! [R: set_Pr448751882837621926eger_o,S3: set_Pr448751882837621926eger_o] :
      ( ord_le2909477399203599485un_o_o(aTP_Lamm_ce(R),aTP_Lamm_ce(S3))
    <=> ord_le8980329558974975238eger_o(R,S3) ) ).

% pred_subset_eq2
tff(fact_6401_pred__subset__eq2,axiom,
    ! [R: set_Pr8218934625190621173um_num,S3: set_Pr8218934625190621173um_num] :
      ( ord_le3306795289226699318_num_o(aTP_Lamm_cf(R),aTP_Lamm_cf(S3))
    <=> ord_le880128212290418581um_num(R,S3) ) ).

% pred_subset_eq2
tff(fact_6402_pred__subset__eq2,axiom,
    ! [R: set_Pr6200539531224447659at_num,S3: set_Pr6200539531224447659at_num] :
      ( ord_le1526520017675560704_num_o(aTP_Lamm_cg(R),aTP_Lamm_cg(S3))
    <=> ord_le8085105155179020875at_num(R,S3) ) ).

% pred_subset_eq2
tff(fact_6403_pred__subset__eq2,axiom,
    ! [R: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat] :
      ( ord_le768339454706049738_nat_o(aTP_Lamm_ch(R),aTP_Lamm_ch(S3))
    <=> ord_le3146513528884898305at_nat(R,S3) ) ).

% pred_subset_eq2
tff(fact_6404_pred__subset__eq2,axiom,
    ! [R: set_Pr958786334691620121nt_int,S3: set_Pr958786334691620121nt_int] :
      ( ord_le77980682512674322_int_o(aTP_Lamm_ci(R),aTP_Lamm_ci(S3))
    <=> ord_le2843351958646193337nt_int(R,S3) ) ).

% pred_subset_eq2
tff(fact_6405_abs__ln__one__plus__x__minus__x__bound,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),aa_real_real(divide_divide_real(one_one_real),aa_num_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(minus_minus_real(aa_real_real(ln_ln_real,aa_real_real(plus_plus_real(one_one_real),Xb))),Xb))),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% abs_ln_one_plus_x_minus_x_bound
tff(fact_6406_or__minus__numerals_I5_J,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))),one_one_int) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit_or_not_num_neg(one),aa_num_num(bitM,Nb)))) ).

% or_minus_numerals(5)
tff(fact_6407_or__minus__numerals_I1_J,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit_or_not_num_neg(one),aa_num_num(bitM,Nb)))) ).

% or_minus_numerals(1)
tff(fact_6408_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_6409_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_6410_abs__abs,axiom,
    ! [A: code_integer] : abs_abs_Code_integer(abs_abs_Code_integer(A)) = abs_abs_Code_integer(A) ).

% abs_abs
tff(fact_6411_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_6412_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_6413_abs__idempotent,axiom,
    ! [A: code_integer] : abs_abs_Code_integer(abs_abs_Code_integer(A)) = abs_abs_Code_integer(A) ).

% abs_idempotent
tff(fact_6414_abs__zero,axiom,
    abs_abs_Code_integer(zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% abs_zero
tff(fact_6415_abs__zero,axiom,
    aa_real_real(abs_abs_real,zero_zero_real) = zero_zero_real ).

% abs_zero
tff(fact_6416_abs__zero,axiom,
    aa_rat_rat(abs_abs_rat,zero_zero_rat) = zero_zero_rat ).

% abs_zero
tff(fact_6417_abs__zero,axiom,
    aa_int_int(abs_abs_int,zero_zero_int) = zero_zero_int ).

% abs_zero
tff(fact_6418_abs__eq__0,axiom,
    ! [A: code_integer] :
      ( ( abs_abs_Code_integer(A) = zero_z3403309356797280102nteger )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% abs_eq_0
tff(fact_6419_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_6420_abs__eq__0,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(abs_abs_rat,A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% abs_eq_0
tff(fact_6421_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_6422_abs__0__eq,axiom,
    ! [A: code_integer] :
      ( ( zero_z3403309356797280102nteger = abs_abs_Code_integer(A) )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% abs_0_eq
tff(fact_6423_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_6424_abs__0__eq,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat = aa_rat_rat(abs_abs_rat,A) )
    <=> ( A = zero_zero_rat ) ) ).

% abs_0_eq
tff(fact_6425_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_6426_abs__0,axiom,
    abs_abs_Code_integer(zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% abs_0
tff(fact_6427_abs__0,axiom,
    aa_complex_complex(abs_abs_complex,zero_zero_complex) = zero_zero_complex ).

% abs_0
tff(fact_6428_abs__0,axiom,
    aa_real_real(abs_abs_real,zero_zero_real) = zero_zero_real ).

% abs_0
tff(fact_6429_abs__0,axiom,
    aa_rat_rat(abs_abs_rat,zero_zero_rat) = zero_zero_rat ).

% abs_0
tff(fact_6430_abs__0,axiom,
    aa_int_int(abs_abs_int,zero_zero_int) = zero_zero_int ).

% abs_0
tff(fact_6431_abs__numeral,axiom,
    ! [Nb: num] : abs_abs_Code_integer(numera6620942414471956472nteger(Nb)) = numera6620942414471956472nteger(Nb) ).

% abs_numeral
tff(fact_6432_abs__numeral,axiom,
    ! [Nb: num] : aa_real_real(abs_abs_real,aa_num_real(numeral_numeral_real,Nb)) = aa_num_real(numeral_numeral_real,Nb) ).

% abs_numeral
tff(fact_6433_abs__numeral,axiom,
    ! [Nb: num] : aa_rat_rat(abs_abs_rat,aa_num_rat(numeral_numeral_rat,Nb)) = aa_num_rat(numeral_numeral_rat,Nb) ).

% abs_numeral
tff(fact_6434_abs__numeral,axiom,
    ! [Nb: num] : aa_int_int(abs_abs_int,aa_num_int(numeral_numeral_int,Nb)) = aa_num_int(numeral_numeral_int,Nb) ).

% abs_numeral
tff(fact_6435_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_6436_abs__mult__self__eq,axiom,
    ! [A: real] : aa_real_real(times_times_real(aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,A)) = aa_real_real(times_times_real(A),A) ).

% abs_mult_self_eq
tff(fact_6437_abs__mult__self__eq,axiom,
    ! [A: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,A)) = aa_rat_rat(times_times_rat(A),A) ).

% abs_mult_self_eq
tff(fact_6438_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_6439_abs__1,axiom,
    abs_abs_Code_integer(one_one_Code_integer) = one_one_Code_integer ).

% abs_1
tff(fact_6440_abs__1,axiom,
    aa_complex_complex(abs_abs_complex,one_one_complex) = one_one_complex ).

% abs_1
tff(fact_6441_abs__1,axiom,
    aa_real_real(abs_abs_real,one_one_real) = one_one_real ).

% abs_1
tff(fact_6442_abs__1,axiom,
    aa_rat_rat(abs_abs_rat,one_one_rat) = one_one_rat ).

% abs_1
tff(fact_6443_abs__1,axiom,
    aa_int_int(abs_abs_int,one_one_int) = one_one_int ).

% abs_1
tff(fact_6444_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_6445_abs__add__abs,axiom,
    ! [A: real,B: real] : aa_real_real(abs_abs_real,aa_real_real(plus_plus_real(aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B))) = aa_real_real(plus_plus_real(aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B)) ).

% abs_add_abs
tff(fact_6446_abs__add__abs,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B))) = aa_rat_rat(plus_plus_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B)) ).

% abs_add_abs
tff(fact_6447_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_6448_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_6449_abs__divide,axiom,
    ! [A: real,B: real] : aa_real_real(abs_abs_real,aa_real_real(divide_divide_real(A),B)) = aa_real_real(divide_divide_real(aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B)) ).

% abs_divide
tff(fact_6450_abs__divide,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(divide_divide_rat(A),B)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B)) ).

% abs_divide
tff(fact_6451_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_6452_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_6453_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_6454_abs__minus,axiom,
    ! [A: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(uminus_uminus_rat,A)) = aa_rat_rat(abs_abs_rat,A) ).

% abs_minus
tff(fact_6455_abs__minus,axiom,
    ! [A: code_integer] : abs_abs_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) = abs_abs_Code_integer(A) ).

% abs_minus
tff(fact_6456_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_6457_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_6458_abs__minus__cancel,axiom,
    ! [A: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(uminus_uminus_rat,A)) = aa_rat_rat(abs_abs_rat,A) ).

% abs_minus_cancel
tff(fact_6459_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_6460_abs__dvd__iff,axiom,
    ! [Mb: real,K: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_real_real(abs_abs_real,Mb)),K)
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Mb),K) ) ).

% abs_dvd_iff
tff(fact_6461_abs__dvd__iff,axiom,
    ! [Mb: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(abs_abs_int,Mb)),K)
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Mb),K) ) ).

% abs_dvd_iff
tff(fact_6462_abs__dvd__iff,axiom,
    ! [Mb: code_integer,K: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,abs_abs_Code_integer(Mb)),K)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Mb),K) ) ).

% abs_dvd_iff
tff(fact_6463_dvd__abs__iff,axiom,
    ! [Mb: real,K: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Mb),aa_real_real(abs_abs_real,K))
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Mb),K) ) ).

% dvd_abs_iff
tff(fact_6464_dvd__abs__iff,axiom,
    ! [Mb: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Mb),aa_int_int(abs_abs_int,K))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Mb),K) ) ).

% dvd_abs_iff
tff(fact_6465_dvd__abs__iff,axiom,
    ! [Mb: code_integer,K: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Mb),abs_abs_Code_integer(K))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Mb),K) ) ).

% dvd_abs_iff
tff(fact_6466_of__int__abs,axiom,
    ! [Xb: int] : aa_int_int(ring_1_of_int_int,aa_int_int(abs_abs_int,Xb)) = aa_int_int(abs_abs_int,aa_int_int(ring_1_of_int_int,Xb)) ).

% of_int_abs
tff(fact_6467_of__int__abs,axiom,
    ! [Xb: int] : aa_int_Code_integer(ring_18347121197199848620nteger,aa_int_int(abs_abs_int,Xb)) = abs_abs_Code_integer(aa_int_Code_integer(ring_18347121197199848620nteger,Xb)) ).

% of_int_abs
tff(fact_6468_of__int__abs,axiom,
    ! [Xb: int] : aa_int_real(ring_1_of_int_real,aa_int_int(abs_abs_int,Xb)) = aa_real_real(abs_abs_real,aa_int_real(ring_1_of_int_real,Xb)) ).

% of_int_abs
tff(fact_6469_of__int__abs,axiom,
    ! [Xb: int] : aa_int_rat(ring_1_of_int_rat,aa_int_int(abs_abs_int,Xb)) = aa_rat_rat(abs_abs_rat,aa_int_rat(ring_1_of_int_rat,Xb)) ).

% of_int_abs
tff(fact_6470_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_6471_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_6472_abs__bool__eq,axiom,
    ! [P: $o] : abs_abs_Code_integer(zero_n356916108424825756nteger((P))) = zero_n356916108424825756nteger((P)) ).

% abs_bool_eq
tff(fact_6473_tanh__0,axiom,
    aa_complex_complex(tanh_complex,zero_zero_complex) = zero_zero_complex ).

% tanh_0
tff(fact_6474_tanh__0,axiom,
    aa_real_real(tanh_real,zero_zero_real) = zero_zero_real ).

% tanh_0
tff(fact_6475_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_6476_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_6477_abs__le__zero__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A)),zero_zero_rat)
    <=> ( A = zero_zero_rat ) ) ).

% abs_le_zero_iff
tff(fact_6478_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_6479_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_6480_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_6481_abs__le__self__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_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_6482_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_6483_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_6484_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_6485_abs__of__nonneg,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( aa_rat_rat(abs_abs_rat,A) = A ) ) ).

% abs_of_nonneg
tff(fact_6486_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_6487_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_6488_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_6489_zero__less__abs__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(abs_abs_rat,A))
    <=> ( A != zero_zero_rat ) ) ).

% zero_less_abs_iff
tff(fact_6490_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_6491_abs__neg__numeral,axiom,
    ! [Nb: num] : aa_real_real(abs_abs_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))) = aa_num_real(numeral_numeral_real,Nb) ).

% abs_neg_numeral
tff(fact_6492_abs__neg__numeral,axiom,
    ! [Nb: num] : aa_int_int(abs_abs_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_num_int(numeral_numeral_int,Nb) ).

% abs_neg_numeral
tff(fact_6493_abs__neg__numeral,axiom,
    ! [Nb: num] : aa_rat_rat(abs_abs_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))) = aa_num_rat(numeral_numeral_rat,Nb) ).

% abs_neg_numeral
tff(fact_6494_abs__neg__numeral,axiom,
    ! [Nb: num] : abs_abs_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb))) = numera6620942414471956472nteger(Nb) ).

% abs_neg_numeral
tff(fact_6495_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_6496_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_6497_abs__neg__one,axiom,
    aa_rat_rat(abs_abs_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)) = one_one_rat ).

% abs_neg_one
tff(fact_6498_abs__neg__one,axiom,
    abs_abs_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = one_one_Code_integer ).

% abs_neg_one
tff(fact_6499_abs__power__minus,axiom,
    ! [A: real,Nb: nat] : aa_real_real(abs_abs_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,A)),Nb)) = aa_real_real(abs_abs_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ).

% abs_power_minus
tff(fact_6500_abs__power__minus,axiom,
    ! [A: int,Nb: nat] : aa_int_int(abs_abs_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A)),Nb)) = aa_int_int(abs_abs_int,aa_nat_int(power_power_int(A),Nb)) ).

% abs_power_minus
tff(fact_6501_abs__power__minus,axiom,
    ! [A: rat,Nb: nat] : aa_rat_rat(abs_abs_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,A)),Nb)) = aa_rat_rat(abs_abs_rat,aa_nat_rat(power_power_rat(A),Nb)) ).

% abs_power_minus
tff(fact_6502_abs__power__minus,axiom,
    ! [A: code_integer,Nb: nat] : abs_abs_Code_integer(aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),Nb)) = abs_abs_Code_integer(aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb)) ).

% abs_power_minus
tff(fact_6503_real__sqrt__abs2,axiom,
    ! [Xb: real] : aa_real_real(sqrt,aa_real_real(times_times_real(Xb),Xb)) = aa_real_real(abs_abs_real,Xb) ).

% real_sqrt_abs2
tff(fact_6504_real__sqrt__mult__self,axiom,
    ! [A: real] : aa_real_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_6505_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(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_6506_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),aa_rat_rat(divide_divide_rat(A),aa_rat_rat(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_6507_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(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_6508_divide__le__0__abs__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(A),aa_rat_rat(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_6509_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_6510_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_6511_abs__of__nonpos,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
     => ( aa_rat_rat(abs_abs_rat,A) = aa_rat_rat(uminus_uminus_rat,A) ) ) ).

% abs_of_nonpos
tff(fact_6512_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_6513_or__nat__numerals_I2_J,axiom,
    ! [Yb: 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,Yb))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Yb)) ).

% or_nat_numerals(2)
tff(fact_6514_or__nat__numerals_I4_J,axiom,
    ! [Xb: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xb))),aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xb)) ).

% or_nat_numerals(4)
tff(fact_6515_artanh__minus__real,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xb)),one_one_real)
     => ( aa_real_real(artanh_real,aa_real_real(uminus_uminus_real,Xb)) = aa_real_real(uminus_uminus_real,aa_real_real(artanh_real,Xb)) ) ) ).

% artanh_minus_real
tff(fact_6516_zero__less__power__abs__iff,axiom,
    ! [A: code_integer,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(abs_abs_Code_integer(A)),Nb))
    <=> ( ( A != zero_z3403309356797280102nteger )
        | ( Nb = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
tff(fact_6517_zero__less__power__abs__iff,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,A)),Nb))
    <=> ( ( A != zero_zero_real )
        | ( Nb = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
tff(fact_6518_zero__less__power__abs__iff,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(aa_rat_rat(abs_abs_rat,A)),Nb))
    <=> ( ( A != zero_zero_rat )
        | ( Nb = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
tff(fact_6519_zero__less__power__abs__iff,axiom,
    ! [A: int,Nb: 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)),Nb))
    <=> ( ( A != zero_zero_int )
        | ( Nb = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
tff(fact_6520_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_6521_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_6522_power2__abs,axiom,
    ! [A: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,A)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_abs
tff(fact_6523_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_6524_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_6525_abs__power2,axiom,
    ! [A: real] : aa_real_real(abs_abs_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% abs_power2
tff(fact_6526_or__nat__numerals_I1_J,axiom,
    ! [Yb: 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,Yb))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Yb)) ).

% or_nat_numerals(1)
tff(fact_6527_or__nat__numerals_I3_J,axiom,
    ! [Xb: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Xb))),aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xb)) ).

% or_nat_numerals(3)
tff(fact_6528_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_6529_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_6530_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(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,A)),aa_num_nat(numeral_numeral_nat,W)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,W)) ) ) ).

% power_even_abs_numeral
tff(fact_6531_real__sqrt__abs,axiom,
    ! [Xb: real] : aa_real_real(sqrt,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_real_real(abs_abs_real,Xb) ).

% real_sqrt_abs
tff(fact_6532_or__minus__numerals_I8_J,axiom,
    ! [Nb: num,Mb: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))),aa_num_int(numeral_numeral_int,Mb)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit_or_not_num_neg(Mb),aa_num_num(bit0,Nb)))) ).

% or_minus_numerals(8)
tff(fact_6533_or__minus__numerals_I4_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit_or_not_num_neg(Mb),aa_num_num(bit0,Nb)))) ).

% or_minus_numerals(4)
tff(fact_6534_or__minus__numerals_I7_J,axiom,
    ! [Nb: num,Mb: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))),aa_num_int(numeral_numeral_int,Mb)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit_or_not_num_neg(Mb),aa_num_num(bitM,Nb)))) ).

% or_minus_numerals(7)
tff(fact_6535_or__minus__numerals_I3_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit_or_not_num_neg(Mb),aa_num_num(bitM,Nb)))) ).

% or_minus_numerals(3)
tff(fact_6536_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_6537_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_6538_abs__ge__self,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(abs_abs_rat,A)) ).

% abs_ge_self
tff(fact_6539_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_6540_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_6541_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_6542_abs__le__D1,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_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_6543_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_6544_abs__eq__0__iff,axiom,
    ! [A: code_integer] :
      ( ( abs_abs_Code_integer(A) = zero_z3403309356797280102nteger )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% abs_eq_0_iff
tff(fact_6545_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_6546_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_6547_abs__eq__0__iff,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(abs_abs_rat,A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% abs_eq_0_iff
tff(fact_6548_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_6549_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_6550_abs__mult,axiom,
    ! [A: real,B: real] : aa_real_real(abs_abs_real,aa_real_real(times_times_real(A),B)) = aa_real_real(times_times_real(aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B)) ).

% abs_mult
tff(fact_6551_abs__mult,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(times_times_rat(A),B)) = aa_rat_rat(times_times_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B)) ).

% abs_mult
tff(fact_6552_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_6553_abs__one,axiom,
    abs_abs_Code_integer(one_one_Code_integer) = one_one_Code_integer ).

% abs_one
tff(fact_6554_abs__one,axiom,
    aa_real_real(abs_abs_real,one_one_real) = one_one_real ).

% abs_one
tff(fact_6555_abs__one,axiom,
    aa_rat_rat(abs_abs_rat,one_one_rat) = one_one_rat ).

% abs_one
tff(fact_6556_abs__one,axiom,
    aa_int_int(abs_abs_int,one_one_int) = one_one_int ).

% abs_one
tff(fact_6557_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_6558_abs__minus__commute,axiom,
    ! [A: real,B: real] : aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(A),B)) = aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(B),A)) ).

% abs_minus_commute
tff(fact_6559_abs__minus__commute,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(A),B)) = aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(B),A)) ).

% abs_minus_commute
tff(fact_6560_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_6561_abs__eq__iff,axiom,
    ! [Xb: real,Yb: real] :
      ( ( aa_real_real(abs_abs_real,Xb) = aa_real_real(abs_abs_real,Yb) )
    <=> ( ( Xb = Yb )
        | ( Xb = aa_real_real(uminus_uminus_real,Yb) ) ) ) ).

% abs_eq_iff
tff(fact_6562_abs__eq__iff,axiom,
    ! [Xb: int,Yb: int] :
      ( ( aa_int_int(abs_abs_int,Xb) = aa_int_int(abs_abs_int,Yb) )
    <=> ( ( Xb = Yb )
        | ( Xb = aa_int_int(uminus_uminus_int,Yb) ) ) ) ).

% abs_eq_iff
tff(fact_6563_abs__eq__iff,axiom,
    ! [Xb: rat,Yb: rat] :
      ( ( aa_rat_rat(abs_abs_rat,Xb) = aa_rat_rat(abs_abs_rat,Yb) )
    <=> ( ( Xb = Yb )
        | ( Xb = aa_rat_rat(uminus_uminus_rat,Yb) ) ) ) ).

% abs_eq_iff
tff(fact_6564_abs__eq__iff,axiom,
    ! [Xb: code_integer,Yb: code_integer] :
      ( ( abs_abs_Code_integer(Xb) = abs_abs_Code_integer(Yb) )
    <=> ( ( Xb = Yb )
        | ( Xb = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Yb) ) ) ) ).

% abs_eq_iff
tff(fact_6565_power__abs,axiom,
    ! [A: code_integer,Nb: nat] : abs_abs_Code_integer(aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb)) = aa_nat_Code_integer(power_8256067586552552935nteger(abs_abs_Code_integer(A)),Nb) ).

% power_abs
tff(fact_6566_power__abs,axiom,
    ! [A: int,Nb: nat] : aa_int_int(abs_abs_int,aa_nat_int(power_power_int(A),Nb)) = aa_nat_int(power_power_int(aa_int_int(abs_abs_int,A)),Nb) ).

% power_abs
tff(fact_6567_power__abs,axiom,
    ! [A: real,Nb: nat] : aa_real_real(abs_abs_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,A)),Nb) ).

% power_abs
tff(fact_6568_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_6569_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_6570_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_6571_or__not__num__neg_Osimps_I1_J,axiom,
    aa_num_num(bit_or_not_num_neg(one),one) = one ).

% or_not_num_neg.simps(1)
tff(fact_6572_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_6573_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_6574_abs__ge__zero,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(abs_abs_rat,A)) ).

% abs_ge_zero
tff(fact_6575_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_6576_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_6577_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_6578_abs__not__less__zero,axiom,
    ! [A: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,A)),zero_zero_rat) ).

% abs_not_less_zero
tff(fact_6579_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_6580_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_6581_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_6582_abs__of__pos,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_rat(abs_abs_rat,A) = A ) ) ).

% abs_of_pos
tff(fact_6583_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_6584_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_6585_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(plus_plus_real(A),B))),aa_real_real(plus_plus_real(aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B))) ).

% abs_triangle_ineq
tff(fact_6586_abs__triangle__ineq,axiom,
    ! [A: rat,B: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(plus_plus_rat(A),B))),aa_rat_rat(plus_plus_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B))) ).

% abs_triangle_ineq
tff(fact_6587_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_6588_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_6589_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(times_times_real(aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B))),aa_real_real(times_times_real(C),D)) ) ) ).

% abs_mult_less
tff(fact_6590_abs__mult__less,axiom,
    ! [A: rat,C: rat,B: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,A)),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,B)),D)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B))),aa_rat_rat(times_times_rat(C),D)) ) ) ).

% abs_mult_less
tff(fact_6591_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_6592_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_6593_abs__triangle__ineq2,axiom,
    ! [A: real,B: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_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(minus_minus_real(A),B))) ).

% abs_triangle_ineq2
tff(fact_6594_abs__triangle__ineq2,axiom,
    ! [A: rat,B: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B))),aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(A),B))) ).

% abs_triangle_ineq2
tff(fact_6595_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_6596_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_6597_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(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(minus_minus_real(A),B))) ).

% abs_triangle_ineq3
tff(fact_6598_abs__triangle__ineq3,axiom,
    ! [A: rat,B: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B)))),aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(A),B))) ).

% abs_triangle_ineq3
tff(fact_6599_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_6600_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_6601_abs__triangle__ineq2__sym,axiom,
    ! [A: real,B: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_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(minus_minus_real(B),A))) ).

% abs_triangle_ineq2_sym
tff(fact_6602_abs__triangle__ineq2__sym,axiom,
    ! [A: rat,B: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B))),aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(B),A))) ).

% abs_triangle_ineq2_sym
tff(fact_6603_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_6604_nonzero__abs__divide,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( aa_real_real(abs_abs_real,aa_real_real(divide_divide_real(A),B)) = aa_real_real(divide_divide_real(aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B)) ) ) ).

% nonzero_abs_divide
tff(fact_6605_nonzero__abs__divide,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( aa_rat_rat(abs_abs_rat,aa_rat_rat(divide_divide_rat(A),B)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B)) ) ) ).

% nonzero_abs_divide
tff(fact_6606_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_6607_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_6608_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,aa_rat_rat(uminus_uminus_rat,A)),B)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A)),B) ) ) ).

% abs_leI
tff(fact_6609_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_6610_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_6611_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_6612_abs__le__D2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A)),B)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A)),B) ) ).

% abs_le_D2
tff(fact_6613_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_6614_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_6615_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_6616_abs__le__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_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,aa_rat_rat(uminus_uminus_rat,A)),B) ) ) ).

% abs_le_iff
tff(fact_6617_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_6618_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_6619_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_6620_abs__ge__minus__self,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(abs_abs_rat,A)) ).

% abs_ge_minus_self
tff(fact_6621_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_6622_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_6623_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_6624_abs__less__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_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,aa_rat_rat(uminus_uminus_rat,A)),B) ) ) ).

% abs_less_iff
tff(fact_6625_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_6626_tanh__real__lt__1,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(tanh_real,Xb)),one_one_real) ).

% tanh_real_lt_1
tff(fact_6627_or__not__num__neg_Osimps_I4_J,axiom,
    ! [Nb: num] : aa_num_num(bit_or_not_num_neg(aa_num_num(bit0,Nb)),one) = aa_num_num(bit0,one) ).

% or_not_num_neg.simps(4)
tff(fact_6628_or__not__num__neg_Osimps_I6_J,axiom,
    ! [Nb: num,Mb: num] : aa_num_num(bit_or_not_num_neg(aa_num_num(bit0,Nb)),aa_num_num(bit1,Mb)) = aa_num_num(bit0,aa_num_num(bit_or_not_num_neg(Nb),Mb)) ).

% or_not_num_neg.simps(6)
tff(fact_6629_or__not__num__neg_Osimps_I3_J,axiom,
    ! [Mb: num] : aa_num_num(bit_or_not_num_neg(one),aa_num_num(bit1,Mb)) = aa_num_num(bit1,Mb) ).

% or_not_num_neg.simps(3)
tff(fact_6630_or__not__num__neg_Osimps_I7_J,axiom,
    ! [Nb: num] : aa_num_num(bit_or_not_num_neg(aa_num_num(bit1,Nb)),one) = one ).

% or_not_num_neg.simps(7)
tff(fact_6631_dense__eq0__I,axiom,
    ! [Xb: 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,Xb)),E) )
     => ( Xb = zero_zero_real ) ) ).

% dense_eq0_I
tff(fact_6632_dense__eq0__I,axiom,
    ! [Xb: 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,aa_rat_rat(abs_abs_rat,Xb)),E) )
     => ( Xb = zero_zero_rat ) ) ).

% dense_eq0_I
tff(fact_6633_or__not__num__neg_Osimps_I5_J,axiom,
    ! [Nb: num,Mb: num] : aa_num_num(bit_or_not_num_neg(aa_num_num(bit0,Nb)),aa_num_num(bit0,Mb)) = aa_num_num(bitM,aa_num_num(bit_or_not_num_neg(Nb),Mb)) ).

% or_not_num_neg.simps(5)
tff(fact_6634_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_6635_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(times_times_real(A),B)) = aa_real_real(times_times_real(aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B)) ) ) ).

% abs_eq_mult
tff(fact_6636_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) ) )
     => ( aa_rat_rat(abs_abs_rat,aa_rat_rat(times_times_rat(A),B)) = aa_rat_rat(times_times_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B)) ) ) ).

% abs_eq_mult
tff(fact_6637_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_6638_abs__mult__pos,axiom,
    ! [Xb: code_integer,Yb: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Xb)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,abs_abs_Code_integer(Yb)),Xb) = abs_abs_Code_integer(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Yb),Xb)) ) ) ).

% abs_mult_pos
tff(fact_6639_abs__mult__pos,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_real(times_times_real(aa_real_real(abs_abs_real,Yb)),Xb) = aa_real_real(abs_abs_real,aa_real_real(times_times_real(Yb),Xb)) ) ) ).

% abs_mult_pos
tff(fact_6640_abs__mult__pos,axiom,
    ! [Xb: rat,Yb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xb)
     => ( aa_rat_rat(times_times_rat(aa_rat_rat(abs_abs_rat,Yb)),Xb) = aa_rat_rat(abs_abs_rat,aa_rat_rat(times_times_rat(Yb),Xb)) ) ) ).

% abs_mult_pos
tff(fact_6641_abs__mult__pos,axiom,
    ! [Xb: int,Yb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,Yb)),Xb) = aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(times_times_int,Yb),Xb)) ) ) ).

% abs_mult_pos
tff(fact_6642_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_6643_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_6644_abs__minus__le__zero,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(abs_abs_rat,A))),zero_zero_rat) ).

% abs_minus_le_zero
tff(fact_6645_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_6646_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_6647_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_6648_eq__abs__iff_H,axiom,
    ! [A: rat,B: rat] :
      ( ( A = aa_rat_rat(abs_abs_rat,B) )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
        & ( ( B = A )
          | ( B = aa_rat_rat(uminus_uminus_rat,A) ) ) ) ) ).

% eq_abs_iff'
tff(fact_6649_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_6650_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_6651_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_6652_abs__eq__iff_H,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_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 = aa_rat_rat(uminus_uminus_rat,B) ) ) ) ) ).

% abs_eq_iff'
tff(fact_6653_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_6654_zero__le__power__abs,axiom,
    ! [A: code_integer,Nb: nat] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(abs_abs_Code_integer(A)),Nb)) ).

% zero_le_power_abs
tff(fact_6655_zero__le__power__abs,axiom,
    ! [A: real,Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,A)),Nb)) ).

% zero_le_power_abs
tff(fact_6656_zero__le__power__abs,axiom,
    ! [A: rat,Nb: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(aa_rat_rat(abs_abs_rat,A)),Nb)) ).

% zero_le_power_abs
tff(fact_6657_zero__le__power__abs,axiom,
    ! [A: int,Nb: 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)),Nb)) ).

% zero_le_power_abs
tff(fact_6658_abs__div__pos,axiom,
    ! [Yb: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Yb)
     => ( aa_real_real(divide_divide_real(aa_real_real(abs_abs_real,Xb)),Yb) = aa_real_real(abs_abs_real,aa_real_real(divide_divide_real(Xb),Yb)) ) ) ).

% abs_div_pos
tff(fact_6659_abs__div__pos,axiom,
    ! [Yb: rat,Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Yb)
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(abs_abs_rat,Xb)),Yb) = aa_rat_rat(abs_abs_rat,aa_rat_rat(divide_divide_rat(Xb),Yb)) ) ) ).

% abs_div_pos
tff(fact_6660_abs__if__raw,axiom,
    ! [X: real] :
      aa_real_real(abs_abs_real,X) = $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,X),zero_zero_real),aa_real_real(uminus_uminus_real,X),X) ).

% abs_if_raw
tff(fact_6661_abs__if__raw,axiom,
    ! [X: int] :
      aa_int_int(abs_abs_int,X) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,X),zero_zero_int),aa_int_int(uminus_uminus_int,X),X) ).

% abs_if_raw
tff(fact_6662_abs__if__raw,axiom,
    ! [X: rat] :
      aa_rat_rat(abs_abs_rat,X) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),zero_zero_rat),aa_rat_rat(uminus_uminus_rat,X),X) ).

% abs_if_raw
tff(fact_6663_abs__if__raw,axiom,
    ! [X: code_integer] :
      abs_abs_Code_integer(X) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X),zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,X),X) ).

% abs_if_raw
tff(fact_6664_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_6665_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_6666_abs__of__neg,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_rat(abs_abs_rat,A) = aa_rat_rat(uminus_uminus_rat,A) ) ) ).

% abs_of_neg
tff(fact_6667_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_6668_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_6669_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_6670_abs__if,axiom,
    ! [A: rat] :
      aa_rat_rat(abs_abs_rat,A) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat),aa_rat_rat(uminus_uminus_rat,A),A) ).

% abs_if
tff(fact_6671_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_6672_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_6673_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(minus_minus_real(aa_real_real(plus_plus_real(A),B)),aa_real_real(plus_plus_real(C),D)))),aa_real_real(plus_plus_real(aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(A),C))),aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(B),D)))) ).

% abs_diff_triangle_ineq
tff(fact_6674_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,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(A),B)),aa_rat_rat(plus_plus_rat(C),D)))),aa_rat_rat(plus_plus_rat(aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(A),C))),aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(B),D)))) ).

% abs_diff_triangle_ineq
tff(fact_6675_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_6676_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_6677_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(minus_minus_real(A),B))),aa_real_real(plus_plus_real(aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B))) ).

% abs_triangle_ineq4
tff(fact_6678_abs__triangle__ineq4,axiom,
    ! [A: rat,B: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(A),B))),aa_rat_rat(plus_plus_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B))) ).

% abs_triangle_ineq4
tff(fact_6679_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_6680_abs__diff__le__iff,axiom,
    ! [Xb: code_integer,A: code_integer,R2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(Xb,A))),R2)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,minus_8373710615458151222nteger(A,R2)),Xb)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Xb),plus_p5714425477246183910nteger(A,R2)) ) ) ).

% abs_diff_le_iff
tff(fact_6681_abs__diff__le__iff,axiom,
    ! [Xb: real,A: real,R2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(Xb),A))),R2)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(A),R2)),Xb)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(plus_plus_real(A),R2)) ) ) ).

% abs_diff_le_iff
tff(fact_6682_abs__diff__le__iff,axiom,
    ! [Xb: rat,A: rat,R2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(Xb),A))),R2)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(A),R2)),Xb)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),aa_rat_rat(plus_plus_rat(A),R2)) ) ) ).

% abs_diff_le_iff
tff(fact_6683_abs__diff__le__iff,axiom,
    ! [Xb: int,A: int,R2: 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,Xb),A))),R2)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),R2)),Xb)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),R2)) ) ) ).

% abs_diff_le_iff
tff(fact_6684_abs__diff__less__iff,axiom,
    ! [Xb: code_integer,A: code_integer,R2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(Xb,A))),R2)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,minus_8373710615458151222nteger(A,R2)),Xb)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Xb),plus_p5714425477246183910nteger(A,R2)) ) ) ).

% abs_diff_less_iff
tff(fact_6685_abs__diff__less__iff,axiom,
    ! [Xb: real,A: real,R2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(Xb),A))),R2)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(minus_minus_real(A),R2)),Xb)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(plus_plus_real(A),R2)) ) ) ).

% abs_diff_less_iff
tff(fact_6686_abs__diff__less__iff,axiom,
    ! [Xb: rat,A: rat,R2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(Xb),A))),R2)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(A),R2)),Xb)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),aa_rat_rat(plus_plus_rat(A),R2)) ) ) ).

% abs_diff_less_iff
tff(fact_6687_abs__diff__less__iff,axiom,
    ! [Xb: int,A: int,R2: 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,Xb),A))),R2)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),R2)),Xb)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),R2)) ) ) ).

% abs_diff_less_iff
tff(fact_6688_or__not__num__neg_Osimps_I9_J,axiom,
    ! [Nb: num,Mb: num] : aa_num_num(bit_or_not_num_neg(aa_num_num(bit1,Nb)),aa_num_num(bit1,Mb)) = aa_num_num(bitM,aa_num_num(bit_or_not_num_neg(Nb),Mb)) ).

% or_not_num_neg.simps(9)
tff(fact_6689_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_6690_sin__bound__lemma,axiom,
    ! [Xb: real,Yb: real,U: real,V: real] :
      ( ( Xb = Yb )
     => ( 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(minus_minus_real(aa_real_real(plus_plus_real(Xb),U)),Yb))),V) ) ) ).

% sin_bound_lemma
tff(fact_6691_tanh__real__gt__neg1,axiom,
    ! [Xb: 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,Xb)) ).

% tanh_real_gt_neg1
tff(fact_6692_abs__add__one__gt__zero,axiom,
    ! [Xb: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),plus_p5714425477246183910nteger(one_one_Code_integer,abs_abs_Code_integer(Xb))) ).

% abs_add_one_gt_zero
tff(fact_6693_abs__add__one__gt__zero,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(plus_plus_real(one_one_real),aa_real_real(abs_abs_real,Xb))) ).

% abs_add_one_gt_zero
tff(fact_6694_abs__add__one__gt__zero,axiom,
    ! [Xb: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(one_one_rat),aa_rat_rat(abs_abs_rat,Xb))) ).

% abs_add_one_gt_zero
tff(fact_6695_abs__add__one__gt__zero,axiom,
    ! [Xb: 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,Xb))) ).

% abs_add_one_gt_zero
tff(fact_6696_or__not__num__neg_Osimps_I2_J,axiom,
    ! [Mb: num] : aa_num_num(bit_or_not_num_neg(one),aa_num_num(bit0,Mb)) = aa_num_num(bit1,Mb) ).

% or_not_num_neg.simps(2)
tff(fact_6697_of__int__leD,axiom,
    ! [Nb: int,Xb: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(aa_int_Code_integer(ring_18347121197199848620nteger,Nb))),Xb)
     => ( ( Nb = zero_zero_int )
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),Xb) ) ) ).

% of_int_leD
tff(fact_6698_of__int__leD,axiom,
    ! [Nb: int,Xb: 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,Nb))),Xb)
     => ( ( Nb = zero_zero_int )
        | aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xb) ) ) ).

% of_int_leD
tff(fact_6699_of__int__leD,axiom,
    ! [Nb: int,Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_int_rat(ring_1_of_int_rat,Nb))),Xb)
     => ( ( Nb = zero_zero_int )
        | aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),Xb) ) ) ).

% of_int_leD
tff(fact_6700_of__int__leD,axiom,
    ! [Nb: int,Xb: 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,Nb))),Xb)
     => ( ( Nb = zero_zero_int )
        | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),Xb) ) ) ).

% of_int_leD
tff(fact_6701_of__int__lessD,axiom,
    ! [Nb: int,Xb: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,abs_abs_Code_integer(aa_int_Code_integer(ring_18347121197199848620nteger,Nb))),Xb)
     => ( ( Nb = zero_zero_int )
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),Xb) ) ) ).

% of_int_lessD
tff(fact_6702_of__int__lessD,axiom,
    ! [Nb: int,Xb: 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,Nb))),Xb)
     => ( ( Nb = zero_zero_int )
        | aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb) ) ) ).

% of_int_lessD
tff(fact_6703_of__int__lessD,axiom,
    ! [Nb: int,Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_int_rat(ring_1_of_int_rat,Nb))),Xb)
     => ( ( Nb = zero_zero_int )
        | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),Xb) ) ) ).

% of_int_lessD
tff(fact_6704_of__int__lessD,axiom,
    ! [Nb: int,Xb: 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,Nb))),Xb)
     => ( ( Nb = zero_zero_int )
        | aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),Xb) ) ) ).

% of_int_lessD
tff(fact_6705_or__not__num__neg_Osimps_I8_J,axiom,
    ! [Nb: num,Mb: num] : aa_num_num(bit_or_not_num_neg(aa_num_num(bit1,Nb)),aa_num_num(bit0,Mb)) = aa_num_num(bitM,aa_num_num(bit_or_not_num_neg(Nb),Mb)) ).

% or_not_num_neg.simps(8)
tff(fact_6706_round__diff__minimal,axiom,
    ! [Z: real,Mb: int] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_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(minus_minus_real(Z),aa_int_real(ring_1_of_int_real,Mb)))) ).

% round_diff_minimal
tff(fact_6707_round__diff__minimal,axiom,
    ! [Z: rat,Mb: int] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(Z),aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim7778729529865785530nd_rat,Z))))),aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(Z),aa_int_rat(ring_1_of_int_rat,Mb)))) ).

% round_diff_minimal
tff(fact_6708_bot__empty__eq2,axiom,
    ! [X: code_integer,Xa: $o] :
      ( aa_o_o(aa_Cod6379977994291289310un_o_o(bot_bo897209107778963633un_o_o,X),(Xa))
    <=> member1379723562493234055eger_o(produc6677183202524767010eger_o(X,(Xa)),bot_bo5379713665208646970eger_o) ) ).

% bot_empty_eq2
tff(fact_6709_bot__empty__eq2,axiom,
    ! [X: num,Xa: num] :
      ( aa_num_o(aa_num_fun_num_o(bot_bo3657453288633029378_num_o,X),Xa)
    <=> member7279096912039735102um_num(product_Pair_num_num(X,Xa),bot_bo9056780473022590049um_num) ) ).

% bot_empty_eq2
tff(fact_6710_bot__empty__eq2,axiom,
    ! [X: nat,Xa: num] :
      ( aa_num_o(aa_nat_fun_num_o(bot_bo1877178017081890764_num_o,X),Xa)
    <=> member9148766508732265716at_num(product_Pair_nat_num(X,Xa),bot_bo7038385379056416535at_num) ) ).

% bot_empty_eq2
tff(fact_6711_bot__empty__eq2,axiom,
    ! [X: nat,Xa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bot_bo1118997454112379798_nat_o,X),Xa)
    <=> member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Xa),bot_bo2099793752762293965at_nat) ) ).

% bot_empty_eq2
tff(fact_6712_bot__empty__eq2,axiom,
    ! [X: int,Xa: int] :
      ( aa_int_o(aa_int_fun_int_o(bot_bo428638681919004382_int_o,X),Xa)
    <=> member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X),Xa),bot_bo1796632182523588997nt_int) ) ).

% bot_empty_eq2
tff(fact_6713_pred__equals__eq2,axiom,
    ! [R: set_Pr448751882837621926eger_o,S3: set_Pr448751882837621926eger_o] :
      ( ! [X4: code_integer,Xa3: $o] :
          ( member1379723562493234055eger_o(produc6677183202524767010eger_o(X4,(Xa3)),R)
        <=> member1379723562493234055eger_o(produc6677183202524767010eger_o(X4,(Xa3)),S3) )
    <=> ( R = S3 ) ) ).

% pred_equals_eq2
tff(fact_6714_pred__equals__eq2,axiom,
    ! [R: set_Pr8218934625190621173um_num,S3: set_Pr8218934625190621173um_num] :
      ( ! [X4: num,Xa3: num] :
          ( member7279096912039735102um_num(product_Pair_num_num(X4,Xa3),R)
        <=> member7279096912039735102um_num(product_Pair_num_num(X4,Xa3),S3) )
    <=> ( R = S3 ) ) ).

% pred_equals_eq2
tff(fact_6715_pred__equals__eq2,axiom,
    ! [R: set_Pr6200539531224447659at_num,S3: set_Pr6200539531224447659at_num] :
      ( ! [X4: nat,Xa3: num] :
          ( member9148766508732265716at_num(product_Pair_nat_num(X4,Xa3),R)
        <=> member9148766508732265716at_num(product_Pair_nat_num(X4,Xa3),S3) )
    <=> ( R = S3 ) ) ).

% pred_equals_eq2
tff(fact_6716_pred__equals__eq2,axiom,
    ! [R: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat] :
      ( ! [X4: nat,Xa3: nat] :
          ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X4),Xa3),R)
        <=> member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X4),Xa3),S3) )
    <=> ( R = S3 ) ) ).

% pred_equals_eq2
tff(fact_6717_pred__equals__eq2,axiom,
    ! [R: set_Pr958786334691620121nt_int,S3: set_Pr958786334691620121nt_int] :
      ( ! [X4: int,Xa3: int] :
          ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X4),Xa3),R)
        <=> member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X4),Xa3),S3) )
    <=> ( R = S3 ) ) ).

% pred_equals_eq2
tff(fact_6718_abs__le__square__iff,axiom,
    ! [Xb: code_integer,Yb: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(Xb)),abs_abs_Code_integer(Yb))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_Code_integer(power_8256067586552552935nteger(Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% abs_le_square_iff
tff(fact_6719_abs__le__square__iff,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),aa_real_real(abs_abs_real,Yb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% abs_le_square_iff
tff(fact_6720_abs__le__square__iff,axiom,
    ! [Xb: rat,Yb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,Xb)),aa_rat_rat(abs_abs_rat,Yb))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% abs_le_square_iff
tff(fact_6721_abs__le__square__iff,axiom,
    ! [Xb: int,Yb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,Xb)),aa_int_int(abs_abs_int,Yb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% abs_le_square_iff
tff(fact_6722_abs__square__eq__1,axiom,
    ! [Xb: code_integer] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_Code_integer )
    <=> ( abs_abs_Code_integer(Xb) = one_one_Code_integer ) ) ).

% abs_square_eq_1
tff(fact_6723_abs__square__eq__1,axiom,
    ! [Xb: rat] :
      ( ( aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_rat )
    <=> ( aa_rat_rat(abs_abs_rat,Xb) = one_one_rat ) ) ).

% abs_square_eq_1
tff(fact_6724_abs__square__eq__1,axiom,
    ! [Xb: int] :
      ( ( aa_nat_int(power_power_int(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_int )
    <=> ( aa_int_int(abs_abs_int,Xb) = one_one_int ) ) ).

% abs_square_eq_1
tff(fact_6725_abs__square__eq__1,axiom,
    ! [Xb: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_real )
    <=> ( aa_real_real(abs_abs_real,Xb) = one_one_real ) ) ).

% abs_square_eq_1
tff(fact_6726_power__even__abs,axiom,
    ! [Nb: 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))),Nb)
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(abs_abs_Code_integer(A)),Nb) = aa_nat_Code_integer(power_8256067586552552935nteger(A),Nb) ) ) ).

% power_even_abs
tff(fact_6727_power__even__abs,axiom,
    ! [Nb: 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))),Nb)
     => ( aa_nat_int(power_power_int(aa_int_int(abs_abs_int,A)),Nb) = aa_nat_int(power_power_int(A),Nb) ) ) ).

% power_even_abs
tff(fact_6728_power__even__abs,axiom,
    ! [Nb: 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))),Nb)
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,A)),Nb) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb) ) ) ).

% power_even_abs
tff(fact_6729_or__not__num__neg_Oelims,axiom,
    ! [Xb: num,Xaa: num,Yb: num] :
      ( ( aa_num_num(bit_or_not_num_neg(Xb),Xaa) = Yb )
     => ( ( ( Xb = one )
         => ( ( Xaa = one )
           => ( Yb != one ) ) )
       => ( ( ( Xb = one )
           => ! [M2: num] :
                ( ( Xaa = aa_num_num(bit0,M2) )
               => ( Yb != aa_num_num(bit1,M2) ) ) )
         => ( ( ( Xb = one )
             => ! [M2: num] :
                  ( ( Xaa = aa_num_num(bit1,M2) )
                 => ( Yb != aa_num_num(bit1,M2) ) ) )
           => ( ( ? [N: num] : Xb = aa_num_num(bit0,N)
               => ( ( Xaa = one )
                 => ( Yb != aa_num_num(bit0,one) ) ) )
             => ( ! [N: num] :
                    ( ( Xb = aa_num_num(bit0,N) )
                   => ! [M2: num] :
                        ( ( Xaa = aa_num_num(bit0,M2) )
                       => ( Yb != aa_num_num(bitM,aa_num_num(bit_or_not_num_neg(N),M2)) ) ) )
               => ( ! [N: num] :
                      ( ( Xb = aa_num_num(bit0,N) )
                     => ! [M2: num] :
                          ( ( Xaa = aa_num_num(bit1,M2) )
                         => ( Yb != aa_num_num(bit0,aa_num_num(bit_or_not_num_neg(N),M2)) ) ) )
                 => ( ( ? [N: num] : Xb = aa_num_num(bit1,N)
                     => ( ( Xaa = one )
                       => ( Yb != one ) ) )
                   => ( ! [N: num] :
                          ( ( Xb = aa_num_num(bit1,N) )
                         => ! [M2: num] :
                              ( ( Xaa = aa_num_num(bit0,M2) )
                             => ( Yb != aa_num_num(bitM,aa_num_num(bit_or_not_num_neg(N),M2)) ) ) )
                     => ~ ! [N: num] :
                            ( ( Xb = aa_num_num(bit1,N) )
                           => ! [M2: num] :
                                ( ( Xaa = aa_num_num(bit1,M2) )
                               => ( Yb != aa_num_num(bitM,aa_num_num(bit_or_not_num_neg(N),M2)) ) ) ) ) ) ) ) ) ) ) ) ) ).

% or_not_num_neg.elims
tff(fact_6730_power2__le__iff__abs__le,axiom,
    ! [Yb: code_integer,Xb: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Yb)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_Code_integer(power_8256067586552552935nteger(Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(Xb)),Yb) ) ) ).

% power2_le_iff_abs_le
tff(fact_6731_power2__le__iff__abs__le,axiom,
    ! [Yb: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Yb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Yb),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,Xb)),Yb) ) ) ).

% power2_le_iff_abs_le
tff(fact_6732_power2__le__iff__abs__le,axiom,
    ! [Yb: rat,Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Yb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,Xb)),Yb) ) ) ).

% power2_le_iff_abs_le
tff(fact_6733_power2__le__iff__abs__le,axiom,
    ! [Yb: int,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Yb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Yb),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,Xb)),Yb) ) ) ).

% power2_le_iff_abs_le
tff(fact_6734_abs__square__le__1,axiom,
    ! [Xb: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(Xb),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(Xb)),one_one_Code_integer) ) ).

% abs_square_le_1
tff(fact_6735_abs__square__le__1,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),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,Xb)),one_one_real) ) ).

% abs_square_le_1
tff(fact_6736_abs__square__le__1,axiom,
    ! [Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(Xb),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,aa_rat_rat(abs_abs_rat,Xb)),one_one_rat) ) ).

% abs_square_le_1
tff(fact_6737_abs__square__le__1,axiom,
    ! [Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(Xb),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,Xb)),one_one_int) ) ).

% abs_square_le_1
tff(fact_6738_abs__square__less__1,axiom,
    ! [Xb: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(Xb),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(Xb)),one_one_Code_integer) ) ).

% abs_square_less_1
tff(fact_6739_abs__square__less__1,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),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,Xb)),one_one_real) ) ).

% abs_square_less_1
tff(fact_6740_abs__square__less__1,axiom,
    ! [Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(Xb),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,aa_rat_rat(abs_abs_rat,Xb)),one_one_rat) ) ).

% abs_square_less_1
tff(fact_6741_abs__square__less__1,axiom,
    ! [Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(Xb),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,Xb)),one_one_int) ) ).

% abs_square_less_1
tff(fact_6742_power__mono__even,axiom,
    ! [Nb: 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))),Nb)
     => ( 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),Nb)),aa_nat_Code_integer(power_8256067586552552935nteger(B),Nb)) ) ) ).

% power_mono_even
tff(fact_6743_power__mono__even,axiom,
    ! [Nb: 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))),Nb)
     => ( 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(aa_real_fun_nat_real(power_power_real,A),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb)) ) ) ).

% power_mono_even
tff(fact_6744_power__mono__even,axiom,
    ! [Nb: 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))),Nb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B))
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),Nb)),aa_nat_rat(power_power_rat(B),Nb)) ) ) ).

% power_mono_even
tff(fact_6745_power__mono__even,axiom,
    ! [Nb: 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))),Nb)
     => ( 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),Nb)),aa_nat_int(power_power_int(B),Nb)) ) ) ).

% power_mono_even
tff(fact_6746_sqrt__ge__absD,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),aa_real_real(sqrt,Yb))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Yb) ) ).

% sqrt_ge_absD
tff(fact_6747_real__sqrt__ge__abs1,axiom,
    ! [Xb: real,Yb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),aa_real_real(sqrt,aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% real_sqrt_ge_abs1
tff(fact_6748_real__sqrt__ge__abs2,axiom,
    ! [Yb: real,Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Yb)),aa_real_real(sqrt,aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% real_sqrt_ge_abs2
tff(fact_6749_sqrt__sum__squares__le__sum__abs,axiom,
    ! [Xb: real,Yb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(plus_plus_real(aa_real_real(abs_abs_real,Xb)),aa_real_real(abs_abs_real,Yb))) ).

% sqrt_sum_squares_le_sum_abs
tff(fact_6750_subrelI,axiom,
    ! [R2: set_Pr448751882837621926eger_o,S2: set_Pr448751882837621926eger_o] :
      ( ! [X3: code_integer,Y: $o] :
          ( member1379723562493234055eger_o(produc6677183202524767010eger_o(X3,(Y)),R2)
         => member1379723562493234055eger_o(produc6677183202524767010eger_o(X3,(Y)),S2) )
     => ord_le8980329558974975238eger_o(R2,S2) ) ).

% subrelI
tff(fact_6751_subrelI,axiom,
    ! [R2: set_Pr8218934625190621173um_num,S2: set_Pr8218934625190621173um_num] :
      ( ! [X3: num,Y: num] :
          ( member7279096912039735102um_num(product_Pair_num_num(X3,Y),R2)
         => member7279096912039735102um_num(product_Pair_num_num(X3,Y),S2) )
     => ord_le880128212290418581um_num(R2,S2) ) ).

% subrelI
tff(fact_6752_subrelI,axiom,
    ! [R2: set_Pr6200539531224447659at_num,S2: set_Pr6200539531224447659at_num] :
      ( ! [X3: nat,Y: num] :
          ( member9148766508732265716at_num(product_Pair_nat_num(X3,Y),R2)
         => member9148766508732265716at_num(product_Pair_nat_num(X3,Y),S2) )
     => ord_le8085105155179020875at_num(R2,S2) ) ).

% subrelI
tff(fact_6753_subrelI,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
      ( ! [X3: nat,Y: nat] :
          ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),Y),R2)
         => member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),Y),S2) )
     => ord_le3146513528884898305at_nat(R2,S2) ) ).

% subrelI
tff(fact_6754_subrelI,axiom,
    ! [R2: set_Pr958786334691620121nt_int,S2: set_Pr958786334691620121nt_int] :
      ( ! [X3: int,Y: int] :
          ( member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X3),Y),R2)
         => member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,X3),Y),S2) )
     => ord_le2843351958646193337nt_int(R2,S2) ) ).

% subrelI
tff(fact_6755_or__Suc__0__eq,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,Nb),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),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))),Nb))) ).

% or_Suc_0_eq
tff(fact_6756_Suc__0__or__eq,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(suc,zero_zero_nat)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),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))),Nb))) ).

% Suc_0_or_eq
tff(fact_6757_or__nat__rec,axiom,
    ! [Mb: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,Mb),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_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))),Mb)
            | ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) ))),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,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% or_nat_rec
tff(fact_6758_or__nat__unfold,axiom,
    ! [Mb: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,Mb),Nb) = $ite(
        Mb = zero_zero_nat,
        Nb,
        $ite(Nb = zero_zero_nat,Mb,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,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),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,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ) ).

% or_nat_unfold
tff(fact_6759_of__int__round__abs__le,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(aa_int_real(ring_1_of_int_real,aa_real_int(archim8280529875227126926d_real,Xb))),Xb))),aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ).

% of_int_round_abs_le
tff(fact_6760_of__int__round__abs__le,axiom,
    ! [Xb: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim7778729529865785530nd_rat,Xb))),Xb))),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))) ).

% of_int_round_abs_le
tff(fact_6761_round__unique_H,axiom,
    ! [Xb: real,Nb: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(Xb),aa_int_real(ring_1_of_int_real,Nb)))),aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
     => ( aa_real_int(archim8280529875227126926d_real,Xb) = Nb ) ) ).

% round_unique'
tff(fact_6762_round__unique_H,axiom,
    ! [Xb: rat,Nb: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(Xb),aa_int_rat(ring_1_of_int_rat,Nb)))),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))
     => ( aa_rat_int(archim7778729529865785530nd_rat,Xb) = Nb ) ) ).

% round_unique'
tff(fact_6763_cos__x__y__le__one,axiom,
    ! [Xb: real,Yb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(divide_divide_real(Xb),aa_real_real(sqrt,aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),one_one_real) ).

% cos_x_y_le_one
tff(fact_6764_real__sqrt__sum__squares__less,axiom,
    ! [Xb: real,U: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xb)),aa_real_real(divide_divide_real(U),aa_real_real(sqrt,aa_num_real(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,Yb)),aa_real_real(divide_divide_real(U),aa_real_real(sqrt,aa_num_real(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(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),U) ) ) ).

% real_sqrt_sum_squares_less
tff(fact_6765_pred__subset__eq,axiom,
    ! [R: set_complex,S3: set_complex] :
      ( ord_le7644322562001619171plex_o(aTP_Lamm_a(R),aTP_Lamm_a(S3))
    <=> ord_le211207098394363844omplex(R,S3) ) ).

% pred_subset_eq
tff(fact_6766_pred__subset__eq,axiom,
    ! [R: set_real,S3: set_real] :
      ( ord_le3605727316730466277real_o(aTP_Lamm_aa(R),aTP_Lamm_aa(S3))
    <=> ord_less_eq_set_real(R,S3) ) ).

% pred_subset_eq
tff(fact_6767_pred__subset__eq,axiom,
    ! [R: set_set_nat,S3: set_set_nat] :
      ( ord_le6468474429079055883_nat_o(aTP_Lamm_ac(R),aTP_Lamm_ac(S3))
    <=> ord_le6893508408891458716et_nat(R,S3) ) ).

% pred_subset_eq
tff(fact_6768_pred__subset__eq,axiom,
    ! [R: set_int,S3: set_int] :
      ( ord_le5418975059635765733_int_o(aTP_Lamm_ae(R),aTP_Lamm_ae(S3))
    <=> ord_less_eq_set_int(R,S3) ) ).

% pred_subset_eq
tff(fact_6769_pred__subset__eq,axiom,
    ! [R: set_nat,S3: set_nat] :
      ( ord_le7240017405068453697_nat_o(aTP_Lamm_ad(R),aTP_Lamm_ad(S3))
    <=> aa_set_nat_o(ord_less_eq_set_nat(R),S3) ) ).

% pred_subset_eq
tff(fact_6770_abs__ln__one__plus__x__minus__x__bound__nonneg,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(aa_real_real(ln_ln_real,aa_real_real(plus_plus_real(one_one_real),Xb))),Xb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ).

% abs_ln_one_plus_x_minus_x_bound_nonneg
tff(fact_6771_abs__sqrt__wlog,axiom,
    ! [P: fun_Co2081214437782444048eger_o,Xb: code_integer] :
      ( ! [X3: code_integer] :
          ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),X3)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(P,X3),aa_nat_Code_integer(power_8256067586552552935nteger(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(P,abs_abs_Code_integer(Xb)),aa_nat_Code_integer(power_8256067586552552935nteger(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% abs_sqrt_wlog
tff(fact_6772_abs__sqrt__wlog,axiom,
    ! [P: fun_real_fun_real_o,Xb: real] :
      ( ! [X3: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X3)
         => aa_real_o(aa_real_fun_real_o(P,X3),aa_nat_real(aa_real_fun_nat_real(power_power_real,X3),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,Xb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% abs_sqrt_wlog
tff(fact_6773_abs__sqrt__wlog,axiom,
    ! [P: fun_rat_fun_rat_o,Xb: rat] :
      ( ! [X3: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),X3)
         => aa_rat_o(aa_rat_fun_rat_o(P,X3),aa_nat_rat(power_power_rat(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) )
     => aa_rat_o(aa_rat_fun_rat_o(P,aa_rat_rat(abs_abs_rat,Xb)),aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% abs_sqrt_wlog
tff(fact_6774_abs__sqrt__wlog,axiom,
    ! [P: fun_int_fun_int_o,Xb: int] :
      ( ! [X3: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X3)
         => aa_int_o(aa_int_fun_int_o(P,X3),aa_nat_int(power_power_int(X3),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,Xb)),aa_nat_int(power_power_int(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% abs_sqrt_wlog
tff(fact_6775_arctan__double,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xb)),one_one_real)
     => ( aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(arctan,Xb)) = aa_real_real(arctan,aa_real_real(divide_divide_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Xb)),aa_real_real(minus_minus_real(one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ).

% arctan_double
tff(fact_6776_arctan__half,axiom,
    ! [Xb: real] : aa_real_real(arctan,Xb) = aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(arctan,aa_real_real(divide_divide_real(Xb),aa_real_real(plus_plus_real(one_one_real),aa_real_real(sqrt,aa_real_real(plus_plus_real(one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))) ).

% arctan_half
tff(fact_6777_int__ge__less__than__def,axiom,
    ! [D: int] : int_ge_less_than(D) = collec213857154873943460nt_int(produc4947309494688390418_int_o(aTP_Lamm_cj(D))) ).

% int_ge_less_than_def
tff(fact_6778_int__ge__less__than2__def,axiom,
    ! [D: int] : int_ge_less_than2(D) = collec213857154873943460nt_int(produc4947309494688390418_int_o(aTP_Lamm_ck(D))) ).

% int_ge_less_than2_def
tff(fact_6779_signed__take__bit__eq__take__bit__minus,axiom,
    ! [Nb: nat,K: int] : aa_int_int(bit_ri631733984087533419it_int(Nb),K) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Nb)),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,Nb))),aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(bit_se1146084159140164899it_int(K),Nb)))) ).

% signed_take_bit_eq_take_bit_minus
tff(fact_6780_bit__0__eq,axiom,
    bit_se1146084159140164899it_int(zero_zero_int) = bot_bot_fun_nat_o ).

% bit_0_eq
tff(fact_6781_bit__0__eq,axiom,
    aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,zero_zero_nat) = bot_bot_fun_nat_o ).

% bit_0_eq
tff(fact_6782_zdvd1__eq,axiom,
    ! [Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xb),one_one_int)
    <=> ( aa_int_int(abs_abs_int,Xb) = one_one_int ) ) ).

% zdvd1_eq
tff(fact_6783_bit__numeral__Bit0__Suc__iff,axiom,
    ! [Mb: num,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,Mb))),aa_nat_nat(suc,Nb))
    <=> aa_nat_o(bit_se1146084159140164899it_int(aa_num_int(numeral_numeral_int,Mb)),Nb) ) ).

% bit_numeral_Bit0_Suc_iff
tff(fact_6784_bit__numeral__Bit0__Suc__iff,axiom,
    ! [Mb: num,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Mb))),aa_nat_nat(suc,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_num_nat(numeral_numeral_nat,Mb)),Nb) ) ).

% bit_numeral_Bit0_Suc_iff
tff(fact_6785_bit__numeral__Bit1__Suc__iff,axiom,
    ! [Mb: num,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,Mb))),aa_nat_nat(suc,Nb))
    <=> aa_nat_o(bit_se1146084159140164899it_int(aa_num_int(numeral_numeral_int,Mb)),Nb) ) ).

% bit_numeral_Bit1_Suc_iff
tff(fact_6786_bit__numeral__Bit1__Suc__iff,axiom,
    ! [Mb: num,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Mb))),aa_nat_nat(suc,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_num_nat(numeral_numeral_nat,Mb)),Nb) ) ).

% bit_numeral_Bit1_Suc_iff
tff(fact_6787_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_6788_signed__take__bit__nonnegative__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(bit_ri631733984087533419it_int(Nb),K))
    <=> ~ aa_nat_o(bit_se1146084159140164899it_int(K),Nb) ) ).

% signed_take_bit_nonnegative_iff
tff(fact_6789_signed__take__bit__negative__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_ri631733984087533419it_int(Nb),K)),zero_zero_int)
    <=> aa_nat_o(bit_se1146084159140164899it_int(K),Nb) ) ).

% signed_take_bit_negative_iff
tff(fact_6790_bit__numeral__simps_I2_J,axiom,
    ! [W: num,Nb: num] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,W))),aa_num_nat(numeral_numeral_nat,Nb))
    <=> aa_nat_o(bit_se1146084159140164899it_int(aa_num_int(numeral_numeral_int,W)),aa_num_nat(pred_numeral,Nb)) ) ).

% bit_numeral_simps(2)
tff(fact_6791_bit__numeral__simps_I2_J,axiom,
    ! [W: num,Nb: 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,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_num_nat(numeral_numeral_nat,W)),aa_num_nat(pred_numeral,Nb)) ) ).

% bit_numeral_simps(2)
tff(fact_6792_bit__minus__numeral__Bit0__Suc__iff,axiom,
    ! [W: num,Nb: 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,Nb))
    <=> aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))),Nb) ) ).

% bit_minus_numeral_Bit0_Suc_iff
tff(fact_6793_bit__numeral__simps_I3_J,axiom,
    ! [W: num,Nb: num] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,W))),aa_num_nat(numeral_numeral_nat,Nb))
    <=> aa_nat_o(bit_se1146084159140164899it_int(aa_num_int(numeral_numeral_int,W)),aa_num_nat(pred_numeral,Nb)) ) ).

% bit_numeral_simps(3)
tff(fact_6794_bit__numeral__simps_I3_J,axiom,
    ! [W: num,Nb: 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,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_num_nat(numeral_numeral_nat,W)),aa_num_nat(pred_numeral,Nb)) ) ).

% bit_numeral_simps(3)
tff(fact_6795_bit__minus__numeral__Bit1__Suc__iff,axiom,
    ! [W: num,Nb: 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,Nb))
    <=> ~ aa_nat_o(bit_se1146084159140164899it_int(aa_num_int(numeral_numeral_int,W)),Nb) ) ).

% bit_minus_numeral_Bit1_Suc_iff
tff(fact_6796_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_6797_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_6798_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_6799_bit__minus__numeral__int_I1_J,axiom,
    ! [W: num,Nb: 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,Nb))
    <=> aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))),aa_num_nat(pred_numeral,Nb)) ) ).

% bit_minus_numeral_int(1)
tff(fact_6800_bit__minus__numeral__int_I2_J,axiom,
    ! [W: num,Nb: 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,Nb))
    <=> ~ aa_nat_o(bit_se1146084159140164899it_int(aa_num_int(numeral_numeral_int,W)),aa_num_nat(pred_numeral,Nb)) ) ).

% bit_minus_numeral_int(2)
tff(fact_6801_bit__mod__2__iff,axiom,
    ! [A: code_integer,Nb: nat] :
      ( aa_nat_o(bit_se9216721137139052372nteger(modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))),Nb)
    <=> ( ( Nb = 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_6802_bit__mod__2__iff,axiom,
    ! [A: int,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),Nb)
    <=> ( ( Nb = 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_6803_bit__mod__2__iff,axiom,
    ! [A: nat,Nb: 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)))),Nb)
    <=> ( ( Nb = 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_6804_bit__numeral__iff,axiom,
    ! [Mb: num,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_num_int(numeral_numeral_int,Mb)),Nb)
    <=> aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_num_nat(numeral_numeral_nat,Mb)),Nb) ) ).

% bit_numeral_iff
tff(fact_6805_bit__numeral__iff,axiom,
    ! [Mb: num,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_num_nat(numeral_numeral_nat,Mb)),Nb)
    <=> aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_num_nat(numeral_numeral_nat,Mb)),Nb) ) ).

% bit_numeral_iff
tff(fact_6806_bit__disjunctive__add__iff,axiom,
    ! [A: int,B: int,Nb: 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)),Nb)
      <=> ( aa_nat_o(bit_se1146084159140164899it_int(A),Nb)
          | aa_nat_o(bit_se1146084159140164899it_int(B),Nb) ) ) ) ).

% bit_disjunctive_add_iff
tff(fact_6807_bit__disjunctive__add__iff,axiom,
    ! [A: nat,B: nat,Nb: 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)),Nb)
      <=> ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),Nb)
          | aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,B),Nb) ) ) ) ).

% bit_disjunctive_add_iff
tff(fact_6808_bit__or__iff,axiom,
    ! [A: int,B: int,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,A),B)),Nb)
    <=> ( aa_nat_o(bit_se1146084159140164899it_int(A),Nb)
        | aa_nat_o(bit_se1146084159140164899it_int(B),Nb) ) ) ).

% bit_or_iff
tff(fact_6809_bit__or__iff,axiom,
    ! [A: nat,B: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,A),B)),Nb)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),Nb)
        | aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,B),Nb) ) ) ).

% bit_or_iff
tff(fact_6810_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_6811_bit__unset__bit__iff,axiom,
    ! [Mb: nat,A: int,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,Mb),A)),Nb)
    <=> ( aa_nat_o(bit_se1146084159140164899it_int(A),Nb)
        & ( Mb != Nb ) ) ) ).

% bit_unset_bit_iff
tff(fact_6812_bit__unset__bit__iff,axiom,
    ! [Mb: nat,A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,Mb),A)),Nb)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),Nb)
        & ( Mb != Nb ) ) ) ).

% bit_unset_bit_iff
tff(fact_6813_bit__or__int__iff,axiom,
    ! [K: int,L: int,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,K),L)),Nb)
    <=> ( aa_nat_o(bit_se1146084159140164899it_int(K),Nb)
        | aa_nat_o(bit_se1146084159140164899it_int(L),Nb) ) ) ).

% bit_or_int_iff
tff(fact_6814_not__bit__1__Suc,axiom,
    ! [Nb: nat] : ~ aa_nat_o(bit_se1146084159140164899it_int(one_one_int),aa_nat_nat(suc,Nb)) ).

% not_bit_1_Suc
tff(fact_6815_not__bit__1__Suc,axiom,
    ! [Nb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,one_one_nat),aa_nat_nat(suc,Nb)) ).

% not_bit_1_Suc
tff(fact_6816_bit__1__iff,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(one_one_int),Nb)
    <=> ( Nb = zero_zero_nat ) ) ).

% bit_1_iff
tff(fact_6817_bit__1__iff,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,one_one_nat),Nb)
    <=> ( Nb = zero_zero_nat ) ) ).

% bit_1_iff
tff(fact_6818_bit__numeral__simps_I1_J,axiom,
    ! [Nb: num] : ~ aa_nat_o(bit_se1146084159140164899it_int(one_one_int),aa_num_nat(numeral_numeral_nat,Nb)) ).

% bit_numeral_simps(1)
tff(fact_6819_bit__numeral__simps_I1_J,axiom,
    ! [Nb: num] : ~ aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,Nb)) ).

% bit_numeral_simps(1)
tff(fact_6820_disjunctive__add,axiom,
    ! [A: int,B: int] :
      ( ! [N: nat] :
          ( ~ aa_nat_o(bit_se1146084159140164899it_int(A),N)
          | ~ aa_nat_o(bit_se1146084159140164899it_int(B),N) )
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,A),B) ) ) ).

% disjunctive_add
tff(fact_6821_disjunctive__add,axiom,
    ! [A: nat,B: 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_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,A),B) ) ) ).

% disjunctive_add
tff(fact_6822_bit__take__bit__iff,axiom,
    ! [Mb: nat,A: int,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(bit_se2923211474154528505it_int(Mb),A)),Nb)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb)
        & aa_nat_o(bit_se1146084159140164899it_int(A),Nb) ) ) ).

% bit_take_bit_iff
tff(fact_6823_bit__take__bit__iff,axiom,
    ! [Mb: nat,A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Mb),A)),Nb)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb)
        & aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),Nb) ) ) ).

% bit_take_bit_iff
tff(fact_6824_bit__of__bool__iff,axiom,
    ! [B: $o,Nb: nat] :
      ( aa_nat_o(bit_se9216721137139052372nteger(zero_n356916108424825756nteger((B))),Nb)
    <=> ( (B)
        & ( Nb = zero_zero_nat ) ) ) ).

% bit_of_bool_iff
tff(fact_6825_bit__of__bool__iff,axiom,
    ! [B: $o,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_o_int(zero_n2684676970156552555ol_int,(B))),Nb)
    <=> ( (B)
        & ( Nb = zero_zero_nat ) ) ) ).

% bit_of_bool_iff
tff(fact_6826_bit__of__bool__iff,axiom,
    ! [B: $o,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,zero_n2687167440665602831ol_nat((B))),Nb)
    <=> ( (B)
        & ( Nb = zero_zero_nat ) ) ) ).

% bit_of_bool_iff
tff(fact_6827_signed__take__bit__eq__if__positive,axiom,
    ! [A: int,Nb: nat] :
      ( ~ aa_nat_o(bit_se1146084159140164899it_int(A),Nb)
     => ( aa_int_int(bit_ri631733984087533419it_int(Nb),A) = aa_int_int(bit_se2923211474154528505it_int(Nb),A) ) ) ).

% signed_take_bit_eq_if_positive
tff(fact_6828_abs__zmult__eq__1,axiom,
    ! [Mb: int,Nb: int] :
      ( ( aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(times_times_int,Mb),Nb)) = one_one_int )
     => ( aa_int_int(abs_abs_int,Mb) = one_one_int ) ) ).

% abs_zmult_eq_1
tff(fact_6829_abs__div,axiom,
    ! [Yb: int,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Yb),Xb)
     => ( aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,Xb),Yb)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(abs_abs_int,Xb)),aa_int_int(abs_abs_int,Yb)) ) ) ).

% abs_div
tff(fact_6830_bit__not__int__iff_H,axiom,
    ! [K: int,Nb: 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)),Nb)
    <=> ~ aa_nat_o(bit_se1146084159140164899it_int(K),Nb) ) ).

% bit_not_int_iff'
tff(fact_6831_zabs__def,axiom,
    ! [I3: int] :
      aa_int_int(abs_abs_int,I3) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,I3),zero_zero_int),aa_int_int(uminus_uminus_int,I3),I3) ).

% zabs_def
tff(fact_6832_dvd__imp__le__int,axiom,
    ! [I3: int,D: int] :
      ( ( I3 != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),I3)
       => 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,I3)) ) ) ).

% dvd_imp_le_int
tff(fact_6833_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(aa_int_fun_int_int(modulo_modulo_int,K),L))),aa_int_int(abs_abs_int,L)) ) ).

% abs_mod_less
tff(fact_6834_flip__bit__eq__if,axiom,
    ! [Nb: nat,A: int] :
      aa_int_int(bit_se2159334234014336723it_int(Nb),A) = aa_int_int(aa_nat_fun_int_int($ite(aa_nat_o(bit_se1146084159140164899it_int(A),Nb),bit_se4203085406695923979it_int,bit_se7879613467334960850it_int),Nb),A) ).

% flip_bit_eq_if
tff(fact_6835_flip__bit__eq__if,axiom,
    ! [Nb: nat,A: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,Nb),A) = aa_nat_nat(aa_nat_fun_nat_nat($ite(aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),Nb),bit_se4205575877204974255it_nat,bit_se7882103937844011126it_nat),Nb),A) ).

% flip_bit_eq_if
tff(fact_6836_bit__imp__take__bit__positive,axiom,
    ! [Nb: nat,Mb: nat,K: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb)
     => ( aa_nat_o(bit_se1146084159140164899it_int(K),Nb)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(bit_se2923211474154528505it_int(Mb),K)) ) ) ).

% bit_imp_take_bit_positive
tff(fact_6837_bit__concat__bit__iff,axiom,
    ! [Mb: nat,K: int,L: int,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(aa_int_fun_int_int(bit_concat_bit(Mb),K),L)),Nb)
    <=> ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb)
          & aa_nat_o(bit_se1146084159140164899it_int(K),Nb) )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
          & aa_nat_o(bit_se1146084159140164899it_int(L),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)) ) ) ) ).

% bit_concat_bit_iff
tff(fact_6838_zdvd__mult__cancel1,axiom,
    ! [Mb: int,Nb: int] :
      ( ( Mb != 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,Mb),Nb)),Mb)
      <=> ( aa_int_int(abs_abs_int,Nb) = one_one_int ) ) ) ).

% zdvd_mult_cancel1
tff(fact_6839_signed__take__bit__eq__concat__bit,axiom,
    ! [Nb: nat,K: int] : aa_int_int(bit_ri631733984087533419it_int(Nb),K) = aa_int_int(aa_int_fun_int_int(bit_concat_bit(Nb),K),aa_int_int(uminus_uminus_int,aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(bit_se1146084159140164899it_int(K),Nb)))) ).

% signed_take_bit_eq_concat_bit
tff(fact_6840_exp__eq__0__imp__not__bit,axiom,
    ! [Nb: nat,A: int] :
      ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb) = zero_zero_int )
     => ~ aa_nat_o(bit_se1146084159140164899it_int(A),Nb) ) ).

% exp_eq_0_imp_not_bit
tff(fact_6841_exp__eq__0__imp__not__bit,axiom,
    ! [Nb: 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))),Nb) = zero_zero_nat )
     => ~ aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),Nb) ) ).

% exp_eq_0_imp_not_bit
tff(fact_6842_bit__Suc,axiom,
    ! [A: int,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(A),aa_nat_nat(suc,Nb))
    <=> 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)))),Nb) ) ).

% bit_Suc
tff(fact_6843_bit__Suc,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),aa_nat_nat(suc,Nb))
    <=> 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)))),Nb) ) ).

% bit_Suc
tff(fact_6844_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_6845_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_6846_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_6847_stable__imp__bit__iff__odd,axiom,
    ! [A: code_integer,Nb: nat] :
      ( ( divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) = A )
     => ( aa_nat_o(bit_se9216721137139052372nteger(A),Nb)
      <=> ~ 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_6848_stable__imp__bit__iff__odd,axiom,
    ! [A: int,Nb: 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),Nb)
      <=> ~ 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_6849_stable__imp__bit__iff__odd,axiom,
    ! [A: nat,Nb: 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),Nb)
      <=> ~ 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_6850_int__bit__bound,axiom,
    ! [K: int] :
      ~ ! [N: nat] :
          ( ! [M: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M)
             => ( aa_nat_o(bit_se1146084159140164899it_int(K),M)
              <=> 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_6851_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_6852_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_6853_bit__iff__odd,axiom,
    ! [A: code_integer,Nb: nat] :
      ( aa_nat_o(bit_se9216721137139052372nteger(A),Nb)
    <=> ~ 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))),Nb))) ) ).

% bit_iff_odd
tff(fact_6854_bit__iff__odd,axiom,
    ! [A: int,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(A),Nb)
    <=> ~ 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))),Nb))) ) ).

% bit_iff_odd
tff(fact_6855_bit__iff__odd,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),Nb)
    <=> ~ 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))),Nb))) ) ).

% bit_iff_odd
tff(fact_6856_bit__int__def,axiom,
    ! [K: int,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(K),Nb)
    <=> ~ 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))),Nb))) ) ).

% bit_int_def
tff(fact_6857_nat__intermed__int__val,axiom,
    ! [Mb: nat,Nb: nat,F: fun_nat_int,K: int] :
      ( ! [I2: nat] :
          ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),I2)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),Nb) )
         => 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,Mb),Nb)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,Mb)),K)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),aa_nat_int(F,Nb))
           => ? [I2: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),I2)
                & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),Nb)
                & ( aa_nat_int(F,I2) = K ) ) ) ) ) ) ).

% nat_intermed_int_val
tff(fact_6858_decr__lemma,axiom,
    ! [D: int,Xb: 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,Xb),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,Xb),Z))),one_one_int)),D))),Z) ) ).

% decr_lemma
tff(fact_6859_incr__lemma,axiom,
    ! [D: int,Z: int,Xb: 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,Xb),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,Xb),Z))),one_one_int)),D))) ) ).

% incr_lemma
tff(fact_6860_even__bit__succ__iff,axiom,
    ! [A: code_integer,Nb: 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)),Nb)
      <=> ( aa_nat_o(bit_se9216721137139052372nteger(A),Nb)
          | ( Nb = zero_zero_nat ) ) ) ) ).

% even_bit_succ_iff
tff(fact_6861_even__bit__succ__iff,axiom,
    ! [A: int,Nb: 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)),Nb)
      <=> ( aa_nat_o(bit_se1146084159140164899it_int(A),Nb)
          | ( Nb = zero_zero_nat ) ) ) ) ).

% even_bit_succ_iff
tff(fact_6862_even__bit__succ__iff,axiom,
    ! [A: nat,Nb: 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)),Nb)
      <=> ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),Nb)
          | ( Nb = zero_zero_nat ) ) ) ) ).

% even_bit_succ_iff
tff(fact_6863_odd__bit__iff__bit__pred,axiom,
    ! [A: code_integer,Nb: 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),Nb)
      <=> ( aa_nat_o(bit_se9216721137139052372nteger(minus_8373710615458151222nteger(A,one_one_Code_integer)),Nb)
          | ( Nb = zero_zero_nat ) ) ) ) ).

% odd_bit_iff_bit_pred
tff(fact_6864_odd__bit__iff__bit__pred,axiom,
    ! [A: int,Nb: 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),Nb)
      <=> ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,A),one_one_int)),Nb)
          | ( Nb = zero_zero_nat ) ) ) ) ).

% odd_bit_iff_bit_pred
tff(fact_6865_odd__bit__iff__bit__pred,axiom,
    ! [A: nat,Nb: 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),Nb)
      <=> ( 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)),Nb)
          | ( Nb = zero_zero_nat ) ) ) ) ).

% odd_bit_iff_bit_pred
tff(fact_6866_nat__ivt__aux,axiom,
    ! [Nb: nat,F: fun_nat_int,K: int] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),Nb)
         => 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,Nb))
         => ? [I2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),Nb)
              & ( aa_nat_int(F,I2) = K ) ) ) ) ) ).

% nat_ivt_aux
tff(fact_6867_bit__sum__mult__2__cases,axiom,
    ! [A: code_integer,B: code_integer,Nb: 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))),Nb)
      <=> $ite(Nb = 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)),Nb)) ) ) ).

% bit_sum_mult_2_cases
tff(fact_6868_bit__sum__mult__2__cases,axiom,
    ! [A: int,B: int,Nb: 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))),Nb)
      <=> $ite(Nb = 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)),Nb)) ) ) ).

% bit_sum_mult_2_cases
tff(fact_6869_bit__sum__mult__2__cases,axiom,
    ! [A: nat,B: nat,Nb: 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))),Nb)
      <=> $ite(Nb = 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)),Nb)) ) ) ).

% bit_sum_mult_2_cases
tff(fact_6870_bit__rec,axiom,
    ! [A: code_integer,Nb: nat] :
      ( aa_nat_o(bit_se9216721137139052372nteger(A),Nb)
    <=> $ite(Nb = 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,Nb),one_one_nat))) ) ).

% bit_rec
tff(fact_6871_bit__rec,axiom,
    ! [A: int,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(A),Nb)
    <=> $ite(Nb = 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,Nb),one_one_nat))) ) ).

% bit_rec
tff(fact_6872_bit__rec,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),Nb)
    <=> $ite(Nb = 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,Nb),one_one_nat))) ) ).

% bit_rec
tff(fact_6873_nat0__intermed__int__val,axiom,
    ! [Nb: nat,F: fun_nat_int,K: int] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),Nb)
         => 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,Nb))
         => ? [I2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),Nb)
              & ( aa_nat_int(F,I2) = K ) ) ) ) ) ).

% nat0_intermed_int_val
tff(fact_6874_set__bit__eq,axiom,
    ! [Nb: nat,K: int] : aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,Nb),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),Nb))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))) ).

% set_bit_eq
tff(fact_6875_unset__bit__eq,axiom,
    ! [Nb: nat,K: int] : aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,Nb),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),Nb))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))) ).

% unset_bit_eq
tff(fact_6876_arctan__add,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Yb)),one_one_real)
       => ( aa_real_real(plus_plus_real(aa_real_real(arctan,Xb)),aa_real_real(arctan,Yb)) = aa_real_real(arctan,aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(Xb),Yb)),aa_real_real(minus_minus_real(one_one_real),aa_real_real(times_times_real(Xb),Yb)))) ) ) ) ).

% arctan_add
tff(fact_6877_take__bit__Suc__from__most,axiom,
    ! [Nb: nat,K: int] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Nb)),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))),Nb)),aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(bit_se1146084159140164899it_int(K),Nb)))),aa_int_int(bit_se2923211474154528505it_int(Nb),K)) ).

% take_bit_Suc_from_most
tff(fact_6878_Sum__Icc__int,axiom,
    ! [Mb: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Mb),Nb)
     => ( groups4538972089207619220nt_int(aTP_Lamm_as,set_or1266510415728281911st_int(Mb,Nb)) = 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,Nb),aa_int_int(aa_int_fun_int_int(plus_plus_int,Nb),one_one_int))),aa_int_int(aa_int_fun_int_int(times_times_int,Mb),aa_int_int(aa_int_fun_int_int(minus_minus_int,Mb),one_one_int)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ) ) ).

% Sum_Icc_int
tff(fact_6879_mask__numeral,axiom,
    ! [Nb: num] : aa_nat_nat(bit_se2002935070580805687sk_nat,aa_num_nat(numeral_numeral_nat,Nb)) = 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,Nb)))) ).

% mask_numeral
tff(fact_6880_mask__numeral,axiom,
    ! [Nb: num] : aa_nat_int(bit_se2000444600071755411sk_int,aa_num_nat(numeral_numeral_nat,Nb)) = 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,Nb)))) ).

% mask_numeral
tff(fact_6881_tanh__real__altdef,axiom,
    ! [Xb: real] : aa_real_real(tanh_real,Xb) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(one_one_real),aa_real_real(exp_real,aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),Xb)))),aa_real_real(plus_plus_real(one_one_real),aa_real_real(exp_real,aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),Xb)))) ).

% tanh_real_altdef
tff(fact_6882_and__int__unfold,axiom,
    ! [K: int,L: int] :
      aa_int_int(aa_int_fun_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(aa_int_fun_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(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(aa_int_fun_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_6883_machin,axiom,
    aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_real_real(minus_minus_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_real_real(arctan,aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit1,aa_num_num(bit0,one))))))),aa_real_real(arctan,aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,one))))))))))) ).

% machin
tff(fact_6884_machin__Euler,axiom,
    aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_real_real(arctan,aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit1,aa_num_num(bit1,one))))))),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(arctan,aa_real_real(divide_divide_real(aa_num_real(numeral_numeral_real,aa_num_num(bit1,one))),aa_num_real(numeral_numeral_real,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,one))))))))))) = aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))) ).

% machin_Euler
tff(fact_6885_and_Oidem,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,A),A) = A ).

% and.idem
tff(fact_6886_and_Oidem,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),A) = A ).

% and.idem
tff(fact_6887_and_Oleft__idem,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,A),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,A),B)) = aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,A),B) ).

% and.left_idem
tff(fact_6888_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_6889_and_Oright__idem,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,A),B)),B) = aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,A),B) ).

% and.right_idem
tff(fact_6890_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_6891_mask__nat__positive__iff,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(bit_se2002935070580805687sk_nat,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).

% mask_nat_positive_iff
tff(fact_6892_bit_Oconj__zero__right,axiom,
    ! [Xb: int] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Xb),zero_zero_int) = zero_zero_int ).

% bit.conj_zero_right
tff(fact_6893_bit_Oconj__zero__left,axiom,
    ! [Xb: int] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,zero_zero_int),Xb) = zero_zero_int ).

% bit.conj_zero_left
tff(fact_6894_zero__and__eq,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,zero_zero_int),A) = zero_zero_int ).

% zero_and_eq
tff(fact_6895_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_6896_and__zero__eq,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,A),zero_zero_int) = zero_zero_int ).

% and_zero_eq
tff(fact_6897_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_6898_take__bit__and,axiom,
    ! [Nb: nat,A: int,B: int] : aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,A),B)) = aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(bit_se2923211474154528505it_int(Nb),A)),aa_int_int(bit_se2923211474154528505it_int(Nb),B)) ).

% take_bit_and
tff(fact_6899_take__bit__and,axiom,
    ! [Nb: nat,A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),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,Nb),A)),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),B)) ).

% take_bit_and
tff(fact_6900_exp__zero,axiom,
    aa_complex_complex(exp_complex,zero_zero_complex) = one_one_complex ).

% exp_zero
tff(fact_6901_exp__zero,axiom,
    aa_real_real(exp_real,zero_zero_real) = one_one_real ).

% exp_zero
tff(fact_6902_and_Oleft__neutral,axiom,
    ! [A: code_integer] : bit_se3949692690581998587nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),A) = A ).

% and.left_neutral
tff(fact_6903_and_Oleft__neutral,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(uminus_uminus_int,one_one_int)),A) = A ).

% and.left_neutral
tff(fact_6904_and_Oright__neutral,axiom,
    ! [A: code_integer] : bit_se3949692690581998587nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = A ).

% and.right_neutral
tff(fact_6905_and_Oright__neutral,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,A),aa_int_int(uminus_uminus_int,one_one_int)) = A ).

% and.right_neutral
tff(fact_6906_bit_Oconj__one__right,axiom,
    ! [Xb: code_integer] : bit_se3949692690581998587nteger(Xb,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = Xb ).

% bit.conj_one_right
tff(fact_6907_bit_Oconj__one__right,axiom,
    ! [Xb: int] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Xb),aa_int_int(uminus_uminus_int,one_one_int)) = Xb ).

% bit.conj_one_right
tff(fact_6908_mask__eq__0__iff,axiom,
    ! [Nb: nat] :
      ( ( aa_nat_nat(bit_se2002935070580805687sk_nat,Nb) = zero_zero_nat )
    <=> ( Nb = zero_zero_nat ) ) ).

% mask_eq_0_iff
tff(fact_6909_mask__eq__0__iff,axiom,
    ! [Nb: nat] :
      ( ( aa_nat_int(bit_se2000444600071755411sk_int,Nb) = zero_zero_int )
    <=> ( Nb = zero_zero_nat ) ) ).

% mask_eq_0_iff
tff(fact_6910_mask__0,axiom,
    aa_nat_nat(bit_se2002935070580805687sk_nat,zero_zero_nat) = zero_zero_nat ).

% mask_0
tff(fact_6911_mask__0,axiom,
    aa_nat_int(bit_se2000444600071755411sk_int,zero_zero_nat) = zero_zero_int ).

% mask_0
tff(fact_6912_exp__eq__one__iff,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(exp_real,Xb) = one_one_real )
    <=> ( Xb = zero_zero_real ) ) ).

% exp_eq_one_iff
tff(fact_6913_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(aa_int_fun_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_6914_and__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_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_6915_and__numerals_I8_J,axiom,
    ! [Xb: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Xb))),one_one_int) = one_one_int ).

% and_numerals(8)
tff(fact_6916_and__numerals_I8_J,axiom,
    ! [Xb: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xb))),one_one_nat) = one_one_nat ).

% and_numerals(8)
tff(fact_6917_and__numerals_I2_J,axiom,
    ! [Yb: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,one_one_int),aa_num_int(numeral_numeral_int,aa_num_num(bit1,Yb))) = one_one_int ).

% and_numerals(2)
tff(fact_6918_and__numerals_I2_J,axiom,
    ! [Yb: 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,Yb))) = one_one_nat ).

% and_numerals(2)
tff(fact_6919_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_6920_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_6921_exp__less__one__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(exp_real,Xb)),one_one_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real) ) ).

% exp_less_one_iff
tff(fact_6922_one__less__exp__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(exp_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb) ) ).

% one_less_exp_iff
tff(fact_6923_exp__le__one__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,Xb)),one_one_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real) ) ).

% exp_le_one_iff
tff(fact_6924_one__le__exp__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(exp_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb) ) ).

% one_le_exp_iff
tff(fact_6925_take__bit__minus__one__eq__mask,axiom,
    ! [Nb: nat] : bit_se1745604003318907178nteger(Nb,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_nat_Code_integer(bit_se2119862282449309892nteger,Nb) ).

% take_bit_minus_one_eq_mask
tff(fact_6926_take__bit__minus__one__eq__mask,axiom,
    ! [Nb: nat] : aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(uminus_uminus_int,one_one_int)) = aa_nat_int(bit_se2000444600071755411sk_int,Nb) ).

% take_bit_minus_one_eq_mask
tff(fact_6927_of__int__sum,axiom,
    ! [F: fun_complex_int,A3: set_complex] : aa_int_complex(ring_17405671764205052669omplex,groups5690904116761175830ex_int(F,A3)) = groups7754918857620584856omplex(aTP_Lamm_cl(F),A3) ).

% of_int_sum
tff(fact_6928_of__int__sum,axiom,
    ! [F: fun_nat_int,A3: set_nat] : aa_int_real(ring_1_of_int_real,groups3539618377306564664at_int(F,A3)) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_cm(F)),A3) ).

% of_int_sum
tff(fact_6929_of__int__sum,axiom,
    ! [F: fun_int_int,A3: set_int] : aa_int_real(ring_1_of_int_real,groups4538972089207619220nt_int(F,A3)) = groups8778361861064173332t_real(aTP_Lamm_cn(F),A3) ).

% of_int_sum
tff(fact_6930_of__int__sum,axiom,
    ! [F: fun_int_int,A3: set_int] : aa_int_rat(ring_1_of_int_rat,groups4538972089207619220nt_int(F,A3)) = groups3906332499630173760nt_rat(aTP_Lamm_co(F),A3) ).

% of_int_sum
tff(fact_6931_of__int__sum,axiom,
    ! [F: fun_int_int,A3: set_int] : aa_int_int(ring_1_of_int_int,groups4538972089207619220nt_int(F,A3)) = groups4538972089207619220nt_int(aTP_Lamm_cp(F),A3) ).

% of_int_sum
tff(fact_6932_and__numerals_I1_J,axiom,
    ! [Yb: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,one_one_int),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Yb))) = zero_zero_int ).

% and_numerals(1)
tff(fact_6933_and__numerals_I1_J,axiom,
    ! [Yb: 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,Yb))) = zero_zero_nat ).

% and_numerals(1)
tff(fact_6934_and__numerals_I5_J,axiom,
    ! [Xb: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Xb))),one_one_int) = zero_zero_int ).

% and_numerals(5)
tff(fact_6935_and__numerals_I5_J,axiom,
    ! [Xb: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Xb))),one_one_nat) = zero_zero_nat ).

% and_numerals(5)
tff(fact_6936_and__numerals_I3_J,axiom,
    ! [Xb: num,Yb: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Xb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Yb))) = 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(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,Xb)),aa_num_int(numeral_numeral_int,Yb))) ).

% and_numerals(3)
tff(fact_6937_and__numerals_I3_J,axiom,
    ! [Xb: num,Yb: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Xb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Yb))) = 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,Xb)),aa_num_nat(numeral_numeral_nat,Yb))) ).

% and_numerals(3)
tff(fact_6938_and__minus__numerals_I6_J,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))),one_one_int) = one_one_int ).

% and_minus_numerals(6)
tff(fact_6939_and__minus__numerals_I2_J,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))) = one_one_int ).

% and_minus_numerals(2)
tff(fact_6940_and__numerals_I4_J,axiom,
    ! [Xb: num,Yb: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Xb))),aa_num_int(numeral_numeral_int,aa_num_num(bit1,Yb))) = 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(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,Xb)),aa_num_int(numeral_numeral_int,Yb))) ).

% and_numerals(4)
tff(fact_6941_and__numerals_I4_J,axiom,
    ! [Xb: num,Yb: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Xb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Yb))) = 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,Xb)),aa_num_nat(numeral_numeral_nat,Yb))) ).

% and_numerals(4)
tff(fact_6942_and__numerals_I6_J,axiom,
    ! [Xb: num,Yb: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Xb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Yb))) = 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(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,Xb)),aa_num_int(numeral_numeral_int,Yb))) ).

% and_numerals(6)
tff(fact_6943_and__numerals_I6_J,axiom,
    ! [Xb: num,Yb: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Yb))) = 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,Xb)),aa_num_nat(numeral_numeral_nat,Yb))) ).

% and_numerals(6)
tff(fact_6944_and__minus__numerals_I1_J,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))) = zero_zero_int ).

% and_minus_numerals(1)
tff(fact_6945_and__minus__numerals_I5_J,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))),one_one_int) = zero_zero_int ).

% and_minus_numerals(5)
tff(fact_6946_and__numerals_I7_J,axiom,
    ! [Xb: num,Yb: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Xb))),aa_num_int(numeral_numeral_int,aa_num_num(bit1,Yb))) = 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(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,Xb)),aa_num_int(numeral_numeral_int,Yb)))) ).

% and_numerals(7)
tff(fact_6947_and__numerals_I7_J,axiom,
    ! [Xb: num,Yb: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Yb))) = 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,Xb)),aa_num_nat(numeral_numeral_nat,Yb)))) ).

% and_numerals(7)
tff(fact_6948_of__int__and__eq,axiom,
    ! [K: int,L: int] : aa_int_int(ring_1_of_int_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K),L)) = aa_int_int(aa_int_fun_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_6949_of__int__mask__eq,axiom,
    ! [Nb: nat] : aa_int_int(ring_1_of_int_int,aa_nat_int(bit_se2000444600071755411sk_int,Nb)) = aa_nat_int(bit_se2000444600071755411sk_int,Nb) ).

% of_int_mask_eq
tff(fact_6950_take__bit__eq__mask,axiom,
    ! [Nb: nat,A: int] : aa_int_int(bit_se2923211474154528505it_int(Nb),A) = aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,A),aa_nat_int(bit_se2000444600071755411sk_int,Nb)) ).

% take_bit_eq_mask
tff(fact_6951_take__bit__eq__mask,axiom,
    ! [Nb: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),A) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),aa_nat_nat(bit_se2002935070580805687sk_nat,Nb)) ).

% take_bit_eq_mask
tff(fact_6952_and_Oassoc,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,A),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,B),C)) ).

% and.assoc
tff(fact_6953_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_6954_and_Ocommute,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,A),B) = aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,B),A) ).

% and.commute
tff(fact_6955_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_6956_and_Oleft__commute,axiom,
    ! [B: int,A: int,C: int] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,B),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,A),C)) = aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,A),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,B),C)) ).

% and.left_commute
tff(fact_6957_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_6958_bit__and__iff,axiom,
    ! [A: int,B: int,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,A),B)),Nb)
    <=> ( aa_nat_o(bit_se1146084159140164899it_int(A),Nb)
        & aa_nat_o(bit_se1146084159140164899it_int(B),Nb) ) ) ).

% bit_and_iff
tff(fact_6959_bit__and__iff,axiom,
    ! [A: nat,B: nat,Nb: 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)),Nb)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),Nb)
        & aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,B),Nb) ) ) ).

% bit_and_iff
tff(fact_6960_exp__not__eq__zero,axiom,
    ! [Xb: complex] : aa_complex_complex(exp_complex,Xb) != zero_zero_complex ).

% exp_not_eq_zero
tff(fact_6961_exp__not__eq__zero,axiom,
    ! [Xb: real] : aa_real_real(exp_real,Xb) != zero_zero_real ).

% exp_not_eq_zero
tff(fact_6962_bit_Oconj__disj__distrib,axiom,
    ! [Xb: int,Yb: int,Z: int] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Xb),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,Yb),Z)) = aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Xb),Yb)),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Xb),Z)) ).

% bit.conj_disj_distrib
tff(fact_6963_bit_Odisj__conj__distrib,axiom,
    ! [Xb: int,Yb: int,Z: int] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,Xb),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Yb),Z)) = aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,Xb),Yb)),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,Xb),Z)) ).

% bit.disj_conj_distrib
tff(fact_6964_bit_Oconj__disj__distrib2,axiom,
    ! [Yb: int,Z: int,Xb: int] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,Yb),Z)),Xb) = aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Yb),Xb)),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Z),Xb)) ).

% bit.conj_disj_distrib2
tff(fact_6965_bit_Odisj__conj__distrib2,axiom,
    ! [Yb: int,Z: int,Xb: int] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Yb),Z)),Xb) = aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,Yb),Xb)),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,Z),Xb)) ).

% bit.disj_conj_distrib2
tff(fact_6966_exp__times__arg__commute,axiom,
    ! [A3: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(exp_complex,A3)),A3) = aa_complex_complex(times_times_complex(A3),aa_complex_complex(exp_complex,A3)) ).

% exp_times_arg_commute
tff(fact_6967_exp__times__arg__commute,axiom,
    ! [A3: real] : aa_real_real(times_times_real(aa_real_real(exp_real,A3)),A3) = aa_real_real(times_times_real(A3),aa_real_real(exp_real,A3)) ).

% exp_times_arg_commute
tff(fact_6968_bit__and__int__iff,axiom,
    ! [K: int,L: int,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K),L)),Nb)
    <=> ( aa_nat_o(bit_se1146084159140164899it_int(K),Nb)
        & aa_nat_o(bit_se1146084159140164899it_int(L),Nb) ) ) ).

% bit_and_int_iff
tff(fact_6969_less__eq__mask,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_nat_nat(bit_se2002935070580805687sk_nat,Nb)) ).

% less_eq_mask
tff(fact_6970_mod__sum__eq,axiom,
    ! [F: fun_int_int,A: int,A3: set_int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,groups4538972089207619220nt_int(aa_int_fun_int_int(aTP_Lamm_cq(F),A),A3)),A) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,groups4538972089207619220nt_int(F,A3)),A) ).

% mod_sum_eq
tff(fact_6971_mod__sum__eq,axiom,
    ! [F: fun_nat_nat,A: nat,A3: 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_cr(F),A)),A3)),A) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A3)),A) ).

% mod_sum_eq
tff(fact_6972_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_6973_and__eq__minus__1__iff,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_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_6974_not__bit__Suc__0__Suc,axiom,
    ! [Nb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(suc,Nb)) ).

% not_bit_Suc_0_Suc
tff(fact_6975_bit__Suc__0__iff,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(suc,zero_zero_nat)),Nb)
    <=> ( Nb = zero_zero_nat ) ) ).

% bit_Suc_0_iff
tff(fact_6976_AND__lower,axiom,
    ! [Xb: int,Yb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Xb),Yb)) ) ).

% AND_lower
tff(fact_6977_AND__upper1,axiom,
    ! [Xb: int,Yb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Xb),Yb)),Xb) ) ).

% AND_upper1
tff(fact_6978_AND__upper2,axiom,
    ! [Yb: int,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Yb)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Xb),Yb)),Yb) ) ).

% AND_upper2
tff(fact_6979_AND__upper1_H,axiom,
    ! [Yb: int,Z: int,Yaa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Yb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Yb),Z)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Yb),Yaa)),Z) ) ) ).

% AND_upper1'
tff(fact_6980_AND__upper2_H,axiom,
    ! [Yb: int,Z: int,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Yb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Yb),Z)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Xb),Yb)),Z) ) ) ).

% AND_upper2'
tff(fact_6981_mult__exp__exp,axiom,
    ! [Xb: complex,Yb: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(exp_complex,Xb)),aa_complex_complex(exp_complex,Yb)) = aa_complex_complex(exp_complex,aa_complex_complex(plus_plus_complex(Xb),Yb)) ).

% mult_exp_exp
tff(fact_6982_mult__exp__exp,axiom,
    ! [Xb: real,Yb: real] : aa_real_real(times_times_real(aa_real_real(exp_real,Xb)),aa_real_real(exp_real,Yb)) = aa_real_real(exp_real,aa_real_real(plus_plus_real(Xb),Yb)) ).

% mult_exp_exp
tff(fact_6983_exp__add__commuting,axiom,
    ! [Xb: complex,Yb: complex] :
      ( ( aa_complex_complex(times_times_complex(Xb),Yb) = aa_complex_complex(times_times_complex(Yb),Xb) )
     => ( aa_complex_complex(exp_complex,aa_complex_complex(plus_plus_complex(Xb),Yb)) = aa_complex_complex(times_times_complex(aa_complex_complex(exp_complex,Xb)),aa_complex_complex(exp_complex,Yb)) ) ) ).

% exp_add_commuting
tff(fact_6984_exp__add__commuting,axiom,
    ! [Xb: real,Yb: real] :
      ( ( aa_real_real(times_times_real(Xb),Yb) = aa_real_real(times_times_real(Yb),Xb) )
     => ( aa_real_real(exp_real,aa_real_real(plus_plus_real(Xb),Yb)) = aa_real_real(times_times_real(aa_real_real(exp_real,Xb)),aa_real_real(exp_real,Yb)) ) ) ).

% exp_add_commuting
tff(fact_6985_exp__diff,axiom,
    ! [Xb: complex,Yb: complex] : aa_complex_complex(exp_complex,aa_complex_complex(minus_minus_complex(Xb),Yb)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(exp_complex,Xb)),aa_complex_complex(exp_complex,Yb)) ).

% exp_diff
tff(fact_6986_exp__diff,axiom,
    ! [Xb: real,Yb: real] : aa_real_real(exp_real,aa_real_real(minus_minus_real(Xb),Yb)) = aa_real_real(divide_divide_real(aa_real_real(exp_real,Xb)),aa_real_real(exp_real,Yb)) ).

% exp_diff
tff(fact_6987_plus__and__or,axiom,
    ! [Xb: int,Yb: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Xb),Yb)),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,Xb),Yb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Yb) ).

% plus_and_or
tff(fact_6988_mask__nonnegative__int,axiom,
    ! [Nb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(bit_se2000444600071755411sk_int,Nb)) ).

% mask_nonnegative_int
tff(fact_6989_not__mask__negative__int,axiom,
    ! [Nb: nat] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(bit_se2000444600071755411sk_int,Nb)),zero_zero_int) ).

% not_mask_negative_int
tff(fact_6990_not__bit__Suc__0__numeral,axiom,
    ! [Nb: 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,Nb)) ).

% not_bit_Suc_0_numeral
tff(fact_6991_exp__gt__one,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(exp_real,Xb)) ) ).

% exp_gt_one
tff(fact_6992_exp__ge__add__one__self,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(one_one_real),Xb)),aa_real_real(exp_real,Xb)) ).

% exp_ge_add_one_self
tff(fact_6993_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(aa_int_fun_int_int(bit_se725231765392027082nd_int,K),L)),K) ) ).

% and_less_eq
tff(fact_6994_AND__upper1_H_H,axiom,
    ! [Yb: int,Z: int,Yaa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Yb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Yb),Z)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Yb),Yaa)),Z) ) ) ).

% AND_upper1''
tff(fact_6995_AND__upper2_H_H,axiom,
    ! [Yb: int,Z: int,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Yb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Yb),Z)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Xb),Yb)),Z) ) ) ).

% AND_upper2''
tff(fact_6996_exp__minus__inverse,axiom,
    ! [Xb: real] : aa_real_real(times_times_real(aa_real_real(exp_real,Xb)),aa_real_real(exp_real,aa_real_real(uminus_uminus_real,Xb))) = one_one_real ).

% exp_minus_inverse
tff(fact_6997_exp__minus__inverse,axiom,
    ! [Xb: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(exp_complex,Xb)),aa_complex_complex(exp_complex,aa_complex_complex(uminus1482373934393186551omplex,Xb))) = one_one_complex ).

% exp_minus_inverse
tff(fact_6998_less__mask,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(bit_se2002935070580805687sk_nat,Nb)) ) ).

% less_mask
tff(fact_6999_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_7000_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(aa_int_fun_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_7001_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_7002_bit_Ocomplement__unique,axiom,
    ! [A: code_integer,Xb: code_integer,Yb: code_integer] :
      ( ( bit_se3949692690581998587nteger(A,Xb) = zero_z3403309356797280102nteger )
     => ( ( bit_se1080825931792720795nteger(A,Xb) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) )
       => ( ( bit_se3949692690581998587nteger(A,Yb) = zero_z3403309356797280102nteger )
         => ( ( bit_se1080825931792720795nteger(A,Yb) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) )
           => ( Xb = Yb ) ) ) ) ) ).

% bit.complement_unique
tff(fact_7003_bit_Ocomplement__unique,axiom,
    ! [A: int,Xb: int,Yb: int] :
      ( ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,A),Xb) = zero_zero_int )
     => ( ( aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,A),Xb) = aa_int_int(uminus_uminus_int,one_one_int) )
       => ( ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,A),Yb) = zero_zero_int )
         => ( ( aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,A),Yb) = aa_int_int(uminus_uminus_int,one_one_int) )
           => ( Xb = Yb ) ) ) ) ) ).

% bit.complement_unique
tff(fact_7004_exp__ge__add__one__self__aux,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(one_one_real),Xb)),aa_real_real(exp_real,Xb)) ) ).

% exp_ge_add_one_self_aux
tff(fact_7005_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(aa_int_fun_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_7006_lemma__exp__total,axiom,
    ! [Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Yb)
     => ? [X3: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X3)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),aa_real_real(minus_minus_real(Yb),one_one_real))
          & ( aa_real_real(exp_real,X3) = Yb ) ) ) ).

% lemma_exp_total
tff(fact_7007_ln__x__over__x__mono,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,one_one_real)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Yb)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(aa_real_real(ln_ln_real,Yb)),Yb)),aa_real_real(divide_divide_real(aa_real_real(ln_ln_real,Xb)),Xb)) ) ) ).

% ln_x_over_x_mono
tff(fact_7008_pi__less__4,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))) ).

% pi_less_4
tff(fact_7009_pi__ge__two,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi) ).

% pi_ge_two
tff(fact_7010_pi__half__neq__two,axiom,
    aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) != aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)) ).

% pi_half_neq_two
tff(fact_7011_one__and__eq,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,one_one_int),A) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% one_and_eq
tff(fact_7012_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_7013_and__one__eq,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,A),one_one_int) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% and_one_eq
tff(fact_7014_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_7015_exp__le,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,one_one_real)),aa_num_real(numeral_numeral_real,aa_num_num(bit1,one))) ).

% exp_le
tff(fact_7016_take__bit__eq__mask__iff,axiom,
    ! [Nb: nat,K: int] :
      ( ( aa_int_int(bit_se2923211474154528505it_int(Nb),K) = aa_nat_int(bit_se2000444600071755411sk_int,Nb) )
    <=> ( aa_int_int(bit_se2923211474154528505it_int(Nb),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_7017_pi__half__neq__zero,axiom,
    aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) != zero_zero_real ).

% pi_half_neq_zero
tff(fact_7018_pi__half__less__two,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% pi_half_less_two
tff(fact_7019_pi__half__le__two,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% pi_half_le_two
tff(fact_7020_tanh__altdef,axiom,
    ! [Xb: real] : aa_real_real(tanh_real,Xb) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(exp_real,Xb)),aa_real_real(exp_real,aa_real_real(uminus_uminus_real,Xb)))),aa_real_real(plus_plus_real(aa_real_real(exp_real,Xb)),aa_real_real(exp_real,aa_real_real(uminus_uminus_real,Xb)))) ).

% tanh_altdef
tff(fact_7021_tanh__altdef,axiom,
    ! [Xb: complex] : aa_complex_complex(tanh_complex,Xb) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(exp_complex,Xb)),aa_complex_complex(exp_complex,aa_complex_complex(uminus1482373934393186551omplex,Xb)))),aa_complex_complex(plus_plus_complex(aa_complex_complex(exp_complex,Xb)),aa_complex_complex(exp_complex,aa_complex_complex(uminus1482373934393186551omplex,Xb)))) ).

% tanh_altdef
tff(fact_7022_and__exp__eq__0__iff__not__bit,axiom,
    ! [A: int,Nb: nat] :
      ( ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) = zero_zero_int )
    <=> ~ aa_nat_o(bit_se1146084159140164899it_int(A),Nb) ) ).

% and_exp_eq_0_iff_not_bit
tff(fact_7023_and__exp__eq__0__iff__not__bit,axiom,
    ! [A: nat,Nb: 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))),Nb)) = zero_zero_nat )
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),Nb) ) ).

% and_exp_eq_0_iff_not_bit
tff(fact_7024_Suc__mask__eq__exp,axiom,
    ! [Nb: nat] : aa_nat_nat(suc,aa_nat_nat(bit_se2002935070580805687sk_nat,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) ).

% Suc_mask_eq_exp
tff(fact_7025_exp__half__le2,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% exp_half_le2
tff(fact_7026_mask__nat__less__exp,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(bit_se2002935070580805687sk_nat,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% mask_nat_less_exp
tff(fact_7027_bit__nat__def,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,Mb),Nb)
    <=> ~ 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,Mb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) ) ).

% bit_nat_def
tff(fact_7028_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_7029_exp__double,axiom,
    ! [Z: real] : aa_real_real(exp_real,aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Z)) = aa_nat_real(aa_real_fun_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_7030_pi__half__gt__zero,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ).

% pi_half_gt_zero
tff(fact_7031_pi__half__ge__zero,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ).

% pi_half_ge_zero
tff(fact_7032_m2pi__less__pi,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))),pi) ).

% m2pi_less_pi
tff(fact_7033_semiring__bit__operations__class_Oeven__mask__iff,axiom,
    ! [Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_nat_Code_integer(bit_se2119862282449309892nteger,Nb))
    <=> ( Nb = zero_zero_nat ) ) ).

% semiring_bit_operations_class.even_mask_iff
tff(fact_7034_semiring__bit__operations__class_Oeven__mask__iff,axiom,
    ! [Nb: 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,Nb))
    <=> ( Nb = zero_zero_nat ) ) ).

% semiring_bit_operations_class.even_mask_iff
tff(fact_7035_semiring__bit__operations__class_Oeven__mask__iff,axiom,
    ! [Nb: 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,Nb))
    <=> ( Nb = zero_zero_nat ) ) ).

% semiring_bit_operations_class.even_mask_iff
tff(fact_7036_arctan__ubound,axiom,
    ! [Yb: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arctan,Yb)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ).

% arctan_ubound
tff(fact_7037_arctan__one,axiom,
    aa_real_real(arctan,one_one_real) = aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))) ).

% arctan_one
tff(fact_7038_mask__nat__def,axiom,
    ! [Nb: nat] : aa_nat_nat(bit_se2002935070580805687sk_nat,Nb) = 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))),Nb)),one_one_nat) ).

% mask_nat_def
tff(fact_7039_mask__half__int,axiom,
    ! [Nb: nat] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(bit_se2000444600071755411sk_int,Nb)),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,Nb),one_one_nat)) ).

% mask_half_int
tff(fact_7040_mask__int__def,axiom,
    ! [Nb: nat] : aa_nat_int(bit_se2000444600071755411sk_int,Nb) = 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))),Nb)),one_one_int) ).

% mask_int_def
tff(fact_7041_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(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),zero_zero_real) ).

% minus_pi_half_less_zero
tff(fact_7042_arctan__bounded,axiom,
    ! [Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arctan,Yb))
      & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arctan,Yb)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ).

% arctan_bounded
tff(fact_7043_arctan__lbound,axiom,
    ! [Yb: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arctan,Yb)) ).

% arctan_lbound
tff(fact_7044_mask__eq__exp__minus__1,axiom,
    ! [Nb: nat] : aa_nat_nat(bit_se2002935070580805687sk_nat,Nb) = 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))),Nb)),one_one_nat) ).

% mask_eq_exp_minus_1
tff(fact_7045_mask__eq__exp__minus__1,axiom,
    ! [Nb: nat] : aa_nat_int(bit_se2000444600071755411sk_int,Nb) = 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))),Nb)),one_one_int) ).

% mask_eq_exp_minus_1
tff(fact_7046_mask__Suc__exp,axiom,
    ! [Nb: nat] : aa_nat_nat(bit_se2002935070580805687sk_nat,aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),aa_nat_nat(bit_se2002935070580805687sk_nat,Nb)) ).

% mask_Suc_exp
tff(fact_7047_mask__Suc__exp,axiom,
    ! [Nb: nat] : aa_nat_int(bit_se2000444600071755411sk_int,aa_nat_nat(suc,Nb)) = aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)),aa_nat_int(bit_se2000444600071755411sk_int,Nb)) ).

% mask_Suc_exp
tff(fact_7048_exp__bound,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,Xb)),aa_real_real(plus_plus_real(aa_real_real(plus_plus_real(one_one_real),Xb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ).

% exp_bound
tff(fact_7049_and__int__rec,axiom,
    ! [K: int,L: int] :
      aa_int_int(aa_int_fun_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(aa_int_fun_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_7050_real__exp__bound__lemma,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(divide_divide_real(one_one_real),aa_num_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,Xb)),aa_real_real(plus_plus_real(one_one_real),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Xb))) ) ) ).

% real_exp_bound_lemma
tff(fact_7051_mask__Suc__double,axiom,
    ! [Nb: nat] : aa_nat_nat(bit_se2002935070580805687sk_nat,aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_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,Nb))) ).

% mask_Suc_double
tff(fact_7052_mask__Suc__double,axiom,
    ! [Nb: nat] : aa_nat_int(bit_se2000444600071755411sk_int,aa_nat_nat(suc,Nb)) = aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_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,Nb))) ).

% mask_Suc_double
tff(fact_7053_take__bit__eq__mask__iff__exp__dvd,axiom,
    ! [Nb: nat,K: int] :
      ( ( aa_int_int(bit_se2923211474154528505it_int(Nb),K) = aa_nat_int(bit_se2000444600071755411sk_int,Nb) )
    <=> 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))),Nb)),aa_int_int(aa_int_fun_int_int(plus_plus_int,K),one_one_int)) ) ).

% take_bit_eq_mask_iff_exp_dvd
tff(fact_7054_exp__lower__Taylor__quadratic,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(aa_real_real(plus_plus_real(one_one_real),Xb)),aa_real_real(divide_divide_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(exp_real,Xb)) ) ).

% exp_lower_Taylor_quadratic
tff(fact_7055_sum__abs__ge__zero,axiom,
    ! [F: fun_int_int,A3: set_int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),groups4538972089207619220nt_int(aTP_Lamm_cs(F),A3)) ).

% sum_abs_ge_zero
tff(fact_7056_sum__abs__ge__zero,axiom,
    ! [F: fun_nat_real,A3: 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)),A3)) ).

% sum_abs_ge_zero
tff(fact_7057_sum__abs,axiom,
    ! [F: fun_int_int,A3: set_int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,groups4538972089207619220nt_int(F,A3))),groups4538972089207619220nt_int(aTP_Lamm_cs(F),A3)) ).

% sum_abs
tff(fact_7058_sum__abs,axiom,
    ! [F: fun_nat_real,A3: 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),A3))),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_ct(F)),A3)) ).

% sum_abs
tff(fact_7059_sum_Oempty,axiom,
    ! [G: fun_nat_complex] : aa_set_nat_complex(groups2073611262835488442omplex(G),bot_bot_set_nat) = zero_zero_complex ).

% sum.empty
tff(fact_7060_sum_Oempty,axiom,
    ! [G: fun_nat_rat] : groups2906978787729119204at_rat(G,bot_bot_set_nat) = zero_zero_rat ).

% sum.empty
tff(fact_7061_sum_Oempty,axiom,
    ! [G: fun_nat_int] : groups3539618377306564664at_int(G,bot_bot_set_nat) = zero_zero_int ).

% sum.empty
tff(fact_7062_sum_Oempty,axiom,
    ! [G: fun_int_complex] : groups3049146728041665814omplex(G,bot_bot_set_int) = zero_zero_complex ).

% sum.empty
tff(fact_7063_sum_Oempty,axiom,
    ! [G: fun_int_real] : groups8778361861064173332t_real(G,bot_bot_set_int) = zero_zero_real ).

% sum.empty
tff(fact_7064_sum_Oempty,axiom,
    ! [G: fun_int_rat] : groups3906332499630173760nt_rat(G,bot_bot_set_int) = zero_zero_rat ).

% sum.empty
tff(fact_7065_sum_Oempty,axiom,
    ! [G: fun_int_nat] : groups4541462559716669496nt_nat(G,bot_bot_set_int) = zero_zero_nat ).

% sum.empty
tff(fact_7066_sum_Oempty,axiom,
    ! [G: fun_real_complex] : groups5754745047067104278omplex(G,bot_bot_set_real) = zero_zero_complex ).

% sum.empty
tff(fact_7067_sum_Oempty,axiom,
    ! [G: fun_real_real] : groups8097168146408367636l_real(G,bot_bot_set_real) = zero_zero_real ).

% sum.empty
tff(fact_7068_sum_Oempty,axiom,
    ! [G: fun_real_rat] : groups1300246762558778688al_rat(G,bot_bot_set_real) = zero_zero_rat ).

% sum.empty
tff(fact_7069_convex__sum__bound__le,axiom,
    ! [I6: set_complex,Xb: fun_co8153928105989009288nteger,A: fun_co8153928105989009288nteger,B: code_integer,Delta: code_integer] :
      ( ! [I2: complex] :
          ( member_complex(I2,I6)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_com4217951221296830281nteger(Xb,I2)) )
     => ( ( groups6621422865394947399nteger(Xb,I6) = one_one_Code_integer )
       => ( ! [I2: complex] :
              ( member_complex(I2,I6)
             => 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_cu(Xb),A),I6),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_7070_convex__sum__bound__le,axiom,
    ! [I6: set_real,Xb: fun_re6978948838062957830nteger,A: fun_re6978948838062957830nteger,B: code_integer,Delta: code_integer] :
      ( ! [I2: real] :
          ( member_real(I2,I6)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_real_Code_integer(Xb,I2)) )
     => ( ( groups7713935264441627589nteger(Xb,I6) = one_one_Code_integer )
       => ( ! [I2: real] :
              ( member_real(I2,I6)
             => 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_cv(Xb),A),I6),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_7071_convex__sum__bound__le,axiom,
    ! [I6: set_nat,Xb: fun_nat_Code_integer,A: fun_nat_Code_integer,B: code_integer,Delta: code_integer] :
      ( ! [I2: nat] :
          ( aa_set_nat_o(member_nat(I2),I6)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(Xb,I2)) )
     => ( ( groups7501900531339628137nteger(Xb,I6) = one_one_Code_integer )
       => ( ! [I2: nat] :
              ( aa_set_nat_o(member_nat(I2),I6)
             => 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_cw(Xb),A),I6),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_7072_convex__sum__bound__le,axiom,
    ! [I6: set_int,Xb: fun_int_Code_integer,A: fun_int_Code_integer,B: code_integer,Delta: code_integer] :
      ( ! [I2: int] :
          ( member_int(I2,I6)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(Xb,I2)) )
     => ( ( groups7873554091576472773nteger(Xb,I6) = one_one_Code_integer )
       => ( ! [I2: int] :
              ( member_int(I2,I6)
             => 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_cx(Xb),A),I6),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_7073_convex__sum__bound__le,axiom,
    ! [I6: set_complex,Xb: fun_complex_real,A: fun_complex_real,B: real,Delta: real] :
      ( ! [I2: complex] :
          ( member_complex(I2,I6)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(Xb,I2)) )
     => ( ( groups5808333547571424918x_real(Xb,I6) = one_one_real )
       => ( ! [I2: complex] :
              ( member_complex(I2,I6)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(aa_complex_real(A,I2)),B))),Delta) )
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(groups5808333547571424918x_real(aa_fun8933909628229591020x_real(aTP_Lamm_cy(Xb),A),I6)),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_7074_convex__sum__bound__le,axiom,
    ! [I6: set_real,Xb: fun_real_real,A: fun_real_real,B: real,Delta: real] :
      ( ! [I2: real] :
          ( member_real(I2,I6)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(Xb,I2)) )
     => ( ( groups8097168146408367636l_real(Xb,I6) = one_one_real )
       => ( ! [I2: real] :
              ( member_real(I2,I6)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(aa_real_real(A,I2)),B))),Delta) )
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(groups8097168146408367636l_real(aa_fun6946558556880649704l_real(aTP_Lamm_cz(Xb),A),I6)),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_7075_convex__sum__bound__le,axiom,
    ! [I6: set_int,Xb: fun_int_real,A: fun_int_real,B: real,Delta: real] :
      ( ! [I2: int] :
          ( member_int(I2,I6)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(Xb,I2)) )
     => ( ( groups8778361861064173332t_real(Xb,I6) = one_one_real )
       => ( ! [I2: int] :
              ( member_int(I2,I6)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(aa_int_real(A,I2)),B))),Delta) )
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(groups8778361861064173332t_real(aa_fun3252798400008884712t_real(aTP_Lamm_da(Xb),A),I6)),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_7076_convex__sum__bound__le,axiom,
    ! [I6: set_complex,Xb: fun_complex_rat,A: fun_complex_rat,B: rat,Delta: rat] :
      ( ! [I2: complex] :
          ( member_complex(I2,I6)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(Xb,I2)) )
     => ( ( groups5058264527183730370ex_rat(Xb,I6) = one_one_rat )
       => ( ! [I2: complex] :
              ( member_complex(I2,I6)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(aa_complex_rat(A,I2)),B))),Delta) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(groups5058264527183730370ex_rat(aa_fun7622890401989837636ex_rat(aTP_Lamm_db(Xb),A),I6)),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_7077_convex__sum__bound__le,axiom,
    ! [I6: set_real,Xb: fun_real_rat,A: fun_real_rat,B: rat,Delta: rat] :
      ( ! [I2: real] :
          ( member_real(I2,I6)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(Xb,I2)) )
     => ( ( groups1300246762558778688al_rat(Xb,I6) = one_one_rat )
       => ( ! [I2: real] :
              ( member_real(I2,I6)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(aa_real_rat(A,I2)),B))),Delta) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(groups1300246762558778688al_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_dc(Xb),A),I6)),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_7078_convex__sum__bound__le,axiom,
    ! [I6: set_nat,Xb: fun_nat_rat,A: fun_nat_rat,B: rat,Delta: rat] :
      ( ! [I2: nat] :
          ( aa_set_nat_o(member_nat(I2),I6)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(Xb,I2)) )
     => ( ( groups2906978787729119204at_rat(Xb,I6) = one_one_rat )
       => ( ! [I2: nat] :
              ( aa_set_nat_o(member_nat(I2),I6)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(aa_nat_rat(A,I2)),B))),Delta) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_dd(Xb),A),I6)),B))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_7079_sum_Oneutral__const,axiom,
    ! [A3: set_int] : groups4538972089207619220nt_int(aTP_Lamm_an,A3) = zero_zero_int ).

% sum.neutral_const
tff(fact_7080_sum_Oneutral__const,axiom,
    ! [A3: set_complex] : groups7754918857620584856omplex(aTP_Lamm_aj,A3) = zero_zero_complex ).

% sum.neutral_const
tff(fact_7081_sum_Oneutral__const,axiom,
    ! [A3: set_nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_am),A3) = zero_zero_nat ).

% sum.neutral_const
tff(fact_7082_sum_Oneutral__const,axiom,
    ! [A3: set_nat] : aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_de),A3) = zero_zero_real ).

% sum.neutral_const
tff(fact_7083_and__int_Oelims,axiom,
    ! [Xb: int,Xaa: int,Yb: int] :
      ( ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Xb),Xaa) = Yb )
     => ( Yb = $ite(
            ( member_int(Xb,insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
            & member_int(Xaa,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))),Xb)
                & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xaa) ))),
            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))),Xb)
                  & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xaa) ))),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(bit_se725231765392027082nd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,Xb),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,Xaa),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) ) ).

% and_int.elims
tff(fact_7084_insert__subset,axiom,
    ! [Xb: complex,A3: set_complex,B4: set_complex] :
      ( ord_le211207098394363844omplex(insert_complex(Xb,A3),B4)
    <=> ( member_complex(Xb,B4)
        & ord_le211207098394363844omplex(A3,B4) ) ) ).

% insert_subset
tff(fact_7085_insert__subset,axiom,
    ! [Xb: real,A3: set_real,B4: set_real] :
      ( ord_less_eq_set_real(insert_real(Xb,A3),B4)
    <=> ( member_real(Xb,B4)
        & ord_less_eq_set_real(A3,B4) ) ) ).

% insert_subset
tff(fact_7086_insert__subset,axiom,
    ! [Xb: set_nat,A3: set_set_nat,B4: set_set_nat] :
      ( ord_le6893508408891458716et_nat(insert_set_nat(Xb,A3),B4)
    <=> ( member_set_nat(Xb,B4)
        & ord_le6893508408891458716et_nat(A3,B4) ) ) ).

% insert_subset
tff(fact_7087_insert__subset,axiom,
    ! [Xb: int,A3: set_int,B4: set_int] :
      ( ord_less_eq_set_int(insert_int(Xb,A3),B4)
    <=> ( member_int(Xb,B4)
        & ord_less_eq_set_int(A3,B4) ) ) ).

% insert_subset
tff(fact_7088_insert__subset,axiom,
    ! [Xb: nat,A3: set_nat,B4: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(insert_nat(Xb,A3)),B4)
    <=> ( aa_set_nat_o(member_nat(Xb),B4)
        & aa_set_nat_o(ord_less_eq_set_nat(A3),B4) ) ) ).

% insert_subset
tff(fact_7089_singleton__insert__inj__eq_H,axiom,
    ! [A: int,A3: set_int,B: int] :
      ( ( insert_int(A,A3) = insert_int(B,bot_bot_set_int) )
    <=> ( ( A = B )
        & ord_less_eq_set_int(A3,insert_int(B,bot_bot_set_int)) ) ) ).

% singleton_insert_inj_eq'
tff(fact_7090_singleton__insert__inj__eq_H,axiom,
    ! [A: real,A3: set_real,B: real] :
      ( ( insert_real(A,A3) = insert_real(B,bot_bot_set_real) )
    <=> ( ( A = B )
        & ord_less_eq_set_real(A3,insert_real(B,bot_bot_set_real)) ) ) ).

% singleton_insert_inj_eq'
tff(fact_7091_singleton__insert__inj__eq_H,axiom,
    ! [A: nat,A3: set_nat,B: nat] :
      ( ( insert_nat(A,A3) = insert_nat(B,bot_bot_set_nat) )
    <=> ( ( A = B )
        & aa_set_nat_o(ord_less_eq_set_nat(A3),insert_nat(B,bot_bot_set_nat)) ) ) ).

% singleton_insert_inj_eq'
tff(fact_7092_singleton__insert__inj__eq,axiom,
    ! [B: int,A: int,A3: set_int] :
      ( ( insert_int(B,bot_bot_set_int) = insert_int(A,A3) )
    <=> ( ( A = B )
        & ord_less_eq_set_int(A3,insert_int(B,bot_bot_set_int)) ) ) ).

% singleton_insert_inj_eq
tff(fact_7093_singleton__insert__inj__eq,axiom,
    ! [B: real,A: real,A3: set_real] :
      ( ( insert_real(B,bot_bot_set_real) = insert_real(A,A3) )
    <=> ( ( A = B )
        & ord_less_eq_set_real(A3,insert_real(B,bot_bot_set_real)) ) ) ).

% singleton_insert_inj_eq
tff(fact_7094_singleton__insert__inj__eq,axiom,
    ! [B: nat,A: nat,A3: set_nat] :
      ( ( insert_nat(B,bot_bot_set_nat) = insert_nat(A,A3) )
    <=> ( ( A = B )
        & aa_set_nat_o(ord_less_eq_set_nat(A3),insert_nat(B,bot_bot_set_nat)) ) ) ).

% singleton_insert_inj_eq
tff(fact_7095_and__nat__numerals_I3_J,axiom,
    ! [Xb: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Xb))),aa_nat_nat(suc,zero_zero_nat)) = zero_zero_nat ).

% and_nat_numerals(3)
tff(fact_7096_and__nat__numerals_I1_J,axiom,
    ! [Yb: 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,Yb))) = zero_zero_nat ).

% and_nat_numerals(1)
tff(fact_7097_subset__Compl__singleton,axiom,
    ! [A3: set_complex,B: complex] :
      ( ord_le211207098394363844omplex(A3,uminus8566677241136511917omplex(insert_complex(B,bot_bot_set_complex)))
    <=> ~ member_complex(B,A3) ) ).

% subset_Compl_singleton
tff(fact_7098_subset__Compl__singleton,axiom,
    ! [A3: set_set_nat,B: set_nat] :
      ( ord_le6893508408891458716et_nat(A3,uminus613421341184616069et_nat(insert_set_nat(B,bot_bot_set_set_nat)))
    <=> ~ member_set_nat(B,A3) ) ).

% subset_Compl_singleton
tff(fact_7099_subset__Compl__singleton,axiom,
    ! [A3: set_int,B: int] :
      ( ord_less_eq_set_int(A3,uminus1532241313380277803et_int(insert_int(B,bot_bot_set_int)))
    <=> ~ member_int(B,A3) ) ).

% subset_Compl_singleton
tff(fact_7100_subset__Compl__singleton,axiom,
    ! [A3: set_real,B: real] :
      ( ord_less_eq_set_real(A3,uminus612125837232591019t_real(insert_real(B,bot_bot_set_real)))
    <=> ~ member_real(B,A3) ) ).

% subset_Compl_singleton
tff(fact_7101_subset__Compl__singleton,axiom,
    ! [A3: set_nat,B: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A3),uminus5710092332889474511et_nat(insert_nat(B,bot_bot_set_nat)))
    <=> ~ aa_set_nat_o(member_nat(B),A3) ) ).

% subset_Compl_singleton
tff(fact_7102_sum_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_complex,Mb: nat,Nb: nat] :
      aa_set_nat_complex(groups2073611262835488442omplex(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),Mb),zero_zero_complex,aa_complex_complex(plus_plus_complex(aa_set_nat_complex(groups2073611262835488442omplex(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))),aa_nat_complex(G,aa_nat_nat(suc,Nb)))) ).

% sum.cl_ivl_Suc
tff(fact_7103_sum_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_rat,Mb: nat,Nb: nat] :
      groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),Mb),zero_zero_rat,aa_rat_rat(plus_plus_rat(groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))),aa_nat_rat(G,aa_nat_nat(suc,Nb)))) ).

% sum.cl_ivl_Suc
tff(fact_7104_sum_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_int,Mb: nat,Nb: nat] :
      groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),Mb),zero_zero_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))),aa_nat_int(G,aa_nat_nat(suc,Nb)))) ).

% sum.cl_ivl_Suc
tff(fact_7105_sum_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_nat,Mb: nat,Nb: nat] :
      aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),Mb),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(Mb),Nb))),aa_nat_nat(G,aa_nat_nat(suc,Nb)))) ).

% sum.cl_ivl_Suc
tff(fact_7106_sum_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_real,Mb: nat,Nb: nat] :
      aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),Mb),zero_zero_real,aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))),aa_nat_real(G,aa_nat_nat(suc,Nb)))) ).

% sum.cl_ivl_Suc
tff(fact_7107_and__nat__numerals_I4_J,axiom,
    ! [Xb: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xb))),aa_nat_nat(suc,zero_zero_nat)) = one_one_nat ).

% and_nat_numerals(4)
tff(fact_7108_and__nat__numerals_I2_J,axiom,
    ! [Yb: 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,Yb))) = one_one_nat ).

% and_nat_numerals(2)
tff(fact_7109_set__replicate,axiom,
    ! [Nb: nat,Xb: vEBT_VEBT] :
      ( ( Nb != zero_zero_nat )
     => ( set_VEBT_VEBT2(replicate_VEBT_VEBT(Nb,Xb)) = insert_VEBT_VEBT(Xb,bot_bo8194388402131092736T_VEBT) ) ) ).

% set_replicate
tff(fact_7110_set__replicate,axiom,
    ! [Nb: nat,Xb: nat] :
      ( ( Nb != zero_zero_nat )
     => ( set_nat2(replicate_nat(Nb,Xb)) = insert_nat(Xb,bot_bot_set_nat) ) ) ).

% set_replicate
tff(fact_7111_set__replicate,axiom,
    ! [Nb: nat,Xb: int] :
      ( ( Nb != zero_zero_nat )
     => ( set_int2(replicate_int(Nb,Xb)) = insert_int(Xb,bot_bot_set_int) ) ) ).

% set_replicate
tff(fact_7112_set__replicate,axiom,
    ! [Nb: nat,Xb: real] :
      ( ( Nb != zero_zero_nat )
     => ( set_real2(replicate_real(Nb,Xb)) = insert_real(Xb,bot_bot_set_real) ) ) ).

% set_replicate
tff(fact_7113_Suc__0__and__eq,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(suc,zero_zero_nat)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% Suc_0_and_eq
tff(fact_7114_and__Suc__0__eq,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,Nb),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% and_Suc_0_eq
tff(fact_7115_insert__mono,axiom,
    ! [C5: set_int,D4: set_int,A: int] :
      ( ord_less_eq_set_int(C5,D4)
     => ord_less_eq_set_int(insert_int(A,C5),insert_int(A,D4)) ) ).

% insert_mono
tff(fact_7116_insert__mono,axiom,
    ! [C5: set_real,D4: set_real,A: real] :
      ( ord_less_eq_set_real(C5,D4)
     => ord_less_eq_set_real(insert_real(A,C5),insert_real(A,D4)) ) ).

% insert_mono
tff(fact_7117_insert__mono,axiom,
    ! [C5: set_nat,D4: set_nat,A: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(C5),D4)
     => aa_set_nat_o(ord_less_eq_set_nat(insert_nat(A,C5)),insert_nat(A,D4)) ) ).

% insert_mono
tff(fact_7118_subset__insert,axiom,
    ! [Xb: complex,A3: set_complex,B4: set_complex] :
      ( ~ member_complex(Xb,A3)
     => ( ord_le211207098394363844omplex(A3,insert_complex(Xb,B4))
      <=> ord_le211207098394363844omplex(A3,B4) ) ) ).

% subset_insert
tff(fact_7119_subset__insert,axiom,
    ! [Xb: real,A3: set_real,B4: set_real] :
      ( ~ member_real(Xb,A3)
     => ( ord_less_eq_set_real(A3,insert_real(Xb,B4))
      <=> ord_less_eq_set_real(A3,B4) ) ) ).

% subset_insert
tff(fact_7120_subset__insert,axiom,
    ! [Xb: set_nat,A3: set_set_nat,B4: set_set_nat] :
      ( ~ member_set_nat(Xb,A3)
     => ( ord_le6893508408891458716et_nat(A3,insert_set_nat(Xb,B4))
      <=> ord_le6893508408891458716et_nat(A3,B4) ) ) ).

% subset_insert
tff(fact_7121_subset__insert,axiom,
    ! [Xb: int,A3: set_int,B4: set_int] :
      ( ~ member_int(Xb,A3)
     => ( ord_less_eq_set_int(A3,insert_int(Xb,B4))
      <=> ord_less_eq_set_int(A3,B4) ) ) ).

% subset_insert
tff(fact_7122_subset__insert,axiom,
    ! [Xb: nat,A3: set_nat,B4: set_nat] :
      ( ~ aa_set_nat_o(member_nat(Xb),A3)
     => ( aa_set_nat_o(ord_less_eq_set_nat(A3),insert_nat(Xb,B4))
      <=> aa_set_nat_o(ord_less_eq_set_nat(A3),B4) ) ) ).

% subset_insert
tff(fact_7123_subset__insertI,axiom,
    ! [B4: set_int,A: int] : ord_less_eq_set_int(B4,insert_int(A,B4)) ).

% subset_insertI
tff(fact_7124_subset__insertI,axiom,
    ! [B4: set_real,A: real] : ord_less_eq_set_real(B4,insert_real(A,B4)) ).

% subset_insertI
tff(fact_7125_subset__insertI,axiom,
    ! [B4: set_nat,A: nat] : aa_set_nat_o(ord_less_eq_set_nat(B4),insert_nat(A,B4)) ).

% subset_insertI
tff(fact_7126_subset__insertI2,axiom,
    ! [A3: set_int,B4: set_int,B: int] :
      ( ord_less_eq_set_int(A3,B4)
     => ord_less_eq_set_int(A3,insert_int(B,B4)) ) ).

% subset_insertI2
tff(fact_7127_subset__insertI2,axiom,
    ! [A3: set_real,B4: set_real,B: real] :
      ( ord_less_eq_set_real(A3,B4)
     => ord_less_eq_set_real(A3,insert_real(B,B4)) ) ).

% subset_insertI2
tff(fact_7128_subset__insertI2,axiom,
    ! [A3: set_nat,B4: set_nat,B: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A3),B4)
     => aa_set_nat_o(ord_less_eq_set_nat(A3),insert_nat(B,B4)) ) ).

% subset_insertI2
tff(fact_7129_sum__cong__Suc,axiom,
    ! [A3: set_nat,F: fun_nat_nat,G: fun_nat_nat] :
      ( ~ aa_set_nat_o(member_nat(zero_zero_nat),A3)
     => ( ! [X3: nat] :
            ( aa_set_nat_o(member_nat(aa_nat_nat(suc,X3)),A3)
           => ( aa_nat_nat(F,aa_nat_nat(suc,X3)) = aa_nat_nat(G,aa_nat_nat(suc,X3)) ) )
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(F),A3) = aa_set_nat_nat(groups3542108847815614940at_nat(G),A3) ) ) ) ).

% sum_cong_Suc
tff(fact_7130_sum__cong__Suc,axiom,
    ! [A3: set_nat,F: fun_nat_real,G: fun_nat_real] :
      ( ~ aa_set_nat_o(member_nat(zero_zero_nat),A3)
     => ( ! [X3: nat] :
            ( aa_set_nat_o(member_nat(aa_nat_nat(suc,X3)),A3)
           => ( aa_nat_real(F,aa_nat_nat(suc,X3)) = aa_nat_real(G,aa_nat_nat(suc,X3)) ) )
       => ( aa_set_nat_real(groups6591440286371151544t_real(F),A3) = aa_set_nat_real(groups6591440286371151544t_real(G),A3) ) ) ) ).

% sum_cong_Suc
tff(fact_7131_sum__subtractf__nat,axiom,
    ! [A3: set_complex,G: fun_complex_nat,F: fun_complex_nat] :
      ( ! [X3: complex] :
          ( member_complex(X3,A3)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_complex_nat(G,X3)),aa_complex_nat(F,X3)) )
     => ( groups5693394587270226106ex_nat(aa_fun2465681967280144180ex_nat(aTP_Lamm_df(G),F),A3) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups5693394587270226106ex_nat(F,A3)),groups5693394587270226106ex_nat(G,A3)) ) ) ).

% sum_subtractf_nat
tff(fact_7132_sum__subtractf__nat,axiom,
    ! [A3: set_real,G: fun_real_nat,F: fun_real_nat] :
      ( ! [X3: real] :
          ( member_real(X3,A3)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_real_nat(G,X3)),aa_real_nat(F,X3)) )
     => ( groups1935376822645274424al_nat(aa_fun211602392931233840al_nat(aTP_Lamm_dg(G),F),A3) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups1935376822645274424al_nat(F,A3)),groups1935376822645274424al_nat(G,A3)) ) ) ).

% sum_subtractf_nat
tff(fact_7133_sum__subtractf__nat,axiom,
    ! [A3: set_set_nat,G: fun_set_nat_nat,F: fun_set_nat_nat] :
      ( ! [X3: set_nat] :
          ( member_set_nat(X3,A3)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(G,X3)),aa_set_nat_nat(F,X3)) )
     => ( groups8294997508430121362at_nat(aa_fun3386010016858254564at_nat(aTP_Lamm_dh(G),F),A3) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups8294997508430121362at_nat(F,A3)),groups8294997508430121362at_nat(G,A3)) ) ) ).

% sum_subtractf_nat
tff(fact_7134_sum__subtractf__nat,axiom,
    ! [A3: set_int,G: fun_int_nat,F: fun_int_nat] :
      ( ! [X3: int] :
          ( member_int(X3,A3)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(G,X3)),aa_int_nat(F,X3)) )
     => ( groups4541462559716669496nt_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_di(G),F),A3) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups4541462559716669496nt_nat(F,A3)),groups4541462559716669496nt_nat(G,A3)) ) ) ).

% sum_subtractf_nat
tff(fact_7135_sum__subtractf__nat,axiom,
    ! [A3: set_nat,G: fun_nat_nat,F: fun_nat_nat] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A3)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(G,X3)),aa_nat_nat(F,X3)) )
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_dj(G),F)),A3) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A3)),aa_set_nat_nat(groups3542108847815614940at_nat(G),A3)) ) ) ).

% sum_subtractf_nat
tff(fact_7136_sum__SucD,axiom,
    ! [F: fun_nat_nat,A3: set_nat,Nb: nat] :
      ( ( aa_set_nat_nat(groups3542108847815614940at_nat(F),A3) = aa_nat_nat(suc,Nb) )
     => ? [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A3)
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(F,X3)) ) ) ).

% sum_SucD
tff(fact_7137_subset__singleton__iff,axiom,
    ! [X6: set_int,A: int] :
      ( ord_less_eq_set_int(X6,insert_int(A,bot_bot_set_int))
    <=> ( ( X6 = bot_bot_set_int )
        | ( X6 = insert_int(A,bot_bot_set_int) ) ) ) ).

% subset_singleton_iff
tff(fact_7138_subset__singleton__iff,axiom,
    ! [X6: set_real,A: real] :
      ( ord_less_eq_set_real(X6,insert_real(A,bot_bot_set_real))
    <=> ( ( X6 = bot_bot_set_real )
        | ( X6 = insert_real(A,bot_bot_set_real) ) ) ) ).

% subset_singleton_iff
tff(fact_7139_subset__singleton__iff,axiom,
    ! [X6: set_nat,A: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(X6),insert_nat(A,bot_bot_set_nat))
    <=> ( ( X6 = bot_bot_set_nat )
        | ( X6 = insert_nat(A,bot_bot_set_nat) ) ) ) ).

% subset_singleton_iff
tff(fact_7140_subset__singletonD,axiom,
    ! [A3: set_int,Xb: int] :
      ( ord_less_eq_set_int(A3,insert_int(Xb,bot_bot_set_int))
     => ( ( A3 = bot_bot_set_int )
        | ( A3 = insert_int(Xb,bot_bot_set_int) ) ) ) ).

% subset_singletonD
tff(fact_7141_subset__singletonD,axiom,
    ! [A3: set_real,Xb: real] :
      ( ord_less_eq_set_real(A3,insert_real(Xb,bot_bot_set_real))
     => ( ( A3 = bot_bot_set_real )
        | ( A3 = insert_real(Xb,bot_bot_set_real) ) ) ) ).

% subset_singletonD
tff(fact_7142_subset__singletonD,axiom,
    ! [A3: set_nat,Xb: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A3),insert_nat(Xb,bot_bot_set_nat))
     => ( ( A3 = bot_bot_set_nat )
        | ( A3 = insert_nat(Xb,bot_bot_set_nat) ) ) ) ).

% subset_singletonD
tff(fact_7143_subset__Diff__insert,axiom,
    ! [A3: set_complex,B4: set_complex,Xb: complex,C5: set_complex] :
      ( ord_le211207098394363844omplex(A3,minus_811609699411566653omplex(B4,insert_complex(Xb,C5)))
    <=> ( ord_le211207098394363844omplex(A3,minus_811609699411566653omplex(B4,C5))
        & ~ member_complex(Xb,A3) ) ) ).

% subset_Diff_insert
tff(fact_7144_subset__Diff__insert,axiom,
    ! [A3: set_real,B4: set_real,Xb: real,C5: set_real] :
      ( ord_less_eq_set_real(A3,minus_minus_set_real(B4,insert_real(Xb,C5)))
    <=> ( ord_less_eq_set_real(A3,minus_minus_set_real(B4,C5))
        & ~ member_real(Xb,A3) ) ) ).

% subset_Diff_insert
tff(fact_7145_subset__Diff__insert,axiom,
    ! [A3: set_set_nat,B4: set_set_nat,Xb: set_nat,C5: set_set_nat] :
      ( ord_le6893508408891458716et_nat(A3,minus_2163939370556025621et_nat(B4,insert_set_nat(Xb,C5)))
    <=> ( ord_le6893508408891458716et_nat(A3,minus_2163939370556025621et_nat(B4,C5))
        & ~ member_set_nat(Xb,A3) ) ) ).

% subset_Diff_insert
tff(fact_7146_subset__Diff__insert,axiom,
    ! [A3: set_int,B4: set_int,Xb: int,C5: set_int] :
      ( ord_less_eq_set_int(A3,minus_minus_set_int(B4,insert_int(Xb,C5)))
    <=> ( ord_less_eq_set_int(A3,minus_minus_set_int(B4,C5))
        & ~ member_int(Xb,A3) ) ) ).

% subset_Diff_insert
tff(fact_7147_subset__Diff__insert,axiom,
    ! [A3: set_nat,B4: set_nat,Xb: nat,C5: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A3),minus_minus_set_nat(B4,insert_nat(Xb,C5)))
    <=> ( aa_set_nat_o(ord_less_eq_set_nat(A3),minus_minus_set_nat(B4,C5))
        & ~ aa_set_nat_o(member_nat(Xb),A3) ) ) ).

% subset_Diff_insert
tff(fact_7148_sum__nth__roots,axiom,
    ! [Nb: nat,C: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),Nb)
     => ( groups7754918857620584856omplex(aTP_Lamm_ao,collect_complex(aa_com5808801942499186519plex_o(aTP_Lamm_dk(Nb),C))) = zero_zero_complex ) ) ).

% sum_nth_roots
tff(fact_7149_sum__roots__unity,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),Nb)
     => ( groups7754918857620584856omplex(aTP_Lamm_ao,collect_complex(aTP_Lamm_dl(Nb))) = zero_zero_complex ) ) ).

% sum_roots_unity
tff(fact_7150_sum_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: fun_nat_nat,Mb: nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb)),aa_nat_nat(suc,Nb))) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_dm(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) ).

% sum.shift_bounds_cl_Suc_ivl
tff(fact_7151_sum_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: fun_nat_real,Mb: nat,Nb: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb)),aa_nat_nat(suc,Nb))) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_dn(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) ).

% sum.shift_bounds_cl_Suc_ivl
tff(fact_7152_sum_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: fun_nat_nat,Mb: nat,K: nat,Nb: 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,Mb),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),K))) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_do(G),K)),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) ).

% sum.shift_bounds_cl_nat_ivl
tff(fact_7153_sum_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: fun_nat_real,Mb: nat,K: nat,Nb: 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,Mb),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),K))) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_dp(G),K)),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) ).

% sum.shift_bounds_cl_nat_ivl
tff(fact_7154_Diff__single__insert,axiom,
    ! [A3: set_int,Xb: int,B4: set_int] :
      ( ord_less_eq_set_int(minus_minus_set_int(A3,insert_int(Xb,bot_bot_set_int)),B4)
     => ord_less_eq_set_int(A3,insert_int(Xb,B4)) ) ).

% Diff_single_insert
tff(fact_7155_Diff__single__insert,axiom,
    ! [A3: set_real,Xb: real,B4: set_real] :
      ( ord_less_eq_set_real(minus_minus_set_real(A3,insert_real(Xb,bot_bot_set_real)),B4)
     => ord_less_eq_set_real(A3,insert_real(Xb,B4)) ) ).

% Diff_single_insert
tff(fact_7156_Diff__single__insert,axiom,
    ! [A3: set_nat,Xb: nat,B4: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(minus_minus_set_nat(A3,insert_nat(Xb,bot_bot_set_nat))),B4)
     => aa_set_nat_o(ord_less_eq_set_nat(A3),insert_nat(Xb,B4)) ) ).

% Diff_single_insert
tff(fact_7157_subset__insert__iff,axiom,
    ! [A3: set_complex,Xb: complex,B4: set_complex] :
      ( ord_le211207098394363844omplex(A3,insert_complex(Xb,B4))
    <=> $ite(member_complex(Xb,A3),ord_le211207098394363844omplex(minus_811609699411566653omplex(A3,insert_complex(Xb,bot_bot_set_complex)),B4),ord_le211207098394363844omplex(A3,B4)) ) ).

% subset_insert_iff
tff(fact_7158_subset__insert__iff,axiom,
    ! [A3: set_set_nat,Xb: set_nat,B4: set_set_nat] :
      ( ord_le6893508408891458716et_nat(A3,insert_set_nat(Xb,B4))
    <=> $ite(member_set_nat(Xb,A3),ord_le6893508408891458716et_nat(minus_2163939370556025621et_nat(A3,insert_set_nat(Xb,bot_bot_set_set_nat)),B4),ord_le6893508408891458716et_nat(A3,B4)) ) ).

% subset_insert_iff
tff(fact_7159_subset__insert__iff,axiom,
    ! [A3: set_int,Xb: int,B4: set_int] :
      ( ord_less_eq_set_int(A3,insert_int(Xb,B4))
    <=> $ite(member_int(Xb,A3),ord_less_eq_set_int(minus_minus_set_int(A3,insert_int(Xb,bot_bot_set_int)),B4),ord_less_eq_set_int(A3,B4)) ) ).

% subset_insert_iff
tff(fact_7160_subset__insert__iff,axiom,
    ! [A3: set_real,Xb: real,B4: set_real] :
      ( ord_less_eq_set_real(A3,insert_real(Xb,B4))
    <=> $ite(member_real(Xb,A3),ord_less_eq_set_real(minus_minus_set_real(A3,insert_real(Xb,bot_bot_set_real)),B4),ord_less_eq_set_real(A3,B4)) ) ).

% subset_insert_iff
tff(fact_7161_subset__insert__iff,axiom,
    ! [A3: set_nat,Xb: nat,B4: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A3),insert_nat(Xb,B4))
    <=> $ite(aa_set_nat_o(member_nat(Xb),A3),aa_set_nat_o(ord_less_eq_set_nat(minus_minus_set_nat(A3,insert_nat(Xb,bot_bot_set_nat))),B4),aa_set_nat_o(ord_less_eq_set_nat(A3),B4)) ) ).

% subset_insert_iff
tff(fact_7162_set__update__subset__insert,axiom,
    ! [Xs: list_real,I3: nat,Xb: real] : ord_less_eq_set_real(set_real2(list_update_real(Xs,I3,Xb)),insert_real(Xb,set_real2(Xs))) ).

% set_update_subset_insert
tff(fact_7163_set__update__subset__insert,axiom,
    ! [Xs: list_int,I3: nat,Xb: int] : ord_less_eq_set_int(set_int2(list_update_int(Xs,I3,Xb)),insert_int(Xb,set_int2(Xs))) ).

% set_update_subset_insert
tff(fact_7164_set__update__subset__insert,axiom,
    ! [Xs: list_VEBT_VEBT,I3: nat,Xb: vEBT_VEBT] : ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(Xs,I3,Xb)),insert_VEBT_VEBT(Xb,set_VEBT_VEBT2(Xs))) ).

% set_update_subset_insert
tff(fact_7165_set__update__subset__insert,axiom,
    ! [Xs: list_nat,I3: nat,Xb: nat] : aa_set_nat_o(ord_less_eq_set_nat(set_nat2(list_update_nat(Xs,I3,Xb))),insert_nat(Xb,set_nat2(Xs))) ).

% set_update_subset_insert
tff(fact_7166_sum__power__add,axiom,
    ! [Xb: complex,Mb: nat,I6: set_nat] : aa_set_nat_complex(groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aTP_Lamm_dq(Xb),Mb)),I6) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(Xb),Mb)),aa_set_nat_complex(groups2073611262835488442omplex(power_power_complex(Xb)),I6)) ).

% sum_power_add
tff(fact_7167_sum__power__add,axiom,
    ! [Xb: rat,Mb: nat,I6: set_nat] : groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aTP_Lamm_dr(Xb),Mb),I6) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(Xb),Mb)),groups2906978787729119204at_rat(power_power_rat(Xb),I6)) ).

% sum_power_add
tff(fact_7168_sum__power__add,axiom,
    ! [Xb: int,Mb: nat,I6: set_nat] : groups3539618377306564664at_int(aa_nat_fun_nat_int(aTP_Lamm_ds(Xb),Mb),I6) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(Xb),Mb)),groups3539618377306564664at_int(power_power_int(Xb),I6)) ).

% sum_power_add
tff(fact_7169_sum__power__add,axiom,
    ! [Xb: real,Mb: nat,I6: set_nat] : aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_dt(Xb),Mb)),I6) = aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Mb)),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,Xb)),I6)) ).

% sum_power_add
tff(fact_7170_sum_OatLeastAtMost__rev,axiom,
    ! [G: fun_nat_nat,Nb: nat,Mb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Nb),Mb)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_du(G),Nb),Mb)),aa_nat_set_nat(set_or1269000886237332187st_nat(Nb),Mb)) ).

% sum.atLeastAtMost_rev
tff(fact_7171_sum_OatLeastAtMost__rev,axiom,
    ! [G: fun_nat_real,Nb: nat,Mb: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Nb),Mb)) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_nat4085805313368488148t_real(aTP_Lamm_dv(G),Nb),Mb)),aa_nat_set_nat(set_or1269000886237332187st_nat(Nb),Mb)) ).

% sum.atLeastAtMost_rev
tff(fact_7172_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_7173_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_7174_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_7175_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_7176_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_7177_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_rat,Nb: nat] : groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,Nb))) = aa_rat_rat(plus_plus_rat(groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb))),aa_nat_rat(G,aa_nat_nat(suc,Nb))) ).

% sum.atLeast0_atMost_Suc
tff(fact_7178_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_int,Nb: nat] : groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,Nb))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb))),aa_nat_int(G,aa_nat_nat(suc,Nb))) ).

% sum.atLeast0_atMost_Suc
tff(fact_7179_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,Nb))) = 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),Nb))),aa_nat_nat(G,aa_nat_nat(suc,Nb))) ).

% sum.atLeast0_atMost_Suc
tff(fact_7180_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_real,Nb: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,Nb))) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb))),aa_nat_real(G,aa_nat_nat(suc,Nb))) ).

% sum.atLeast0_atMost_Suc
tff(fact_7181_sum_Onat__ivl__Suc_H,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb))
     => ( groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(suc,Nb))) = aa_rat_rat(plus_plus_rat(aa_nat_rat(G,aa_nat_nat(suc,Nb))),groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_7182_sum_Onat__ivl__Suc_H,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb))
     => ( groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(suc,Nb))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,aa_nat_nat(suc,Nb))),groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_7183_sum_Onat__ivl__Suc_H,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb))
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G,aa_nat_nat(suc,Nb))),aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_7184_sum_Onat__ivl__Suc_H,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb))
     => ( aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(suc,Nb))) = aa_real_real(plus_plus_real(aa_nat_real(G,aa_nat_nat(suc,Nb))),aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_7185_sum_OatLeast__Suc__atMost,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) = aa_rat_rat(plus_plus_rat(aa_nat_rat(G,Mb)),groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb)),Nb))) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_7186_sum_OatLeast__Suc__atMost,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,Mb)),groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb)),Nb))) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_7187_sum_OatLeast__Suc__atMost,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G,Mb)),aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb)),Nb))) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_7188_sum_OatLeast__Suc__atMost,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) = aa_real_real(plus_plus_real(aa_nat_real(G,Mb)),aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb)),Nb))) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_7189_psubset__insert__iff,axiom,
    ! [A3: set_complex,Xb: complex,B4: set_complex] :
      ( ord_less_set_complex(A3,insert_complex(Xb,B4))
    <=> $ite(
          member_complex(Xb,B4),
          ord_less_set_complex(A3,B4),
          $ite(member_complex(Xb,A3),ord_less_set_complex(minus_811609699411566653omplex(A3,insert_complex(Xb,bot_bot_set_complex)),B4),ord_le211207098394363844omplex(A3,B4)) ) ) ).

% psubset_insert_iff
tff(fact_7190_psubset__insert__iff,axiom,
    ! [A3: set_set_nat,Xb: set_nat,B4: set_set_nat] :
      ( ord_less_set_set_nat(A3,insert_set_nat(Xb,B4))
    <=> $ite(
          member_set_nat(Xb,B4),
          ord_less_set_set_nat(A3,B4),
          $ite(member_set_nat(Xb,A3),ord_less_set_set_nat(minus_2163939370556025621et_nat(A3,insert_set_nat(Xb,bot_bot_set_set_nat)),B4),ord_le6893508408891458716et_nat(A3,B4)) ) ) ).

% psubset_insert_iff
tff(fact_7191_psubset__insert__iff,axiom,
    ! [A3: set_int,Xb: int,B4: set_int] :
      ( ord_less_set_int(A3,insert_int(Xb,B4))
    <=> $ite(
          member_int(Xb,B4),
          ord_less_set_int(A3,B4),
          $ite(member_int(Xb,A3),ord_less_set_int(minus_minus_set_int(A3,insert_int(Xb,bot_bot_set_int)),B4),ord_less_eq_set_int(A3,B4)) ) ) ).

% psubset_insert_iff
tff(fact_7192_psubset__insert__iff,axiom,
    ! [A3: set_real,Xb: real,B4: set_real] :
      ( ord_less_set_real(A3,insert_real(Xb,B4))
    <=> $ite(
          member_real(Xb,B4),
          ord_less_set_real(A3,B4),
          $ite(member_real(Xb,A3),ord_less_set_real(minus_minus_set_real(A3,insert_real(Xb,bot_bot_set_real)),B4),ord_less_eq_set_real(A3,B4)) ) ) ).

% psubset_insert_iff
tff(fact_7193_psubset__insert__iff,axiom,
    ! [A3: set_nat,Xb: nat,B4: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(A3),insert_nat(Xb,B4))
    <=> $ite(
          aa_set_nat_o(member_nat(Xb),B4),
          aa_set_nat_o(ord_less_set_nat(A3),B4),
          $ite(aa_set_nat_o(member_nat(Xb),A3),aa_set_nat_o(ord_less_set_nat(minus_minus_set_nat(A3,insert_nat(Xb,bot_bot_set_nat))),B4),aa_set_nat_o(ord_less_eq_set_nat(A3),B4)) ) ) ).

% psubset_insert_iff
tff(fact_7194_set__replicate__Suc,axiom,
    ! [Nb: nat,Xb: vEBT_VEBT] : set_VEBT_VEBT2(replicate_VEBT_VEBT(aa_nat_nat(suc,Nb),Xb)) = insert_VEBT_VEBT(Xb,bot_bo8194388402131092736T_VEBT) ).

% set_replicate_Suc
tff(fact_7195_set__replicate__Suc,axiom,
    ! [Nb: nat,Xb: nat] : set_nat2(replicate_nat(aa_nat_nat(suc,Nb),Xb)) = insert_nat(Xb,bot_bot_set_nat) ).

% set_replicate_Suc
tff(fact_7196_set__replicate__Suc,axiom,
    ! [Nb: nat,Xb: int] : set_int2(replicate_int(aa_nat_nat(suc,Nb),Xb)) = insert_int(Xb,bot_bot_set_int) ).

% set_replicate_Suc
tff(fact_7197_set__replicate__Suc,axiom,
    ! [Nb: nat,Xb: real] : set_real2(replicate_real(aa_nat_nat(suc,Nb),Xb)) = insert_real(Xb,bot_bot_set_real) ).

% set_replicate_Suc
tff(fact_7198_set__replicate__conv__if,axiom,
    ! [Nb: nat,Xb: vEBT_VEBT] :
      set_VEBT_VEBT2(replicate_VEBT_VEBT(Nb,Xb)) = $ite(Nb = zero_zero_nat,bot_bo8194388402131092736T_VEBT,insert_VEBT_VEBT(Xb,bot_bo8194388402131092736T_VEBT)) ).

% set_replicate_conv_if
tff(fact_7199_set__replicate__conv__if,axiom,
    ! [Nb: nat,Xb: nat] :
      set_nat2(replicate_nat(Nb,Xb)) = $ite(Nb = zero_zero_nat,bot_bot_set_nat,insert_nat(Xb,bot_bot_set_nat)) ).

% set_replicate_conv_if
tff(fact_7200_set__replicate__conv__if,axiom,
    ! [Nb: nat,Xb: int] :
      set_int2(replicate_int(Nb,Xb)) = $ite(Nb = zero_zero_nat,bot_bot_set_int,insert_int(Xb,bot_bot_set_int)) ).

% set_replicate_conv_if
tff(fact_7201_set__replicate__conv__if,axiom,
    ! [Nb: nat,Xb: real] :
      set_real2(replicate_real(Nb,Xb)) = $ite(Nb = zero_zero_nat,bot_bot_set_real,insert_real(Xb,bot_bot_set_real)) ).

% set_replicate_conv_if
tff(fact_7202_sum_OSuc__reindex__ivl,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_rat_rat(plus_plus_rat(groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))),aa_nat_rat(G,aa_nat_nat(suc,Nb))) = aa_rat_rat(plus_plus_rat(aa_nat_rat(G,Mb)),groups2906978787729119204at_rat(aTP_Lamm_dw(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_7203_sum_OSuc__reindex__ivl,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))),aa_nat_int(G,aa_nat_nat(suc,Nb))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,Mb)),groups3539618377306564664at_int(aTP_Lamm_dx(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_7204_sum_OSuc__reindex__ivl,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( 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(Mb),Nb))),aa_nat_nat(G,aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G,Mb)),aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_dm(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_7205_sum_OSuc__reindex__ivl,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))),aa_nat_real(G,aa_nat_nat(suc,Nb))) = aa_real_real(plus_plus_real(aa_nat_real(G,Mb)),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_dn(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_7206_sum__Suc__diff,axiom,
    ! [Mb: nat,Nb: nat,F: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb))
     => ( groups2906978787729119204at_rat(aTP_Lamm_dy(F),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) = aa_rat_rat(minus_minus_rat(aa_nat_rat(F,aa_nat_nat(suc,Nb))),aa_nat_rat(F,Mb)) ) ) ).

% sum_Suc_diff
tff(fact_7207_sum__Suc__diff,axiom,
    ! [Mb: nat,Nb: nat,F: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb))
     => ( groups3539618377306564664at_int(aTP_Lamm_dz(F),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,aa_nat_nat(suc,Nb))),aa_nat_int(F,Mb)) ) ) ).

% sum_Suc_diff
tff(fact_7208_sum__Suc__diff,axiom,
    ! [Mb: nat,Nb: nat,F: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb))
     => ( aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_ea(F)),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) = aa_real_real(minus_minus_real(aa_nat_real(F,aa_nat_nat(suc,Nb))),aa_nat_real(F,Mb)) ) ) ).

% sum_Suc_diff
tff(fact_7209_atLeastAtMostPlus1__int__conv,axiom,
    ! [Mb: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Mb),aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),Nb))
     => ( set_or1266510415728281911st_int(Mb,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),Nb)) = insert_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),Nb),set_or1266510415728281911st_int(Mb,Nb)) ) ) ).

% atLeastAtMostPlus1_int_conv
tff(fact_7210_simp__from__to,axiom,
    ! [I3: int,J: int] :
      set_or1266510415728281911st_int(I3,J) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,J),I3),bot_bot_set_int,insert_int(I3,set_or1266510415728281911st_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,I3),one_one_int),J))) ).

% simp_from_to
tff(fact_7211_sum_Oub__add__nat,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_rat,P2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))
     => ( groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2))) = aa_rat_rat(plus_plus_rat(groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))),groups2906978787729119204at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2)))) ) ) ).

% sum.ub_add_nat
tff(fact_7212_sum_Oub__add__nat,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_int,P2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))
     => ( groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))),groups3539618377306564664at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2)))) ) ) ).

% sum.ub_add_nat
tff(fact_7213_sum_Oub__add__nat,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_nat,P2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2))) = 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(Mb),Nb))),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,Nb),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2)))) ) ) ).

% sum.ub_add_nat
tff(fact_7214_sum_Oub__add__nat,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_real,P2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))
     => ( aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2))) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))),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,Nb),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2)))) ) ) ).

% sum.ub_add_nat
tff(fact_7215_sum__natinterval__diff,axiom,
    ! [F: fun_nat_complex,Mb: nat,Nb: nat] :
      aa_set_nat_complex(groups2073611262835488442omplex(aTP_Lamm_eb(F)),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb),aa_complex_complex(minus_minus_complex(aa_nat_complex(F,Mb)),aa_nat_complex(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))),zero_zero_complex) ).

% sum_natinterval_diff
tff(fact_7216_sum__natinterval__diff,axiom,
    ! [F: fun_nat_rat,Mb: nat,Nb: nat] :
      groups2906978787729119204at_rat(aTP_Lamm_ec(F),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb),aa_rat_rat(minus_minus_rat(aa_nat_rat(F,Mb)),aa_nat_rat(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))),zero_zero_rat) ).

% sum_natinterval_diff
tff(fact_7217_sum__natinterval__diff,axiom,
    ! [F: fun_nat_int,Mb: nat,Nb: nat] :
      groups3539618377306564664at_int(aTP_Lamm_ed(F),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,Mb)),aa_nat_int(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))),zero_zero_int) ).

% sum_natinterval_diff
tff(fact_7218_sum__natinterval__diff,axiom,
    ! [F: fun_nat_real,Mb: nat,Nb: nat] :
      aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_ee(F)),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb),aa_real_real(minus_minus_real(aa_nat_real(F,Mb)),aa_nat_real(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))),zero_zero_real) ).

% sum_natinterval_diff
tff(fact_7219_sum__telescope_H_H,axiom,
    ! [Mb: nat,Nb: nat,F: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( groups2906978787729119204at_rat(aTP_Lamm_ef(F),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb)),Nb)) = aa_rat_rat(minus_minus_rat(aa_nat_rat(F,Nb)),aa_nat_rat(F,Mb)) ) ) ).

% sum_telescope''
tff(fact_7220_sum__telescope_H_H,axiom,
    ! [Mb: nat,Nb: nat,F: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( groups3539618377306564664at_int(aTP_Lamm_eg(F),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb)),Nb)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,Nb)),aa_nat_int(F,Mb)) ) ) ).

% sum_telescope''
tff(fact_7221_sum__telescope_H_H,axiom,
    ! [Mb: nat,Nb: nat,F: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_eh(F)),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb)),Nb)) = aa_real_real(minus_minus_real(aa_nat_real(F,Nb)),aa_nat_real(F,Mb)) ) ) ).

% sum_telescope''
tff(fact_7222_sum_Oneutral,axiom,
    ! [A3: set_int,G: fun_int_int] :
      ( ! [X3: int] :
          ( member_int(X3,A3)
         => ( aa_int_int(G,X3) = zero_zero_int ) )
     => ( groups4538972089207619220nt_int(G,A3) = zero_zero_int ) ) ).

% sum.neutral
tff(fact_7223_sum_Oneutral,axiom,
    ! [A3: set_complex,G: fun_complex_complex] :
      ( ! [X3: complex] :
          ( member_complex(X3,A3)
         => ( aa_complex_complex(G,X3) = zero_zero_complex ) )
     => ( groups7754918857620584856omplex(G,A3) = zero_zero_complex ) ) ).

% sum.neutral
tff(fact_7224_sum_Oneutral,axiom,
    ! [A3: set_nat,G: fun_nat_nat] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A3)
         => ( aa_nat_nat(G,X3) = zero_zero_nat ) )
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),A3) = zero_zero_nat ) ) ).

% sum.neutral
tff(fact_7225_sum_Oneutral,axiom,
    ! [A3: set_nat,G: fun_nat_real] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A3)
         => ( aa_nat_real(G,X3) = zero_zero_real ) )
     => ( aa_set_nat_real(groups6591440286371151544t_real(G),A3) = zero_zero_real ) ) ).

% sum.neutral
tff(fact_7226_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_real_complex,A3: set_real] :
      ( ( groups5754745047067104278omplex(G,A3) != zero_zero_complex )
     => ~ ! [A4: real] :
            ( member_real(A4,A3)
           => ( aa_real_complex(G,A4) = zero_zero_complex ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_7227_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_nat_complex,A3: set_nat] :
      ( ( aa_set_nat_complex(groups2073611262835488442omplex(G),A3) != zero_zero_complex )
     => ~ ! [A4: nat] :
            ( aa_set_nat_o(member_nat(A4),A3)
           => ( aa_nat_complex(G,A4) = zero_zero_complex ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_7228_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_int_complex,A3: set_int] :
      ( ( groups3049146728041665814omplex(G,A3) != zero_zero_complex )
     => ~ ! [A4: int] :
            ( member_int(A4,A3)
           => ( aa_int_complex(G,A4) = zero_zero_complex ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_7229_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_complex_real,A3: set_complex] :
      ( ( groups5808333547571424918x_real(G,A3) != zero_zero_real )
     => ~ ! [A4: complex] :
            ( member_complex(A4,A3)
           => ( aa_complex_real(G,A4) = zero_zero_real ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_7230_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_real_real,A3: set_real] :
      ( ( groups8097168146408367636l_real(G,A3) != zero_zero_real )
     => ~ ! [A4: real] :
            ( member_real(A4,A3)
           => ( aa_real_real(G,A4) = zero_zero_real ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_7231_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_int_real,A3: set_int] :
      ( ( groups8778361861064173332t_real(G,A3) != zero_zero_real )
     => ~ ! [A4: int] :
            ( member_int(A4,A3)
           => ( aa_int_real(G,A4) = zero_zero_real ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_7232_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_complex_rat,A3: set_complex] :
      ( ( groups5058264527183730370ex_rat(G,A3) != zero_zero_rat )
     => ~ ! [A4: complex] :
            ( member_complex(A4,A3)
           => ( aa_complex_rat(G,A4) = zero_zero_rat ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_7233_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_real_rat,A3: set_real] :
      ( ( groups1300246762558778688al_rat(G,A3) != zero_zero_rat )
     => ~ ! [A4: real] :
            ( member_real(A4,A3)
           => ( aa_real_rat(G,A4) = zero_zero_rat ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_7234_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_nat_rat,A3: set_nat] :
      ( ( groups2906978787729119204at_rat(G,A3) != zero_zero_rat )
     => ~ ! [A4: nat] :
            ( aa_set_nat_o(member_nat(A4),A3)
           => ( aa_nat_rat(G,A4) = zero_zero_rat ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_7235_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_int_rat,A3: set_int] :
      ( ( groups3906332499630173760nt_rat(G,A3) != zero_zero_rat )
     => ~ ! [A4: int] :
            ( member_int(A4,A3)
           => ( aa_int_rat(G,A4) = zero_zero_rat ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_7236_mask__eq__sum__exp,axiom,
    ! [Nb: 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))),Nb)),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_ei,Nb))) ).

% mask_eq_sum_exp
tff(fact_7237_mask__eq__sum__exp,axiom,
    ! [Nb: 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))),Nb)),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_ei,Nb))) ).

% mask_eq_sum_exp
tff(fact_7238_sum__gp__multiplied,axiom,
    ! [Mb: nat,Nb: nat,Xb: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(one_one_complex),Xb)),aa_set_nat_complex(groups2073611262835488442omplex(power_power_complex(Xb)),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))) = aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(Xb),Mb)),aa_nat_complex(power_power_complex(Xb),aa_nat_nat(suc,Nb))) ) ) ).

% sum_gp_multiplied
tff(fact_7239_sum__gp__multiplied,axiom,
    ! [Mb: nat,Nb: nat,Xb: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(one_one_rat),Xb)),groups2906978787729119204at_rat(power_power_rat(Xb),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))) = aa_rat_rat(minus_minus_rat(aa_nat_rat(power_power_rat(Xb),Mb)),aa_nat_rat(power_power_rat(Xb),aa_nat_nat(suc,Nb))) ) ) ).

% sum_gp_multiplied
tff(fact_7240_sum__gp__multiplied,axiom,
    ! [Mb: nat,Nb: nat,Xb: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( 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),Xb)),groups3539618377306564664at_int(power_power_int(Xb),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(Xb),Mb)),aa_nat_int(power_power_int(Xb),aa_nat_nat(suc,Nb))) ) ) ).

% sum_gp_multiplied
tff(fact_7241_sum__gp__multiplied,axiom,
    ! [Mb: nat,Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_real_real(times_times_real(aa_real_real(minus_minus_real(one_one_real),Xb)),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,Xb)),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))) = aa_real_real(minus_minus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Mb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_nat_nat(suc,Nb))) ) ) ).

% sum_gp_multiplied
tff(fact_7242_sum_Oin__pairs,axiom,
    ! [G: fun_nat_rat,Mb: nat,Nb: 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))),Mb)),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))),Nb)))) = groups2906978787729119204at_rat(aTP_Lamm_ej(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) ).

% sum.in_pairs
tff(fact_7243_sum_Oin__pairs,axiom,
    ! [G: fun_nat_int,Mb: nat,Nb: 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))),Mb)),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))),Nb)))) = groups3539618377306564664at_int(aTP_Lamm_ek(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) ).

% sum.in_pairs
tff(fact_7244_sum_Oin__pairs,axiom,
    ! [G: fun_nat_nat,Mb: nat,Nb: 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))),Mb)),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))),Nb)))) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_el(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) ).

% sum.in_pairs
tff(fact_7245_sum_Oin__pairs,axiom,
    ! [G: fun_nat_real,Mb: nat,Nb: 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))),Mb)),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))),Nb)))) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_em(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) ).

% sum.in_pairs
tff(fact_7246_and__nat__unfold,axiom,
    ! [Mb: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,Mb),Nb) = $ite(
        ( ( Mb = zero_zero_nat )
        | ( Nb = 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,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),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,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ).

% and_nat_unfold
tff(fact_7247_and__nat__rec,axiom,
    ! [Mb: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,Mb),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_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))),Mb)
            & ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) ))),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,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% and_nat_rec
tff(fact_7248_mask__eq__sum__exp__nat,axiom,
    ! [Nb: 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))),Nb)),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_ei,Nb))) ).

% mask_eq_sum_exp_nat
tff(fact_7249_gauss__sum__nat,axiom,
    ! [Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_ar),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(suc,Nb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% gauss_sum_nat
tff(fact_7250_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_7251_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_7252_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_7253_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_7254_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_7255_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_7256_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_7257_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_7258_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_7259_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_7260_sum__distrib__left,axiom,
    ! [R2: int,F: fun_int_int,A3: set_int] : aa_int_int(aa_int_fun_int_int(times_times_int,R2),groups4538972089207619220nt_int(F,A3)) = groups4538972089207619220nt_int(aa_fun6280345759651049192nt_int(aTP_Lamm_en(R2),F),A3) ).

% sum_distrib_left
tff(fact_7261_sum__distrib__left,axiom,
    ! [R2: complex,F: fun_complex_complex,A3: set_complex] : aa_complex_complex(times_times_complex(R2),groups7754918857620584856omplex(F,A3)) = groups7754918857620584856omplex(aa_fun3187328805393345264omplex(aTP_Lamm_eo(R2),F),A3) ).

% sum_distrib_left
tff(fact_7262_sum__distrib__left,axiom,
    ! [R2: nat,F: fun_nat_nat,A3: set_nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R2),aa_set_nat_nat(groups3542108847815614940at_nat(F),A3)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_ep(R2),F)),A3) ).

% sum_distrib_left
tff(fact_7263_sum__distrib__left,axiom,
    ! [R2: real,F: fun_nat_real,A3: set_nat] : aa_real_real(times_times_real(R2),aa_set_nat_real(groups6591440286371151544t_real(F),A3)) = aa_set_nat_real(groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_eq(R2),F)),A3) ).

% sum_distrib_left
tff(fact_7264_sum__distrib__right,axiom,
    ! [F: fun_int_int,A3: set_int,R2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,groups4538972089207619220nt_int(F,A3)),R2) = groups4538972089207619220nt_int(aa_int_fun_int_int(aTP_Lamm_er(F),R2),A3) ).

% sum_distrib_right
tff(fact_7265_sum__distrib__right,axiom,
    ! [F: fun_complex_complex,A3: set_complex,R2: complex] : aa_complex_complex(times_times_complex(groups7754918857620584856omplex(F,A3)),R2) = groups7754918857620584856omplex(aa_com3342990543769683525omplex(aTP_Lamm_es(F),R2),A3) ).

% sum_distrib_right
tff(fact_7266_sum__distrib__right,axiom,
    ! [F: fun_nat_nat,A3: set_nat,R2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A3)),R2) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_et(F),R2)),A3) ).

% sum_distrib_right
tff(fact_7267_sum__distrib__right,axiom,
    ! [F: fun_nat_real,A3: set_nat,R2: real] : aa_real_real(times_times_real(aa_set_nat_real(groups6591440286371151544t_real(F),A3)),R2) = aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_eu(F),R2)),A3) ).

% sum_distrib_right
tff(fact_7268_sum__product,axiom,
    ! [F: fun_int_int,A3: set_int,G: fun_int_int,B4: set_int] : aa_int_int(aa_int_fun_int_int(times_times_int,groups4538972089207619220nt_int(F,A3)),groups4538972089207619220nt_int(G,B4)) = groups4538972089207619220nt_int(aa_set7310040872927613301nt_int(aa_fun8076508010538293575nt_int(aTP_Lamm_ew(F),G),B4),A3) ).

% sum_product
tff(fact_7269_sum__product,axiom,
    ! [F: fun_complex_complex,A3: set_complex,G: fun_complex_complex,B4: set_complex] : aa_complex_complex(times_times_complex(groups7754918857620584856omplex(F,A3)),groups7754918857620584856omplex(G,B4)) = groups7754918857620584856omplex(aa_set7743095458595710203omplex(aa_fun1583801116089702097omplex(aTP_Lamm_ey(F),G),B4),A3) ).

% sum_product
tff(fact_7270_sum__product,axiom,
    ! [F: fun_nat_nat,A3: set_nat,G: fun_nat_nat,B4: set_nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A3)),aa_set_nat_nat(groups3542108847815614940at_nat(G),B4)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_set6537093552120046177at_nat(aa_fun4561837934398617467at_nat(aTP_Lamm_fa(F),G),B4)),A3) ).

% sum_product
tff(fact_7271_sum__product,axiom,
    ! [F: fun_nat_real,A3: set_nat,G: fun_nat_real,B4: set_nat] : aa_real_real(times_times_real(aa_set_nat_real(groups6591440286371151544t_real(F),A3)),aa_set_nat_real(groups6591440286371151544t_real(G),B4)) = aa_set_nat_real(groups6591440286371151544t_real(aa_set1166323925324055613t_real(aa_fun1400677770452585267t_real(aTP_Lamm_fc(F),G),B4)),A3) ).

% sum_product
tff(fact_7272_sum_Odistrib,axiom,
    ! [G: fun_int_int,H: fun_int_int,A3: set_int] : groups4538972089207619220nt_int(aa_fun6280345759651049192nt_int(aTP_Lamm_fd(G),H),A3) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups4538972089207619220nt_int(G,A3)),groups4538972089207619220nt_int(H,A3)) ).

% sum.distrib
tff(fact_7273_sum_Odistrib,axiom,
    ! [G: fun_complex_complex,H: fun_complex_complex,A3: set_complex] : groups7754918857620584856omplex(aa_fun3187328805393345264omplex(aTP_Lamm_fe(G),H),A3) = aa_complex_complex(plus_plus_complex(groups7754918857620584856omplex(G,A3)),groups7754918857620584856omplex(H,A3)) ).

% sum.distrib
tff(fact_7274_sum_Odistrib,axiom,
    ! [G: fun_nat_nat,H: fun_nat_nat,A3: set_nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_ff(G),H)),A3) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),A3)),aa_set_nat_nat(groups3542108847815614940at_nat(H),A3)) ).

% sum.distrib
tff(fact_7275_sum_Odistrib,axiom,
    ! [G: fun_nat_real,H: fun_nat_real,A3: set_nat] : aa_set_nat_real(groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_fg(G),H)),A3) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(G),A3)),aa_set_nat_real(groups6591440286371151544t_real(H),A3)) ).

% sum.distrib
tff(fact_7276_sum__divide__distrib,axiom,
    ! [F: fun_complex_complex,A3: set_complex,R2: complex] : aa_complex_complex(divide1717551699836669952omplex(groups7754918857620584856omplex(F,A3)),R2) = groups7754918857620584856omplex(aa_com3342990543769683525omplex(aTP_Lamm_fh(F),R2),A3) ).

% sum_divide_distrib
tff(fact_7277_sum__divide__distrib,axiom,
    ! [F: fun_nat_real,A3: set_nat,R2: real] : aa_real_real(divide_divide_real(aa_set_nat_real(groups6591440286371151544t_real(F),A3)),R2) = aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_fi(F),R2)),A3) ).

% sum_divide_distrib
tff(fact_7278_arith__series__nat,axiom,
    ! [A: nat,D: nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_fj(A),D)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb)) = 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,Nb)),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,Nb),D)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% arith_series_nat
tff(fact_7279_Sum__Icc__nat,axiom,
    ! [Mb: nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_ar),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) = 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,Nb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% Sum_Icc_nat
tff(fact_7280_sum__nonneg,axiom,
    ! [A3: set_complex,F: fun_complex_real] :
      ( ! [X3: complex] :
          ( member_complex(X3,A3)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(F,X3)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),groups5808333547571424918x_real(F,A3)) ) ).

% sum_nonneg
tff(fact_7281_sum__nonneg,axiom,
    ! [A3: set_real,F: fun_real_real] :
      ( ! [X3: real] :
          ( member_real(X3,A3)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,X3)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),groups8097168146408367636l_real(F,A3)) ) ).

% sum_nonneg
tff(fact_7282_sum__nonneg,axiom,
    ! [A3: set_int,F: fun_int_real] :
      ( ! [X3: int] :
          ( member_int(X3,A3)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(F,X3)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),groups8778361861064173332t_real(F,A3)) ) ).

% sum_nonneg
tff(fact_7283_sum__nonneg,axiom,
    ! [A3: set_complex,F: fun_complex_rat] :
      ( ! [X3: complex] :
          ( member_complex(X3,A3)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,X3)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),groups5058264527183730370ex_rat(F,A3)) ) ).

% sum_nonneg
tff(fact_7284_sum__nonneg,axiom,
    ! [A3: set_real,F: fun_real_rat] :
      ( ! [X3: real] :
          ( member_real(X3,A3)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,X3)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),groups1300246762558778688al_rat(F,A3)) ) ).

% sum_nonneg
tff(fact_7285_sum__nonneg,axiom,
    ! [A3: set_nat,F: fun_nat_rat] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A3)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,X3)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),groups2906978787729119204at_rat(F,A3)) ) ).

% sum_nonneg
tff(fact_7286_sum__nonneg,axiom,
    ! [A3: set_int,F: fun_int_rat] :
      ( ! [X3: int] :
          ( member_int(X3,A3)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,X3)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),groups3906332499630173760nt_rat(F,A3)) ) ).

% sum_nonneg
tff(fact_7287_sum__nonneg,axiom,
    ! [A3: set_complex,F: fun_complex_nat] :
      ( ! [X3: complex] :
          ( member_complex(X3,A3)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_complex_nat(F,X3)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),groups5693394587270226106ex_nat(F,A3)) ) ).

% sum_nonneg
tff(fact_7288_sum__nonneg,axiom,
    ! [A3: set_real,F: fun_real_nat] :
      ( ! [X3: real] :
          ( member_real(X3,A3)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,X3)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),groups1935376822645274424al_nat(F,A3)) ) ).

% sum_nonneg
tff(fact_7289_sum__nonneg,axiom,
    ! [A3: set_int,F: fun_int_nat] :
      ( ! [X3: int] :
          ( member_int(X3,A3)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F,X3)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),groups4541462559716669496nt_nat(F,A3)) ) ).

% sum_nonneg
tff(fact_7290_sum__nonpos,axiom,
    ! [A3: set_complex,F: fun_complex_real] :
      ( ! [X3: complex] :
          ( member_complex(X3,A3)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(F,X3)),zero_zero_real) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups5808333547571424918x_real(F,A3)),zero_zero_real) ) ).

% sum_nonpos
tff(fact_7291_sum__nonpos,axiom,
    ! [A3: set_real,F: fun_real_real] :
      ( ! [X3: real] :
          ( member_real(X3,A3)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,X3)),zero_zero_real) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups8097168146408367636l_real(F,A3)),zero_zero_real) ) ).

% sum_nonpos
tff(fact_7292_sum__nonpos,axiom,
    ! [A3: set_int,F: fun_int_real] :
      ( ! [X3: int] :
          ( member_int(X3,A3)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(F,X3)),zero_zero_real) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups8778361861064173332t_real(F,A3)),zero_zero_real) ) ).

% sum_nonpos
tff(fact_7293_sum__nonpos,axiom,
    ! [A3: set_complex,F: fun_complex_rat] :
      ( ! [X3: complex] :
          ( member_complex(X3,A3)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_complex_rat(F,X3)),zero_zero_rat) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups5058264527183730370ex_rat(F,A3)),zero_zero_rat) ) ).

% sum_nonpos
tff(fact_7294_sum__nonpos,axiom,
    ! [A3: set_real,F: fun_real_rat] :
      ( ! [X3: real] :
          ( member_real(X3,A3)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_real_rat(F,X3)),zero_zero_rat) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups1300246762558778688al_rat(F,A3)),zero_zero_rat) ) ).

% sum_nonpos
tff(fact_7295_sum__nonpos,axiom,
    ! [A3: set_nat,F: fun_nat_rat] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A3)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X3)),zero_zero_rat) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups2906978787729119204at_rat(F,A3)),zero_zero_rat) ) ).

% sum_nonpos
tff(fact_7296_sum__nonpos,axiom,
    ! [A3: set_int,F: fun_int_rat] :
      ( ! [X3: int] :
          ( member_int(X3,A3)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,X3)),zero_zero_rat) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups3906332499630173760nt_rat(F,A3)),zero_zero_rat) ) ).

% sum_nonpos
tff(fact_7297_sum__nonpos,axiom,
    ! [A3: set_complex,F: fun_complex_nat] :
      ( ! [X3: complex] :
          ( member_complex(X3,A3)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_complex_nat(F,X3)),zero_zero_nat) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups5693394587270226106ex_nat(F,A3)),zero_zero_nat) ) ).

% sum_nonpos
tff(fact_7298_sum__nonpos,axiom,
    ! [A3: set_real,F: fun_real_nat] :
      ( ! [X3: real] :
          ( member_real(X3,A3)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_real_nat(F,X3)),zero_zero_nat) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups1935376822645274424al_nat(F,A3)),zero_zero_nat) ) ).

% sum_nonpos
tff(fact_7299_sum__nonpos,axiom,
    ! [A3: set_int,F: fun_int_nat] :
      ( ! [X3: int] :
          ( member_int(X3,A3)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,X3)),zero_zero_nat) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups4541462559716669496nt_nat(F,A3)),zero_zero_nat) ) ).

% sum_nonpos
tff(fact_7300_and__int_Osimps,axiom,
    ! [K: int,L: int] :
      aa_int_int(aa_int_fun_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(aa_int_fun_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_7301_and__int_Opsimps,axiom,
    ! [K: int,L: int] :
      ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(bit_and_int_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,K),L))
     => ( aa_int_int(aa_int_fun_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(aa_int_fun_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_7302_and__int_Opelims,axiom,
    ! [Xb: int,Xaa: int,Yb: int] :
      ( ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Xb),Xaa) = Yb )
     => ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(bit_and_int_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Xb),Xaa))
       => ~ ( ( Yb = $ite(
                  ( member_int(Xb,insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
                  & member_int(Xaa,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))),Xb)
                      & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xaa) ))),
                  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))),Xb)
                        & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xaa) ))),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(bit_se725231765392027082nd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,Xb),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,Xaa),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) )
           => ~ aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(bit_and_int_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Xb),Xaa)) ) ) ) ).

% and_int.pelims
tff(fact_7303_sum__gp,axiom,
    ! [Xb: complex,Mb: nat,Nb: nat] :
      aa_set_nat_complex(groups2073611262835488442omplex(power_power_complex(Xb)),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb),
        zero_zero_complex,
        $ite(Xb = 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,Nb),one_one_nat)),Mb)),aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(Xb),Mb)),aa_nat_complex(power_power_complex(Xb),aa_nat_nat(suc,Nb)))),aa_complex_complex(minus_minus_complex(one_one_complex),Xb))) ) ).

% sum_gp
tff(fact_7304_sum__gp,axiom,
    ! [Xb: rat,Mb: nat,Nb: nat] :
      groups2906978787729119204at_rat(power_power_rat(Xb),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb),
        zero_zero_rat,
        $ite(Xb = 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,Nb),one_one_nat)),Mb)),aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_nat_rat(power_power_rat(Xb),Mb)),aa_nat_rat(power_power_rat(Xb),aa_nat_nat(suc,Nb)))),aa_rat_rat(minus_minus_rat(one_one_rat),Xb))) ) ).

% sum_gp
tff(fact_7305_sum__gp,axiom,
    ! [Xb: real,Mb: nat,Nb: nat] :
      aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,Xb)),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb),
        zero_zero_real,
        $ite(Xb = 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,Nb),one_one_nat)),Mb)),aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Mb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_nat_nat(suc,Nb)))),aa_real_real(minus_minus_real(one_one_real),Xb))) ) ).

% sum_gp
tff(fact_7306_cot__less__zero,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(aa_real_real(uminus_uminus_real,pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cot_real,Xb)),zero_zero_real) ) ) ).

% cot_less_zero
tff(fact_7307_cos__npi__int,axiom,
    ! [Nb: int] :
      aa_real_real(cos_real,aa_real_real(times_times_real(pi),aa_int_real(ring_1_of_int_real,Nb))) = $ite(aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb),one_one_real,aa_real_real(uminus_uminus_real,one_one_real)) ).

% cos_npi_int
tff(fact_7308_of__nat__eq__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,Mb) = aa_nat_int(semiri1314217659103216013at_int,Nb) )
    <=> ( Mb = Nb ) ) ).

% of_nat_eq_iff
tff(fact_7309_of__nat__eq__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,Mb) = aa_nat_real(semiri5074537144036343181t_real,Nb) )
    <=> ( Mb = Nb ) ) ).

% of_nat_eq_iff
tff(fact_7310_of__nat__eq__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,Mb) = aa_nat_nat(semiri1316708129612266289at_nat,Nb) )
    <=> ( Mb = Nb ) ) ).

% of_nat_eq_iff
tff(fact_7311_of__nat__eq__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_rat(semiri681578069525770553at_rat,Mb) = aa_nat_rat(semiri681578069525770553at_rat,Nb) )
    <=> ( Mb = Nb ) ) ).

% of_nat_eq_iff
tff(fact_7312_int__eq__iff__numeral,axiom,
    ! [Mb: nat,V: num] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,Mb) = aa_num_int(numeral_numeral_int,V) )
    <=> ( Mb = aa_num_nat(numeral_numeral_nat,V) ) ) ).

% int_eq_iff_numeral
tff(fact_7313_abs__of__nat,axiom,
    ! [Nb: nat] : abs_abs_Code_integer(aa_nat_Code_integer(semiri4939895301339042750nteger,Nb)) = aa_nat_Code_integer(semiri4939895301339042750nteger,Nb) ).

% abs_of_nat
tff(fact_7314_abs__of__nat,axiom,
    ! [Nb: nat] : aa_int_int(abs_abs_int,aa_nat_int(semiri1314217659103216013at_int,Nb)) = aa_nat_int(semiri1314217659103216013at_int,Nb) ).

% abs_of_nat
tff(fact_7315_abs__of__nat,axiom,
    ! [Nb: nat] : aa_real_real(abs_abs_real,aa_nat_real(semiri5074537144036343181t_real,Nb)) = aa_nat_real(semiri5074537144036343181t_real,Nb) ).

% abs_of_nat
tff(fact_7316_abs__of__nat,axiom,
    ! [Nb: nat] : aa_rat_rat(abs_abs_rat,aa_nat_rat(semiri681578069525770553at_rat,Nb)) = aa_nat_rat(semiri681578069525770553at_rat,Nb) ).

% abs_of_nat
tff(fact_7317_negative__eq__positive,axiom,
    ! [Nb: nat,Mb: nat] :
      ( ( aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,Nb)) = aa_nat_int(semiri1314217659103216013at_int,Mb) )
    <=> ( ( Nb = zero_zero_nat )
        & ( Mb = zero_zero_nat ) ) ) ).

% negative_eq_positive
tff(fact_7318_of__int__of__nat__eq,axiom,
    ! [Nb: nat] : aa_int_int(ring_1_of_int_int,aa_nat_int(semiri1314217659103216013at_int,Nb)) = aa_nat_int(semiri1314217659103216013at_int,Nb) ).

% of_int_of_nat_eq
tff(fact_7319_of__int__of__nat__eq,axiom,
    ! [Nb: nat] : aa_int_real(ring_1_of_int_real,aa_nat_int(semiri1314217659103216013at_int,Nb)) = aa_nat_real(semiri5074537144036343181t_real,Nb) ).

% of_int_of_nat_eq
tff(fact_7320_of__int__of__nat__eq,axiom,
    ! [Nb: nat] : aa_int_rat(ring_1_of_int_rat,aa_nat_int(semiri1314217659103216013at_int,Nb)) = aa_nat_rat(semiri681578069525770553at_rat,Nb) ).

% of_int_of_nat_eq
tff(fact_7321_negative__zle,axiom,
    ! [Nb: nat,Mb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,Nb))),aa_nat_int(semiri1314217659103216013at_int,Mb)) ).

% negative_zle
tff(fact_7322_int__dvd__int__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),Nb) ) ).

% int_dvd_int_iff
tff(fact_7323_cot__zero,axiom,
    aa_complex_complex(cot_complex,zero_zero_complex) = zero_zero_complex ).

% cot_zero
tff(fact_7324_cot__zero,axiom,
    aa_real_real(cot_real,zero_zero_real) = zero_zero_real ).

% cot_zero
tff(fact_7325_of__nat__eq__0__iff,axiom,
    ! [Mb: nat] :
      ( ( aa_nat_complex(semiri8010041392384452111omplex,Mb) = zero_zero_complex )
    <=> ( Mb = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_7326_of__nat__eq__0__iff,axiom,
    ! [Mb: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,Mb) = zero_zero_int )
    <=> ( Mb = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_7327_of__nat__eq__0__iff,axiom,
    ! [Mb: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,Mb) = zero_zero_real )
    <=> ( Mb = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_7328_of__nat__eq__0__iff,axiom,
    ! [Mb: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,Mb) = zero_zero_nat )
    <=> ( Mb = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_7329_of__nat__eq__0__iff,axiom,
    ! [Mb: nat] :
      ( ( aa_nat_rat(semiri681578069525770553at_rat,Mb) = zero_zero_rat )
    <=> ( Mb = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_7330_of__nat__0__eq__iff,axiom,
    ! [Nb: nat] :
      ( ( zero_zero_complex = aa_nat_complex(semiri8010041392384452111omplex,Nb) )
    <=> ( zero_zero_nat = Nb ) ) ).

% of_nat_0_eq_iff
tff(fact_7331_of__nat__0__eq__iff,axiom,
    ! [Nb: nat] :
      ( ( zero_zero_int = aa_nat_int(semiri1314217659103216013at_int,Nb) )
    <=> ( zero_zero_nat = Nb ) ) ).

% of_nat_0_eq_iff
tff(fact_7332_of__nat__0__eq__iff,axiom,
    ! [Nb: nat] :
      ( ( zero_zero_real = aa_nat_real(semiri5074537144036343181t_real,Nb) )
    <=> ( zero_zero_nat = Nb ) ) ).

% of_nat_0_eq_iff
tff(fact_7333_of__nat__0__eq__iff,axiom,
    ! [Nb: nat] :
      ( ( zero_zero_nat = aa_nat_nat(semiri1316708129612266289at_nat,Nb) )
    <=> ( zero_zero_nat = Nb ) ) ).

% of_nat_0_eq_iff
tff(fact_7334_of__nat__0__eq__iff,axiom,
    ! [Nb: nat] :
      ( ( zero_zero_rat = aa_nat_rat(semiri681578069525770553at_rat,Nb) )
    <=> ( zero_zero_nat = Nb ) ) ).

% of_nat_0_eq_iff
tff(fact_7335_of__nat__0,axiom,
    aa_nat_complex(semiri8010041392384452111omplex,zero_zero_nat) = zero_zero_complex ).

% of_nat_0
tff(fact_7336_of__nat__0,axiom,
    aa_nat_int(semiri1314217659103216013at_int,zero_zero_nat) = zero_zero_int ).

% of_nat_0
tff(fact_7337_of__nat__0,axiom,
    aa_nat_real(semiri5074537144036343181t_real,zero_zero_nat) = zero_zero_real ).

% of_nat_0
tff(fact_7338_of__nat__0,axiom,
    aa_nat_nat(semiri1316708129612266289at_nat,zero_zero_nat) = zero_zero_nat ).

% of_nat_0
tff(fact_7339_of__nat__0,axiom,
    aa_nat_rat(semiri681578069525770553at_rat,zero_zero_nat) = zero_zero_rat ).

% of_nat_0
tff(fact_7340_of__nat__less__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% of_nat_less_iff
tff(fact_7341_of__nat__less__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_nat_real(semiri5074537144036343181t_real,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% of_nat_less_iff
tff(fact_7342_of__nat__less__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),aa_nat_nat(semiri1316708129612266289at_nat,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% of_nat_less_iff
tff(fact_7343_of__nat__less__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,Mb)),aa_nat_rat(semiri681578069525770553at_rat,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% of_nat_less_iff
tff(fact_7344_of__nat__le__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_nat_real(semiri5074537144036343181t_real,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% of_nat_le_iff
tff(fact_7345_of__nat__le__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,Mb)),aa_nat_rat(semiri681578069525770553at_rat,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% of_nat_le_iff
tff(fact_7346_of__nat__le__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),aa_nat_nat(semiri1316708129612266289at_nat,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% of_nat_le_iff
tff(fact_7347_of__nat__le__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% of_nat_le_iff
tff(fact_7348_of__nat__numeral,axiom,
    ! [Nb: num] : aa_nat_complex(semiri8010041392384452111omplex,aa_num_nat(numeral_numeral_nat,Nb)) = numera6690914467698888265omplex(Nb) ).

% of_nat_numeral
tff(fact_7349_of__nat__numeral,axiom,
    ! [Nb: num] : aa_nat_int(semiri1314217659103216013at_int,aa_num_nat(numeral_numeral_nat,Nb)) = aa_num_int(numeral_numeral_int,Nb) ).

% of_nat_numeral
tff(fact_7350_of__nat__numeral,axiom,
    ! [Nb: num] : aa_nat_real(semiri5074537144036343181t_real,aa_num_nat(numeral_numeral_nat,Nb)) = aa_num_real(numeral_numeral_real,Nb) ).

% of_nat_numeral
tff(fact_7351_of__nat__numeral,axiom,
    ! [Nb: num] : aa_nat_nat(semiri1316708129612266289at_nat,aa_num_nat(numeral_numeral_nat,Nb)) = aa_num_nat(numeral_numeral_nat,Nb) ).

% of_nat_numeral
tff(fact_7352_of__nat__numeral,axiom,
    ! [Nb: num] : aa_nat_rat(semiri681578069525770553at_rat,aa_num_nat(numeral_numeral_nat,Nb)) = aa_num_rat(numeral_numeral_rat,Nb) ).

% of_nat_numeral
tff(fact_7353_cos__zero,axiom,
    aa_complex_complex(cos_complex,zero_zero_complex) = one_one_complex ).

% cos_zero
tff(fact_7354_cos__zero,axiom,
    aa_real_real(cos_real,zero_zero_real) = one_one_real ).

% cos_zero
tff(fact_7355_of__nat__add,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb)) ).

% of_nat_add
tff(fact_7356_of__nat__add,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) = aa_real_real(plus_plus_real(aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_nat_real(semiri5074537144036343181t_real,Nb)) ).

% of_nat_add
tff(fact_7357_of__nat__add,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),aa_nat_nat(semiri1316708129612266289at_nat,Nb)) ).

% of_nat_add
tff(fact_7358_of__nat__add,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) = aa_rat_rat(plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,Mb)),aa_nat_rat(semiri681578069525770553at_rat,Nb)) ).

% of_nat_add
tff(fact_7359_of__nat__mult,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb)) ).

% of_nat_mult
tff(fact_7360_of__nat__mult,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)) = aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_nat_real(semiri5074537144036343181t_real,Nb)) ).

% of_nat_mult
tff(fact_7361_of__nat__mult,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),aa_nat_nat(semiri1316708129612266289at_nat,Nb)) ).

% of_nat_mult
tff(fact_7362_of__nat__mult,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)) = aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,Mb)),aa_nat_rat(semiri681578069525770553at_rat,Nb)) ).

% of_nat_mult
tff(fact_7363_of__nat__eq__1__iff,axiom,
    ! [Nb: nat] :
      ( ( aa_nat_complex(semiri8010041392384452111omplex,Nb) = one_one_complex )
    <=> ( Nb = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_7364_of__nat__eq__1__iff,axiom,
    ! [Nb: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,Nb) = one_one_int )
    <=> ( Nb = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_7365_of__nat__eq__1__iff,axiom,
    ! [Nb: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,Nb) = one_one_real )
    <=> ( Nb = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_7366_of__nat__eq__1__iff,axiom,
    ! [Nb: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,Nb) = one_one_nat )
    <=> ( Nb = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_7367_of__nat__eq__1__iff,axiom,
    ! [Nb: nat] :
      ( ( aa_nat_rat(semiri681578069525770553at_rat,Nb) = one_one_rat )
    <=> ( Nb = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_7368_of__nat__1__eq__iff,axiom,
    ! [Nb: nat] :
      ( ( one_one_complex = aa_nat_complex(semiri8010041392384452111omplex,Nb) )
    <=> ( Nb = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_7369_of__nat__1__eq__iff,axiom,
    ! [Nb: nat] :
      ( ( one_one_int = aa_nat_int(semiri1314217659103216013at_int,Nb) )
    <=> ( Nb = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_7370_of__nat__1__eq__iff,axiom,
    ! [Nb: nat] :
      ( ( one_one_real = aa_nat_real(semiri5074537144036343181t_real,Nb) )
    <=> ( Nb = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_7371_of__nat__1__eq__iff,axiom,
    ! [Nb: nat] :
      ( ( one_one_nat = aa_nat_nat(semiri1316708129612266289at_nat,Nb) )
    <=> ( Nb = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_7372_of__nat__1__eq__iff,axiom,
    ! [Nb: nat] :
      ( ( one_one_rat = aa_nat_rat(semiri681578069525770553at_rat,Nb) )
    <=> ( Nb = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_7373_of__nat__1,axiom,
    aa_nat_complex(semiri8010041392384452111omplex,one_one_nat) = one_one_complex ).

% of_nat_1
tff(fact_7374_of__nat__1,axiom,
    aa_nat_int(semiri1314217659103216013at_int,one_one_nat) = one_one_int ).

% of_nat_1
tff(fact_7375_of__nat__1,axiom,
    aa_nat_real(semiri5074537144036343181t_real,one_one_nat) = one_one_real ).

% of_nat_1
tff(fact_7376_of__nat__1,axiom,
    aa_nat_nat(semiri1316708129612266289at_nat,one_one_nat) = one_one_nat ).

% of_nat_1
tff(fact_7377_of__nat__1,axiom,
    aa_nat_rat(semiri681578069525770553at_rat,one_one_nat) = one_one_rat ).

% of_nat_1
tff(fact_7378_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [Xb: nat,B: nat,W: nat] :
      ( ( aa_nat_complex(semiri8010041392384452111omplex,Xb) = aa_nat_complex(power_power_complex(aa_nat_complex(semiri8010041392384452111omplex,B)),W) )
    <=> ( Xb = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
tff(fact_7379_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [Xb: nat,B: nat,W: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,Xb) = aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B)),W) )
    <=> ( Xb = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
tff(fact_7380_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [Xb: nat,B: nat,W: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,Xb) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,B)),W) )
    <=> ( Xb = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
tff(fact_7381_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [Xb: nat,B: nat,W: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,Xb) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B)),W) )
    <=> ( Xb = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
tff(fact_7382_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [Xb: nat,B: nat,W: nat] :
      ( ( aa_nat_rat(semiri681578069525770553at_rat,Xb) = aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B)),W) )
    <=> ( Xb = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
tff(fact_7383_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,Xb: nat] :
      ( ( aa_nat_complex(power_power_complex(aa_nat_complex(semiri8010041392384452111omplex,B)),W) = aa_nat_complex(semiri8010041392384452111omplex,Xb) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) = Xb ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
tff(fact_7384_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,Xb: nat] :
      ( ( aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B)),W) = aa_nat_int(semiri1314217659103216013at_int,Xb) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) = Xb ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
tff(fact_7385_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,Xb: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,B)),W) = aa_nat_real(semiri5074537144036343181t_real,Xb) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) = Xb ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
tff(fact_7386_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,Xb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B)),W) = aa_nat_nat(semiri1316708129612266289at_nat,Xb) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) = Xb ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
tff(fact_7387_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,Xb: nat] :
      ( ( aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B)),W) = aa_nat_rat(semiri681578069525770553at_rat,Xb) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W) = Xb ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
tff(fact_7388_of__nat__power,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Mb),Nb)) = aa_nat_complex(power_power_complex(aa_nat_complex(semiri8010041392384452111omplex,Mb)),Nb) ).

% of_nat_power
tff(fact_7389_of__nat__power,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Mb),Nb)) = aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,Mb)),Nb) ).

% of_nat_power
tff(fact_7390_of__nat__power,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Mb),Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),Nb) ).

% of_nat_power
tff(fact_7391_of__nat__power,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),Nb) ).

% of_nat_power
tff(fact_7392_of__nat__power,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Mb),Nb)) = aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,Mb)),Nb) ).

% of_nat_power
tff(fact_7393_negative__zless,axiom,
    ! [Nb: nat,Mb: 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,Nb)))),aa_nat_int(semiri1314217659103216013at_int,Mb)) ).

% negative_zless
tff(fact_7394_of__nat__of__bool,axiom,
    ! [P: $o] : aa_nat_real(semiri5074537144036343181t_real,zero_n2687167440665602831ol_nat((P))) = zero_n3304061248610475627l_real((P)) ).

% of_nat_of_bool
tff(fact_7395_of__nat__of__bool,axiom,
    ! [P: $o] : aa_nat_rat(semiri681578069525770553at_rat,zero_n2687167440665602831ol_nat((P))) = zero_n2052037380579107095ol_rat((P)) ).

% of_nat_of_bool
tff(fact_7396_of__nat__of__bool,axiom,
    ! [P: $o] : aa_nat_nat(semiri1316708129612266289at_nat,zero_n2687167440665602831ol_nat((P))) = zero_n2687167440665602831ol_nat((P)) ).

% of_nat_of_bool
tff(fact_7397_of__nat__of__bool,axiom,
    ! [P: $o] : aa_nat_int(semiri1314217659103216013at_int,zero_n2687167440665602831ol_nat((P))) = aa_o_int(zero_n2684676970156552555ol_int,(P)) ).

% of_nat_of_bool
tff(fact_7398_of__nat__of__bool,axiom,
    ! [P: $o] : aa_nat_Code_integer(semiri4939895301339042750nteger,zero_n2687167440665602831ol_nat((P))) = zero_n356916108424825756nteger((P)) ).

% of_nat_of_bool
tff(fact_7399_of__nat__sum,axiom,
    ! [F: fun_int_nat,A3: set_int] : aa_nat_int(semiri1314217659103216013at_int,groups4541462559716669496nt_nat(F,A3)) = groups4538972089207619220nt_int(aTP_Lamm_fk(F),A3) ).

% of_nat_sum
tff(fact_7400_of__nat__sum,axiom,
    ! [F: fun_complex_nat,A3: set_complex] : aa_nat_complex(semiri8010041392384452111omplex,groups5693394587270226106ex_nat(F,A3)) = groups7754918857620584856omplex(aTP_Lamm_fl(F),A3) ).

% of_nat_sum
tff(fact_7401_of__nat__sum,axiom,
    ! [F: fun_nat_nat,A3: set_nat] : aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(groups3542108847815614940at_nat(F),A3)) = groups3539618377306564664at_int(aTP_Lamm_fm(F),A3) ).

% of_nat_sum
tff(fact_7402_of__nat__sum,axiom,
    ! [F: fun_nat_nat,A3: set_nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A3)) = groups2906978787729119204at_rat(aTP_Lamm_fn(F),A3) ).

% of_nat_sum
tff(fact_7403_of__nat__sum,axiom,
    ! [F: fun_nat_nat,A3: set_nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A3)) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_fo(F)),A3) ).

% of_nat_sum
tff(fact_7404_of__nat__sum,axiom,
    ! [F: fun_nat_nat,A3: set_nat] : aa_nat_real(semiri5074537144036343181t_real,aa_set_nat_nat(groups3542108847815614940at_nat(F),A3)) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_fp(F)),A3) ).

% of_nat_sum
tff(fact_7405_of__nat__le__0__iff,axiom,
    ! [Mb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),zero_zero_real)
    <=> ( Mb = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_7406_of__nat__le__0__iff,axiom,
    ! [Mb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,Mb)),zero_zero_rat)
    <=> ( Mb = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_7407_of__nat__le__0__iff,axiom,
    ! [Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),zero_zero_nat)
    <=> ( Mb = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_7408_of__nat__le__0__iff,axiom,
    ! [Mb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),zero_zero_int)
    <=> ( Mb = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_7409_of__nat__Suc,axiom,
    ! [Mb: nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,Mb)) = aa_complex_complex(plus_plus_complex(one_one_complex),aa_nat_complex(semiri8010041392384452111omplex,Mb)) ).

% of_nat_Suc
tff(fact_7410_of__nat__Suc,axiom,
    ! [Mb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,Mb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_nat_int(semiri1314217659103216013at_int,Mb)) ).

% of_nat_Suc
tff(fact_7411_of__nat__Suc,axiom,
    ! [Mb: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Mb)) = aa_real_real(plus_plus_real(one_one_real),aa_nat_real(semiri5074537144036343181t_real,Mb)) ).

% of_nat_Suc
tff(fact_7412_of__nat__Suc,axiom,
    ! [Mb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(suc,Mb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(semiri1316708129612266289at_nat,Mb)) ).

% of_nat_Suc
tff(fact_7413_of__nat__Suc,axiom,
    ! [Mb: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,Mb)) = aa_rat_rat(plus_plus_rat(one_one_rat),aa_nat_rat(semiri681578069525770553at_rat,Mb)) ).

% of_nat_Suc
tff(fact_7414_real__of__nat__less__numeral__iff,axiom,
    ! [Nb: nat,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_num_real(numeral_numeral_real,W))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_num_nat(numeral_numeral_nat,W)) ) ).

% real_of_nat_less_numeral_iff
tff(fact_7415_numeral__less__real__of__nat__iff,axiom,
    ! [W: num,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,W)),aa_nat_real(semiri5074537144036343181t_real,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,W)),Nb) ) ).

% numeral_less_real_of_nat_iff
tff(fact_7416_numeral__le__real__of__nat__iff,axiom,
    ! [Nb: num,Mb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,Nb)),aa_nat_real(semiri5074537144036343181t_real,Mb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,Nb)),Mb) ) ).

% numeral_le_real_of_nat_iff
tff(fact_7417_cos__pi,axiom,
    aa_real_real(cos_real,pi) = aa_real_real(uminus_uminus_real,one_one_real) ).

% cos_pi
tff(fact_7418_cos__periodic__pi2,axiom,
    ! [Xb: real] : aa_real_real(cos_real,aa_real_real(plus_plus_real(pi),Xb)) = aa_real_real(uminus_uminus_real,aa_real_real(cos_real,Xb)) ).

% cos_periodic_pi2
tff(fact_7419_cos__periodic__pi,axiom,
    ! [Xb: real] : aa_real_real(cos_real,aa_real_real(plus_plus_real(Xb),pi)) = aa_real_real(uminus_uminus_real,aa_real_real(cos_real,Xb)) ).

% cos_periodic_pi
tff(fact_7420_cot__npi,axiom,
    ! [Nb: nat] : aa_real_real(cot_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),pi)) = zero_zero_real ).

% cot_npi
tff(fact_7421_of__nat__0__less__iff,axiom,
    ! [Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(semiri1314217659103216013at_int,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).

% of_nat_0_less_iff
tff(fact_7422_of__nat__0__less__iff,axiom,
    ! [Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(semiri5074537144036343181t_real,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).

% of_nat_0_less_iff
tff(fact_7423_of__nat__0__less__iff,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(semiri1316708129612266289at_nat,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).

% of_nat_0_less_iff
tff(fact_7424_of__nat__0__less__iff,axiom,
    ! [Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(semiri681578069525770553at_rat,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).

% of_nat_0_less_iff
tff(fact_7425_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [Xb: nat,B: nat,W: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Xb)),aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B)),W))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_7426_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [Xb: nat,B: nat,W: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Xb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,B)),W))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_7427_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [Xb: nat,B: nat,W: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,Xb)),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,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_7428_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [Xb: nat,B: nat,W: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,Xb)),aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B)),W))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_7429_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,Xb: 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,Xb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),Xb) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_7430_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,Xb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,B)),W)),aa_nat_real(semiri5074537144036343181t_real,Xb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),Xb) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_7431_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,Xb: 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,Xb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),Xb) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_7432_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,Xb: 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,Xb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)),Xb) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_7433_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [Xb: nat,B: nat,W: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Xb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,B)),W))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_7434_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [Xb: nat,B: nat,W: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,Xb)),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,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_7435_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [Xb: nat,B: nat,W: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,Xb)),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,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_7436_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [Xb: nat,B: nat,W: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,Xb)),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,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),W)) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_7437_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,Xb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,B)),W)),aa_nat_real(semiri5074537144036343181t_real,Xb))
    <=> 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)),Xb) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_7438_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,Xb: 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,Xb))
    <=> 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)),Xb) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_7439_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,Xb: 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,Xb))
    <=> 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)),Xb) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_7440_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,Xb: 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,Xb))
    <=> 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)),Xb) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_7441_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Yb: nat,Xb: num,Nb: nat] :
      ( ( aa_nat_complex(semiri8010041392384452111omplex,Yb) = aa_nat_complex(power_power_complex(numera6690914467698888265omplex(Xb)),Nb) )
    <=> ( Yb = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_7442_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Yb: nat,Xb: num,Nb: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,Yb) = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb) )
    <=> ( Yb = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_7443_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Yb: nat,Xb: num,Nb: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,Yb) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,Xb)),Nb) )
    <=> ( Yb = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_7444_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Yb: nat,Xb: num,Nb: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,Yb) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb) )
    <=> ( Yb = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_7445_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Yb: nat,Xb: num,Nb: nat] :
      ( ( aa_nat_rat(semiri681578069525770553at_rat,Yb) = aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,Xb)),Nb) )
    <=> ( Yb = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_7446_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,Yb: nat] :
      ( ( aa_nat_complex(power_power_complex(numera6690914467698888265omplex(Xb)),Nb) = aa_nat_complex(semiri8010041392384452111omplex,Yb) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb) = Yb ) ) ).

% numeral_power_eq_of_nat_cancel_iff
tff(fact_7447_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,Yb: nat] :
      ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb) = aa_nat_int(semiri1314217659103216013at_int,Yb) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb) = Yb ) ) ).

% numeral_power_eq_of_nat_cancel_iff
tff(fact_7448_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,Yb: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,Xb)),Nb) = aa_nat_real(semiri5074537144036343181t_real,Yb) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb) = Yb ) ) ).

% numeral_power_eq_of_nat_cancel_iff
tff(fact_7449_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,Yb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb) = aa_nat_nat(semiri1316708129612266289at_nat,Yb) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb) = Yb ) ) ).

% numeral_power_eq_of_nat_cancel_iff
tff(fact_7450_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,Yb: nat] :
      ( ( aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,Xb)),Nb) = aa_nat_rat(semiri681578069525770553at_rat,Yb) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb) = Yb ) ) ).

% numeral_power_eq_of_nat_cancel_iff
tff(fact_7451_of__nat__zero__less__power__iff,axiom,
    ! [Xb: nat,Nb: 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,Xb)),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Xb)
        | ( Nb = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_7452_of__nat__zero__less__power__iff,axiom,
    ! [Xb: nat,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,Xb)),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Xb)
        | ( Nb = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_7453_of__nat__zero__less__power__iff,axiom,
    ! [Xb: nat,Nb: 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,Xb)),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Xb)
        | ( Nb = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_7454_of__nat__zero__less__power__iff,axiom,
    ! [Xb: nat,Nb: 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,Xb)),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Xb)
        | ( Nb = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_7455_cos__npi,axiom,
    ! [Nb: nat] : aa_real_real(cos_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),pi)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Nb) ).

% cos_npi
tff(fact_7456_cos__npi2,axiom,
    ! [Nb: nat] : aa_real_real(cos_real,aa_real_real(times_times_real(pi),aa_nat_real(semiri5074537144036343181t_real,Nb))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Nb) ).

% cos_npi2
tff(fact_7457_even__of__nat,axiom,
    ! [Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_nat_Code_integer(semiri4939895301339042750nteger,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) ) ).

% even_of_nat
tff(fact_7458_even__of__nat,axiom,
    ! [Nb: 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,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) ) ).

% even_of_nat
tff(fact_7459_even__of__nat,axiom,
    ! [Nb: 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,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) ) ).

% even_of_nat
tff(fact_7460_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [Xb: nat,I3: num,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Xb)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,I3)),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I3)),Nb)) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_7461_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [Xb: nat,I3: num,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Xb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,I3)),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I3)),Nb)) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_7462_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [Xb: nat,I3: num,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,Xb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I3)),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I3)),Nb)) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_7463_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [Xb: nat,I3: num,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,Xb)),aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,I3)),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I3)),Nb)) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_7464_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I3: num,Nb: nat,Xb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,I3)),Nb)),aa_nat_int(semiri1314217659103216013at_int,Xb))
    <=> 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,I3)),Nb)),Xb) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_7465_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I3: num,Nb: nat,Xb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,I3)),Nb)),aa_nat_real(semiri5074537144036343181t_real,Xb))
    <=> 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,I3)),Nb)),Xb) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_7466_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I3: num,Nb: nat,Xb: 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,I3)),Nb)),aa_nat_nat(semiri1316708129612266289at_nat,Xb))
    <=> 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,I3)),Nb)),Xb) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_7467_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I3: num,Nb: nat,Xb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,I3)),Nb)),aa_nat_rat(semiri681578069525770553at_rat,Xb))
    <=> 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,I3)),Nb)),Xb) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_7468_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [Xb: nat,I3: num,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Xb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,I3)),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I3)),Nb)) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_7469_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [Xb: nat,I3: num,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,Xb)),aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,I3)),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I3)),Nb)) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_7470_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [Xb: nat,I3: num,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,Xb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I3)),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I3)),Nb)) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_7471_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [Xb: nat,I3: num,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,Xb)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,I3)),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I3)),Nb)) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_7472_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I3: num,Nb: nat,Xb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,I3)),Nb)),aa_nat_real(semiri5074537144036343181t_real,Xb))
    <=> 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,I3)),Nb)),Xb) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_7473_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I3: num,Nb: nat,Xb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,I3)),Nb)),aa_nat_rat(semiri681578069525770553at_rat,Xb))
    <=> 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,I3)),Nb)),Xb) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_7474_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I3: num,Nb: nat,Xb: 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,I3)),Nb)),aa_nat_nat(semiri1316708129612266289at_nat,Xb))
    <=> 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,I3)),Nb)),Xb) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_7475_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I3: num,Nb: nat,Xb: 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,I3)),Nb)),aa_nat_int(semiri1314217659103216013at_int,Xb))
    <=> 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,I3)),Nb)),Xb) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_7476_cos__pi__half,axiom,
    aa_real_real(cos_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = zero_zero_real ).

% cos_pi_half
tff(fact_7477_cos__2npi,axiom,
    ! [Nb: nat] : aa_real_real(cos_real,aa_real_real(times_times_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Nb))),pi)) = one_one_real ).

% cos_2npi
tff(fact_7478_cos__two__pi,axiom,
    aa_real_real(cos_real,aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)) = one_one_real ).

% cos_two_pi
tff(fact_7479_cos__periodic,axiom,
    ! [Xb: real] : aa_real_real(cos_real,aa_real_real(plus_plus_real(Xb),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))) = aa_real_real(cos_real,Xb) ).

% cos_periodic
tff(fact_7480_cos__2pi__minus,axiom,
    ! [Xb: real] : aa_real_real(cos_real,aa_real_real(minus_minus_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)),Xb)) = aa_real_real(cos_real,Xb) ).

% cos_2pi_minus
tff(fact_7481_cot__periodic,axiom,
    ! [Xb: real] : aa_real_real(cot_real,aa_real_real(plus_plus_real(Xb),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))) = aa_real_real(cot_real,Xb) ).

% cot_periodic
tff(fact_7482_cos__int__2pin,axiom,
    ! [Nb: int] : aa_real_real(cos_real,aa_real_real(times_times_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)),aa_int_real(ring_1_of_int_real,Nb))) = one_one_real ).

% cos_int_2pin
tff(fact_7483_cos__3over2__pi,axiom,
    aa_real_real(cos_real,aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_num_real(numeral_numeral_real,aa_num_num(bit1,one))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),pi)) = zero_zero_real ).

% cos_3over2_pi
tff(fact_7484_cos__pi__eq__zero,axiom,
    ! [Mb: nat] : aa_real_real(cos_real,aa_real_real(divide_divide_real(aa_real_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))),Mb))))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = zero_zero_real ).

% cos_pi_eq_zero
tff(fact_7485_real__arch__simple,axiom,
    ! [Xb: real] :
    ? [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_nat_real(semiri5074537144036343181t_real,N)) ).

% real_arch_simple
tff(fact_7486_real__arch__simple,axiom,
    ! [Xb: rat] :
    ? [N: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),aa_nat_rat(semiri681578069525770553at_rat,N)) ).

% real_arch_simple
tff(fact_7487_reals__Archimedean2,axiom,
    ! [Xb: real] :
    ? [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_nat_real(semiri5074537144036343181t_real,N)) ).

% reals_Archimedean2
tff(fact_7488_reals__Archimedean2,axiom,
    ! [Xb: rat] :
    ? [N: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),aa_nat_rat(semiri681578069525770553at_rat,N)) ).

% reals_Archimedean2
tff(fact_7489_mult__of__nat__commute,axiom,
    ! [Xb: nat,Yb: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Xb)),Yb) = aa_int_int(aa_int_fun_int_int(times_times_int,Yb),aa_nat_int(semiri1314217659103216013at_int,Xb)) ).

% mult_of_nat_commute
tff(fact_7490_mult__of__nat__commute,axiom,
    ! [Xb: nat,Yb: real] : aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Xb)),Yb) = aa_real_real(times_times_real(Yb),aa_nat_real(semiri5074537144036343181t_real,Xb)) ).

% mult_of_nat_commute
tff(fact_7491_mult__of__nat__commute,axiom,
    ! [Xb: nat,Yb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Xb)),Yb) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Yb),aa_nat_nat(semiri1316708129612266289at_nat,Xb)) ).

% mult_of_nat_commute
tff(fact_7492_mult__of__nat__commute,axiom,
    ! [Xb: nat,Yb: rat] : aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,Xb)),Yb) = aa_rat_rat(times_times_rat(Yb),aa_nat_rat(semiri681578069525770553at_rat,Xb)) ).

% mult_of_nat_commute
tff(fact_7493_of__nat__and__eq,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,Mb),Nb)) = aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb)) ).

% of_nat_and_eq
tff(fact_7494_of__nat__and__eq,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),aa_nat_nat(semiri1316708129612266289at_nat,Nb)) ).

% of_nat_and_eq
tff(fact_7495_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_7496_int__diff__cases,axiom,
    ! [Z: int] :
      ~ ! [M2: nat,N: nat] : Z != aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,M2)),aa_nat_int(semiri1314217659103216013at_int,N)) ).

% int_diff_cases
tff(fact_7497_of__nat__less__of__int__iff,axiom,
    ! [Nb: nat,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),aa_int_int(ring_1_of_int_int,Xb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),Xb) ) ).

% of_nat_less_of_int_iff
tff(fact_7498_of__nat__less__of__int__iff,axiom,
    ! [Nb: nat,Xb: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_int_real(ring_1_of_int_real,Xb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),Xb) ) ).

% of_nat_less_of_int_iff
tff(fact_7499_of__nat__less__of__int__iff,axiom,
    ! [Nb: nat,Xb: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,Nb)),aa_int_rat(ring_1_of_int_rat,Xb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),Xb) ) ).

% of_nat_less_of_int_iff
tff(fact_7500_of__nat__0__le__iff,axiom,
    ! [Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(semiri5074537144036343181t_real,Nb)) ).

% of_nat_0_le_iff
tff(fact_7501_of__nat__0__le__iff,axiom,
    ! [Nb: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(semiri681578069525770553at_rat,Nb)) ).

% of_nat_0_le_iff
tff(fact_7502_of__nat__0__le__iff,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(semiri1316708129612266289at_nat,Nb)) ).

% of_nat_0_le_iff
tff(fact_7503_of__nat__0__le__iff,axiom,
    ! [Nb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(semiri1314217659103216013at_int,Nb)) ).

% of_nat_0_le_iff
tff(fact_7504_of__nat__less__0__iff,axiom,
    ! [Mb: nat] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),zero_zero_int) ).

% of_nat_less_0_iff
tff(fact_7505_of__nat__less__0__iff,axiom,
    ! [Mb: nat] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),zero_zero_real) ).

% of_nat_less_0_iff
tff(fact_7506_of__nat__less__0__iff,axiom,
    ! [Mb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),zero_zero_nat) ).

% of_nat_less_0_iff
tff(fact_7507_of__nat__less__0__iff,axiom,
    ! [Mb: nat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,Mb)),zero_zero_rat) ).

% of_nat_less_0_iff
tff(fact_7508_of__nat__neq__0,axiom,
    ! [Nb: nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,Nb)) != zero_zero_complex ).

% of_nat_neq_0
tff(fact_7509_of__nat__neq__0,axiom,
    ! [Nb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,Nb)) != zero_zero_int ).

% of_nat_neq_0
tff(fact_7510_of__nat__neq__0,axiom,
    ! [Nb: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Nb)) != zero_zero_real ).

% of_nat_neq_0
tff(fact_7511_of__nat__neq__0,axiom,
    ! [Nb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(suc,Nb)) != zero_zero_nat ).

% of_nat_neq_0
tff(fact_7512_of__nat__neq__0,axiom,
    ! [Nb: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,Nb)) != zero_zero_rat ).

% of_nat_neq_0
tff(fact_7513_div__mult2__eq_H,axiom,
    ! [A: int,Mb: nat,Nb: 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,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb))) = 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,Mb))),aa_nat_int(semiri1314217659103216013at_int,Nb)) ).

% div_mult2_eq'
tff(fact_7514_div__mult2__eq_H,axiom,
    ! [A: nat,Mb: nat,Nb: 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,Mb)),aa_nat_nat(semiri1316708129612266289at_nat,Nb))) = 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,Mb))),aa_nat_nat(semiri1316708129612266289at_nat,Nb)) ).

% div_mult2_eq'
tff(fact_7515_of__nat__less__imp__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% of_nat_less_imp_less
tff(fact_7516_of__nat__less__imp__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_nat_real(semiri5074537144036343181t_real,Nb))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% of_nat_less_imp_less
tff(fact_7517_of__nat__less__imp__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),aa_nat_nat(semiri1316708129612266289at_nat,Nb))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% of_nat_less_imp_less
tff(fact_7518_of__nat__less__imp__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,Mb)),aa_nat_rat(semiri681578069525770553at_rat,Nb))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% of_nat_less_imp_less
tff(fact_7519_less__imp__of__nat__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb)) ) ).

% less_imp_of_nat_less
tff(fact_7520_less__imp__of__nat__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ).

% less_imp_of_nat_less
tff(fact_7521_less__imp__of__nat__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),aa_nat_nat(semiri1316708129612266289at_nat,Nb)) ) ).

% less_imp_of_nat_less
tff(fact_7522_less__imp__of__nat__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,Mb)),aa_nat_rat(semiri681578069525770553at_rat,Nb)) ) ).

% less_imp_of_nat_less
tff(fact_7523_of__nat__mono,axiom,
    ! [I3: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I3),J)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,I3)),aa_nat_real(semiri5074537144036343181t_real,J)) ) ).

% of_nat_mono
tff(fact_7524_of__nat__mono,axiom,
    ! [I3: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I3),J)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,I3)),aa_nat_rat(semiri681578069525770553at_rat,J)) ) ).

% of_nat_mono
tff(fact_7525_of__nat__mono,axiom,
    ! [I3: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I3),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,I3)),aa_nat_nat(semiri1316708129612266289at_nat,J)) ) ).

% of_nat_mono
tff(fact_7526_of__nat__mono,axiom,
    ! [I3: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I3),J)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,I3)),aa_nat_int(semiri1314217659103216013at_int,J)) ) ).

% of_nat_mono
tff(fact_7527_cos__le__one,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(cos_real,Xb)),one_one_real) ).

% cos_le_one
tff(fact_7528_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb)) ).

% unique_euclidean_semiring_with_nat_class.of_nat_div
tff(fact_7529_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),aa_nat_nat(semiri1316708129612266289at_nat,Nb)) ).

% unique_euclidean_semiring_with_nat_class.of_nat_div
tff(fact_7530_of__nat__dvd__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(semiri4939895301339042750nteger,Mb)),aa_nat_Code_integer(semiri4939895301339042750nteger,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),Nb) ) ).

% of_nat_dvd_iff
tff(fact_7531_of__nat__dvd__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),Nb) ) ).

% of_nat_dvd_iff
tff(fact_7532_of__nat__dvd__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),aa_nat_nat(semiri1316708129612266289at_nat,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),Nb) ) ).

% of_nat_dvd_iff
tff(fact_7533_int__ops_I1_J,axiom,
    aa_nat_int(semiri1314217659103216013at_int,zero_zero_nat) = zero_zero_int ).

% int_ops(1)
tff(fact_7534_int__ops_I3_J,axiom,
    ! [Nb: num] : aa_nat_int(semiri1314217659103216013at_int,aa_num_nat(numeral_numeral_nat,Nb)) = aa_num_int(numeral_numeral_int,Nb) ).

% int_ops(3)
tff(fact_7535_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_7536_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_7537_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_7538_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_7539_zle__int,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% zle_int
tff(fact_7540_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_7541_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_7542_of__nat__mod,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb)) ).

% of_nat_mod
tff(fact_7543_of__nat__mod,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),aa_nat_nat(semiri1316708129612266289at_nat,Nb)) ).

% of_nat_mod
tff(fact_7544_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_7545_int__plus,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),aa_nat_int(semiri1314217659103216013at_int,Mb)) ).

% int_plus
tff(fact_7546_zadd__int__left,axiom,
    ! [Mb: nat,Nb: nat,Z: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),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,Mb),Nb))),Z) ).

% zadd_int_left
tff(fact_7547_int__ops_I7_J,axiom,
    ! [A: nat,B: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)) ).

% int_ops(7)
tff(fact_7548_int__ops_I2_J,axiom,
    aa_nat_int(semiri1314217659103216013at_int,one_one_nat) = one_one_int ).

% int_ops(2)
tff(fact_7549_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_7550_not__int__zless__negative,axiom,
    ! [Nb: nat,Mb: nat] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,Mb))) ).

% not_int_zless_negative
tff(fact_7551_int__sum,axiom,
    ! [F: fun_int_nat,A3: set_int] : aa_nat_int(semiri1314217659103216013at_int,groups4541462559716669496nt_nat(F,A3)) = groups4538972089207619220nt_int(aTP_Lamm_fk(F),A3) ).

% int_sum
tff(fact_7552_int__sum,axiom,
    ! [F: fun_nat_nat,A3: set_nat] : aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(groups3542108847815614940at_nat(F),A3)) = groups3539618377306564664at_int(aTP_Lamm_fm(F),A3) ).

% int_sum
tff(fact_7553_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_7554_of__nat__max,axiom,
    ! [Xb: nat,Yb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xb),Yb)) = aa_int_int(aa_int_fun_int_int(ord_max_int,aa_nat_int(semiri1314217659103216013at_int,Xb)),aa_nat_int(semiri1314217659103216013at_int,Yb)) ).

% of_nat_max
tff(fact_7555_of__nat__max,axiom,
    ! [Xb: nat,Yb: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xb),Yb)) = aa_real_real(ord_max_real(aa_nat_real(semiri5074537144036343181t_real,Xb)),aa_nat_real(semiri5074537144036343181t_real,Yb)) ).

% of_nat_max
tff(fact_7556_of__nat__max,axiom,
    ! [Xb: nat,Yb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xb),Yb)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(semiri1316708129612266289at_nat,Xb)),aa_nat_nat(semiri1316708129612266289at_nat,Yb)) ).

% of_nat_max
tff(fact_7557_of__nat__max,axiom,
    ! [Xb: nat,Yb: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xb),Yb)) = aa_rat_rat(ord_max_rat(aa_nat_rat(semiri681578069525770553at_rat,Xb)),aa_nat_rat(semiri681578069525770553at_rat,Yb)) ).

% of_nat_max
tff(fact_7558_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(aa_int_fun_int_int(modulo_modulo_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B)) ).

% zmod_int
tff(fact_7559_take__bit__of__nat,axiom,
    ! [Nb: nat,Mb: nat] : aa_int_int(bit_se2923211474154528505it_int(Nb),aa_nat_int(semiri1314217659103216013at_int,Mb)) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),Mb)) ).

% take_bit_of_nat
tff(fact_7560_take__bit__of__nat,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),aa_nat_nat(semiri1316708129612266289at_nat,Mb)) = aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),Mb)) ).

% take_bit_of_nat
tff(fact_7561_of__nat__or__eq,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,Mb),Nb)) = aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb)) ).

% of_nat_or_eq
tff(fact_7562_of__nat__or__eq,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),aa_nat_nat(semiri1316708129612266289at_nat,Nb)) ).

% of_nat_or_eq
tff(fact_7563_bit__of__nat__iff__bit,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_nat_int(semiri1314217659103216013at_int,Mb)),Nb)
    <=> aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,Mb),Nb) ) ).

% bit_of_nat_iff_bit
tff(fact_7564_bit__of__nat__iff__bit,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),Nb)
    <=> aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,Mb),Nb) ) ).

% bit_of_nat_iff_bit
tff(fact_7565_nat__less__as__int,axiom,
    ! [X: nat,Xa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Xa)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(semiri1314217659103216013at_int,Xa)) ) ).

% nat_less_as_int
tff(fact_7566_of__nat__mask__eq,axiom,
    ! [Nb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(bit_se2002935070580805687sk_nat,Nb)) = aa_nat_nat(bit_se2002935070580805687sk_nat,Nb) ).

% of_nat_mask_eq
tff(fact_7567_of__nat__mask__eq,axiom,
    ! [Nb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(bit_se2002935070580805687sk_nat,Nb)) = aa_nat_int(bit_se2000444600071755411sk_int,Nb) ).

% of_nat_mask_eq
tff(fact_7568_nat__leq__as__int,axiom,
    ! [X: nat,Xa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Xa)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(semiri1314217659103216013at_int,Xa)) ) ).

% nat_leq_as_int
tff(fact_7569_ex__less__of__nat__mult,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ? [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,Yb),aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,N)),Xb)) ) ).

% ex_less_of_nat_mult
tff(fact_7570_ex__less__of__nat__mult,axiom,
    ! [Xb: rat,Yb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xb)
     => ? [N: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Yb),aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,N)),Xb)) ) ).

% ex_less_of_nat_mult
tff(fact_7571_cos__ge__minus__one,axiom,
    ! [Xb: 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,Xb)) ).

% cos_ge_minus_one
tff(fact_7572_of__nat__diff,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => ( aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb)) ) ) ).

% of_nat_diff
tff(fact_7573_of__nat__diff,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => ( aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) = aa_real_real(minus_minus_real(aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ) ).

% of_nat_diff
tff(fact_7574_of__nat__diff,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => ( aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),aa_nat_nat(semiri1316708129612266289at_nat,Nb)) ) ) ).

% of_nat_diff
tff(fact_7575_of__nat__diff,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => ( aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) = aa_rat_rat(minus_minus_rat(aa_nat_rat(semiri681578069525770553at_rat,Mb)),aa_nat_rat(semiri681578069525770553at_rat,Nb)) ) ) ).

% of_nat_diff
tff(fact_7576_abs__cos__le__one,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(cos_real,Xb))),one_one_real) ).

% abs_cos_le_one
tff(fact_7577_exp__of__nat__mult,axiom,
    ! [Nb: nat,Xb: complex] : aa_complex_complex(exp_complex,aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,Nb)),Xb)) = aa_nat_complex(power_power_complex(aa_complex_complex(exp_complex,Xb)),Nb) ).

% exp_of_nat_mult
tff(fact_7578_exp__of__nat__mult,axiom,
    ! [Nb: nat,Xb: real] : aa_real_real(exp_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),Xb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(exp_real,Xb)),Nb) ).

% exp_of_nat_mult
tff(fact_7579_exp__of__nat2__mult,axiom,
    ! [Xb: complex,Nb: nat] : aa_complex_complex(exp_complex,aa_complex_complex(times_times_complex(Xb),aa_nat_complex(semiri8010041392384452111omplex,Nb))) = aa_nat_complex(power_power_complex(aa_complex_complex(exp_complex,Xb)),Nb) ).

% exp_of_nat2_mult
tff(fact_7580_exp__of__nat2__mult,axiom,
    ! [Xb: real,Nb: nat] : aa_real_real(exp_real,aa_real_real(times_times_real(Xb),aa_nat_real(semiri5074537144036343181t_real,Nb))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(exp_real,Xb)),Nb) ).

% exp_of_nat2_mult
tff(fact_7581_reals__Archimedean3,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ! [Y3: real] :
        ? [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,Y3),aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,N)),Xb)) ) ).

% reals_Archimedean3
tff(fact_7582_int__cases4,axiom,
    ! [Mb: int] :
      ( ! [N: nat] : Mb != aa_nat_int(semiri1314217659103216013at_int,N)
     => ~ ! [N: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
           => ( Mb != aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N)) ) ) ) ).

% int_cases4
tff(fact_7583_real__of__nat__div4,axiom,
    ! [Nb: nat,Xb: 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,Nb),Xb))),aa_real_real(divide_divide_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(semiri5074537144036343181t_real,Xb))) ).

% real_of_nat_div4
tff(fact_7584_atLeast0__atMost__Suc,axiom,
    ! [Nb: nat] : aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,Nb)) = insert_nat(aa_nat_nat(suc,Nb),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb)) ).

% atLeast0_atMost_Suc
tff(fact_7585_int__zle__neg,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,Mb)))
    <=> ( ( Nb = zero_zero_nat )
        & ( Mb = zero_zero_nat ) ) ) ).

% int_zle_neg
tff(fact_7586_int__Suc,axiom,
    ! [Nb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,Nb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),one_one_int) ).

% int_Suc
tff(fact_7587_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_7588_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_7589_atLeastAtMost__insertL,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( insert_nat(Mb,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb)),Nb)) = aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb) ) ) ).

% atLeastAtMost_insertL
tff(fact_7590_atLeastAtMostSuc__conv,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb))
     => ( aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(suc,Nb)) = insert_nat(aa_nat_nat(suc,Nb),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) ) ) ).

% atLeastAtMostSuc_conv
tff(fact_7591_Icc__eq__insert__lb__nat,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb) = insert_nat(Mb,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb)),Nb)) ) ) ).

% Icc_eq_insert_lb_nat
tff(fact_7592_negative__zle__0,axiom,
    ! [Nb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,Nb))),zero_zero_int) ).

% negative_zle_0
tff(fact_7593_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_7594_real__of__nat__div,axiom,
    ! [D: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),Nb)
     => ( aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),D)) = aa_real_real(divide_divide_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(semiri5074537144036343181t_real,D)) ) ) ).

% real_of_nat_div
tff(fact_7595_cos__one__2pi,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(cos_real,Xb) = one_one_real )
    <=> ( ? [X4: nat] : Xb = aa_real_real(times_times_real(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,X4)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),pi)
        | ? [X4: nat] : Xb = aa_real_real(uminus_uminus_real,aa_real_real(times_times_real(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,X4)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),pi)) ) ) ).

% cos_one_2pi
tff(fact_7596_cos__two__neq__zero,axiom,
    aa_real_real(cos_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) != zero_zero_real ).

% cos_two_neq_zero
tff(fact_7597_mod__mult2__eq_H,axiom,
    ! [A: int,Mb: nat,Nb: nat] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb))) = 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,Mb)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_nat_int(semiri1314217659103216013at_int,Mb))),aa_nat_int(semiri1314217659103216013at_int,Nb)))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_nat_int(semiri1314217659103216013at_int,Mb))) ).

% mod_mult2_eq'
tff(fact_7598_mod__mult2__eq_H,axiom,
    ! [A: nat,Mb: nat,Nb: 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,Mb)),aa_nat_nat(semiri1316708129612266289at_nat,Nb))) = 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,Mb)),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,Mb))),aa_nat_nat(semiri1316708129612266289at_nat,Nb)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(semiri1316708129612266289at_nat,Mb))) ).

% mod_mult2_eq'
tff(fact_7599_field__char__0__class_Oof__nat__div,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_nat_complex(semiri8010041392384452111omplex,Mb)),aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb)))),aa_nat_complex(semiri8010041392384452111omplex,Nb)) ).

% field_char_0_class.of_nat_div
tff(fact_7600_field__char__0__class_Oof__nat__div,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb)))),aa_nat_real(semiri5074537144036343181t_real,Nb)) ).

% field_char_0_class.of_nat_div
tff(fact_7601_field__char__0__class_Oof__nat__div,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_nat_rat(semiri681578069525770553at_rat,Mb)),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb)))),aa_nat_rat(semiri681578069525770553at_rat,Nb)) ).

% field_char_0_class.of_nat_div
tff(fact_7602_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_7603_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_7604_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_7605_nat__less__real__le,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),one_one_real)),aa_nat_real(semiri5074537144036343181t_real,Mb)) ) ).

% nat_less_real_le
tff(fact_7606_nat__le__real__less,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(plus_plus_real(aa_nat_real(semiri5074537144036343181t_real,Mb)),one_one_real)) ) ).

% nat_le_real_less
tff(fact_7607_zmult__zless__mono2__lemma,axiom,
    ! [I3: int,J: int,K: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,I3),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)),I3)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,K)),J)) ) ) ).

% zmult_zless_mono2_lemma
tff(fact_7608_not__zle__0__negative,axiom,
    ! [Nb: 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,Nb)))) ).

% not_zle_0_negative
tff(fact_7609_negD,axiom,
    ! [Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),zero_zero_int)
     => ? [N: nat] : Xb = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N))) ) ).

% negD
tff(fact_7610_negative__zless__0,axiom,
    ! [Nb: 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,Nb)))),zero_zero_int) ).

% negative_zless_0
tff(fact_7611_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_7612_real__of__nat__div__aux,axiom,
    ! [Xb: nat,D: nat] : aa_real_real(divide_divide_real(aa_nat_real(semiri5074537144036343181t_real,Xb)),aa_nat_real(semiri5074537144036343181t_real,D)) = aa_real_real(plus_plus_real(aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xb),D))),aa_real_real(divide_divide_real(aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),D))),aa_nat_real(semiri5074537144036343181t_real,D))) ).

% real_of_nat_div_aux
tff(fact_7613_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(divide_divide_real(one_one_real),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N)))),E2) ) ).

% nat_approx_posE
tff(fact_7614_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,aa_rat_rat(divide_divide_rat(one_one_rat),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,N)))),E2) ) ).

% nat_approx_posE
tff(fact_7615_of__nat__less__two__power,axiom,
    ! [Nb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ).

% of_nat_less_two_power
tff(fact_7616_of__nat__less__two__power,axiom,
    ! [Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Nb)) ).

% of_nat_less_two_power
tff(fact_7617_of__nat__less__two__power,axiom,
    ! [Nb: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,Nb)),aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),Nb)) ).

% of_nat_less_two_power
tff(fact_7618_cos__zero__lemma,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( ( aa_real_real(cos_real,Xb) = 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)
            & ( Xb = aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ) ) ).

% cos_zero_lemma
tff(fact_7619_cos__two__less__zero,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cos_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),zero_zero_real) ).

% cos_two_less_zero
tff(fact_7620_cos__is__zero,axiom,
    ? [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X3)
      & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))
      & ( aa_real_real(cos_real,X3) = 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),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))
            & ( aa_real_real(cos_real,Y3) = zero_zero_real ) )
         => ( Y3 = X3 ) ) ) ).

% cos_is_zero
tff(fact_7621_cos__two__le__zero,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(cos_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),zero_zero_real) ).

% cos_two_le_zero
tff(fact_7622_cos__zero__iff,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(cos_real,Xb) = 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)
            & ( Xb = aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,N4)),aa_real_real(divide_divide_real(pi),aa_num_real(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)
            & ( Xb = aa_real_real(uminus_uminus_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,N4)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ) ) ) ) ).

% cos_zero_iff
tff(fact_7623_inverse__of__nat__le,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => ( ( Nb != zero_zero_nat )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(one_one_real),aa_nat_real(semiri5074537144036343181t_real,Mb))),aa_real_real(divide_divide_real(one_one_real),aa_nat_real(semiri5074537144036343181t_real,Nb))) ) ) ).

% inverse_of_nat_le
tff(fact_7624_inverse__of__nat__le,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => ( ( Nb != zero_zero_nat )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(one_one_rat),aa_nat_rat(semiri681578069525770553at_rat,Mb))),aa_rat_rat(divide_divide_rat(one_one_rat),aa_nat_rat(semiri681578069525770553at_rat,Nb))) ) ) ).

% inverse_of_nat_le
tff(fact_7625_cos__total,axiom,
    ! [Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Yb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Yb),one_one_real)
       => ? [X3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X3)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),pi)
            & ( aa_real_real(cos_real,X3) = Yb )
            & ! [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) = Yb ) )
               => ( Y3 = X3 ) ) ) ) ) ).

% cos_total
tff(fact_7626_exp__divide__power__eq,axiom,
    ! [Nb: nat,Xb: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_complex(power_power_complex(aa_complex_complex(exp_complex,aa_complex_complex(divide1717551699836669952omplex(Xb),aa_nat_complex(semiri8010041392384452111omplex,Nb)))),Nb) = aa_complex_complex(exp_complex,Xb) ) ) ).

% exp_divide_power_eq
tff(fact_7627_exp__divide__power__eq,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(exp_real,aa_real_real(divide_divide_real(Xb),aa_nat_real(semiri5074537144036343181t_real,Nb)))),Nb) = aa_real_real(exp_real,Xb) ) ) ).

% exp_divide_power_eq
tff(fact_7628_real__archimedian__rdiv__eq__0,axiom,
    ! [Xb: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
       => ( ! [M2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M2)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,M2)),Xb)),C) )
         => ( Xb = zero_zero_real ) ) ) ) ).

% real_archimedian_rdiv_eq_0
tff(fact_7629_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_7630_zdiff__int__split,axiom,
    ! [P: fun_int_o,Xb: nat,Yb: nat] :
      ( aa_int_o(P,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xb),Yb)))
    <=> ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Yb),Xb)
         => aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,Xb)),aa_nat_int(semiri1314217659103216013at_int,Yb))) )
        & ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Yb)
         => aa_int_o(P,zero_zero_int) ) ) ) ).

% zdiff_int_split
tff(fact_7631_real__of__nat__div2,axiom,
    ! [Nb: nat,Xb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(semiri5074537144036343181t_real,Xb))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),Xb)))) ).

% real_of_nat_div2
tff(fact_7632_ln__realpow,axiom,
    ! [Xb: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_real(ln_ln_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)) = aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(ln_ln_real,Xb)) ) ) ).

% ln_realpow
tff(fact_7633_real__of__nat__div3,axiom,
    ! [Nb: nat,Xb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(semiri5074537144036343181t_real,Xb))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),Xb)))),one_one_real) ).

% real_of_nat_div3
tff(fact_7634_cos__45,axiom,
    aa_real_real(cos_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one))))) = aa_real_real(divide_divide_real(aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% cos_45
tff(fact_7635_Bernoulli__inequality,axiom,
    ! [Xb: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(one_one_real),aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),Xb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(plus_plus_real(one_one_real),Xb)),Nb)) ) ).

% Bernoulli_inequality
tff(fact_7636_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_7637_cos__times__cos,axiom,
    ! [W: real,Z: real] : aa_real_real(times_times_real(aa_real_real(cos_real,W)),aa_real_real(cos_real,Z)) = aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(aa_real_real(cos_real,aa_real_real(minus_minus_real(W),Z))),aa_real_real(cos_real,aa_real_real(plus_plus_real(W),Z)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% cos_times_cos
tff(fact_7638_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_7639_cos__plus__cos,axiom,
    ! [W: real,Z: real] : aa_real_real(plus_plus_real(aa_real_real(cos_real,W)),aa_real_real(cos_real,Z)) = aa_real_real(times_times_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(cos_real,aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(W),Z)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))),aa_real_real(cos_real,aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(W),Z)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% cos_plus_cos
tff(fact_7640_set__decode__plus__power__2,axiom,
    ! [Nb: nat,Z: nat] :
      ( ~ aa_set_nat_o(member_nat(Nb),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))),Nb)),Z)) = insert_nat(Nb,aa_nat_set_nat(nat_set_decode,Z)) ) ) ).

% set_decode_plus_power_2
tff(fact_7641_cos__double__less__one,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_num_real(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(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Xb))),one_one_real) ) ) ).

% cos_double_less_one
tff(fact_7642_cos__gt__zero,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(divide_divide_real(pi),aa_num_real(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,Xb)) ) ) ).

% cos_gt_zero
tff(fact_7643_cos__60,axiom,
    aa_real_real(cos_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit1,one)))) = aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% cos_60
tff(fact_7644_cos__30,axiom,
    aa_real_real(cos_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,one))))) = aa_real_real(divide_divide_real(aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit1,one)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% cos_30
tff(fact_7645_cos__one__2pi__int,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(cos_real,Xb) = one_one_real )
    <=> ? [X4: int] : Xb = aa_real_real(times_times_real(aa_real_real(times_times_real(aa_int_real(ring_1_of_int_real,X4)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),pi) ) ).

% cos_one_2pi_int
tff(fact_7646_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_7647_cos__double__cos,axiom,
    ! [W: real] : aa_real_real(cos_real,aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),W)) = aa_real_real(minus_minus_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(cos_real,W)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),one_one_real) ).

% cos_double_cos
tff(fact_7648_cos__treble__cos,axiom,
    ! [Xb: complex] : aa_complex_complex(cos_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit1,one))),Xb)) = 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,Xb)),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,Xb))) ).

% cos_treble_cos
tff(fact_7649_cos__treble__cos,axiom,
    ! [Xb: real] : aa_real_real(cos_real,aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit1,one))),Xb)) = aa_real_real(minus_minus_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(cos_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))))),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit1,one))),aa_real_real(cos_real,Xb))) ).

% cos_treble_cos
tff(fact_7650_cos__gt__zero__pi,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(divide_divide_real(pi),aa_num_real(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,Xb)) ) ) ).

% cos_gt_zero_pi
tff(fact_7651_cos__ge__zero,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(divide_divide_real(pi),aa_num_real(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,Xb)) ) ) ).

% cos_ge_zero
tff(fact_7652_and__int_Opinduct,axiom,
    ! [A0: int,A1: int,P: fun_int_fun_int_o] :
      ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(bit_and_int_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A0),A1))
     => ( ! [K2: int,L3: int] :
            ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(bit_and_int_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,K2),L3))
           => ( ( ~ ( member_int(K2,insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
                    & member_int(L3,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,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L3),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) )
             => aa_int_o(aa_int_fun_int_o(P,K2),L3) ) )
       => aa_int_o(aa_int_fun_int_o(P,A0),A1) ) ) ).

% and_int.pinduct
tff(fact_7653_double__arith__series,axiom,
    ! [A: complex,D: complex,Nb: nat] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_fq(A),D)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb))) = aa_complex_complex(times_times_complex(aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,Nb)),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,Nb)),D))) ).

% double_arith_series
tff(fact_7654_double__arith__series,axiom,
    ! [A: int,D: int,Nb: 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_fr(A),D),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb))) = 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,Nb)),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,Nb)),D))) ).

% double_arith_series
tff(fact_7655_double__arith__series,axiom,
    ! [A: rat,D: rat,Nb: nat] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_fs(A),D),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb))) = aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,Nb)),one_one_rat)),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),A)),aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,Nb)),D))) ).

% double_arith_series
tff(fact_7656_double__arith__series,axiom,
    ! [A: nat,D: nat,Nb: 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_ft(A),D)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb))) = 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,Nb)),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,Nb)),D))) ).

% double_arith_series
tff(fact_7657_double__arith__series,axiom,
    ! [A: real,D: real,Nb: nat] : aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_fu(A),D)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb))) = aa_real_real(times_times_real(aa_real_real(plus_plus_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),one_one_real)),aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),A)),aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),D))) ).

% double_arith_series
tff(fact_7658_double__gauss__sum,axiom,
    ! [Nb: 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),Nb))) = aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,Nb)),aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,Nb)),one_one_complex)) ).

% double_gauss_sum
tff(fact_7659_double__gauss__sum,axiom,
    ! [Nb: 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),Nb))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),one_one_int)) ).

% double_gauss_sum
tff(fact_7660_double__gauss__sum,axiom,
    ! [Nb: nat] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),groups2906978787729119204at_rat(semiri681578069525770553at_rat,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb))) = aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,Nb)),aa_rat_rat(plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,Nb)),one_one_rat)) ).

% double_gauss_sum
tff(fact_7661_double__gauss__sum,axiom,
    ! [Nb: 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),Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,Nb)),one_one_nat)) ).

% double_gauss_sum
tff(fact_7662_double__gauss__sum,axiom,
    ! [Nb: nat] : aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_set_nat_real(groups6591440286371151544t_real(semiri5074537144036343181t_real),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb))) = aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(plus_plus_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),one_one_real)) ).

% double_gauss_sum
tff(fact_7663_cos__arctan,axiom,
    ! [Xb: real] : aa_real_real(cos_real,aa_real_real(arctan,Xb)) = aa_real_real(divide_divide_real(one_one_real),aa_real_real(sqrt,aa_real_real(plus_plus_real(one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% cos_arctan
tff(fact_7664_arith__series,axiom,
    ! [A: int,D: int,Nb: nat] : groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_fr(A),D),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb)) = 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,Nb)),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,Nb)),D)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% arith_series
tff(fact_7665_arith__series,axiom,
    ! [A: nat,D: nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ft(A),D)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb)) = 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,Nb)),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,Nb)),D)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% arith_series
tff(fact_7666_gauss__sum,axiom,
    ! [Nb: nat] : groups3539618377306564664at_int(semiri1314217659103216013at_int,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb)) = 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,Nb)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),one_one_int))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% gauss_sum
tff(fact_7667_gauss__sum,axiom,
    ! [Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(semiri1316708129612266289at_nat),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb)) = 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,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,Nb)),one_one_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% gauss_sum
tff(fact_7668_double__gauss__sum__from__Suc__0,axiom,
    ! [Nb: 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)),Nb))) = aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,Nb)),aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,Nb)),one_one_complex)) ).

% double_gauss_sum_from_Suc_0
tff(fact_7669_double__gauss__sum__from__Suc__0,axiom,
    ! [Nb: 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)),Nb))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),one_one_int)) ).

% double_gauss_sum_from_Suc_0
tff(fact_7670_double__gauss__sum__from__Suc__0,axiom,
    ! [Nb: nat] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),groups2906978787729119204at_rat(semiri681578069525770553at_rat,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),Nb))) = aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,Nb)),aa_rat_rat(plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,Nb)),one_one_rat)) ).

% double_gauss_sum_from_Suc_0
tff(fact_7671_double__gauss__sum__from__Suc__0,axiom,
    ! [Nb: 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)),Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,Nb)),one_one_nat)) ).

% double_gauss_sum_from_Suc_0
tff(fact_7672_double__gauss__sum__from__Suc__0,axiom,
    ! [Nb: nat] : aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_set_nat_real(groups6591440286371151544t_real(semiri5074537144036343181t_real),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),Nb))) = aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(plus_plus_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),one_one_real)) ).

% double_gauss_sum_from_Suc_0
tff(fact_7673_Bernoulli__inequality__even,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(one_one_real),aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),Xb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(plus_plus_real(one_one_real),Xb)),Nb)) ) ).

% Bernoulli_inequality_even
tff(fact_7674_sum__gp__offset,axiom,
    ! [Xb: complex,Mb: nat,Nb: nat] :
      aa_set_nat_complex(groups2073611262835488442omplex(power_power_complex(Xb)),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb))) = $ite(Xb = one_one_complex,aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,Nb)),one_one_complex),aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(Xb),Mb)),aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(power_power_complex(Xb),aa_nat_nat(suc,Nb))))),aa_complex_complex(minus_minus_complex(one_one_complex),Xb))) ).

% sum_gp_offset
tff(fact_7675_sum__gp__offset,axiom,
    ! [Xb: rat,Mb: nat,Nb: nat] :
      groups2906978787729119204at_rat(power_power_rat(Xb),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb))) = $ite(Xb = one_one_rat,aa_rat_rat(plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,Nb)),one_one_rat),aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(Xb),Mb)),aa_rat_rat(minus_minus_rat(one_one_rat),aa_nat_rat(power_power_rat(Xb),aa_nat_nat(suc,Nb))))),aa_rat_rat(minus_minus_rat(one_one_rat),Xb))) ).

% sum_gp_offset
tff(fact_7676_sum__gp__offset,axiom,
    ! [Xb: real,Mb: nat,Nb: nat] :
      aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,Xb)),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb))) = $ite(Xb = one_one_real,aa_real_real(plus_plus_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),one_one_real),aa_real_real(divide_divide_real(aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Mb)),aa_real_real(minus_minus_real(one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_nat_nat(suc,Nb))))),aa_real_real(minus_minus_real(one_one_real),Xb))) ).

% sum_gp_offset
tff(fact_7677_exp__ge__one__plus__x__over__n__power__n,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,Nb))),Xb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(plus_plus_real(one_one_real),aa_real_real(divide_divide_real(Xb),aa_nat_real(semiri5074537144036343181t_real,Nb)))),Nb)),aa_real_real(exp_real,Xb)) ) ) ).

% exp_ge_one_plus_x_over_n_power_n
tff(fact_7678_exp__ge__one__minus__x__over__n__power__n,axiom,
    ! [Xb: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_nat_real(semiri5074537144036343181t_real,Nb))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(minus_minus_real(one_one_real),aa_real_real(divide_divide_real(Xb),aa_nat_real(semiri5074537144036343181t_real,Nb)))),Nb)),aa_real_real(exp_real,aa_real_real(uminus_uminus_real,Xb))) ) ) ).

% exp_ge_one_minus_x_over_n_power_n
tff(fact_7679_cot__gt__zero,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(divide_divide_real(pi),aa_num_real(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,Xb)) ) ) ).

% cot_gt_zero
tff(fact_7680_cos__zero__iff__int,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(cos_real,Xb) = zero_zero_real )
    <=> ? [I5: int] :
          ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),I5)
          & ( Xb = aa_real_real(times_times_real(aa_int_real(ring_1_of_int_real,I5)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ) ).

% cos_zero_iff_int
tff(fact_7681_gauss__sum__from__Suc__0,axiom,
    ! [Nb: nat] : groups3539618377306564664at_int(semiri1314217659103216013at_int,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),Nb)) = 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,Nb)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),one_one_int))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% gauss_sum_from_Suc_0
tff(fact_7682_gauss__sum__from__Suc__0,axiom,
    ! [Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(semiri1316708129612266289at_nat),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),Nb)) = 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,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,Nb)),one_one_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% gauss_sum_from_Suc_0
tff(fact_7683_of__nat__code__if,axiom,
    ! [Nb: nat] :
      aa_nat_complex(semiri8010041392384452111omplex,Nb) = $ite(Nb = zero_zero_nat,zero_zero_complex,produc1917071388513777916omplex(aTP_Lamm_fv,divmod_nat(Nb,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% of_nat_code_if
tff(fact_7684_of__nat__code__if,axiom,
    ! [Nb: nat] :
      aa_nat_int(semiri1314217659103216013at_int,Nb) = $ite(Nb = zero_zero_nat,zero_zero_int,aa_Pro4240683299535347697at_int(produc6840382203811409530at_int(aTP_Lamm_fw),divmod_nat(Nb,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% of_nat_code_if
tff(fact_7685_of__nat__code__if,axiom,
    ! [Nb: nat] :
      aa_nat_real(semiri5074537144036343181t_real,Nb) = $ite(Nb = zero_zero_nat,zero_zero_real,aa_Pro6168199777849092593t_real(produc1703576794950452218t_real(aTP_Lamm_fx),divmod_nat(Nb,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% of_nat_code_if
tff(fact_7686_of__nat__code__if,axiom,
    ! [Nb: nat] :
      aa_nat_nat(semiri1316708129612266289at_nat,Nb) = $ite(Nb = zero_zero_nat,zero_zero_nat,aa_Pro4243173770044397973at_nat(produc6842872674320459806at_nat(aTP_Lamm_fy),divmod_nat(Nb,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% of_nat_code_if
tff(fact_7687_of__nat__code__if,axiom,
    ! [Nb: nat] :
      aa_nat_rat(semiri681578069525770553at_rat,Nb) = $ite(Nb = zero_zero_nat,zero_zero_rat,produc6207742614233964070at_rat(aTP_Lamm_fz,divmod_nat(Nb,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% of_nat_code_if
tff(fact_7688_linear__plus__1__le__power,axiom,
    ! [Xb: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),Xb)),one_one_real)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(plus_plus_real(Xb),one_one_real)),Nb)) ) ).

% linear_plus_1_le_power
tff(fact_7689_monoseq__arctan__series,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),one_one_real)
     => topolo6980174941875973593q_real(aa_real_fun_nat_real(aTP_Lamm_ga,Xb)) ) ).

% monoseq_arctan_series
tff(fact_7690_lemma__termdiff3,axiom,
    ! [H: real,Z: real,K5: real,Nb: 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(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(minus_minus_real(aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(plus_plus_real(Z),H)),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),Nb))),H)),aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))))),aa_real_real(times_times_real(aa_real_real(times_times_real(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))),aa_nat_real(aa_real_fun_nat_real(power_power_real,K5),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(real_V7735802525324610683m_real,H))) ) ) ) ).

% lemma_termdiff3
tff(fact_7691_lemma__termdiff3,axiom,
    ! [H: complex,Z: complex,K5: real,Nb: 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)),Nb)),aa_nat_complex(power_power_complex(Z),Nb))),H)),aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,Nb)),aa_nat_complex(power_power_complex(Z),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))))),aa_real_real(times_times_real(aa_real_real(times_times_real(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))),aa_nat_real(aa_real_fun_nat_real(power_power_real,K5),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_complex_real(real_V1022390504157884413omplex,H))) ) ) ) ).

% lemma_termdiff3
tff(fact_7692_ln__series,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))
       => ( aa_real_real(ln_ln_real,Xb) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_gb,Xb)) ) ) ) ).

% ln_series
tff(fact_7693_powser__zero,axiom,
    ! [F: fun_nat_complex] : suminf_complex(aTP_Lamm_gc(F)) = aa_nat_complex(F,zero_zero_nat) ).

% powser_zero
tff(fact_7694_powser__zero,axiom,
    ! [F: fun_nat_real] : suminf_real(aTP_Lamm_gd(F)) = aa_nat_real(F,zero_zero_nat) ).

% powser_zero
tff(fact_7695_int__int__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,Mb) = aa_nat_int(semiri1314217659103216013at_int,Nb) )
    <=> ( Mb = Nb ) ) ).

% int_int_eq
tff(fact_7696_complex__mod__triangle__ineq2,axiom,
    ! [B: complex,A: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_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_7697_monoseq__realpow,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real)
       => topolo6980174941875973593q_real(aa_real_fun_nat_real(power_power_real,Xb)) ) ) ).

% monoseq_realpow
tff(fact_7698_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(divide_divide_real(one_one_real),aa_num_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_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) ).

% exp_bound_half
tff(fact_7699_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(divide_divide_real(one_one_real),aa_num_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_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) ).

% exp_bound_half
tff(fact_7700_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(divide_divide_real(one_one_real),aa_num_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(plus_plus_real(one_one_real),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(real_V7735802525324610683m_real,Z)))) ) ).

% exp_bound_lemma
tff(fact_7701_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(divide_divide_real(one_one_real),aa_num_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(plus_plus_real(one_one_real),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_complex_real(real_V1022390504157884413omplex,Z)))) ) ).

% exp_bound_lemma
tff(fact_7702_pi__series,axiom,
    aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))) = suminf_real(aTP_Lamm_ge) ).

% pi_series
tff(fact_7703_arctan__series,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),one_one_real)
     => ( aa_real_real(arctan,Xb) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_gf,Xb)) ) ) ).

% arctan_series
tff(fact_7704_norm__divide__numeral,axiom,
    ! [A: real,W: num] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(divide_divide_real(A),aa_num_real(numeral_numeral_real,W))) = aa_real_real(divide_divide_real(aa_real_real(real_V7735802525324610683m_real,A)),aa_num_real(numeral_numeral_real,W)) ).

% norm_divide_numeral
tff(fact_7705_norm__divide__numeral,axiom,
    ! [A: complex,W: num] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(divide1717551699836669952omplex(A),numera6690914467698888265omplex(W))) = aa_real_real(divide_divide_real(aa_complex_real(real_V1022390504157884413omplex,A)),aa_num_real(numeral_numeral_real,W)) ).

% norm_divide_numeral
tff(fact_7706_norm__mult__numeral2,axiom,
    ! [A: real,W: num] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(times_times_real(A),aa_num_real(numeral_numeral_real,W))) = aa_real_real(times_times_real(aa_real_real(real_V7735802525324610683m_real,A)),aa_num_real(numeral_numeral_real,W)) ).

% norm_mult_numeral2
tff(fact_7707_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(times_times_real(aa_complex_real(real_V1022390504157884413omplex,A)),aa_num_real(numeral_numeral_real,W)) ).

% norm_mult_numeral2
tff(fact_7708_norm__mult__numeral1,axiom,
    ! [W: num,A: real] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,W)),A)) = aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,W)),aa_real_real(real_V7735802525324610683m_real,A)) ).

% norm_mult_numeral1
tff(fact_7709_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(times_times_real(aa_num_real(numeral_numeral_real,W)),aa_complex_real(real_V1022390504157884413omplex,A)) ).

% norm_mult_numeral1
tff(fact_7710_norm__neg__numeral,axiom,
    ! [W: num] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))) = aa_num_real(numeral_numeral_real,W) ).

% norm_neg_numeral
tff(fact_7711_norm__neg__numeral,axiom,
    ! [W: num] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))) = aa_num_real(numeral_numeral_real,W) ).

% norm_neg_numeral
tff(fact_7712_norm__le__zero__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,Xb)),zero_zero_real)
    <=> ( Xb = zero_zero_real ) ) ).

% norm_le_zero_iff
tff(fact_7713_norm__le__zero__iff,axiom,
    ! [Xb: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Xb)),zero_zero_real)
    <=> ( Xb = zero_zero_complex ) ) ).

% norm_le_zero_iff
tff(fact_7714_norm__eq__zero,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(real_V7735802525324610683m_real,Xb) = zero_zero_real )
    <=> ( Xb = zero_zero_real ) ) ).

% norm_eq_zero
tff(fact_7715_norm__eq__zero,axiom,
    ! [Xb: complex] :
      ( ( aa_complex_real(real_V1022390504157884413omplex,Xb) = zero_zero_real )
    <=> ( Xb = zero_zero_complex ) ) ).

% norm_eq_zero
tff(fact_7716_norm__zero,axiom,
    aa_real_real(real_V7735802525324610683m_real,zero_zero_real) = zero_zero_real ).

% norm_zero
tff(fact_7717_norm__zero,axiom,
    aa_complex_real(real_V1022390504157884413omplex,zero_zero_complex) = zero_zero_real ).

% norm_zero
tff(fact_7718_norm__one,axiom,
    aa_real_real(real_V7735802525324610683m_real,one_one_real) = one_one_real ).

% norm_one
tff(fact_7719_norm__one,axiom,
    aa_complex_real(real_V1022390504157884413omplex,one_one_complex) = one_one_real ).

% norm_one
tff(fact_7720_norm__numeral,axiom,
    ! [W: num] : aa_real_real(real_V7735802525324610683m_real,aa_num_real(numeral_numeral_real,W)) = aa_num_real(numeral_numeral_real,W) ).

% norm_numeral
tff(fact_7721_norm__numeral,axiom,
    ! [W: num] : aa_complex_real(real_V1022390504157884413omplex,numera6690914467698888265omplex(W)) = aa_num_real(numeral_numeral_real,W) ).

% norm_numeral
tff(fact_7722_zero__less__norm__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(real_V7735802525324610683m_real,Xb))
    <=> ( Xb != zero_zero_real ) ) ).

% zero_less_norm_iff
tff(fact_7723_zero__less__norm__iff,axiom,
    ! [Xb: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(real_V1022390504157884413omplex,Xb))
    <=> ( Xb != zero_zero_complex ) ) ).

% zero_less_norm_iff
tff(fact_7724_norm__mult,axiom,
    ! [Xb: real,Yb: real] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(times_times_real(Xb),Yb)) = aa_real_real(times_times_real(aa_real_real(real_V7735802525324610683m_real,Xb)),aa_real_real(real_V7735802525324610683m_real,Yb)) ).

% norm_mult
tff(fact_7725_norm__mult,axiom,
    ! [Xb: complex,Yb: complex] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(times_times_complex(Xb),Yb)) = aa_real_real(times_times_real(aa_complex_real(real_V1022390504157884413omplex,Xb)),aa_complex_real(real_V1022390504157884413omplex,Yb)) ).

% norm_mult
tff(fact_7726_norm__divide,axiom,
    ! [A: real,B: real] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(divide_divide_real(A),B)) = aa_real_real(divide_divide_real(aa_real_real(real_V7735802525324610683m_real,A)),aa_real_real(real_V7735802525324610683m_real,B)) ).

% norm_divide
tff(fact_7727_norm__divide,axiom,
    ! [A: complex,B: complex] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(divide1717551699836669952omplex(A),B)) = aa_real_real(divide_divide_real(aa_complex_real(real_V1022390504157884413omplex,A)),aa_complex_real(real_V1022390504157884413omplex,B)) ).

% norm_divide
tff(fact_7728_norm__power,axiom,
    ! [Xb: real,Nb: nat] : aa_real_real(real_V7735802525324610683m_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(real_V7735802525324610683m_real,Xb)),Nb) ).

% norm_power
tff(fact_7729_norm__power,axiom,
    ! [Xb: complex,Nb: nat] : aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(power_power_complex(Xb),Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(real_V1022390504157884413omplex,Xb)),Nb) ).

% norm_power
tff(fact_7730_norm__uminus__minus,axiom,
    ! [Xb: real,Yb: real] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,Xb)),Yb)) = aa_real_real(real_V7735802525324610683m_real,aa_real_real(plus_plus_real(Xb),Yb)) ).

% norm_uminus_minus
tff(fact_7731_norm__uminus__minus,axiom,
    ! [Xb: complex,Yb: complex] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,Xb)),Yb)) = aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(Xb),Yb)) ).

% norm_uminus_minus
tff(fact_7732_nonzero__norm__divide,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( aa_real_real(real_V7735802525324610683m_real,aa_real_real(divide_divide_real(A),B)) = aa_real_real(divide_divide_real(aa_real_real(real_V7735802525324610683m_real,A)),aa_real_real(real_V7735802525324610683m_real,B)) ) ) ).

% nonzero_norm_divide
tff(fact_7733_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(divide_divide_real(aa_complex_real(real_V1022390504157884413omplex,A)),aa_complex_real(real_V1022390504157884413omplex,B)) ) ) ).

% nonzero_norm_divide
tff(fact_7734_power__eq__imp__eq__norm,axiom,
    ! [W: real,Nb: nat,Z: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,W),Nb) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),Nb) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( aa_real_real(real_V7735802525324610683m_real,W) = aa_real_real(real_V7735802525324610683m_real,Z) ) ) ) ).

% power_eq_imp_eq_norm
tff(fact_7735_power__eq__imp__eq__norm,axiom,
    ! [W: complex,Nb: nat,Z: complex] :
      ( ( aa_nat_complex(power_power_complex(W),Nb) = aa_nat_complex(power_power_complex(Z),Nb) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( aa_complex_real(real_V1022390504157884413omplex,W) = aa_complex_real(real_V1022390504157884413omplex,Z) ) ) ) ).

% power_eq_imp_eq_norm
tff(fact_7736_norm__mult__less,axiom,
    ! [Xb: real,R2: real,Yb: real,S2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Xb)),R2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Yb)),S2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(times_times_real(Xb),Yb))),aa_real_real(times_times_real(R2),S2)) ) ) ).

% norm_mult_less
tff(fact_7737_norm__mult__less,axiom,
    ! [Xb: complex,R2: real,Yb: complex,S2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Xb)),R2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Yb)),S2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(times_times_complex(Xb),Yb))),aa_real_real(times_times_real(R2),S2)) ) ) ).

% norm_mult_less
tff(fact_7738_norm__mult__ineq,axiom,
    ! [Xb: real,Yb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(times_times_real(Xb),Yb))),aa_real_real(times_times_real(aa_real_real(real_V7735802525324610683m_real,Xb)),aa_real_real(real_V7735802525324610683m_real,Yb))) ).

% norm_mult_ineq
tff(fact_7739_norm__mult__ineq,axiom,
    ! [Xb: complex,Yb: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(times_times_complex(Xb),Yb))),aa_real_real(times_times_real(aa_complex_real(real_V1022390504157884413omplex,Xb)),aa_complex_real(real_V1022390504157884413omplex,Yb))) ).

% norm_mult_ineq
tff(fact_7740_norm__triangle__lt,axiom,
    ! [Xb: real,Yb: real,E2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(aa_real_real(real_V7735802525324610683m_real,Xb)),aa_real_real(real_V7735802525324610683m_real,Yb))),E2)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(plus_plus_real(Xb),Yb))),E2) ) ).

% norm_triangle_lt
tff(fact_7741_norm__triangle__lt,axiom,
    ! [Xb: complex,Yb: complex,E2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(plus_plus_real(aa_complex_real(real_V1022390504157884413omplex,Xb)),aa_complex_real(real_V1022390504157884413omplex,Yb))),E2)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(Xb),Yb))),E2) ) ).

% norm_triangle_lt
tff(fact_7742_norm__add__less,axiom,
    ! [Xb: real,R2: real,Yb: real,S2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Xb)),R2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Yb)),S2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(plus_plus_real(Xb),Yb))),aa_real_real(plus_plus_real(R2),S2)) ) ) ).

% norm_add_less
tff(fact_7743_norm__add__less,axiom,
    ! [Xb: complex,R2: real,Yb: complex,S2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Xb)),R2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Yb)),S2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(Xb),Yb))),aa_real_real(plus_plus_real(R2),S2)) ) ) ).

% norm_add_less
tff(fact_7744_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(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(plus_plus_real(aa_real_real(real_V7735802525324610683m_real,A)),C)) ) ).

% norm_add_leD
tff(fact_7745_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(plus_plus_real(aa_complex_real(real_V1022390504157884413omplex,A)),C)) ) ).

% norm_add_leD
tff(fact_7746_norm__triangle__le,axiom,
    ! [Xb: real,Yb: real,E2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(aa_real_real(real_V7735802525324610683m_real,Xb)),aa_real_real(real_V7735802525324610683m_real,Yb))),E2)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(plus_plus_real(Xb),Yb))),E2) ) ).

% norm_triangle_le
tff(fact_7747_norm__triangle__le,axiom,
    ! [Xb: complex,Yb: complex,E2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(aa_complex_real(real_V1022390504157884413omplex,Xb)),aa_complex_real(real_V1022390504157884413omplex,Yb))),E2)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(Xb),Yb))),E2) ) ).

% norm_triangle_le
tff(fact_7748_norm__triangle__ineq,axiom,
    ! [Xb: real,Yb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(plus_plus_real(Xb),Yb))),aa_real_real(plus_plus_real(aa_real_real(real_V7735802525324610683m_real,Xb)),aa_real_real(real_V7735802525324610683m_real,Yb))) ).

% norm_triangle_ineq
tff(fact_7749_norm__triangle__ineq,axiom,
    ! [Xb: complex,Yb: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(Xb),Yb))),aa_real_real(plus_plus_real(aa_complex_real(real_V1022390504157884413omplex,Xb)),aa_complex_real(real_V1022390504157884413omplex,Yb))) ).

% norm_triangle_ineq
tff(fact_7750_norm__triangle__mono,axiom,
    ! [A: real,R2: real,B: real,S2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,A)),R2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,B)),S2)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(plus_plus_real(A),B))),aa_real_real(plus_plus_real(R2),S2)) ) ) ).

% norm_triangle_mono
tff(fact_7751_norm__triangle__mono,axiom,
    ! [A: complex,R2: real,B: complex,S2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,A)),R2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,B)),S2)
       => 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(plus_plus_real(R2),S2)) ) ) ).

% norm_triangle_mono
tff(fact_7752_norm__power__ineq,axiom,
    ! [Xb: real,Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(real_V7735802525324610683m_real,Xb)),Nb)) ).

% norm_power_ineq
tff(fact_7753_norm__power__ineq,axiom,
    ! [Xb: complex,Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(power_power_complex(Xb),Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(real_V1022390504157884413omplex,Xb)),Nb)) ).

% norm_power_ineq
tff(fact_7754_norm__diff__triangle__less,axiom,
    ! [Xb: real,Yb: 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(minus_minus_real(Xb),Yb))),E1)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(minus_minus_real(Yb),Z))),E22)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(minus_minus_real(Xb),Z))),aa_real_real(plus_plus_real(E1),E22)) ) ) ).

% norm_diff_triangle_less
tff(fact_7755_norm__diff__triangle__less,axiom,
    ! [Xb: complex,Yb: 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(Xb),Yb))),E1)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(Yb),Z))),E22)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(Xb),Z))),aa_real_real(plus_plus_real(E1),E22)) ) ) ).

% norm_diff_triangle_less
tff(fact_7756_norm__triangle__le__diff,axiom,
    ! [Xb: real,Yb: real,E2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(aa_real_real(real_V7735802525324610683m_real,Xb)),aa_real_real(real_V7735802525324610683m_real,Yb))),E2)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(minus_minus_real(Xb),Yb))),E2) ) ).

% norm_triangle_le_diff
tff(fact_7757_norm__triangle__le__diff,axiom,
    ! [Xb: complex,Yb: complex,E2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(plus_plus_real(aa_complex_real(real_V1022390504157884413omplex,Xb)),aa_complex_real(real_V1022390504157884413omplex,Yb))),E2)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(Xb),Yb))),E2) ) ).

% norm_triangle_le_diff
tff(fact_7758_norm__diff__triangle__le,axiom,
    ! [Xb: real,Yb: 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(minus_minus_real(Xb),Yb))),E1)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(minus_minus_real(Yb),Z))),E22)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(minus_minus_real(Xb),Z))),aa_real_real(plus_plus_real(E1),E22)) ) ) ).

% norm_diff_triangle_le
tff(fact_7759_norm__diff__triangle__le,axiom,
    ! [Xb: complex,Yb: 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(Xb),Yb))),E1)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(Yb),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(Xb),Z))),aa_real_real(plus_plus_real(E1),E22)) ) ) ).

% norm_diff_triangle_le
tff(fact_7760_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(minus_minus_real(A),B))),aa_real_real(plus_plus_real(aa_real_real(real_V7735802525324610683m_real,A)),aa_real_real(real_V7735802525324610683m_real,B))) ).

% norm_triangle_ineq4
tff(fact_7761_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(plus_plus_real(aa_complex_real(real_V1022390504157884413omplex,A)),aa_complex_real(real_V1022390504157884413omplex,B))) ).

% norm_triangle_ineq4
tff(fact_7762_norm__triangle__sub,axiom,
    ! [Xb: real,Yb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,Xb)),aa_real_real(plus_plus_real(aa_real_real(real_V7735802525324610683m_real,Yb)),aa_real_real(real_V7735802525324610683m_real,aa_real_real(minus_minus_real(Xb),Yb)))) ).

% norm_triangle_sub
tff(fact_7763_norm__triangle__sub,axiom,
    ! [Xb: complex,Yb: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Xb)),aa_real_real(plus_plus_real(aa_complex_real(real_V1022390504157884413omplex,Yb)),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(Xb),Yb)))) ).

% norm_triangle_sub
tff(fact_7764_norm__diff__ineq,axiom,
    ! [A: real,B: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_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(plus_plus_real(A),B))) ).

% norm_diff_ineq
tff(fact_7765_norm__diff__ineq,axiom,
    ! [A: complex,B: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_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_7766_power__eq__1__iff,axiom,
    ! [W: real,Nb: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,W),Nb) = one_one_real )
     => ( ( aa_real_real(real_V7735802525324610683m_real,W) = one_one_real )
        | ( Nb = zero_zero_nat ) ) ) ).

% power_eq_1_iff
tff(fact_7767_power__eq__1__iff,axiom,
    ! [W: complex,Nb: nat] :
      ( ( aa_nat_complex(power_power_complex(W),Nb) = one_one_complex )
     => ( ( aa_complex_real(real_V1022390504157884413omplex,W) = one_one_real )
        | ( Nb = zero_zero_nat ) ) ) ).

% power_eq_1_iff
tff(fact_7768_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(minus_minus_real(aa_real_real(plus_plus_real(A),B)),aa_real_real(plus_plus_real(C),D)))),aa_real_real(plus_plus_real(aa_real_real(real_V7735802525324610683m_real,aa_real_real(minus_minus_real(A),C))),aa_real_real(real_V7735802525324610683m_real,aa_real_real(minus_minus_real(B),D)))) ).

% norm_diff_triangle_ineq
tff(fact_7769_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(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_7770_square__norm__one,axiom,
    ! [Xb: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_real )
     => ( aa_real_real(real_V7735802525324610683m_real,Xb) = one_one_real ) ) ).

% square_norm_one
tff(fact_7771_square__norm__one,axiom,
    ! [Xb: complex] :
      ( ( aa_nat_complex(power_power_complex(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_complex )
     => ( aa_complex_real(real_V1022390504157884413omplex,Xb) = one_one_real ) ) ).

% square_norm_one
tff(fact_7772_norm__power__diff,axiom,
    ! [Z: real,W: real,Mb: 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(minus_minus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),Mb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,W),Mb)))),aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_real_real(real_V7735802525324610683m_real,aa_real_real(minus_minus_real(Z),W)))) ) ) ).

% norm_power_diff
tff(fact_7773_norm__power__diff,axiom,
    ! [Z: complex,W: complex,Mb: 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),Mb)),aa_nat_complex(power_power_complex(W),Mb)))),aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(Z),W)))) ) ) ).

% norm_power_diff
tff(fact_7774_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(aa_real_fun_nat_real(power_power_real,C)) = aa_real_real(divide_divide_real(one_one_real),aa_real_real(minus_minus_real(one_one_real),C)) ) ) ).

% suminf_geometric
tff(fact_7775_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_7776_suminf__zero,axiom,
    suminf_complex(aTP_Lamm_gg) = zero_zero_complex ).

% suminf_zero
tff(fact_7777_suminf__zero,axiom,
    suminf_real(aTP_Lamm_de) = zero_zero_real ).

% suminf_zero
tff(fact_7778_suminf__zero,axiom,
    suminf_nat(aTP_Lamm_am) = zero_zero_nat ).

% suminf_zero
tff(fact_7779_suminf__zero,axiom,
    suminf_int(aTP_Lamm_gh) = zero_zero_int ).

% suminf_zero
tff(fact_7780_sin__cos__npi,axiom,
    ! [Nb: nat] : aa_real_real(sin_real,aa_real_real(divide_divide_real(aa_real_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))),Nb)))),pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Nb) ).

% sin_cos_npi
tff(fact_7781_upto_Opinduct,axiom,
    ! [A0: int,A1: int,P: fun_int_fun_int_o] :
      ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(upto_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A0),A1))
     => ( ! [I2: int,J2: int] :
            ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(upto_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(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_7782_lemma__termdiff2,axiom,
    ! [H: complex,Z: complex,Nb: 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)),Nb)),aa_nat_complex(power_power_complex(Z),Nb))),H)),aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,Nb)),aa_nat_complex(power_power_complex(Z),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),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_gj(H),Z),Nb)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))) ) ) ).

% lemma_termdiff2
tff(fact_7783_lemma__termdiff2,axiom,
    ! [H: rat,Z: rat,Nb: nat] :
      ( ( H != zero_zero_rat )
     => ( aa_rat_rat(minus_minus_rat(aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_nat_rat(power_power_rat(aa_rat_rat(plus_plus_rat(Z),H)),Nb)),aa_nat_rat(power_power_rat(Z),Nb))),H)),aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,Nb)),aa_nat_rat(power_power_rat(Z),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),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_gl(H),Z),Nb),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))) ) ) ).

% lemma_termdiff2
tff(fact_7784_lemma__termdiff2,axiom,
    ! [H: real,Z: real,Nb: nat] :
      ( ( H != zero_zero_real )
     => ( aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(plus_plus_real(Z),H)),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),Nb))),H)),aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))) = aa_real_real(times_times_real(H),aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_gn(H),Z),Nb)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))) ) ) ).

% lemma_termdiff2
tff(fact_7785_lessThan__iff,axiom,
    ! [I3: set_nat,K: set_nat] :
      ( member_set_nat(I3,set_or890127255671739683et_nat(K))
    <=> aa_set_nat_o(ord_less_set_nat(I3),K) ) ).

% lessThan_iff
tff(fact_7786_lessThan__iff,axiom,
    ! [I3: rat,K: rat] :
      ( member_rat(I3,set_ord_lessThan_rat(K))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,I3),K) ) ).

% lessThan_iff
tff(fact_7787_lessThan__iff,axiom,
    ! [I3: num,K: num] :
      ( member_num(I3,set_ord_lessThan_num(K))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,I3),K) ) ).

% lessThan_iff
tff(fact_7788_lessThan__iff,axiom,
    ! [I3: int,K: int] :
      ( member_int(I3,set_ord_lessThan_int(K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,I3),K) ) ).

% lessThan_iff
tff(fact_7789_lessThan__iff,axiom,
    ! [I3: nat,K: nat] :
      ( aa_set_nat_o(member_nat(I3),aa_nat_set_nat(set_ord_lessThan_nat,K))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),K) ) ).

% lessThan_iff
tff(fact_7790_lessThan__iff,axiom,
    ! [I3: real,K: real] :
      ( member_real(I3,set_or5984915006950818249n_real(K))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,I3),K) ) ).

% lessThan_iff
tff(fact_7791_sin__zero,axiom,
    aa_complex_complex(sin_complex,zero_zero_complex) = zero_zero_complex ).

% sin_zero
tff(fact_7792_sin__zero,axiom,
    aa_real_real(sin_real,zero_zero_real) = zero_zero_real ).

% sin_zero
tff(fact_7793_lessThan__subset__iff,axiom,
    ! [Xb: rat,Yb: rat] :
      ( ord_less_eq_set_rat(set_ord_lessThan_rat(Xb),set_ord_lessThan_rat(Yb))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Yb) ) ).

% lessThan_subset_iff
tff(fact_7794_lessThan__subset__iff,axiom,
    ! [Xb: num,Yb: num] :
      ( ord_less_eq_set_num(set_ord_lessThan_num(Xb),set_ord_lessThan_num(Yb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Yb) ) ).

% lessThan_subset_iff
tff(fact_7795_lessThan__subset__iff,axiom,
    ! [Xb: int,Yb: int] :
      ( ord_less_eq_set_int(set_ord_lessThan_int(Xb),set_ord_lessThan_int(Yb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Yb) ) ).

% lessThan_subset_iff
tff(fact_7796_lessThan__subset__iff,axiom,
    ! [Xb: nat,Yb: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(set_ord_lessThan_nat,Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Yb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Yb) ) ).

% lessThan_subset_iff
tff(fact_7797_lessThan__subset__iff,axiom,
    ! [Xb: real,Yb: real] :
      ( ord_less_eq_set_real(set_or5984915006950818249n_real(Xb),set_or5984915006950818249n_real(Yb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Yb) ) ).

% lessThan_subset_iff
tff(fact_7798_lessThan__0,axiom,
    aa_nat_set_nat(set_ord_lessThan_nat,zero_zero_nat) = bot_bot_set_nat ).

% lessThan_0
tff(fact_7799_sum_OlessThan__Suc,axiom,
    ! [G: fun_nat_rat,Nb: nat] : groups2906978787729119204at_rat(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = aa_rat_rat(plus_plus_rat(groups2906978787729119204at_rat(G,aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_nat_rat(G,Nb)) ).

% sum.lessThan_Suc
tff(fact_7800_sum_OlessThan__Suc,axiom,
    ! [G: fun_nat_int,Nb: nat] : groups3539618377306564664at_int(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_nat_int(G,Nb)) ).

% sum.lessThan_Suc
tff(fact_7801_sum_OlessThan__Suc,axiom,
    ! [G: fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = 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,Nb))),aa_nat_nat(G,Nb)) ).

% sum.lessThan_Suc
tff(fact_7802_sum_OlessThan__Suc,axiom,
    ! [G: fun_nat_real,Nb: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_nat_real(G,Nb)) ).

% sum.lessThan_Suc
tff(fact_7803_sin__periodic__pi,axiom,
    ! [Xb: real] : aa_real_real(sin_real,aa_real_real(plus_plus_real(Xb),pi)) = aa_real_real(uminus_uminus_real,aa_real_real(sin_real,Xb)) ).

% sin_periodic_pi
tff(fact_7804_sin__periodic__pi2,axiom,
    ! [Xb: real] : aa_real_real(sin_real,aa_real_real(plus_plus_real(pi),Xb)) = aa_real_real(uminus_uminus_real,aa_real_real(sin_real,Xb)) ).

% sin_periodic_pi2
tff(fact_7805_sin__cos__squared__add3,axiom,
    ! [Xb: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(aa_complex_complex(cos_complex,Xb)),aa_complex_complex(cos_complex,Xb))),aa_complex_complex(times_times_complex(aa_complex_complex(sin_complex,Xb)),aa_complex_complex(sin_complex,Xb))) = one_one_complex ).

% sin_cos_squared_add3
tff(fact_7806_sin__cos__squared__add3,axiom,
    ! [Xb: real] : aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_real_real(cos_real,Xb)),aa_real_real(cos_real,Xb))),aa_real_real(times_times_real(aa_real_real(sin_real,Xb)),aa_real_real(sin_real,Xb))) = one_one_real ).

% sin_cos_squared_add3
tff(fact_7807_sin__npi2,axiom,
    ! [Nb: nat] : aa_real_real(sin_real,aa_real_real(times_times_real(pi),aa_nat_real(semiri5074537144036343181t_real,Nb))) = zero_zero_real ).

% sin_npi2
tff(fact_7808_sin__npi,axiom,
    ! [Nb: nat] : aa_real_real(sin_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),pi)) = zero_zero_real ).

% sin_npi
tff(fact_7809_sin__npi__int,axiom,
    ! [Nb: int] : aa_real_real(sin_real,aa_real_real(times_times_real(pi),aa_int_real(ring_1_of_int_real,Nb))) = zero_zero_real ).

% sin_npi_int
tff(fact_7810_sin__two__pi,axiom,
    aa_real_real(sin_real,aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)) = zero_zero_real ).

% sin_two_pi
tff(fact_7811_sin__pi__half,axiom,
    aa_real_real(sin_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = one_one_real ).

% sin_pi_half
tff(fact_7812_sin__periodic,axiom,
    ! [Xb: real] : aa_real_real(sin_real,aa_real_real(plus_plus_real(Xb),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))) = aa_real_real(sin_real,Xb) ).

% sin_periodic
tff(fact_7813_sin__cos__squared__add,axiom,
    ! [Xb: real] : aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sin_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(cos_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real ).

% sin_cos_squared_add
tff(fact_7814_sin__cos__squared__add,axiom,
    ! [Xb: complex] : aa_complex_complex(plus_plus_complex(aa_nat_complex(power_power_complex(aa_complex_complex(sin_complex,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_complex(power_power_complex(aa_complex_complex(cos_complex,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_complex ).

% sin_cos_squared_add
tff(fact_7815_sin__cos__squared__add2,axiom,
    ! [Xb: real] : aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(cos_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sin_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real ).

% sin_cos_squared_add2
tff(fact_7816_sin__cos__squared__add2,axiom,
    ! [Xb: complex] : aa_complex_complex(plus_plus_complex(aa_nat_complex(power_power_complex(aa_complex_complex(cos_complex,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_complex(power_power_complex(aa_complex_complex(sin_complex,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_complex ).

% sin_cos_squared_add2
tff(fact_7817_sin__2npi,axiom,
    ! [Nb: nat] : aa_real_real(sin_real,aa_real_real(times_times_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Nb))),pi)) = zero_zero_real ).

% sin_2npi
tff(fact_7818_sin__2pi__minus,axiom,
    ! [Xb: real] : aa_real_real(sin_real,aa_real_real(minus_minus_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)),Xb)) = aa_real_real(uminus_uminus_real,aa_real_real(sin_real,Xb)) ).

% sin_2pi_minus
tff(fact_7819_sin__int__2pin,axiom,
    ! [Nb: int] : aa_real_real(sin_real,aa_real_real(times_times_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)),aa_int_real(ring_1_of_int_real,Nb))) = zero_zero_real ).

% sin_int_2pin
tff(fact_7820_sin__3over2__pi,axiom,
    aa_real_real(sin_real,aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_num_real(numeral_numeral_real,aa_num_num(bit1,one))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),pi)) = aa_real_real(uminus_uminus_real,one_one_real) ).

% sin_3over2_pi
tff(fact_7821_lessThan__def,axiom,
    ! [U: set_nat] : set_or890127255671739683et_nat(U) = collect_set_nat(aTP_Lamm_go(U)) ).

% lessThan_def
tff(fact_7822_lessThan__def,axiom,
    ! [U: rat] : set_ord_lessThan_rat(U) = collect_rat(aa_rat_fun_rat_o(aTP_Lamm_gp,U)) ).

% lessThan_def
tff(fact_7823_lessThan__def,axiom,
    ! [U: num] : set_ord_lessThan_num(U) = collect_num(aa_num_fun_num_o(aTP_Lamm_gq,U)) ).

% lessThan_def
tff(fact_7824_lessThan__def,axiom,
    ! [U: int] : set_ord_lessThan_int(U) = collect_int(aa_int_fun_int_o(aTP_Lamm_gr,U)) ).

% lessThan_def
tff(fact_7825_lessThan__def,axiom,
    ! [U: nat] : aa_nat_set_nat(set_ord_lessThan_nat,U) = collect_nat(aa_nat_fun_nat_o(aTP_Lamm_ei,U)) ).

% lessThan_def
tff(fact_7826_lessThan__def,axiom,
    ! [U: real] : set_or5984915006950818249n_real(U) = collect_real(aa_real_fun_real_o(aTP_Lamm_gs,U)) ).

% lessThan_def
tff(fact_7827_sin__le__one,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sin_real,Xb)),one_one_real) ).

% sin_le_one
tff(fact_7828_lessThan__strict__subset__iff,axiom,
    ! [Mb: rat,Nb: rat] :
      ( ord_less_set_rat(set_ord_lessThan_rat(Mb),set_ord_lessThan_rat(Nb))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Mb),Nb) ) ).

% lessThan_strict_subset_iff
tff(fact_7829_lessThan__strict__subset__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( ord_less_set_num(set_ord_lessThan_num(Mb),set_ord_lessThan_num(Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Nb) ) ).

% lessThan_strict_subset_iff
tff(fact_7830_lessThan__strict__subset__iff,axiom,
    ! [Mb: int,Nb: int] :
      ( ord_less_set_int(set_ord_lessThan_int(Mb),set_ord_lessThan_int(Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Mb),Nb) ) ).

% lessThan_strict_subset_iff
tff(fact_7831_lessThan__strict__subset__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_set_nat_o(ord_less_set_nat(aa_nat_set_nat(set_ord_lessThan_nat,Mb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% lessThan_strict_subset_iff
tff(fact_7832_lessThan__strict__subset__iff,axiom,
    ! [Mb: real,Nb: real] :
      ( ord_less_set_real(set_or5984915006950818249n_real(Mb),set_or5984915006950818249n_real(Nb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Mb),Nb) ) ).

% lessThan_strict_subset_iff
tff(fact_7833_polar__Ex,axiom,
    ! [Xb: real,Yb: real] :
    ? [R3: real,A4: real] :
      ( ( Xb = aa_real_real(times_times_real(R3),aa_real_real(cos_real,A4)) )
      & ( Yb = aa_real_real(times_times_real(R3),aa_real_real(sin_real,A4)) ) ) ).

% polar_Ex
tff(fact_7834_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_7835_lessThan__empty__iff,axiom,
    ! [Nb: nat] :
      ( ( aa_nat_set_nat(set_ord_lessThan_nat,Nb) = bot_bot_set_nat )
    <=> ( Nb = zero_zero_nat ) ) ).

% lessThan_empty_iff
tff(fact_7836_cos__one__sin__zero,axiom,
    ! [Xb: complex] :
      ( ( aa_complex_complex(cos_complex,Xb) = one_one_complex )
     => ( aa_complex_complex(sin_complex,Xb) = zero_zero_complex ) ) ).

% cos_one_sin_zero
tff(fact_7837_cos__one__sin__zero,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(cos_real,Xb) = one_one_real )
     => ( aa_real_real(sin_real,Xb) = zero_zero_real ) ) ).

% cos_one_sin_zero
tff(fact_7838_sin__add,axiom,
    ! [Xb: real,Yb: real] : aa_real_real(sin_real,aa_real_real(plus_plus_real(Xb),Yb)) = aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_real_real(sin_real,Xb)),aa_real_real(cos_real,Yb))),aa_real_real(times_times_real(aa_real_real(cos_real,Xb)),aa_real_real(sin_real,Yb))) ).

% sin_add
tff(fact_7839_sin__diff,axiom,
    ! [Xb: real,Yb: real] : aa_real_real(sin_real,aa_real_real(minus_minus_real(Xb),Yb)) = aa_real_real(minus_minus_real(aa_real_real(times_times_real(aa_real_real(sin_real,Xb)),aa_real_real(cos_real,Yb))),aa_real_real(times_times_real(aa_real_real(cos_real,Xb)),aa_real_real(sin_real,Yb))) ).

% sin_diff
tff(fact_7840_sin__ge__minus__one,axiom,
    ! [Xb: 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,Xb)) ).

% sin_ge_minus_one
tff(fact_7841_abs__sin__le__one,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(sin_real,Xb))),one_one_real) ).

% abs_sin_le_one
tff(fact_7842_cot__def,axiom,
    ! [X: complex] : aa_complex_complex(cot_complex,X) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(cos_complex,X)),aa_complex_complex(sin_complex,X)) ).

% cot_def
tff(fact_7843_cot__def,axiom,
    ! [X: real] : aa_real_real(cot_real,X) = aa_real_real(divide_divide_real(aa_real_real(cos_real,X)),aa_real_real(sin_real,X)) ).

% cot_def
tff(fact_7844_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_7845_sum_Onat__diff__reindex,axiom,
    ! [G: fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_gt(G),Nb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ).

% sum.nat_diff_reindex
tff(fact_7846_sum_Onat__diff__reindex,axiom,
    ! [G: fun_nat_real,Nb: nat] : aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_gu(G),Nb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ).

% sum.nat_diff_reindex
tff(fact_7847_sum__diff__distrib,axiom,
    ! [Q: fun_real_nat,P: fun_real_nat,Nb: real] :
      ( ! [X3: real] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_real_nat(Q,X3)),aa_real_nat(P,X3))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups1935376822645274424al_nat(P,set_or5984915006950818249n_real(Nb))),groups1935376822645274424al_nat(Q,set_or5984915006950818249n_real(Nb))) = groups1935376822645274424al_nat(aa_fun211602392931233840al_nat(aTP_Lamm_dg(Q),P),set_or5984915006950818249n_real(Nb)) ) ) ).

% sum_diff_distrib
tff(fact_7848_sum__diff__distrib,axiom,
    ! [Q: fun_nat_nat,P: fun_nat_nat,Nb: nat] :
      ( ! [X3: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(Q,X3)),aa_nat_nat(P,X3))
     => ( 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,Nb))),aa_set_nat_nat(groups3542108847815614940at_nat(Q),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_dj(Q),P)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ) ) ).

% sum_diff_distrib
tff(fact_7849_cos__diff,axiom,
    ! [Xb: real,Yb: real] : aa_real_real(cos_real,aa_real_real(minus_minus_real(Xb),Yb)) = aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_real_real(cos_real,Xb)),aa_real_real(cos_real,Yb))),aa_real_real(times_times_real(aa_real_real(sin_real,Xb)),aa_real_real(sin_real,Yb))) ).

% cos_diff
tff(fact_7850_cos__add,axiom,
    ! [Xb: real,Yb: real] : aa_real_real(cos_real,aa_real_real(plus_plus_real(Xb),Yb)) = aa_real_real(minus_minus_real(aa_real_real(times_times_real(aa_real_real(cos_real,Xb)),aa_real_real(cos_real,Yb))),aa_real_real(times_times_real(aa_real_real(sin_real,Xb)),aa_real_real(sin_real,Yb))) ).

% cos_add
tff(fact_7851_sin__zero__norm__cos__one,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(sin_real,Xb) = zero_zero_real )
     => ( aa_real_real(real_V7735802525324610683m_real,aa_real_real(cos_real,Xb)) = one_one_real ) ) ).

% sin_zero_norm_cos_one
tff(fact_7852_sin__zero__norm__cos__one,axiom,
    ! [Xb: complex] :
      ( ( aa_complex_complex(sin_complex,Xb) = zero_zero_complex )
     => ( aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(cos_complex,Xb)) = one_one_real ) ) ).

% sin_zero_norm_cos_one
tff(fact_7853_sin__zero__abs__cos__one,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(sin_real,Xb) = zero_zero_real )
     => ( aa_real_real(abs_abs_real,aa_real_real(cos_real,Xb)) = one_one_real ) ) ).

% sin_zero_abs_cos_one
tff(fact_7854_sin__zero__iff__int2,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(sin_real,Xb) = zero_zero_real )
    <=> ? [I5: int] : Xb = aa_real_real(times_times_real(aa_int_real(ring_1_of_int_real,I5)),pi) ) ).

% sin_zero_iff_int2
tff(fact_7855_sum_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_rat,Nb: nat] : groups2906978787729119204at_rat(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = aa_rat_rat(plus_plus_rat(aa_nat_rat(G,zero_zero_nat)),groups2906978787729119204at_rat(aTP_Lamm_dw(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% sum.lessThan_Suc_shift
tff(fact_7856_sum_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_int,Nb: nat] : groups3539618377306564664at_int(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,zero_zero_nat)),groups3539618377306564664at_int(aTP_Lamm_dx(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% sum.lessThan_Suc_shift
tff(fact_7857_sum_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = 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_dm(G)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% sum.lessThan_Suc_shift
tff(fact_7858_sum_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_real,Nb: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = aa_real_real(plus_plus_real(aa_nat_real(G,zero_zero_nat)),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_dn(G)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% sum.lessThan_Suc_shift
tff(fact_7859_sumr__diff__mult__const2,axiom,
    ! [F: fun_nat_int,Nb: nat,R2: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,groups3539618377306564664at_int(F,aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),R2)) = groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_gv(F),R2),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ).

% sumr_diff_mult_const2
tff(fact_7860_sumr__diff__mult__const2,axiom,
    ! [F: fun_nat_rat,Nb: nat,R2: rat] : aa_rat_rat(minus_minus_rat(groups2906978787729119204at_rat(F,aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,Nb)),R2)) = groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_gw(F),R2),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ).

% sumr_diff_mult_const2
tff(fact_7861_sumr__diff__mult__const2,axiom,
    ! [F: fun_nat_real,Nb: nat,R2: real] : aa_real_real(minus_minus_real(aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),R2)) = aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_gx(F),R2)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ).

% sumr_diff_mult_const2
tff(fact_7862_sum__lessThan__telescope,axiom,
    ! [F: fun_nat_rat,Mb: nat] : groups2906978787729119204at_rat(aTP_Lamm_dy(F),aa_nat_set_nat(set_ord_lessThan_nat,Mb)) = aa_rat_rat(minus_minus_rat(aa_nat_rat(F,Mb)),aa_nat_rat(F,zero_zero_nat)) ).

% sum_lessThan_telescope
tff(fact_7863_sum__lessThan__telescope,axiom,
    ! [F: fun_nat_int,Mb: nat] : groups3539618377306564664at_int(aTP_Lamm_dz(F),aa_nat_set_nat(set_ord_lessThan_nat,Mb)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,Mb)),aa_nat_int(F,zero_zero_nat)) ).

% sum_lessThan_telescope
tff(fact_7864_sum__lessThan__telescope,axiom,
    ! [F: fun_nat_real,Mb: nat] : aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_ea(F)),aa_nat_set_nat(set_ord_lessThan_nat,Mb)) = aa_real_real(minus_minus_real(aa_nat_real(F,Mb)),aa_nat_real(F,zero_zero_nat)) ).

% sum_lessThan_telescope
tff(fact_7865_sum__lessThan__telescope_H,axiom,
    ! [F: fun_nat_rat,Mb: nat] : groups2906978787729119204at_rat(aTP_Lamm_gy(F),aa_nat_set_nat(set_ord_lessThan_nat,Mb)) = aa_rat_rat(minus_minus_rat(aa_nat_rat(F,zero_zero_nat)),aa_nat_rat(F,Mb)) ).

% sum_lessThan_telescope'
tff(fact_7866_sum__lessThan__telescope_H,axiom,
    ! [F: fun_nat_int,Mb: nat] : groups3539618377306564664at_int(aTP_Lamm_gz(F),aa_nat_set_nat(set_ord_lessThan_nat,Mb)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,zero_zero_nat)),aa_nat_int(F,Mb)) ).

% sum_lessThan_telescope'
tff(fact_7867_sum__lessThan__telescope_H,axiom,
    ! [F: fun_nat_real,Mb: nat] : aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_ha(F)),aa_nat_set_nat(set_ord_lessThan_nat,Mb)) = aa_real_real(minus_minus_real(aa_nat_real(F,zero_zero_nat)),aa_nat_real(F,Mb)) ).

% sum_lessThan_telescope'
tff(fact_7868_sum_OatLeast1__atMost__eq,axiom,
    ! [G: fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),Nb)) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_dm(G)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ).

% sum.atLeast1_atMost_eq
tff(fact_7869_sum_OatLeast1__atMost__eq,axiom,
    ! [G: fun_nat_real,Nb: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),Nb)) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_dn(G)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ).

% sum.atLeast1_atMost_eq
tff(fact_7870_sin__gt__zero__02,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_num_real(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,Xb)) ) ) ).

% sin_gt_zero_02
tff(fact_7871_sin__double,axiom,
    ! [Xb: complex] : aa_complex_complex(sin_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),Xb)) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_complex_complex(sin_complex,Xb))),aa_complex_complex(cos_complex,Xb)) ).

% sin_double
tff(fact_7872_sin__double,axiom,
    ! [Xb: real] : aa_real_real(sin_real,aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Xb)) = aa_real_real(times_times_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(sin_real,Xb))),aa_real_real(cos_real,Xb)) ).

% sin_double
tff(fact_7873_power__diff__1__eq,axiom,
    ! [Xb: complex,Nb: nat] : aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(Xb),Nb)),one_one_complex) = aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(Xb),one_one_complex)),aa_set_nat_complex(groups2073611262835488442omplex(power_power_complex(Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% power_diff_1_eq
tff(fact_7874_power__diff__1__eq,axiom,
    ! [Xb: rat,Nb: nat] : aa_rat_rat(minus_minus_rat(aa_nat_rat(power_power_rat(Xb),Nb)),one_one_rat) = aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(Xb),one_one_rat)),groups2906978787729119204at_rat(power_power_rat(Xb),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% power_diff_1_eq
tff(fact_7875_power__diff__1__eq,axiom,
    ! [Xb: int,Nb: nat] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(Xb),Nb)),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,Xb),one_one_int)),groups3539618377306564664at_int(power_power_int(Xb),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% power_diff_1_eq
tff(fact_7876_power__diff__1__eq,axiom,
    ! [Xb: real,Nb: nat] : aa_real_real(minus_minus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)),one_one_real) = aa_real_real(times_times_real(aa_real_real(minus_minus_real(Xb),one_one_real)),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% power_diff_1_eq
tff(fact_7877_one__diff__power__eq,axiom,
    ! [Xb: complex,Nb: nat] : aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(power_power_complex(Xb),Nb)) = aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(one_one_complex),Xb)),aa_set_nat_complex(groups2073611262835488442omplex(power_power_complex(Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% one_diff_power_eq
tff(fact_7878_one__diff__power__eq,axiom,
    ! [Xb: rat,Nb: nat] : aa_rat_rat(minus_minus_rat(one_one_rat),aa_nat_rat(power_power_rat(Xb),Nb)) = aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(one_one_rat),Xb)),groups2906978787729119204at_rat(power_power_rat(Xb),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% one_diff_power_eq
tff(fact_7879_one__diff__power__eq,axiom,
    ! [Xb: int,Nb: nat] : aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),aa_nat_int(power_power_int(Xb),Nb)) = 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),Xb)),groups3539618377306564664at_int(power_power_int(Xb),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% one_diff_power_eq
tff(fact_7880_one__diff__power__eq,axiom,
    ! [Xb: real,Nb: nat] : aa_real_real(minus_minus_real(one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)) = aa_real_real(times_times_real(aa_real_real(minus_minus_real(one_one_real),Xb)),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% one_diff_power_eq
tff(fact_7881_geometric__sum,axiom,
    ! [Xb: complex,Nb: nat] :
      ( ( Xb != one_one_complex )
     => ( aa_set_nat_complex(groups2073611262835488442omplex(power_power_complex(Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(Xb),Nb)),one_one_complex)),aa_complex_complex(minus_minus_complex(Xb),one_one_complex)) ) ) ).

% geometric_sum
tff(fact_7882_geometric__sum,axiom,
    ! [Xb: rat,Nb: nat] :
      ( ( Xb != one_one_rat )
     => ( groups2906978787729119204at_rat(power_power_rat(Xb),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_nat_rat(power_power_rat(Xb),Nb)),one_one_rat)),aa_rat_rat(minus_minus_rat(Xb),one_one_rat)) ) ) ).

% geometric_sum
tff(fact_7883_geometric__sum,axiom,
    ! [Xb: real,Nb: nat] :
      ( ( Xb != one_one_real )
     => ( aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)),one_one_real)),aa_real_real(minus_minus_real(Xb),one_one_real)) ) ) ).

% geometric_sum
tff(fact_7884_sin__pi__divide__n__ge__0,axiom,
    ! [Nb: nat] :
      ( ( Nb != 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(divide_divide_real(pi),aa_nat_real(semiri5074537144036343181t_real,Nb)))) ) ).

% sin_pi_divide_n_ge_0
tff(fact_7885_sin__45,axiom,
    aa_real_real(sin_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one))))) = aa_real_real(divide_divide_real(aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% sin_45
tff(fact_7886_sin__cos__le1,axiom,
    ! [Xb: real,Yb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_real_real(sin_real,Xb)),aa_real_real(sin_real,Yb))),aa_real_real(times_times_real(aa_real_real(cos_real,Xb)),aa_real_real(cos_real,Yb))))),one_one_real) ).

% sin_cos_le1
tff(fact_7887_sum__gp__strict,axiom,
    ! [Xb: complex,Nb: nat] :
      aa_set_nat_complex(groups2073611262835488442omplex(power_power_complex(Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = $ite(Xb = one_one_complex,aa_nat_complex(semiri8010041392384452111omplex,Nb),aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(power_power_complex(Xb),Nb))),aa_complex_complex(minus_minus_complex(one_one_complex),Xb))) ).

% sum_gp_strict
tff(fact_7888_sum__gp__strict,axiom,
    ! [Xb: rat,Nb: nat] :
      groups2906978787729119204at_rat(power_power_rat(Xb),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = $ite(Xb = one_one_rat,aa_nat_rat(semiri681578069525770553at_rat,Nb),aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(one_one_rat),aa_nat_rat(power_power_rat(Xb),Nb))),aa_rat_rat(minus_minus_rat(one_one_rat),Xb))) ).

% sum_gp_strict
tff(fact_7889_sum__gp__strict,axiom,
    ! [Xb: real,Nb: nat] :
      aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = $ite(Xb = one_one_real,aa_nat_real(semiri5074537144036343181t_real,Nb),aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb))),aa_real_real(minus_minus_real(one_one_real),Xb))) ).

% sum_gp_strict
tff(fact_7890_lemma__termdiff1,axiom,
    ! [Z: complex,H: complex,Mb: nat] : aa_set_nat_complex(groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_hb(Z),H),Mb)),aa_nat_set_nat(set_ord_lessThan_nat,Mb)) = aa_set_nat_complex(groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_hc(Z),H),Mb)),aa_nat_set_nat(set_ord_lessThan_nat,Mb)) ).

% lemma_termdiff1
tff(fact_7891_lemma__termdiff1,axiom,
    ! [Z: rat,H: rat,Mb: nat] : groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_hd(Z),H),Mb),aa_nat_set_nat(set_ord_lessThan_nat,Mb)) = groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_he(Z),H),Mb),aa_nat_set_nat(set_ord_lessThan_nat,Mb)) ).

% lemma_termdiff1
tff(fact_7892_lemma__termdiff1,axiom,
    ! [Z: int,H: int,Mb: nat] : groups3539618377306564664at_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_hf(Z),H),Mb),aa_nat_set_nat(set_ord_lessThan_nat,Mb)) = groups3539618377306564664at_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_hg(Z),H),Mb),aa_nat_set_nat(set_ord_lessThan_nat,Mb)) ).

% lemma_termdiff1
tff(fact_7893_lemma__termdiff1,axiom,
    ! [Z: real,H: real,Mb: nat] : aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_hh(Z),H),Mb)),aa_nat_set_nat(set_ord_lessThan_nat,Mb)) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_hi(Z),H),Mb)),aa_nat_set_nat(set_ord_lessThan_nat,Mb)) ).

% lemma_termdiff1
tff(fact_7894_power__diff__sumr2,axiom,
    ! [Xb: complex,Nb: nat,Yb: complex] : aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(Xb),Nb)),aa_nat_complex(power_power_complex(Yb),Nb)) = aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(Xb),Yb)),aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_hj(Xb),Nb),Yb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% power_diff_sumr2
tff(fact_7895_power__diff__sumr2,axiom,
    ! [Xb: rat,Nb: nat,Yb: rat] : aa_rat_rat(minus_minus_rat(aa_nat_rat(power_power_rat(Xb),Nb)),aa_nat_rat(power_power_rat(Yb),Nb)) = aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(Xb),Yb)),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_nat298952458717073160at_rat(aTP_Lamm_hk(Xb),Nb),Yb),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% power_diff_sumr2
tff(fact_7896_power__diff__sumr2,axiom,
    ! [Xb: int,Nb: nat,Yb: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(Xb),Nb)),aa_nat_int(power_power_int(Yb),Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),Yb)),groups3539618377306564664at_int(aa_int_fun_nat_int(aa_nat6704147566042551216at_int(aTP_Lamm_hl(Xb),Nb),Yb),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% power_diff_sumr2
tff(fact_7897_power__diff__sumr2,axiom,
    ! [Xb: real,Nb: nat,Yb: real] : aa_real_real(minus_minus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Yb),Nb)) = aa_real_real(times_times_real(aa_real_real(minus_minus_real(Xb),Yb)),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_hm(Xb),Nb),Yb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% power_diff_sumr2
tff(fact_7898_diff__power__eq__sum,axiom,
    ! [Xb: complex,Nb: nat,Yb: complex] : aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(Xb),aa_nat_nat(suc,Nb))),aa_nat_complex(power_power_complex(Yb),aa_nat_nat(suc,Nb))) = aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(Xb),Yb)),aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_hn(Xb),Nb),Yb)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb)))) ).

% diff_power_eq_sum
tff(fact_7899_diff__power__eq__sum,axiom,
    ! [Xb: rat,Nb: nat,Yb: rat] : aa_rat_rat(minus_minus_rat(aa_nat_rat(power_power_rat(Xb),aa_nat_nat(suc,Nb))),aa_nat_rat(power_power_rat(Yb),aa_nat_nat(suc,Nb))) = aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(Xb),Yb)),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_nat298952458717073160at_rat(aTP_Lamm_ho(Xb),Nb),Yb),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb)))) ).

% diff_power_eq_sum
tff(fact_7900_diff__power__eq__sum,axiom,
    ! [Xb: int,Nb: nat,Yb: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(Xb),aa_nat_nat(suc,Nb))),aa_nat_int(power_power_int(Yb),aa_nat_nat(suc,Nb))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),Yb)),groups3539618377306564664at_int(aa_int_fun_nat_int(aa_nat6704147566042551216at_int(aTP_Lamm_hp(Xb),Nb),Yb),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb)))) ).

% diff_power_eq_sum
tff(fact_7901_diff__power__eq__sum,axiom,
    ! [Xb: real,Nb: nat,Yb: real] : aa_real_real(minus_minus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_nat_nat(suc,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Yb),aa_nat_nat(suc,Nb))) = aa_real_real(times_times_real(aa_real_real(minus_minus_real(Xb),Yb)),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_hq(Xb),Nb),Yb)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb)))) ).

% diff_power_eq_sum
tff(fact_7902_real__sum__nat__ivl__bounded2,axiom,
    ! [Nb: nat,F: fun_nat_rat,K5: rat,K: nat] :
      ( ! [P4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,P4),Nb)
         => 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,Nb),K)))),aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,Nb)),K5)) ) ) ).

% real_sum_nat_ivl_bounded2
tff(fact_7903_real__sum__nat__ivl__bounded2,axiom,
    ! [Nb: nat,F: fun_nat_int,K5: int,K: nat] :
      ( ! [P4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,P4),Nb)
         => 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,Nb),K)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),K5)) ) ) ).

% real_sum_nat_ivl_bounded2
tff(fact_7904_real__sum__nat__ivl__bounded2,axiom,
    ! [Nb: nat,F: fun_nat_nat,K5: nat,K: nat] :
      ( ! [P4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,P4),Nb)
         => 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,Nb),K)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Nb)),K5)) ) ) ).

% real_sum_nat_ivl_bounded2
tff(fact_7905_real__sum__nat__ivl__bounded2,axiom,
    ! [Nb: nat,F: fun_nat_real,K5: real,K: nat] :
      ( ! [P4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,P4),Nb)
         => 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,Nb),K)))),aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),K5)) ) ) ).

% real_sum_nat_ivl_bounded2
tff(fact_7906_sin__gt__zero2,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(divide_divide_real(pi),aa_num_real(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,Xb)) ) ) ).

% sin_gt_zero2
tff(fact_7907_cos__squared__eq,axiom,
    ! [Xb: complex] : aa_nat_complex(power_power_complex(aa_complex_complex(cos_complex,Xb)),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,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% cos_squared_eq
tff(fact_7908_cos__squared__eq,axiom,
    ! [Xb: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(cos_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(minus_minus_real(one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sin_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% cos_squared_eq
tff(fact_7909_sin__squared__eq,axiom,
    ! [Xb: complex] : aa_nat_complex(power_power_complex(aa_complex_complex(sin_complex,Xb)),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,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% sin_squared_eq
tff(fact_7910_sin__squared__eq,axiom,
    ! [Xb: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sin_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(minus_minus_real(one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(cos_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% sin_squared_eq
tff(fact_7911_sin__lt__zero,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,pi),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(times_times_real(aa_num_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,Xb)),zero_zero_real) ) ) ).

% sin_lt_zero
tff(fact_7912_sin__30,axiom,
    aa_real_real(sin_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,one))))) = aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% sin_30
tff(fact_7913_sin__inj__pi,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(divide_divide_real(pi),aa_num_real(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(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Yb)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Yb),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
           => ( ( aa_real_real(sin_real,Xb) = aa_real_real(sin_real,Yb) )
             => ( Xb = Yb ) ) ) ) ) ) ).

% sin_inj_pi
tff(fact_7914_sin__mono__le__eq,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(divide_divide_real(pi),aa_num_real(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(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Yb)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Yb),aa_real_real(divide_divide_real(pi),aa_num_real(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,Xb)),aa_real_real(sin_real,Yb))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Yb) ) ) ) ) ) ).

% sin_mono_le_eq
tff(fact_7915_sin__monotone__2pi__le,axiom,
    ! [Yb: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Yb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Yb),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(divide_divide_real(pi),aa_num_real(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,Yb)),aa_real_real(sin_real,Xb)) ) ) ) ).

% sin_monotone_2pi_le
tff(fact_7916_sin__60,axiom,
    aa_real_real(sin_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit1,one)))) = aa_real_real(divide_divide_real(aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit1,one)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% sin_60
tff(fact_7917_one__diff__power__eq_H,axiom,
    ! [Xb: complex,Nb: nat] : aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(power_power_complex(Xb),Nb)) = aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(one_one_complex),Xb)),aa_set_nat_complex(groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aTP_Lamm_hr(Xb),Nb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% one_diff_power_eq'
tff(fact_7918_one__diff__power__eq_H,axiom,
    ! [Xb: rat,Nb: nat] : aa_rat_rat(minus_minus_rat(one_one_rat),aa_nat_rat(power_power_rat(Xb),Nb)) = aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(one_one_rat),Xb)),groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aTP_Lamm_hs(Xb),Nb),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% one_diff_power_eq'
tff(fact_7919_one__diff__power__eq_H,axiom,
    ! [Xb: int,Nb: nat] : aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),aa_nat_int(power_power_int(Xb),Nb)) = 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),Xb)),groups3539618377306564664at_int(aa_nat_fun_nat_int(aTP_Lamm_ht(Xb),Nb),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% one_diff_power_eq'
tff(fact_7920_one__diff__power__eq_H,axiom,
    ! [Xb: real,Nb: nat] : aa_real_real(minus_minus_real(one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)) = aa_real_real(times_times_real(aa_real_real(minus_minus_real(one_one_real),Xb)),aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_hu(Xb),Nb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% one_diff_power_eq'
tff(fact_7921_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_7922_cos__diff__cos,axiom,
    ! [W: real,Z: real] : aa_real_real(minus_minus_real(aa_real_real(cos_real,W)),aa_real_real(cos_real,Z)) = aa_real_real(times_times_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(sin_real,aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(W),Z)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))),aa_real_real(sin_real,aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(Z),W)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% cos_diff_cos
tff(fact_7923_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_7924_sin__diff__sin,axiom,
    ! [W: real,Z: real] : aa_real_real(minus_minus_real(aa_real_real(sin_real,W)),aa_real_real(sin_real,Z)) = aa_real_real(times_times_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(sin_real,aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(W),Z)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))),aa_real_real(cos_real,aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(W),Z)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% sin_diff_sin
tff(fact_7925_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_7926_sin__plus__sin,axiom,
    ! [W: real,Z: real] : aa_real_real(plus_plus_real(aa_real_real(sin_real,W)),aa_real_real(sin_real,Z)) = aa_real_real(times_times_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(sin_real,aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(W),Z)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))),aa_real_real(cos_real,aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(W),Z)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% sin_plus_sin
tff(fact_7927_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_7928_cos__times__sin,axiom,
    ! [W: real,Z: real] : aa_real_real(times_times_real(aa_real_real(cos_real,W)),aa_real_real(sin_real,Z)) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(sin_real,aa_real_real(plus_plus_real(W),Z))),aa_real_real(sin_real,aa_real_real(minus_minus_real(W),Z)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% cos_times_sin
tff(fact_7929_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_7930_sin__times__cos,axiom,
    ! [W: real,Z: real] : aa_real_real(times_times_real(aa_real_real(sin_real,W)),aa_real_real(cos_real,Z)) = aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(aa_real_real(sin_real,aa_real_real(plus_plus_real(W),Z))),aa_real_real(sin_real,aa_real_real(minus_minus_real(W),Z)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% sin_times_cos
tff(fact_7931_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_7932_sin__times__sin,axiom,
    ! [W: real,Z: real] : aa_real_real(times_times_real(aa_real_real(sin_real,W)),aa_real_real(sin_real,Z)) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(cos_real,aa_real_real(minus_minus_real(W),Z))),aa_real_real(cos_real,aa_real_real(plus_plus_real(W),Z)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% sin_times_sin
tff(fact_7933_cos__double,axiom,
    ! [Xb: complex] : aa_complex_complex(cos_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),Xb)) = aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(aa_complex_complex(cos_complex,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_complex(power_power_complex(aa_complex_complex(sin_complex,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% cos_double
tff(fact_7934_cos__double,axiom,
    ! [Xb: real] : aa_real_real(cos_real,aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Xb)) = aa_real_real(minus_minus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(cos_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sin_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% cos_double
tff(fact_7935_sin__le__zero,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,pi),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(times_times_real(aa_num_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,Xb)),zero_zero_real) ) ) ).

% sin_le_zero
tff(fact_7936_sin__less__zero,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(aa_real_real(uminus_uminus_real,pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sin_real,Xb)),zero_zero_real) ) ) ).

% sin_less_zero
tff(fact_7937_sin__mono__less__eq,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(divide_divide_real(pi),aa_num_real(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(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Yb)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Yb),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sin_real,Xb)),aa_real_real(sin_real,Yb))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Yb) ) ) ) ) ) ).

% sin_mono_less_eq
tff(fact_7938_sin__monotone__2pi,axiom,
    ! [Yb: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Yb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Yb),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sin_real,Yb)),aa_real_real(sin_real,Xb)) ) ) ) ).

% sin_monotone_2pi
tff(fact_7939_sin__total,axiom,
    ! [Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Yb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Yb),one_one_real)
       => ? [X3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X3)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
            & ( aa_real_real(sin_real,X3) = Yb )
            & ! [Y3: real] :
                ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(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(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
                  & ( aa_real_real(sin_real,Y3) = Yb ) )
               => ( Y3 = X3 ) ) ) ) ) ).

% sin_total
tff(fact_7940_sum__split__even__odd,axiom,
    ! [F: fun_nat_real,G: fun_nat_real,Nb: nat] : aa_set_nat_real(groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_hv(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))),Nb))) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_hw(F)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_hx(G)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% sum_split_even_odd
tff(fact_7941_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_7942_cos__double__sin,axiom,
    ! [W: real] : aa_real_real(cos_real,aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),W)) = aa_real_real(minus_minus_real(one_one_real),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sin_real,W)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% cos_double_sin
tff(fact_7943_sin__pi__divide__n__gt__0,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(sin_real,aa_real_real(divide_divide_real(pi),aa_nat_real(semiri5074537144036343181t_real,Nb)))) ) ).

% sin_pi_divide_n_gt_0
tff(fact_7944_sin__arctan,axiom,
    ! [Xb: real] : aa_real_real(sin_real,aa_real_real(arctan,Xb)) = aa_real_real(divide_divide_real(Xb),aa_real_real(sqrt,aa_real_real(plus_plus_real(one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% sin_arctan
tff(fact_7945_sincos__total__pi,axiom,
    ! [Yb: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Yb)
     => ( ( aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real )
       => ? [T2: real] :
            ( 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),pi)
            & ( Xb = aa_real_real(cos_real,T2) )
            & ( Yb = aa_real_real(sin_real,T2) ) ) ) ) ).

% sincos_total_pi
tff(fact_7946_sin__cos__sqrt,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(sin_real,Xb))
     => ( aa_real_real(sin_real,Xb) = aa_real_real(sqrt,aa_real_real(minus_minus_real(one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(cos_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ).

% sin_cos_sqrt
tff(fact_7947_sin__expansion__lemma,axiom,
    ! [Xb: real,Mb: nat] : aa_real_real(sin_real,aa_real_real(plus_plus_real(Xb),aa_real_real(divide_divide_real(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Mb))),pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) = aa_real_real(cos_real,aa_real_real(plus_plus_real(Xb),aa_real_real(divide_divide_real(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Mb)),pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% sin_expansion_lemma
tff(fact_7948_sin__zero__iff__int,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(sin_real,Xb) = zero_zero_real )
    <=> ? [I5: int] :
          ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),I5)
          & ( Xb = aa_real_real(times_times_real(aa_int_real(ring_1_of_int_real,I5)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ) ).

% sin_zero_iff_int
tff(fact_7949_sin__zero__lemma,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( ( aa_real_real(sin_real,Xb) = 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)
            & ( Xb = aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ) ) ).

% sin_zero_lemma
tff(fact_7950_sin__zero__iff,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(sin_real,Xb) = 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)
            & ( Xb = aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,N4)),aa_real_real(divide_divide_real(pi),aa_num_real(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)
            & ( Xb = aa_real_real(uminus_uminus_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,N4)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ) ) ) ) ).

% sin_zero_iff
tff(fact_7951_cos__expansion__lemma,axiom,
    ! [Xb: real,Mb: nat] : aa_real_real(cos_real,aa_real_real(plus_plus_real(Xb),aa_real_real(divide_divide_real(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Mb))),pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) = aa_real_real(uminus_uminus_real,aa_real_real(sin_real,aa_real_real(plus_plus_real(Xb),aa_real_real(divide_divide_real(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Mb)),pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))) ).

% cos_expansion_lemma
tff(fact_7952_sincos__total__pi__half,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Yb)
       => ( ( aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real )
         => ? [T2: real] :
              ( 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),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
              & ( Xb = aa_real_real(cos_real,T2) )
              & ( Yb = aa_real_real(sin_real,T2) ) ) ) ) ) ).

% sincos_total_pi_half
tff(fact_7953_sincos__total__2pi__le,axiom,
    ! [Xb: real,Yb: real] :
      ( ( aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real )
     => ? [T2: real] :
          ( 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),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))
          & ( Xb = aa_real_real(cos_real,T2) )
          & ( Yb = aa_real_real(sin_real,T2) ) ) ) ).

% sincos_total_2pi_le
tff(fact_7954_sincos__total__2pi,axiom,
    ! [Xb: real,Yb: real] :
      ( ( aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real )
     => ~ ! [T2: real] :
            ( 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_real,T2),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))
             => ( ( Xb = aa_real_real(cos_real,T2) )
               => ( Yb != aa_real_real(sin_real,T2) ) ) ) ) ) ).

% sincos_total_2pi
tff(fact_7955_sum__bounds__lt__plus1,axiom,
    ! [F: fun_nat_nat,Mm: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_dm(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_7956_sum__bounds__lt__plus1,axiom,
    ! [F: fun_nat_real,Mm: nat] : aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_dn(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_7957_sumr__cos__zero__one,axiom,
    ! [Nb: nat] : aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_hy),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = one_one_real ).

% sumr_cos_zero_one
tff(fact_7958_summable__arctan__series,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),one_one_real)
     => summable_real(aa_real_fun_nat_real(aTP_Lamm_gf,Xb)) ) ).

% summable_arctan_series
tff(fact_7959_tan__double,axiom,
    ! [Xb: complex] :
      ( ( aa_complex_complex(cos_complex,Xb) != zero_zero_complex )
     => ( ( aa_complex_complex(cos_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),Xb)) != zero_zero_complex )
       => ( aa_complex_complex(tan_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),Xb)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_complex_complex(tan_complex,Xb))),aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(power_power_complex(aa_complex_complex(tan_complex,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ) ).

% tan_double
tff(fact_7960_tan__double,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(cos_real,Xb) != zero_zero_real )
     => ( ( aa_real_real(cos_real,aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Xb)) != zero_zero_real )
       => ( aa_real_real(tan_real,aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Xb)) = aa_real_real(divide_divide_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(tan_real,Xb))),aa_real_real(minus_minus_real(one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(tan_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ) ).

% tan_double
tff(fact_7961_sin__tan,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xb)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
     => ( aa_real_real(sin_real,Xb) = aa_real_real(divide_divide_real(aa_real_real(tan_real,Xb)),aa_real_real(sqrt,aa_real_real(plus_plus_real(one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(tan_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ).

% sin_tan
tff(fact_7962_tan__zero,axiom,
    aa_complex_complex(tan_complex,zero_zero_complex) = zero_zero_complex ).

% tan_zero
tff(fact_7963_tan__zero,axiom,
    aa_real_real(tan_real,zero_zero_real) = zero_zero_real ).

% tan_zero
tff(fact_7964_summable__single,axiom,
    ! [I3: nat,F: fun_nat_complex] : summable_complex(aa_fun2955221450537884980omplex(aTP_Lamm_hz(I3),F)) ).

% summable_single
tff(fact_7965_summable__single,axiom,
    ! [I3: nat,F: fun_nat_real] : summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_ia(I3),F)) ).

% summable_single
tff(fact_7966_summable__single,axiom,
    ! [I3: nat,F: fun_nat_nat] : summable_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_ib(I3),F)) ).

% summable_single
tff(fact_7967_summable__single,axiom,
    ! [I3: nat,F: fun_nat_int] : summable_int(aa_fun5565043122120104496at_int(aTP_Lamm_ic(I3),F)) ).

% summable_single
tff(fact_7968_summable__zero,axiom,
    summable_complex(aTP_Lamm_gg) ).

% summable_zero
tff(fact_7969_summable__zero,axiom,
    summable_real(aTP_Lamm_de) ).

% summable_zero
tff(fact_7970_summable__zero,axiom,
    summable_nat(aTP_Lamm_am) ).

% summable_zero
tff(fact_7971_summable__zero,axiom,
    summable_int(aTP_Lamm_gh) ).

% summable_zero
tff(fact_7972_summable__iff__shift,axiom,
    ! [F: fun_nat_real,K: nat] :
      ( summable_real(aa_nat_fun_nat_real(aTP_Lamm_dp(F),K))
    <=> summable_real(F) ) ).

% summable_iff_shift
tff(fact_7973_tan__periodic__pi,axiom,
    ! [Xb: real] : aa_real_real(tan_real,aa_real_real(plus_plus_real(Xb),pi)) = aa_real_real(tan_real,Xb) ).

% tan_periodic_pi
tff(fact_7974_cos__coeff__0,axiom,
    aa_nat_real(cos_coeff,zero_zero_nat) = one_one_real ).

% cos_coeff_0
tff(fact_7975_summable__cmult__iff,axiom,
    ! [C: complex,F: fun_nat_complex] :
      ( summable_complex(aa_fun2955221450537884980omplex(aTP_Lamm_id(C),F))
    <=> ( ( C = zero_zero_complex )
        | summable_complex(F) ) ) ).

% summable_cmult_iff
tff(fact_7976_summable__cmult__iff,axiom,
    ! [C: real,F: fun_nat_real] :
      ( summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_eq(C),F))
    <=> ( ( C = zero_zero_real )
        | summable_real(F) ) ) ).

% summable_cmult_iff
tff(fact_7977_summable__divide__iff,axiom,
    ! [F: fun_nat_complex,C: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_ie(F),C))
    <=> ( ( C = zero_zero_complex )
        | summable_complex(F) ) ) ).

% summable_divide_iff
tff(fact_7978_summable__divide__iff,axiom,
    ! [F: fun_nat_real,C: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_fi(F),C))
    <=> ( ( C = zero_zero_real )
        | summable_real(F) ) ) ).

% summable_divide_iff
tff(fact_7979_tan__npi,axiom,
    ! [Nb: nat] : aa_real_real(tan_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),pi)) = zero_zero_real ).

% tan_npi
tff(fact_7980_tan__periodic__n,axiom,
    ! [Xb: real,Nb: num] : aa_real_real(tan_real,aa_real_real(plus_plus_real(Xb),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,Nb)),pi))) = aa_real_real(tan_real,Xb) ).

% tan_periodic_n
tff(fact_7981_tan__periodic__nat,axiom,
    ! [Xb: real,Nb: nat] : aa_real_real(tan_real,aa_real_real(plus_plus_real(Xb),aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),pi))) = aa_real_real(tan_real,Xb) ).

% tan_periodic_nat
tff(fact_7982_tan__periodic__int,axiom,
    ! [Xb: real,I3: int] : aa_real_real(tan_real,aa_real_real(plus_plus_real(Xb),aa_real_real(times_times_real(aa_int_real(ring_1_of_int_real,I3)),pi))) = aa_real_real(tan_real,Xb) ).

% tan_periodic_int
tff(fact_7983_summable__geometric__iff,axiom,
    ! [C: real] :
      ( summable_real(aa_real_fun_nat_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_7984_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_7985_tan__periodic,axiom,
    ! [Xb: real] : aa_real_real(tan_real,aa_real_real(plus_plus_real(Xb),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))) = aa_real_real(tan_real,Xb) ).

% tan_periodic
tff(fact_7986_summable__const__iff,axiom,
    ! [C: complex] :
      ( summable_complex(aTP_Lamm_if(C))
    <=> ( C = zero_zero_complex ) ) ).

% summable_const_iff
tff(fact_7987_summable__const__iff,axiom,
    ! [C: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_ig,C))
    <=> ( C = zero_zero_real ) ) ).

% summable_const_iff
tff(fact_7988_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_7989_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_7990_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_7991_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_7992_summable__mult2,axiom,
    ! [F: fun_nat_real,C: real] :
      ( summable_real(F)
     => summable_real(aa_real_fun_nat_real(aTP_Lamm_eu(F),C)) ) ).

% summable_mult2
tff(fact_7993_summable__mult,axiom,
    ! [F: fun_nat_real,C: real] :
      ( summable_real(F)
     => summable_real(aa_real_fun_nat_real(aTP_Lamm_ih(F),C)) ) ).

% summable_mult
tff(fact_7994_summable__add,axiom,
    ! [F: fun_nat_real,G: fun_nat_real] :
      ( summable_real(F)
     => ( summable_real(G)
       => summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_fg(F),G)) ) ) ).

% summable_add
tff(fact_7995_summable__add,axiom,
    ! [F: fun_nat_nat,G: fun_nat_nat] :
      ( summable_nat(F)
     => ( summable_nat(G)
       => summable_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_ff(F),G)) ) ) ).

% summable_add
tff(fact_7996_summable__add,axiom,
    ! [F: fun_nat_int,G: fun_nat_int] :
      ( summable_int(F)
     => ( summable_int(G)
       => summable_int(aa_fun5565043122120104496at_int(aTP_Lamm_ii(F),G)) ) ) ).

% summable_add
tff(fact_7997_summable__divide,axiom,
    ! [F: fun_nat_complex,C: complex] :
      ( summable_complex(F)
     => summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_ie(F),C)) ) ).

% summable_divide
tff(fact_7998_summable__divide,axiom,
    ! [F: fun_nat_real,C: real] :
      ( summable_real(F)
     => summable_real(aa_real_fun_nat_real(aTP_Lamm_fi(F),C)) ) ).

% summable_divide
tff(fact_7999_summable__Suc__iff,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(aTP_Lamm_dn(F))
    <=> summable_real(F) ) ).

% summable_Suc_iff
tff(fact_8000_summable__ignore__initial__segment,axiom,
    ! [F: fun_nat_real,K: nat] :
      ( summable_real(F)
     => summable_real(aa_nat_fun_nat_real(aTP_Lamm_dp(F),K)) ) ).

% summable_ignore_initial_segment
tff(fact_8001_powser__insidea,axiom,
    ! [F: fun_nat_real,Xb: real,Z: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_ij(F),Xb))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Z)),aa_real_real(real_V7735802525324610683m_real,Xb))
       => summable_real(aa_real_fun_nat_real(aTP_Lamm_ik(F),Z)) ) ) ).

% powser_insidea
tff(fact_8002_powser__insidea,axiom,
    ! [F: fun_nat_complex,Xb: complex,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_il(F),Xb))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_complex_real(real_V1022390504157884413omplex,Xb))
       => summable_real(aa_com20334259897932645t_real(aTP_Lamm_im(F),Z)) ) ) ).

% powser_insidea
tff(fact_8003_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_8004_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_8005_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_8006_summable__mult__D,axiom,
    ! [C: complex,F: fun_nat_complex] :
      ( summable_complex(aa_fun2955221450537884980omplex(aTP_Lamm_id(C),F))
     => ( ( C != zero_zero_complex )
       => summable_complex(F) ) ) ).

% summable_mult_D
tff(fact_8007_summable__mult__D,axiom,
    ! [C: real,F: fun_nat_real] :
      ( summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_eq(C),F))
     => ( ( C != zero_zero_real )
       => summable_real(F) ) ) ).

% summable_mult_D
tff(fact_8008_summable__zero__power,axiom,
    summable_int(power_power_int(zero_zero_int)) ).

% summable_zero_power
tff(fact_8009_summable__zero__power,axiom,
    summable_real(aa_real_fun_nat_real(power_power_real,zero_zero_real)) ).

% summable_zero_power
tff(fact_8010_summable__zero__power,axiom,
    summable_complex(power_power_complex(zero_zero_complex)) ).

% summable_zero_power
tff(fact_8011_suminf__mult2,axiom,
    ! [F: fun_nat_real,C: real] :
      ( summable_real(F)
     => ( aa_real_real(times_times_real(suminf_real(F)),C) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_eu(F),C)) ) ) ).

% suminf_mult2
tff(fact_8012_suminf__mult,axiom,
    ! [F: fun_nat_real,C: real] :
      ( summable_real(F)
     => ( suminf_real(aa_real_fun_nat_real(aTP_Lamm_ih(F),C)) = aa_real_real(times_times_real(C),suminf_real(F)) ) ) ).

% suminf_mult
tff(fact_8013_suminf__add,axiom,
    ! [F: fun_nat_real,G: fun_nat_real] :
      ( summable_real(F)
     => ( summable_real(G)
       => ( aa_real_real(plus_plus_real(suminf_real(F)),suminf_real(G)) = suminf_real(aa_fun1839206346889086768t_real(aTP_Lamm_fg(F),G)) ) ) ) ).

% suminf_add
tff(fact_8014_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_ff(F),G)) ) ) ) ).

% suminf_add
tff(fact_8015_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_ii(F),G)) ) ) ) ).

% suminf_add
tff(fact_8016_suminf__divide,axiom,
    ! [F: fun_nat_complex,C: complex] :
      ( summable_complex(F)
     => ( suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_ie(F),C)) = aa_complex_complex(divide1717551699836669952omplex(suminf_complex(F)),C) ) ) ).

% suminf_divide
tff(fact_8017_suminf__divide,axiom,
    ! [F: fun_nat_real,C: real] :
      ( summable_real(F)
     => ( suminf_real(aa_real_fun_nat_real(aTP_Lamm_fi(F),C)) = aa_real_real(divide_divide_real(suminf_real(F)),C) ) ) ).

% suminf_divide
tff(fact_8018_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_8019_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_8020_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_8021_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_8022_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_8023_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_8024_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_8025_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_8026_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_8027_summable__zero__power_H,axiom,
    ! [F: fun_nat_complex] : summable_complex(aTP_Lamm_gc(F)) ).

% summable_zero_power'
tff(fact_8028_summable__zero__power_H,axiom,
    ! [F: fun_nat_real] : summable_real(aTP_Lamm_gd(F)) ).

% summable_zero_power'
tff(fact_8029_summable__zero__power_H,axiom,
    ! [F: fun_nat_int] : summable_int(aTP_Lamm_in(F)) ).

% summable_zero_power'
tff(fact_8030_summable__0__powser,axiom,
    ! [F: fun_nat_complex] : summable_complex(aTP_Lamm_gc(F)) ).

% summable_0_powser
tff(fact_8031_summable__0__powser,axiom,
    ! [F: fun_nat_real] : summable_real(aTP_Lamm_gd(F)) ).

% summable_0_powser
tff(fact_8032_summable__powser__split__head,axiom,
    ! [F: fun_nat_complex,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_io(F),Z))
    <=> summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_il(F),Z)) ) ).

% summable_powser_split_head
tff(fact_8033_summable__powser__split__head,axiom,
    ! [F: fun_nat_real,Z: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_ip(F),Z))
    <=> summable_real(aa_real_fun_nat_real(aTP_Lamm_ij(F),Z)) ) ).

% summable_powser_split_head
tff(fact_8034_powser__split__head_I3_J,axiom,
    ! [F: fun_nat_complex,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_il(F),Z))
     => summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_io(F),Z)) ) ).

% powser_split_head(3)
tff(fact_8035_powser__split__head_I3_J,axiom,
    ! [F: fun_nat_real,Z: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_ij(F),Z))
     => summable_real(aa_real_fun_nat_real(aTP_Lamm_ip(F),Z)) ) ).

% powser_split_head(3)
tff(fact_8036_summable__powser__ignore__initial__segment,axiom,
    ! [F: fun_nat_complex,Mb: nat,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_iq(F),Mb),Z))
    <=> summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_il(F),Z)) ) ).

% summable_powser_ignore_initial_segment
tff(fact_8037_summable__powser__ignore__initial__segment,axiom,
    ! [F: fun_nat_real,Mb: nat,Z: real] :
      ( summable_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_ir(F),Mb),Z))
    <=> summable_real(aa_real_fun_nat_real(aTP_Lamm_ij(F),Z)) ) ).

% summable_powser_ignore_initial_segment
tff(fact_8038_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_is(F)) ) ) ).

% summable_norm_comparison_test
tff(fact_8039_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_8040_suminf__pos2,axiom,
    ! [F: fun_nat_real,I3: 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,I3))
         => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),suminf_real(F)) ) ) ) ).

% suminf_pos2
tff(fact_8041_suminf__pos2,axiom,
    ! [F: fun_nat_nat,I3: 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,I3))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),suminf_nat(F)) ) ) ) ).

% suminf_pos2
tff(fact_8042_suminf__pos2,axiom,
    ! [F: fun_nat_int,I3: 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,I3))
         => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),suminf_int(F)) ) ) ) ).

% suminf_pos2
tff(fact_8043_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))
        <=> ? [I5: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(F,I5)) ) ) ) ).

% suminf_pos_iff
tff(fact_8044_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))
        <=> ? [I5: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(F,I5)) ) ) ) ).

% suminf_pos_iff
tff(fact_8045_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))
        <=> ? [I5: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(F,I5)) ) ) ) ).

% suminf_pos_iff
tff(fact_8046_suminf__le__const,axiom,
    ! [F: fun_nat_int,Xb: 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))),Xb)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,suminf_int(F)),Xb) ) ) ).

% suminf_le_const
tff(fact_8047_suminf__le__const,axiom,
    ! [F: fun_nat_nat,Xb: 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))),Xb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,suminf_nat(F)),Xb) ) ) ).

% suminf_le_const
tff(fact_8048_suminf__le__const,axiom,
    ! [F: fun_nat_real,Xb: 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))),Xb)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,suminf_real(F)),Xb) ) ) ).

% suminf_le_const
tff(fact_8049_tan__def,axiom,
    ! [X: complex] : aa_complex_complex(tan_complex,X) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(sin_complex,X)),aa_complex_complex(cos_complex,X)) ).

% tan_def
tff(fact_8050_tan__def,axiom,
    ! [X: real] : aa_real_real(tan_real,X) = aa_real_real(divide_divide_real(aa_real_real(sin_real,X)),aa_real_real(cos_real,X)) ).

% tan_def
tff(fact_8051_powser__inside,axiom,
    ! [F: fun_nat_real,Xb: real,Z: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_ij(F),Xb))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Z)),aa_real_real(real_V7735802525324610683m_real,Xb))
       => summable_real(aa_real_fun_nat_real(aTP_Lamm_ij(F),Z)) ) ) ).

% powser_inside
tff(fact_8052_powser__inside,axiom,
    ! [F: fun_nat_complex,Xb: complex,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_il(F),Xb))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_complex_real(real_V1022390504157884413omplex,Xb))
       => summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_il(F),Z)) ) ) ).

% powser_inside
tff(fact_8053_summableI__nonneg__bounded,axiom,
    ! [F: fun_nat_int,Xb: 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))),Xb)
       => summable_int(F) ) ) ).

% summableI_nonneg_bounded
tff(fact_8054_summableI__nonneg__bounded,axiom,
    ! [F: fun_nat_nat,Xb: 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))),Xb)
       => summable_nat(F) ) ) ).

% summableI_nonneg_bounded
tff(fact_8055_summableI__nonneg__bounded,axiom,
    ! [F: fun_nat_real,Xb: 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))),Xb)
       => summable_real(F) ) ) ).

% summableI_nonneg_bounded
tff(fact_8056_complete__algebra__summable__geometric,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Xb)),one_one_real)
     => summable_real(aa_real_fun_nat_real(power_power_real,Xb)) ) ).

% complete_algebra_summable_geometric
tff(fact_8057_complete__algebra__summable__geometric,axiom,
    ! [Xb: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Xb)),one_one_real)
     => summable_complex(power_power_complex(Xb)) ) ).

% complete_algebra_summable_geometric
tff(fact_8058_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(aa_real_fun_nat_real(power_power_real,C)) ) ).

% summable_geometric
tff(fact_8059_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_8060_suminf__split__head,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(F)
     => ( suminf_real(aTP_Lamm_dn(F)) = aa_real_real(minus_minus_real(suminf_real(F)),aa_nat_real(F,zero_zero_nat)) ) ) ).

% suminf_split_head
tff(fact_8061_suminf__split__initial__segment,axiom,
    ! [F: fun_nat_real,K: nat] :
      ( summable_real(F)
     => ( suminf_real(F) = aa_real_real(plus_plus_real(suminf_real(aa_nat_fun_nat_real(aTP_Lamm_dp(F),K))),aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_ord_lessThan_nat,K))) ) ) ).

% suminf_split_initial_segment
tff(fact_8062_suminf__minus__initial__segment,axiom,
    ! [F: fun_nat_real,K: nat] :
      ( summable_real(F)
     => ( suminf_real(aa_nat_fun_nat_real(aTP_Lamm_dp(F),K)) = aa_real_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_8063_sum__less__suminf,axiom,
    ! [F: fun_nat_int,Nb: nat] :
      ( summable_int(F)
     => ( ! [M2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M2)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(F,M2)) )
       => aa_int_o(aa_int_fun_int_o(ord_less_int,groups3539618377306564664at_int(F,aa_nat_set_nat(set_ord_lessThan_nat,Nb))),suminf_int(F)) ) ) ).

% sum_less_suminf
tff(fact_8064_sum__less__suminf,axiom,
    ! [F: fun_nat_nat,Nb: nat] :
      ( summable_nat(F)
     => ( ! [M2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(F,M2)) )
       => 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,Nb))),suminf_nat(F)) ) ) ).

% sum_less_suminf
tff(fact_8065_sum__less__suminf,axiom,
    ! [F: fun_nat_real,Nb: nat] :
      ( summable_real(F)
     => ( ! [M2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M2)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(F,M2)) )
       => 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,Nb))),suminf_real(F)) ) ) ).

% sum_less_suminf
tff(fact_8066_powser__split__head_I1_J,axiom,
    ! [F: fun_nat_complex,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_il(F),Z))
     => ( suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_il(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_io(F),Z))),Z)) ) ) ).

% powser_split_head(1)
tff(fact_8067_powser__split__head_I1_J,axiom,
    ! [F: fun_nat_real,Z: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_ij(F),Z))
     => ( suminf_real(aa_real_fun_nat_real(aTP_Lamm_ij(F),Z)) = aa_real_real(plus_plus_real(aa_nat_real(F,zero_zero_nat)),aa_real_real(times_times_real(suminf_real(aa_real_fun_nat_real(aTP_Lamm_ip(F),Z))),Z)) ) ) ).

% powser_split_head(1)
tff(fact_8068_powser__split__head_I2_J,axiom,
    ! [F: fun_nat_complex,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_il(F),Z))
     => ( aa_complex_complex(times_times_complex(suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_io(F),Z))),Z) = aa_complex_complex(minus_minus_complex(suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_il(F),Z))),aa_nat_complex(F,zero_zero_nat)) ) ) ).

% powser_split_head(2)
tff(fact_8069_powser__split__head_I2_J,axiom,
    ! [F: fun_nat_real,Z: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_ij(F),Z))
     => ( aa_real_real(times_times_real(suminf_real(aa_real_fun_nat_real(aTP_Lamm_ip(F),Z))),Z) = aa_real_real(minus_minus_real(suminf_real(aa_real_fun_nat_real(aTP_Lamm_ij(F),Z))),aa_nat_real(F,zero_zero_nat)) ) ) ).

% powser_split_head(2)
tff(fact_8070_tan__45,axiom,
    aa_real_real(tan_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one))))) = one_one_real ).

% tan_45
tff(fact_8071_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] :
              ~ ! [M: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N6),M)
                 => ! [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(M),N7)))),E2) ) ) ) ).

% summable_partial_sum_bound
tff(fact_8072_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] :
              ~ ! [M: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N6),M)
                 => ! [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(M),N7)))),E2) ) ) ) ).

% summable_partial_sum_bound
tff(fact_8073_suminf__exist__split,axiom,
    ! [R2: real,F: fun_nat_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R2)
     => ( 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_dp(F),N7)))),R2) ) ) ) ).

% suminf_exist_split
tff(fact_8074_suminf__exist__split,axiom,
    ! [R2: real,F: fun_nat_complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R2)
     => ( 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_it(F),N7)))),R2) ) ) ) ).

% suminf_exist_split
tff(fact_8075_tan__60,axiom,
    aa_real_real(tan_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit1,one)))) = aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit1,one))) ).

% tan_60
tff(fact_8076_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_ij(F),Z)) ) ) ) ) ).

% summable_power_series
tff(fact_8077_Abel__lemma,axiom,
    ! [R2: real,R0: real,A: fun_nat_complex,M6: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),R2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,R2),R0)
       => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(A,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,R0),N))),M6)
         => summable_real(aa_fun7054779078796324018t_real(aTP_Lamm_iu(R2),A)) ) ) ) ).

% Abel_lemma
tff(fact_8078_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(times_times_real(C),aa_real_real(real_V7735802525324610683m_real,aa_nat_real(F,N)))) )
       => summable_real(F) ) ) ).

% summable_ratio_test
tff(fact_8079_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(times_times_real(C),aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(F,N)))) )
       => summable_complex(F) ) ) ).

% summable_ratio_test
tff(fact_8080_sum__less__suminf2,axiom,
    ! [F: fun_nat_int,Nb: nat,I3: nat] :
      ( summable_int(F)
     => ( ! [M2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M2)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(F,M2)) )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),I3)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(F,I3))
           => aa_int_o(aa_int_fun_int_o(ord_less_int,groups3539618377306564664at_int(F,aa_nat_set_nat(set_ord_lessThan_nat,Nb))),suminf_int(F)) ) ) ) ) ).

% sum_less_suminf2
tff(fact_8081_sum__less__suminf2,axiom,
    ! [F: fun_nat_nat,Nb: nat,I3: nat] :
      ( summable_nat(F)
     => ( ! [M2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,M2)) )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),I3)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(F,I3))
           => 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,Nb))),suminf_nat(F)) ) ) ) ) ).

% sum_less_suminf2
tff(fact_8082_sum__less__suminf2,axiom,
    ! [F: fun_nat_real,Nb: nat,I3: nat] :
      ( summable_real(F)
     => ( ! [M2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M2)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,M2)) )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),I3)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(F,I3))
           => 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,Nb))),suminf_real(F)) ) ) ) ) ).

% sum_less_suminf2
tff(fact_8083_lemma__tan__total,axiom,
    ! [Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Yb)
     => ? [X3: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X3)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,X3),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
          & aa_real_o(aa_real_fun_real_o(ord_less_real,Yb),aa_real_real(tan_real,X3)) ) ) ).

% lemma_tan_total
tff(fact_8084_tan__gt__zero,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(divide_divide_real(pi),aa_num_real(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,Xb)) ) ) ).

% tan_gt_zero
tff(fact_8085_lemma__tan__total1,axiom,
    ! [Yb: real] :
    ? [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X3)
      & aa_real_o(aa_real_fun_real_o(ord_less_real,X3),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
      & ( aa_real_real(tan_real,X3) = Yb ) ) ).

% lemma_tan_total1
tff(fact_8086_tan__mono__lt__eq,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(divide_divide_real(pi),aa_num_real(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(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Yb)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Yb),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(tan_real,Xb)),aa_real_real(tan_real,Yb))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Yb) ) ) ) ) ) ).

% tan_mono_lt_eq
tff(fact_8087_tan__monotone_H,axiom,
    ! [Yb: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Yb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Yb),aa_real_real(divide_divide_real(pi),aa_num_real(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(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xb)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Yb),Xb)
            <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(tan_real,Yb)),aa_real_real(tan_real,Xb)) ) ) ) ) ) ).

% tan_monotone'
tff(fact_8088_tan__monotone,axiom,
    ! [Yb: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Yb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Yb),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(tan_real,Yb)),aa_real_real(tan_real,Xb)) ) ) ) ).

% tan_monotone
tff(fact_8089_tan__total,axiom,
    ! [Yb: real] :
    ? [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X3)
      & aa_real_o(aa_real_fun_real_o(ord_less_real,X3),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
      & ( aa_real_real(tan_real,X3) = Yb )
      & ! [Y3: real] :
          ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y3)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,Y3),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
            & ( aa_real_real(tan_real,Y3) = Yb ) )
         => ( Y3 = X3 ) ) ) ).

% tan_total
tff(fact_8090_tan__minus__45,axiom,
    aa_real_real(tan_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))))) = aa_real_real(uminus_uminus_real,one_one_real) ).

% tan_minus_45
tff(fact_8091_tan__inverse,axiom,
    ! [Yb: real] : aa_real_real(divide_divide_real(one_one_real),aa_real_real(tan_real,Yb)) = aa_real_real(tan_real,aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),Yb)) ).

% tan_inverse
tff(fact_8092_add__tan__eq,axiom,
    ! [Xb: complex,Yb: complex] :
      ( ( aa_complex_complex(cos_complex,Xb) != zero_zero_complex )
     => ( ( aa_complex_complex(cos_complex,Yb) != zero_zero_complex )
       => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(tan_complex,Xb)),aa_complex_complex(tan_complex,Yb)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(sin_complex,aa_complex_complex(plus_plus_complex(Xb),Yb))),aa_complex_complex(times_times_complex(aa_complex_complex(cos_complex,Xb)),aa_complex_complex(cos_complex,Yb))) ) ) ) ).

% add_tan_eq
tff(fact_8093_add__tan__eq,axiom,
    ! [Xb: real,Yb: real] :
      ( ( aa_real_real(cos_real,Xb) != zero_zero_real )
     => ( ( aa_real_real(cos_real,Yb) != zero_zero_real )
       => ( aa_real_real(plus_plus_real(aa_real_real(tan_real,Xb)),aa_real_real(tan_real,Yb)) = aa_real_real(divide_divide_real(aa_real_real(sin_real,aa_real_real(plus_plus_real(Xb),Yb))),aa_real_real(times_times_real(aa_real_real(cos_real,Xb)),aa_real_real(cos_real,Yb))) ) ) ) ).

% add_tan_eq
tff(fact_8094_tan__cot_H,axiom,
    ! [Xb: real] : aa_real_real(tan_real,aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),Xb)) = aa_real_real(cot_real,Xb) ).

% tan_cot'
tff(fact_8095_tan__pos__pi2__le,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(divide_divide_real(pi),aa_num_real(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,Xb)) ) ) ).

% tan_pos_pi2_le
tff(fact_8096_tan__total__pos,axiom,
    ! [Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Yb)
     => ? [X3: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X3)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,X3),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
          & ( aa_real_real(tan_real,X3) = Yb ) ) ) ).

% tan_total_pos
tff(fact_8097_tan__less__zero,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(divide_divide_real(aa_real_real(uminus_uminus_real,pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(tan_real,Xb)),zero_zero_real) ) ) ).

% tan_less_zero
tff(fact_8098_tan__mono__le__eq,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(divide_divide_real(pi),aa_num_real(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(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Yb)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Yb),aa_real_real(divide_divide_real(pi),aa_num_real(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,Xb)),aa_real_real(tan_real,Yb))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Yb) ) ) ) ) ) ).

% tan_mono_le_eq
tff(fact_8099_tan__mono__le,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Yb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Yb),aa_real_real(divide_divide_real(pi),aa_num_real(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,Xb)),aa_real_real(tan_real,Yb)) ) ) ) ).

% tan_mono_le
tff(fact_8100_tan__bound__pi2,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xb)),aa_real_real(divide_divide_real(pi),aa_num_real(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,Xb))),one_one_real) ) ).

% tan_bound_pi2
tff(fact_8101_tan__30,axiom,
    aa_real_real(tan_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,one))))) = aa_real_real(divide_divide_real(one_one_real),aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit1,one)))) ).

% tan_30
tff(fact_8102_arctan,axiom,
    ! [Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arctan,Yb))
      & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arctan,Yb)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
      & ( aa_real_real(tan_real,aa_real_real(arctan,Yb)) = Yb ) ) ).

% arctan
tff(fact_8103_arctan__tan,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => ( aa_real_real(arctan,aa_real_real(tan_real,Xb)) = Xb ) ) ) ).

% arctan_tan
tff(fact_8104_arctan__unique,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => ( ( aa_real_real(tan_real,Xb) = Yb )
         => ( aa_real_real(arctan,Yb) = Xb ) ) ) ) ).

% arctan_unique
tff(fact_8105_tan__add,axiom,
    ! [Xb: complex,Yb: complex] :
      ( ( aa_complex_complex(cos_complex,Xb) != zero_zero_complex )
     => ( ( aa_complex_complex(cos_complex,Yb) != zero_zero_complex )
       => ( ( aa_complex_complex(cos_complex,aa_complex_complex(plus_plus_complex(Xb),Yb)) != zero_zero_complex )
         => ( aa_complex_complex(tan_complex,aa_complex_complex(plus_plus_complex(Xb),Yb)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(tan_complex,Xb)),aa_complex_complex(tan_complex,Yb))),aa_complex_complex(minus_minus_complex(one_one_complex),aa_complex_complex(times_times_complex(aa_complex_complex(tan_complex,Xb)),aa_complex_complex(tan_complex,Yb)))) ) ) ) ) ).

% tan_add
tff(fact_8106_tan__add,axiom,
    ! [Xb: real,Yb: real] :
      ( ( aa_real_real(cos_real,Xb) != zero_zero_real )
     => ( ( aa_real_real(cos_real,Yb) != zero_zero_real )
       => ( ( aa_real_real(cos_real,aa_real_real(plus_plus_real(Xb),Yb)) != zero_zero_real )
         => ( aa_real_real(tan_real,aa_real_real(plus_plus_real(Xb),Yb)) = aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(aa_real_real(tan_real,Xb)),aa_real_real(tan_real,Yb))),aa_real_real(minus_minus_real(one_one_real),aa_real_real(times_times_real(aa_real_real(tan_real,Xb)),aa_real_real(tan_real,Yb)))) ) ) ) ) ).

% tan_add
tff(fact_8107_tan__diff,axiom,
    ! [Xb: complex,Yb: complex] :
      ( ( aa_complex_complex(cos_complex,Xb) != zero_zero_complex )
     => ( ( aa_complex_complex(cos_complex,Yb) != zero_zero_complex )
       => ( ( aa_complex_complex(cos_complex,aa_complex_complex(minus_minus_complex(Xb),Yb)) != zero_zero_complex )
         => ( aa_complex_complex(tan_complex,aa_complex_complex(minus_minus_complex(Xb),Yb)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(tan_complex,Xb)),aa_complex_complex(tan_complex,Yb))),aa_complex_complex(plus_plus_complex(one_one_complex),aa_complex_complex(times_times_complex(aa_complex_complex(tan_complex,Xb)),aa_complex_complex(tan_complex,Yb)))) ) ) ) ) ).

% tan_diff
tff(fact_8108_tan__diff,axiom,
    ! [Xb: real,Yb: real] :
      ( ( aa_real_real(cos_real,Xb) != zero_zero_real )
     => ( ( aa_real_real(cos_real,Yb) != zero_zero_real )
       => ( ( aa_real_real(cos_real,aa_real_real(minus_minus_real(Xb),Yb)) != zero_zero_real )
         => ( aa_real_real(tan_real,aa_real_real(minus_minus_real(Xb),Yb)) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(tan_real,Xb)),aa_real_real(tan_real,Yb))),aa_real_real(plus_plus_real(one_one_real),aa_real_real(times_times_real(aa_real_real(tan_real,Xb)),aa_real_real(tan_real,Yb)))) ) ) ) ) ).

% tan_diff
tff(fact_8109_lemma__tan__add1,axiom,
    ! [Xb: complex,Yb: complex] :
      ( ( aa_complex_complex(cos_complex,Xb) != zero_zero_complex )
     => ( ( aa_complex_complex(cos_complex,Yb) != zero_zero_complex )
       => ( aa_complex_complex(minus_minus_complex(one_one_complex),aa_complex_complex(times_times_complex(aa_complex_complex(tan_complex,Xb)),aa_complex_complex(tan_complex,Yb))) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(cos_complex,aa_complex_complex(plus_plus_complex(Xb),Yb))),aa_complex_complex(times_times_complex(aa_complex_complex(cos_complex,Xb)),aa_complex_complex(cos_complex,Yb))) ) ) ) ).

% lemma_tan_add1
tff(fact_8110_lemma__tan__add1,axiom,
    ! [Xb: real,Yb: real] :
      ( ( aa_real_real(cos_real,Xb) != zero_zero_real )
     => ( ( aa_real_real(cos_real,Yb) != zero_zero_real )
       => ( aa_real_real(minus_minus_real(one_one_real),aa_real_real(times_times_real(aa_real_real(tan_real,Xb)),aa_real_real(tan_real,Yb))) = aa_real_real(divide_divide_real(aa_real_real(cos_real,aa_real_real(plus_plus_real(Xb),Yb))),aa_real_real(times_times_real(aa_real_real(cos_real,Xb)),aa_real_real(cos_real,Yb))) ) ) ) ).

% lemma_tan_add1
tff(fact_8111_tan__total__pi4,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xb)),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(divide_divide_real(pi),aa_num_real(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(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))))
          & ( aa_real_real(tan_real,Z2) = Xb ) ) ) ).

% tan_total_pi4
tff(fact_8112_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(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_8113_tan__half,axiom,
    ! [Xb: complex] : aa_complex_complex(tan_complex,Xb) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(sin_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),Xb))),aa_complex_complex(plus_plus_complex(aa_complex_complex(cos_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),Xb))),one_one_complex)) ).

% tan_half
tff(fact_8114_tan__half,axiom,
    ! [Xb: real] : aa_real_real(tan_real,Xb) = aa_real_real(divide_divide_real(aa_real_real(sin_real,aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Xb))),aa_real_real(plus_plus_real(aa_real_real(cos_real,aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Xb))),one_one_real)) ).

% tan_half
tff(fact_8115_cos__tan,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xb)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
     => ( aa_real_real(cos_real,Xb) = aa_real_real(divide_divide_real(one_one_real),aa_real_real(sqrt,aa_real_real(plus_plus_real(one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(tan_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ).

% cos_tan
tff(fact_8116_Maclaurin__minus__cos__expansion,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real)
       => ? [T2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),T2)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,T2),zero_zero_real)
            & ( aa_real_real(cos_real,Xb) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_iv,Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(cos_real,aa_real_real(plus_plus_real(T2),aa_real_real(times_times_real(aa_real_real(times_times_real(aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,Nb))),pi)))),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb))) ) ) ) ) ).

% Maclaurin_minus_cos_expansion
tff(fact_8117_Maclaurin__cos__expansion2,axiom,
    ! [Xb: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ? [T2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),T2)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,T2),Xb)
            & ( aa_real_real(cos_real,Xb) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_iv,Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(cos_real,aa_real_real(plus_plus_real(T2),aa_real_real(times_times_real(aa_real_real(times_times_real(aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,Nb))),pi)))),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb))) ) ) ) ) ).

% Maclaurin_cos_expansion2
tff(fact_8118_Maclaurin__cos__expansion,axiom,
    ! [Xb: real,Nb: nat] :
    ? [T2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T2)),aa_real_real(abs_abs_real,Xb))
      & ( aa_real_real(cos_real,Xb) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_iv,Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(cos_real,aa_real_real(plus_plus_real(T2),aa_real_real(times_times_real(aa_real_real(times_times_real(aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,Nb))),pi)))),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb))) ) ) ).

% Maclaurin_cos_expansion
tff(fact_8119_complex__unimodular__polar,axiom,
    ! [Z: complex] :
      ( ( aa_complex_real(real_V1022390504157884413omplex,Z) = one_one_real )
     => ~ ! [T2: real] :
            ( 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_real,T2),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))
             => ( Z != aa_real_complex(complex2(aa_real_real(cos_real,T2)),aa_real_real(sin_real,T2)) ) ) ) ) ).

% complex_unimodular_polar
tff(fact_8120_Maclaurin__exp__lt,axiom,
    ! [Xb: real,Nb: nat] :
      ( ( Xb != zero_zero_real )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ? [T2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(abs_abs_real,T2))
            & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,T2)),aa_real_real(abs_abs_real,Xb))
            & ( aa_real_real(exp_real,Xb) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_iw,Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(exp_real,T2)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb))) ) ) ) ) ).

% Maclaurin_exp_lt
tff(fact_8121_fact__0,axiom,
    aa_nat_complex(semiri5044797733671781792omplex,zero_zero_nat) = one_one_complex ).

% fact_0
tff(fact_8122_fact__0,axiom,
    aa_nat_rat(semiri773545260158071498ct_rat,zero_zero_nat) = one_one_rat ).

% fact_0
tff(fact_8123_fact__0,axiom,
    aa_nat_int(semiri1406184849735516958ct_int,zero_zero_nat) = one_one_int ).

% fact_0
tff(fact_8124_fact__0,axiom,
    aa_nat_real(semiri2265585572941072030t_real,zero_zero_nat) = one_one_real ).

% fact_0
tff(fact_8125_fact__0,axiom,
    aa_nat_nat(semiri1408675320244567234ct_nat,zero_zero_nat) = one_one_nat ).

% fact_0
tff(fact_8126_fact__1,axiom,
    aa_nat_complex(semiri5044797733671781792omplex,one_one_nat) = one_one_complex ).

% fact_1
tff(fact_8127_fact__1,axiom,
    aa_nat_rat(semiri773545260158071498ct_rat,one_one_nat) = one_one_rat ).

% fact_1
tff(fact_8128_fact__1,axiom,
    aa_nat_int(semiri1406184849735516958ct_int,one_one_nat) = one_one_int ).

% fact_1
tff(fact_8129_fact__1,axiom,
    aa_nat_real(semiri2265585572941072030t_real,one_one_nat) = one_one_real ).

% fact_1
tff(fact_8130_fact__1,axiom,
    aa_nat_nat(semiri1408675320244567234ct_nat,one_one_nat) = one_one_nat ).

% fact_1
tff(fact_8131_fact__Suc__0,axiom,
    aa_nat_complex(semiri5044797733671781792omplex,aa_nat_nat(suc,zero_zero_nat)) = one_one_complex ).

% fact_Suc_0
tff(fact_8132_fact__Suc__0,axiom,
    aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(suc,zero_zero_nat)) = one_one_rat ).

% fact_Suc_0
tff(fact_8133_fact__Suc__0,axiom,
    aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(suc,zero_zero_nat)) = one_one_int ).

% fact_Suc_0
tff(fact_8134_fact__Suc__0,axiom,
    aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(suc,zero_zero_nat)) = one_one_real ).

% fact_Suc_0
tff(fact_8135_fact__Suc__0,axiom,
    aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(suc,zero_zero_nat)) = one_one_nat ).

% fact_Suc_0
tff(fact_8136_fact__Suc,axiom,
    ! [Nb: nat] : aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(suc,Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,Nb))),aa_nat_int(semiri1406184849735516958ct_int,Nb)) ).

% fact_Suc
tff(fact_8137_fact__Suc,axiom,
    ! [Nb: nat] : aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(suc,Nb)) = aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,Nb))),aa_nat_rat(semiri773545260158071498ct_rat,Nb)) ).

% fact_Suc
tff(fact_8138_fact__Suc,axiom,
    ! [Nb: nat] : aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(suc,Nb)) = aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Nb))),aa_nat_real(semiri2265585572941072030t_real,Nb)) ).

% fact_Suc
tff(fact_8139_fact__Suc,axiom,
    ! [Nb: nat] : aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(suc,Nb))),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ).

% fact_Suc
tff(fact_8140_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_8141_fact__2,axiom,
    aa_nat_rat(semiri773545260158071498ct_rat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)) ).

% fact_2
tff(fact_8142_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_8143_fact__2,axiom,
    aa_nat_real(semiri2265585572941072030t_real,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)) ).

% fact_2
tff(fact_8144_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_8145_norm__cos__sin,axiom,
    ! [Ta: real] : aa_complex_real(real_V1022390504157884413omplex,aa_real_complex(complex2(aa_real_real(cos_real,Ta)),aa_real_real(sin_real,Ta))) = one_one_real ).

% norm_cos_sin
tff(fact_8146_fact__nonzero,axiom,
    ! [Nb: nat] : aa_nat_complex(semiri5044797733671781792omplex,Nb) != zero_zero_complex ).

% fact_nonzero
tff(fact_8147_fact__nonzero,axiom,
    ! [Nb: nat] : aa_nat_rat(semiri773545260158071498ct_rat,Nb) != zero_zero_rat ).

% fact_nonzero
tff(fact_8148_fact__nonzero,axiom,
    ! [Nb: nat] : aa_nat_int(semiri1406184849735516958ct_int,Nb) != zero_zero_int ).

% fact_nonzero
tff(fact_8149_fact__nonzero,axiom,
    ! [Nb: nat] : aa_nat_real(semiri2265585572941072030t_real,Nb) != zero_zero_real ).

% fact_nonzero
tff(fact_8150_fact__nonzero,axiom,
    ! [Nb: nat] : aa_nat_nat(semiri1408675320244567234ct_nat,Nb) != zero_zero_nat ).

% fact_nonzero
tff(fact_8151_fact__ge__zero,axiom,
    ! [Nb: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(semiri773545260158071498ct_rat,Nb)) ).

% fact_ge_zero
tff(fact_8152_fact__ge__zero,axiom,
    ! [Nb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(semiri1406184849735516958ct_int,Nb)) ).

% fact_ge_zero
tff(fact_8153_fact__ge__zero,axiom,
    ! [Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(semiri2265585572941072030t_real,Nb)) ).

% fact_ge_zero
tff(fact_8154_fact__ge__zero,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ).

% fact_ge_zero
tff(fact_8155_fact__gt__zero,axiom,
    ! [Nb: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(semiri773545260158071498ct_rat,Nb)) ).

% fact_gt_zero
tff(fact_8156_fact__gt__zero,axiom,
    ! [Nb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(semiri1406184849735516958ct_int,Nb)) ).

% fact_gt_zero
tff(fact_8157_fact__gt__zero,axiom,
    ! [Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(semiri2265585572941072030t_real,Nb)) ).

% fact_gt_zero
tff(fact_8158_fact__gt__zero,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ).

% fact_gt_zero
tff(fact_8159_fact__not__neg,axiom,
    ! [Nb: nat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri773545260158071498ct_rat,Nb)),zero_zero_rat) ).

% fact_not_neg
tff(fact_8160_fact__not__neg,axiom,
    ! [Nb: nat] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1406184849735516958ct_int,Nb)),zero_zero_int) ).

% fact_not_neg
tff(fact_8161_fact__not__neg,axiom,
    ! [Nb: nat] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri2265585572941072030t_real,Nb)),zero_zero_real) ).

% fact_not_neg
tff(fact_8162_fact__not__neg,axiom,
    ! [Nb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Nb)),zero_zero_nat) ).

% fact_not_neg
tff(fact_8163_fact__ge__1,axiom,
    ! [Nb: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_nat_rat(semiri773545260158071498ct_rat,Nb)) ).

% fact_ge_1
tff(fact_8164_fact__ge__1,axiom,
    ! [Nb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_nat_int(semiri1406184849735516958ct_int,Nb)) ).

% fact_ge_1
tff(fact_8165_fact__ge__1,axiom,
    ! [Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_nat_real(semiri2265585572941072030t_real,Nb)) ).

% fact_ge_1
tff(fact_8166_fact__ge__1,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ).

% fact_ge_1
tff(fact_8167_fact__mono,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri773545260158071498ct_rat,Mb)),aa_nat_rat(semiri773545260158071498ct_rat,Nb)) ) ).

% fact_mono
tff(fact_8168_fact__mono,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1406184849735516958ct_int,Mb)),aa_nat_int(semiri1406184849735516958ct_int,Nb)) ) ).

% fact_mono
tff(fact_8169_fact__mono,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri2265585572941072030t_real,Mb)),aa_nat_real(semiri2265585572941072030t_real,Nb)) ) ).

% fact_mono
tff(fact_8170_fact__mono,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Mb)),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ) ).

% fact_mono
tff(fact_8171_fact__dvd,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(semiri1406184849735516958ct_int,Nb)),aa_nat_int(semiri1406184849735516958ct_int,Mb)) ) ).

% fact_dvd
tff(fact_8172_fact__dvd,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(semiri3624122377584611663nteger,Nb)),aa_nat_Code_integer(semiri3624122377584611663nteger,Mb)) ) ).

% fact_dvd
tff(fact_8173_fact__dvd,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_nat_real(semiri2265585572941072030t_real,Nb)),aa_nat_real(semiri2265585572941072030t_real,Mb)) ) ).

% fact_dvd
tff(fact_8174_fact__dvd,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Nb)),aa_nat_nat(semiri1408675320244567234ct_nat,Mb)) ) ).

% fact_dvd
tff(fact_8175_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(plus_plus_real(A),C)),aa_real_real(plus_plus_real(B),D)) ).

% complex_add
tff(fact_8176_fact__less__mono,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri773545260158071498ct_rat,Mb)),aa_nat_rat(semiri773545260158071498ct_rat,Nb)) ) ) ).

% fact_less_mono
tff(fact_8177_fact__less__mono,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1406184849735516958ct_int,Mb)),aa_nat_int(semiri1406184849735516958ct_int,Nb)) ) ) ).

% fact_less_mono
tff(fact_8178_fact__less__mono,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri2265585572941072030t_real,Mb)),aa_nat_real(semiri2265585572941072030t_real,Nb)) ) ) ).

% fact_less_mono
tff(fact_8179_fact__less__mono,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Mb)),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ) ) ).

% fact_less_mono
tff(fact_8180_fact__fact__dvd__fact,axiom,
    ! [K: nat,Nb: 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,Nb))),aa_nat_Code_integer(semiri3624122377584611663nteger,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Nb))) ).

% fact_fact_dvd_fact
tff(fact_8181_fact__fact__dvd__fact,axiom,
    ! [K: nat,Nb: 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,Nb))),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Nb))) ).

% fact_fact_dvd_fact
tff(fact_8182_fact__fact__dvd__fact,axiom,
    ! [K: nat,Nb: 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,Nb))),aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Nb))) ).

% fact_fact_dvd_fact
tff(fact_8183_fact__fact__dvd__fact,axiom,
    ! [K: nat,Nb: nat] : aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_real_real(times_times_real(aa_nat_real(semiri2265585572941072030t_real,K)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Nb))) ).

% fact_fact_dvd_fact
tff(fact_8184_fact__fact__dvd__fact,axiom,
    ! [K: nat,Nb: 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,Nb))),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Nb))) ).

% fact_fact_dvd_fact
tff(fact_8185_fact__mod,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_nat_int(semiri1406184849735516958ct_int,Nb)),aa_nat_int(semiri1406184849735516958ct_int,Mb)) = zero_zero_int ) ) ).

% fact_mod
tff(fact_8186_fact__mod,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Nb)),aa_nat_nat(semiri1408675320244567234ct_nat,Mb)) = zero_zero_nat ) ) ).

% fact_mod
tff(fact_8187_fact__le__power,axiom,
    ! [Nb: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri773545260158071498ct_rat,Nb)),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Nb),Nb))) ).

% fact_le_power
tff(fact_8188_fact__le__power,axiom,
    ! [Nb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1406184849735516958ct_int,Nb)),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Nb),Nb))) ).

% fact_le_power
tff(fact_8189_fact__le__power,axiom,
    ! [Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri2265585572941072030t_real,Nb)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Nb),Nb))) ).

% fact_le_power
tff(fact_8190_fact__le__power,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Nb)),aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Nb),Nb))) ).

% fact_le_power
tff(fact_8191_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(minus_minus_real(aa_real_real(times_times_real(A),C)),aa_real_real(times_times_real(B),D))),aa_real_real(plus_plus_real(aa_real_real(times_times_real(A),D)),aa_real_real(times_times_real(B),C))) ).

% complex_mult
tff(fact_8192_one__complex_Ocode,axiom,
    one_one_complex = aa_real_complex(complex2(one_one_real),zero_zero_real) ).

% one_complex.code
tff(fact_8193_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_8194_choose__dvd,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => 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,Nb),K)))),aa_nat_Code_integer(semiri3624122377584611663nteger,Nb)) ) ).

% choose_dvd
tff(fact_8195_choose__dvd,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => 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,Nb),K)))),aa_nat_rat(semiri773545260158071498ct_rat,Nb)) ) ).

% choose_dvd
tff(fact_8196_choose__dvd,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => 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,Nb),K)))),aa_nat_int(semiri1406184849735516958ct_int,Nb)) ) ).

% choose_dvd
tff(fact_8197_choose__dvd,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_real_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,Nb),K)))),aa_nat_real(semiri2265585572941072030t_real,Nb)) ) ).

% choose_dvd
tff(fact_8198_choose__dvd,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => 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,Nb),K)))),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ) ).

% choose_dvd
tff(fact_8199_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_8200_fact__numeral,axiom,
    ! [K: num] : aa_nat_rat(semiri773545260158071498ct_rat,aa_num_nat(numeral_numeral_nat,K)) = aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,K)),aa_nat_rat(semiri773545260158071498ct_rat,aa_num_nat(pred_numeral,K))) ).

% fact_numeral
tff(fact_8201_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_8202_fact__numeral,axiom,
    ! [K: num] : aa_nat_real(semiri2265585572941072030t_real,aa_num_nat(numeral_numeral_nat,K)) = aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,K)),aa_nat_real(semiri2265585572941072030t_real,aa_num_nat(pred_numeral,K))) ).

% fact_numeral
tff(fact_8203_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_8204_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_8205_square__fact__le__2__fact,axiom,
    ! [Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(times_times_real(aa_nat_real(semiri2265585572941072030t_real,Nb)),aa_nat_real(semiri2265585572941072030t_real,Nb))),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))),Nb))) ).

% square_fact_le_2_fact
tff(fact_8206_fact__num__eq__if,axiom,
    ! [Mb: nat] :
      aa_nat_complex(semiri5044797733671781792omplex,Mb) = $ite(Mb = zero_zero_nat,one_one_complex,aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,Mb)),aa_nat_complex(semiri5044797733671781792omplex,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)))) ).

% fact_num_eq_if
tff(fact_8207_fact__num__eq__if,axiom,
    ! [Mb: nat] :
      aa_nat_int(semiri1406184849735516958ct_int,Mb) = $ite(Mb = zero_zero_nat,one_one_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)))) ).

% fact_num_eq_if
tff(fact_8208_fact__num__eq__if,axiom,
    ! [Mb: nat] :
      aa_nat_rat(semiri773545260158071498ct_rat,Mb) = $ite(Mb = zero_zero_nat,one_one_rat,aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,Mb)),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)))) ).

% fact_num_eq_if
tff(fact_8209_fact__num__eq__if,axiom,
    ! [Mb: nat] :
      aa_nat_real(semiri2265585572941072030t_real,Mb) = $ite(Mb = zero_zero_nat,one_one_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)))) ).

% fact_num_eq_if
tff(fact_8210_fact__num__eq__if,axiom,
    ! [Mb: nat] :
      aa_nat_nat(semiri1408675320244567234ct_nat,Mb) = $ite(Mb = zero_zero_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)))) ).

% fact_num_eq_if
tff(fact_8211_fact__reduce,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_int(semiri1406184849735516958ct_int,Nb) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))) ) ) ).

% fact_reduce
tff(fact_8212_fact__reduce,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_rat(semiri773545260158071498ct_rat,Nb) = aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,Nb)),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))) ) ) ).

% fact_reduce
tff(fact_8213_fact__reduce,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_real(semiri2265585572941072030t_real,Nb) = aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))) ) ) ).

% fact_reduce
tff(fact_8214_fact__reduce,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_nat(semiri1408675320244567234ct_nat,Nb) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Nb)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))) ) ) ).

% fact_reduce
tff(fact_8215_complex__norm,axiom,
    ! [Xb: real,Yb: real] : aa_complex_real(real_V1022390504157884413omplex,aa_real_complex(complex2(Xb),Yb)) = aa_real_real(sqrt,aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% complex_norm
tff(fact_8216_Maclaurin__zero,axiom,
    ! [Xb: real,Nb: nat,Diff: fun_na5723442673449079972x_real] :
      ( ( Xb = zero_zero_real )
     => ( ( Nb != zero_zero_nat )
       => ( aa_set_nat_real(groups6591440286371151544t_real(aa_fun5746040925223427547t_real(aTP_Lamm_ix(Xb),Diff)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = aa_complex_real(aa_nat480766339200631653x_real(Diff,zero_zero_nat),zero_zero_complex) ) ) ) ).

% Maclaurin_zero
tff(fact_8217_Maclaurin__zero,axiom,
    ! [Xb: real,Nb: nat,Diff: fun_na6724076748692969250l_real] :
      ( ( Xb = zero_zero_real )
     => ( ( Nb != zero_zero_nat )
       => ( aa_set_nat_real(groups6591440286371151544t_real(aa_fun1113820866546176217t_real(aTP_Lamm_iy(Xb),Diff)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = aa_real_real(aa_nat_fun_real_real(Diff,zero_zero_nat),zero_zero_real) ) ) ) ).

% Maclaurin_zero
tff(fact_8218_Maclaurin__zero,axiom,
    ! [Xb: real,Nb: nat,Diff: fun_nat_fun_rat_real] :
      ( ( Xb = zero_zero_real )
     => ( ( Nb != zero_zero_nat )
       => ( aa_set_nat_real(groups6591440286371151544t_real(aa_fun507281639146497029t_real(aTP_Lamm_iz(Xb),Diff)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = aa_rat_real(aa_nat_fun_rat_real(Diff,zero_zero_nat),zero_zero_rat) ) ) ) ).

% Maclaurin_zero
tff(fact_8219_Maclaurin__zero,axiom,
    ! [Xb: real,Nb: nat,Diff: fun_nat_fun_nat_real] :
      ( ( Xb = zero_zero_real )
     => ( ( Nb != zero_zero_nat )
       => ( aa_set_nat_real(groups6591440286371151544t_real(aa_fun7039663905532832253t_real(aTP_Lamm_ja(Xb),Diff)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = aa_nat_real(aa_nat_fun_nat_real(Diff,zero_zero_nat),zero_zero_nat) ) ) ) ).

% Maclaurin_zero
tff(fact_8220_Maclaurin__zero,axiom,
    ! [Xb: real,Nb: nat,Diff: fun_nat_fun_int_real] :
      ( ( Xb = zero_zero_real )
     => ( ( Nb != zero_zero_nat )
       => ( aa_set_nat_real(groups6591440286371151544t_real(aa_fun241848180832411225t_real(aTP_Lamm_jb(Xb),Diff)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = aa_int_real(aa_nat_fun_int_real(Diff,zero_zero_nat),zero_zero_int) ) ) ) ).

% Maclaurin_zero
tff(fact_8221_Maclaurin__lemma,axiom,
    ! [H: real,F: fun_real_real,J: fun_nat_real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H)
     => ? [B7: real] : aa_real_real(F,H) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_jc(H),J)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(times_times_real(B7),aa_real_real(divide_divide_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,H),Nb)),aa_nat_real(semiri2265585572941072030t_real,Nb)))) ) ).

% Maclaurin_lemma
tff(fact_8222_Maclaurin__exp__le,axiom,
    ! [Xb: real,Nb: nat] :
    ? [T2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T2)),aa_real_real(abs_abs_real,Xb))
      & ( aa_real_real(exp_real,Xb) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_iw,Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(exp_real,T2)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb))) ) ) ).

% Maclaurin_exp_le
tff(fact_8223_cos__coeff__def,axiom,
    ! [X: nat] :
      aa_nat_real(cos_coeff,X) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),X),aa_real_real(divide_divide_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_real(semiri2265585572941072030t_real,X)),zero_zero_real) ).

% cos_coeff_def
tff(fact_8224_Maclaurin__sin__expansion3,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ? [T2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),T2)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,T2),Xb)
            & ( aa_real_real(sin_real,Xb) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_jd,Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(sin_real,aa_real_real(plus_plus_real(T2),aa_real_real(times_times_real(aa_real_real(times_times_real(aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,Nb))),pi)))),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb))) ) ) ) ) ).

% Maclaurin_sin_expansion3
tff(fact_8225_Maclaurin__sin__expansion4,axiom,
    ! [Xb: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ? [T2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),T2)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T2),Xb)
          & ( aa_real_real(sin_real,Xb) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_jd,Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(sin_real,aa_real_real(plus_plus_real(T2),aa_real_real(times_times_real(aa_real_real(times_times_real(aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,Nb))),pi)))),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb))) ) ) ) ).

% Maclaurin_sin_expansion4
tff(fact_8226_Maclaurin__sin__expansion2,axiom,
    ! [Xb: real,Nb: nat] :
    ? [T2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T2)),aa_real_real(abs_abs_real,Xb))
      & ( aa_real_real(sin_real,Xb) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_jd,Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(sin_real,aa_real_real(plus_plus_real(T2),aa_real_real(times_times_real(aa_real_real(times_times_real(aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,Nb))),pi)))),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb))) ) ) ).

% Maclaurin_sin_expansion2
tff(fact_8227_Maclaurin__sin__expansion,axiom,
    ! [Xb: real,Nb: nat] :
    ? [T2: real] : aa_real_real(sin_real,Xb) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_jd,Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(sin_real,aa_real_real(plus_plus_real(T2),aa_real_real(times_times_real(aa_real_real(times_times_real(aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,Nb))),pi)))),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb))) ).

% Maclaurin_sin_expansion
tff(fact_8228_sin__coeff__def,axiom,
    ! [X: nat] :
      aa_nat_real(sin_coeff,X) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),X),zero_zero_real,aa_real_real(divide_divide_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X),aa_nat_nat(suc,zero_zero_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_real(semiri2265585572941072030t_real,X))) ).

% sin_coeff_def
tff(fact_8229_sin__coeff__0,axiom,
    aa_nat_real(sin_coeff,zero_zero_nat) = zero_zero_real ).

% sin_coeff_0
tff(fact_8230_fact__mono__nat,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Mb)),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ) ).

% fact_mono_nat
tff(fact_8231_fact__ge__self,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ).

% fact_ge_self
tff(fact_8232_fact__less__mono__nat,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Mb)),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ) ) ).

% fact_less_mono_nat
tff(fact_8233_fact__ge__Suc__0__nat,axiom,
    ! [Nb: 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,Nb)) ).

% fact_ge_Suc_0_nat
tff(fact_8234_dvd__fact,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ) ) ).

% dvd_fact
tff(fact_8235_fact__diff__Suc,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(suc,Mb))
     => ( aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,Mb)),Nb)) = 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,Mb)),Nb)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb))) ) ) ).

% fact_diff_Suc
tff(fact_8236_fact__div__fact__le__pow,axiom,
    ! [R2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,R2),Nb)
     => 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,Nb)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),R2)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Nb),R2)) ) ).

% fact_div_fact_le_pow
tff(fact_8237_sin__coeff__Suc,axiom,
    ! [Nb: nat] : aa_nat_real(sin_coeff,aa_nat_nat(suc,Nb)) = aa_real_real(divide_divide_real(aa_nat_real(cos_coeff,Nb)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Nb))) ).

% sin_coeff_Suc
tff(fact_8238_cos__coeff__Suc,axiom,
    ! [Nb: nat] : aa_nat_real(cos_coeff,aa_nat_nat(suc,Nb)) = aa_real_real(divide_divide_real(aa_real_real(uminus_uminus_real,aa_nat_real(sin_coeff,Nb))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Nb))) ).

% cos_coeff_Suc
tff(fact_8239_sin__paired,axiom,
    ! [Xb: real] : aa_real_o(sums_real(aa_real_fun_nat_real(aTP_Lamm_je,Xb)),aa_real_real(sin_real,Xb)) ).

% sin_paired
tff(fact_8240_cos__arcsin,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real)
       => ( aa_real_real(cos_real,aa_real_real(arcsin,Xb)) = aa_real_real(sqrt,aa_real_real(minus_minus_real(one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ) ).

% cos_arcsin
tff(fact_8241_sin__arccos__abs,axiom,
    ! [Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Yb)),one_one_real)
     => ( aa_real_real(sin_real,aa_real_real(arccos,Yb)) = aa_real_real(sqrt,aa_real_real(minus_minus_real(one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Yb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ).

% sin_arccos_abs
tff(fact_8242_sin__arccos,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real)
       => ( aa_real_real(sin_real,aa_real_real(arccos,Xb)) = aa_real_real(sqrt,aa_real_real(minus_minus_real(one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ) ).

% sin_arccos
tff(fact_8243_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(aa_real_fun_nat_real(aTP_Lamm_jf,Z)),aa_real_real(divide_divide_real(one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(minus_minus_real(one_one_real),Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% geometric_deriv_sums
tff(fact_8244_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_jg(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_8245_sums__zero,axiom,
    aa_complex_o(sums_complex(aTP_Lamm_gg),zero_zero_complex) ).

% sums_zero
tff(fact_8246_sums__zero,axiom,
    aa_real_o(sums_real(aTP_Lamm_de),zero_zero_real) ).

% sums_zero
tff(fact_8247_sums__zero,axiom,
    aa_nat_o(sums_nat(aTP_Lamm_am),zero_zero_nat) ).

% sums_zero
tff(fact_8248_sums__zero,axiom,
    aa_int_o(sums_int(aTP_Lamm_gh),zero_zero_int) ).

% sums_zero
tff(fact_8249_arccos__1,axiom,
    aa_real_real(arccos,one_one_real) = zero_zero_real ).

% arccos_1
tff(fact_8250_arccos__minus__1,axiom,
    aa_real_real(arccos,aa_real_real(uminus_uminus_real,one_one_real)) = pi ).

% arccos_minus_1
tff(fact_8251_cos__arccos,axiom,
    ! [Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Yb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Yb),one_one_real)
       => ( aa_real_real(cos_real,aa_real_real(arccos,Yb)) = Yb ) ) ) ).

% cos_arccos
tff(fact_8252_sin__arcsin,axiom,
    ! [Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Yb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Yb),one_one_real)
       => ( aa_real_real(sin_real,aa_real_real(arcsin,Yb)) = Yb ) ) ) ).

% sin_arcsin
tff(fact_8253_powser__sums__zero__iff,axiom,
    ! [A: fun_nat_complex,Xb: complex] :
      ( aa_complex_o(sums_complex(aTP_Lamm_gc(A)),Xb)
    <=> ( aa_nat_complex(A,zero_zero_nat) = Xb ) ) ).

% powser_sums_zero_iff
tff(fact_8254_powser__sums__zero__iff,axiom,
    ! [A: fun_nat_real,Xb: real] :
      ( aa_real_o(sums_real(aTP_Lamm_gd(A)),Xb)
    <=> ( aa_nat_real(A,zero_zero_nat) = Xb ) ) ).

% powser_sums_zero_iff
tff(fact_8255_arccos__0,axiom,
    aa_real_real(arccos,zero_zero_real) = aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% arccos_0
tff(fact_8256_arcsin__1,axiom,
    aa_real_real(arcsin,one_one_real) = aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% arcsin_1
tff(fact_8257_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(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ).

% arcsin_minus_1
tff(fact_8258_sums__le,axiom,
    ! [F: fun_nat_real,G: fun_nat_real,S2: real,Ta: 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),S2)
       => ( aa_real_o(sums_real(G),Ta)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,S2),Ta) ) ) ) ).

% sums_le
tff(fact_8259_sums__le,axiom,
    ! [F: fun_nat_nat,G: fun_nat_nat,S2: nat,Ta: 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),S2)
       => ( aa_nat_o(sums_nat(G),Ta)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,S2),Ta) ) ) ) ).

% sums_le
tff(fact_8260_sums__le,axiom,
    ! [F: fun_nat_int,G: fun_nat_int,S2: int,Ta: 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),S2)
       => ( aa_int_o(sums_int(G),Ta)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,S2),Ta) ) ) ) ).

% sums_le
tff(fact_8261_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_8262_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_8263_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_8264_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_8265_sums__single,axiom,
    ! [I3: nat,F: fun_nat_complex] : aa_complex_o(sums_complex(aa_fun2955221450537884980omplex(aTP_Lamm_hz(I3),F)),aa_nat_complex(F,I3)) ).

% sums_single
tff(fact_8266_sums__single,axiom,
    ! [I3: nat,F: fun_nat_real] : aa_real_o(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_ia(I3),F)),aa_nat_real(F,I3)) ).

% sums_single
tff(fact_8267_sums__single,axiom,
    ! [I3: nat,F: fun_nat_nat] : aa_nat_o(sums_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_ib(I3),F)),aa_nat_nat(F,I3)) ).

% sums_single
tff(fact_8268_sums__single,axiom,
    ! [I3: nat,F: fun_nat_int] : aa_int_o(sums_int(aa_fun5565043122120104496at_int(aTP_Lamm_ic(I3),F)),aa_nat_int(F,I3)) ).

% sums_single
tff(fact_8269_sums__mult,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_ih(F),C)),aa_real_real(times_times_real(C),A)) ) ).

% sums_mult
tff(fact_8270_sums__mult2,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_eu(F),C)),aa_real_real(times_times_real(A),C)) ) ).

% sums_mult2
tff(fact_8271_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_fg(F),G)),aa_real_real(plus_plus_real(A),B)) ) ) ).

% sums_add
tff(fact_8272_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_ff(F),G)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).

% sums_add
tff(fact_8273_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_ii(F),G)),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ) ).

% sums_add
tff(fact_8274_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_ie(F),C)),aa_complex_complex(divide1717551699836669952omplex(A),C)) ) ).

% sums_divide
tff(fact_8275_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_fi(F),C)),aa_real_real(divide_divide_real(A),C)) ) ).

% sums_divide
tff(fact_8276_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_id(C),F)),aa_complex_complex(times_times_complex(C),D))
      <=> aa_complex_o(sums_complex(F),D) ) ) ).

% sums_mult_iff
tff(fact_8277_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_eq(C),F)),aa_real_real(times_times_real(C),D))
      <=> aa_real_o(sums_real(F),D) ) ) ).

% sums_mult_iff
tff(fact_8278_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_jh(C),F)),aa_complex_complex(times_times_complex(D),C))
      <=> aa_complex_o(sums_complex(F),D) ) ) ).

% sums_mult2_iff
tff(fact_8279_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_ji(C),F)),aa_real_real(times_times_real(D),C))
      <=> aa_real_o(sums_real(F),D) ) ) ).

% sums_mult2_iff
tff(fact_8280_sums__mult__D,axiom,
    ! [C: complex,F: fun_nat_complex,A: complex] :
      ( aa_complex_o(sums_complex(aa_fun2955221450537884980omplex(aTP_Lamm_id(C),F)),A)
     => ( ( C != zero_zero_complex )
       => aa_complex_o(sums_complex(F),aa_complex_complex(divide1717551699836669952omplex(A),C)) ) ) ).

% sums_mult_D
tff(fact_8281_sums__mult__D,axiom,
    ! [C: real,F: fun_nat_real,A: real] :
      ( aa_real_o(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_eq(C),F)),A)
     => ( ( C != zero_zero_real )
       => aa_real_o(sums_real(F),aa_real_real(divide_divide_real(A),C)) ) ) ).

% sums_mult_D
tff(fact_8282_sums__Suc__imp,axiom,
    ! [F: fun_nat_complex,S2: complex] :
      ( ( aa_nat_complex(F,zero_zero_nat) = zero_zero_complex )
     => ( aa_complex_o(sums_complex(aTP_Lamm_jj(F)),S2)
       => aa_complex_o(sums_complex(F),S2) ) ) ).

% sums_Suc_imp
tff(fact_8283_sums__Suc__imp,axiom,
    ! [F: fun_nat_real,S2: real] :
      ( ( aa_nat_real(F,zero_zero_nat) = zero_zero_real )
     => ( aa_real_o(sums_real(aTP_Lamm_dn(F)),S2)
       => aa_real_o(sums_real(F),S2) ) ) ).

% sums_Suc_imp
tff(fact_8284_sums__Suc__iff,axiom,
    ! [F: fun_nat_real,S2: real] :
      ( aa_real_o(sums_real(aTP_Lamm_dn(F)),S2)
    <=> aa_real_o(sums_real(F),aa_real_real(plus_plus_real(S2),aa_nat_real(F,zero_zero_nat))) ) ).

% sums_Suc_iff
tff(fact_8285_sums__Suc,axiom,
    ! [F: fun_nat_real,L: real] :
      ( aa_real_o(sums_real(aTP_Lamm_dn(F)),L)
     => aa_real_o(sums_real(F),aa_real_real(plus_plus_real(L),aa_nat_real(F,zero_zero_nat))) ) ).

% sums_Suc
tff(fact_8286_sums__Suc,axiom,
    ! [F: fun_nat_nat,L: nat] :
      ( aa_nat_o(sums_nat(aTP_Lamm_dm(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_8287_sums__Suc,axiom,
    ! [F: fun_nat_int,L: int] :
      ( aa_int_o(sums_int(aTP_Lamm_dx(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_8288_sums__zero__iff__shift,axiom,
    ! [Nb: nat,F: fun_nat_complex,S2: complex] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),Nb)
         => ( aa_nat_complex(F,I2) = zero_zero_complex ) )
     => ( aa_complex_o(sums_complex(aa_fun2955221450537884980omplex(aTP_Lamm_jk(Nb),F)),S2)
      <=> aa_complex_o(sums_complex(F),S2) ) ) ).

% sums_zero_iff_shift
tff(fact_8289_sums__zero__iff__shift,axiom,
    ! [Nb: nat,F: fun_nat_real,S2: real] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),Nb)
         => ( aa_nat_real(F,I2) = zero_zero_real ) )
     => ( aa_real_o(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_jl(Nb),F)),S2)
      <=> aa_real_o(sums_real(F),S2) ) ) ).

% sums_zero_iff_shift
tff(fact_8290_arccos__le__arccos,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Yb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Yb),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arccos,Yb)),aa_real_real(arccos,Xb)) ) ) ) ).

% arccos_le_arccos
tff(fact_8291_arccos__eq__iff,axiom,
    ! [Xb: real,Yb: real] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),one_one_real)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Yb)),one_one_real) )
     => ( ( aa_real_real(arccos,Xb) = aa_real_real(arccos,Yb) )
      <=> ( Xb = Yb ) ) ) ).

% arccos_eq_iff
tff(fact_8292_arccos__le__mono,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Yb)),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arccos,Xb)),aa_real_real(arccos,Yb))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Yb),Xb) ) ) ) ).

% arccos_le_mono
tff(fact_8293_arcsin__le__arcsin,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Yb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Yb),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arcsin,Xb)),aa_real_real(arcsin,Yb)) ) ) ) ).

% arcsin_le_arcsin
tff(fact_8294_arcsin__minus,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real)
       => ( aa_real_real(arcsin,aa_real_real(uminus_uminus_real,Xb)) = aa_real_real(uminus_uminus_real,aa_real_real(arcsin,Xb)) ) ) ) ).

% arcsin_minus
tff(fact_8295_arcsin__eq__iff,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Yb)),one_one_real)
       => ( ( aa_real_real(arcsin,Xb) = aa_real_real(arcsin,Yb) )
        <=> ( Xb = Yb ) ) ) ) ).

% arcsin_eq_iff
tff(fact_8296_arcsin__le__mono,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Yb)),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arcsin,Xb)),aa_real_real(arcsin,Yb))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Yb) ) ) ) ).

% arcsin_le_mono
tff(fact_8297_powser__sums__if,axiom,
    ! [Mb: nat,Z: complex] : aa_complex_o(sums_complex(aa_com8181001469867462503omplex(aTP_Lamm_jm(Mb),Z)),aa_nat_complex(power_power_complex(Z),Mb)) ).

% powser_sums_if
tff(fact_8298_powser__sums__if,axiom,
    ! [Mb: nat,Z: real] : aa_real_o(sums_real(aa_real_fun_nat_real(aTP_Lamm_jn(Mb),Z)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),Mb)) ).

% powser_sums_if
tff(fact_8299_powser__sums__if,axiom,
    ! [Mb: nat,Z: int] : aa_int_o(sums_int(aa_int_fun_nat_int(aTP_Lamm_jo(Mb),Z)),aa_nat_int(power_power_int(Z),Mb)) ).

% powser_sums_if
tff(fact_8300_powser__sums__zero,axiom,
    ! [A: fun_nat_complex] : aa_complex_o(sums_complex(aTP_Lamm_gc(A)),aa_nat_complex(A,zero_zero_nat)) ).

% powser_sums_zero
tff(fact_8301_powser__sums__zero,axiom,
    ! [A: fun_nat_real] : aa_real_o(sums_real(aTP_Lamm_gd(A)),aa_nat_real(A,zero_zero_nat)) ).

% powser_sums_zero
tff(fact_8302_sums__iff__shift,axiom,
    ! [F: fun_nat_real,Nb: nat,S2: real] :
      ( aa_real_o(sums_real(aa_nat_fun_nat_real(aTP_Lamm_dp(F),Nb)),S2)
    <=> aa_real_o(sums_real(F),aa_real_real(plus_plus_real(S2),aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_ord_lessThan_nat,Nb)))) ) ).

% sums_iff_shift
tff(fact_8303_sums__split__initial__segment,axiom,
    ! [F: fun_nat_real,S2: real,Nb: nat] :
      ( aa_real_o(sums_real(F),S2)
     => aa_real_o(sums_real(aa_nat_fun_nat_real(aTP_Lamm_dp(F),Nb)),aa_real_real(minus_minus_real(S2),aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_ord_lessThan_nat,Nb)))) ) ).

% sums_split_initial_segment
tff(fact_8304_sums__iff__shift_H,axiom,
    ! [F: fun_nat_real,Nb: nat,S2: real] :
      ( aa_real_o(sums_real(aa_nat_fun_nat_real(aTP_Lamm_dp(F),Nb)),aa_real_real(minus_minus_real(S2),aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_ord_lessThan_nat,Nb))))
    <=> aa_real_o(sums_real(F),S2) ) ).

% sums_iff_shift'
tff(fact_8305_arccos__lbound,axiom,
    ! [Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Yb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Yb),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(arccos,Yb)) ) ) ).

% arccos_lbound
tff(fact_8306_arccos__less__arccos,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Yb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Yb),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arccos,Yb)),aa_real_real(arccos,Xb)) ) ) ) ).

% arccos_less_arccos
tff(fact_8307_arccos__less__mono,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Yb)),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arccos,Xb)),aa_real_real(arccos,Yb))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Yb),Xb) ) ) ) ).

% arccos_less_mono
tff(fact_8308_arccos__ubound,axiom,
    ! [Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Yb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Yb),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arccos,Yb)),pi) ) ) ).

% arccos_ubound
tff(fact_8309_arcsin__less__arcsin,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Yb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Yb),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arcsin,Xb)),aa_real_real(arcsin,Yb)) ) ) ) ).

% arcsin_less_arcsin
tff(fact_8310_arcsin__less__mono,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Yb)),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arcsin,Xb)),aa_real_real(arcsin,Yb))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Yb) ) ) ) ).

% arcsin_less_mono
tff(fact_8311_cos__arccos__abs,axiom,
    ! [Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Yb)),one_one_real)
     => ( aa_real_real(cos_real,aa_real_real(arccos,Yb)) = Yb ) ) ).

% cos_arccos_abs
tff(fact_8312_arccos__lt__bounded,axiom,
    ! [Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Yb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Yb),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(arccos,Yb))
          & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arccos,Yb)),pi) ) ) ) ).

% arccos_lt_bounded
tff(fact_8313_arccos__bounded,axiom,
    ! [Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Yb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Yb),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(arccos,Yb))
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arccos,Yb)),pi) ) ) ) ).

% arccos_bounded
tff(fact_8314_sin__arccos__nonzero,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),one_one_real)
       => ( aa_real_real(sin_real,aa_real_real(arccos,Xb)) != zero_zero_real ) ) ) ).

% sin_arccos_nonzero
tff(fact_8315_arccos__minus,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real)
       => ( aa_real_real(arccos,aa_real_real(uminus_uminus_real,Xb)) = aa_real_real(minus_minus_real(pi),aa_real_real(arccos,Xb)) ) ) ) ).

% arccos_minus
tff(fact_8316_cos__arcsin__nonzero,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),one_one_real)
       => ( aa_real_real(cos_real,aa_real_real(arcsin,Xb)) != zero_zero_real ) ) ) ).

% cos_arcsin_nonzero
tff(fact_8317_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(aa_real_fun_nat_real(power_power_real,C)),aa_real_real(divide_divide_real(one_one_real),aa_real_real(minus_minus_real(one_one_real),C))) ) ).

% geometric_sums
tff(fact_8318_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_8319_power__half__series,axiom,
    aa_real_o(sums_real(aTP_Lamm_jp),one_one_real) ).

% power_half_series
tff(fact_8320_arccos,axiom,
    ! [Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Yb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Yb),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(arccos,Yb))
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arccos,Yb)),pi)
          & ( aa_real_real(cos_real,aa_real_real(arccos,Yb)) = Yb ) ) ) ) ).

% arccos
tff(fact_8321_arccos__minus__abs,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),one_one_real)
     => ( aa_real_real(arccos,aa_real_real(uminus_uminus_real,Xb)) = aa_real_real(minus_minus_real(pi),aa_real_real(arccos,Xb)) ) ) ).

% arccos_minus_abs
tff(fact_8322_sums__if_H,axiom,
    ! [G: fun_nat_real,Xb: real] :
      ( aa_real_o(sums_real(G),Xb)
     => aa_real_o(sums_real(aTP_Lamm_jq(G)),Xb) ) ).

% sums_if'
tff(fact_8323_sums__if,axiom,
    ! [G: fun_nat_real,Xb: real,F: fun_nat_real,Yb: real] :
      ( aa_real_o(sums_real(G),Xb)
     => ( aa_real_o(sums_real(F),Yb)
       => aa_real_o(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_jr(G),F)),aa_real_real(plus_plus_real(Xb),Yb)) ) ) ).

% sums_if
tff(fact_8324_arccos__le__pi2,axiom,
    ! [Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Yb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Yb),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arccos,Yb)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ).

% arccos_le_pi2
tff(fact_8325_arcsin__lt__bounded,axiom,
    ! [Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Yb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Yb),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arcsin,Yb))
          & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arcsin,Yb)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ) ).

% arcsin_lt_bounded
tff(fact_8326_arcsin__lbound,axiom,
    ! [Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Yb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Yb),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arcsin,Yb)) ) ) ).

% arcsin_lbound
tff(fact_8327_arcsin__ubound,axiom,
    ! [Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Yb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Yb),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arcsin,Yb)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ).

% arcsin_ubound
tff(fact_8328_arcsin__bounded,axiom,
    ! [Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Yb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Yb),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arcsin,Yb))
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arcsin,Yb)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ) ).

% arcsin_bounded
tff(fact_8329_arcsin__sin,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => ( aa_real_real(arcsin,aa_real_real(sin_real,Xb)) = Xb ) ) ) ).

% arcsin_sin
tff(fact_8330_cos__paired,axiom,
    ! [Xb: real] : aa_real_o(sums_real(aa_real_fun_nat_real(aTP_Lamm_js,Xb)),aa_real_real(cos_real,Xb)) ).

% cos_paired
tff(fact_8331_le__arcsin__iff,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(aa_real_real(uminus_uminus_real,pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),Yb)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Yb),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Yb),aa_real_real(arcsin,Xb))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sin_real,Yb)),Xb) ) ) ) ) ) ).

% le_arcsin_iff
tff(fact_8332_arcsin__le__iff,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(aa_real_real(uminus_uminus_real,pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),Yb)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Yb),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arcsin,Xb)),Yb)
            <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(sin_real,Yb)) ) ) ) ) ) ).

% arcsin_le_iff
tff(fact_8333_arcsin__pi,axiom,
    ! [Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Yb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Yb),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arcsin,Yb))
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arcsin,Yb)),pi)
          & ( aa_real_real(sin_real,aa_real_real(arcsin,Yb)) = Yb ) ) ) ) ).

% arcsin_pi
tff(fact_8334_arcsin,axiom,
    ! [Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Yb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Yb),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arcsin,Yb))
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arcsin,Yb)),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
          & ( aa_real_real(sin_real,aa_real_real(arcsin,Yb)) = Yb ) ) ) ) ).

% arcsin
tff(fact_8335_arccos__cos__eq__abs__2pi,axiom,
    ! [Theta: real] :
      ~ ! [K2: int] : aa_real_real(arccos,aa_real_real(cos_real,Theta)) != aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(Theta),aa_real_real(times_times_real(aa_int_real(ring_1_of_int_real,K2)),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)))) ).

% arccos_cos_eq_abs_2pi
tff(fact_8336_diffs__equiv,axiom,
    ! [C: fun_nat_complex,Xb: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_jt(C),Xb))
     => aa_complex_o(sums_complex(aa_com8181001469867462503omplex(aTP_Lamm_ju(C),Xb)),suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_jt(C),Xb))) ) ).

% diffs_equiv
tff(fact_8337_diffs__equiv,axiom,
    ! [C: fun_nat_real,Xb: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_jv(C),Xb))
     => aa_real_o(sums_real(aa_real_fun_nat_real(aTP_Lamm_jw(C),Xb)),suminf_real(aa_real_fun_nat_real(aTP_Lamm_jv(C),Xb))) ) ).

% diffs_equiv
tff(fact_8338_monoI1,axiom,
    ! [X6: fun_nat_real] :
      ( ! [M2: nat,N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(X6,M2)),aa_nat_real(X6,N)) )
     => topolo6980174941875973593q_real(X6) ) ).

% monoI1
tff(fact_8339_monoI1,axiom,
    ! [X6: fun_nat_set_nat] :
      ( ! [M2: nat,N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N)
         => aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(X6,M2)),aa_nat_set_nat(X6,N)) )
     => topolo7278393974255667507et_nat(X6) ) ).

% monoI1
tff(fact_8340_monoI1,axiom,
    ! [X6: fun_nat_rat] :
      ( ! [M2: nat,N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(X6,M2)),aa_nat_rat(X6,N)) )
     => topolo4267028734544971653eq_rat(X6) ) ).

% monoI1
tff(fact_8341_monoI1,axiom,
    ! [X6: fun_nat_num] :
      ( ! [M2: nat,N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N)
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(X6,M2)),aa_nat_num(X6,N)) )
     => topolo1459490580787246023eq_num(X6) ) ).

% monoI1
tff(fact_8342_monoI1,axiom,
    ! [X6: fun_nat_nat] :
      ( ! [M2: nat,N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(X6,M2)),aa_nat_nat(X6,N)) )
     => topolo4902158794631467389eq_nat(X6) ) ).

% monoI1
tff(fact_8343_monoI1,axiom,
    ! [X6: fun_nat_int] :
      ( ! [M2: nat,N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(X6,M2)),aa_nat_int(X6,N)) )
     => topolo4899668324122417113eq_int(X6) ) ).

% monoI1
tff(fact_8344_monoI2,axiom,
    ! [X6: fun_nat_real] :
      ( ! [M2: nat,N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(X6,N)),aa_nat_real(X6,M2)) )
     => topolo6980174941875973593q_real(X6) ) ).

% monoI2
tff(fact_8345_monoI2,axiom,
    ! [X6: fun_nat_set_nat] :
      ( ! [M2: nat,N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N)
         => aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(X6,N)),aa_nat_set_nat(X6,M2)) )
     => topolo7278393974255667507et_nat(X6) ) ).

% monoI2
tff(fact_8346_monoI2,axiom,
    ! [X6: fun_nat_rat] :
      ( ! [M2: nat,N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(X6,N)),aa_nat_rat(X6,M2)) )
     => topolo4267028734544971653eq_rat(X6) ) ).

% monoI2
tff(fact_8347_monoI2,axiom,
    ! [X6: fun_nat_num] :
      ( ! [M2: nat,N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N)
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(X6,N)),aa_nat_num(X6,M2)) )
     => topolo1459490580787246023eq_num(X6) ) ).

% monoI2
tff(fact_8348_monoI2,axiom,
    ! [X6: fun_nat_nat] :
      ( ! [M2: nat,N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(X6,N)),aa_nat_nat(X6,M2)) )
     => topolo4902158794631467389eq_nat(X6) ) ).

% monoI2
tff(fact_8349_monoI2,axiom,
    ! [X6: fun_nat_int] :
      ( ! [M2: nat,N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(X6,N)),aa_nat_int(X6,M2)) )
     => topolo4899668324122417113eq_int(X6) ) ).

% monoI2
tff(fact_8350_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_8351_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_8352_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_8353_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(aa_num_fun_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(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(X6,N4)),aa_nat_num(X6,M5)) ) ) ) ).

% monoseq_def
tff(fact_8354_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_8355_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_8356_pochhammer__double,axiom,
    ! [Z: complex,Nb: 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))),Nb)) = 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))),Nb)))),aa_nat_complex(comm_s2602460028002588243omplex(Z),Nb))),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))))),Nb)) ).

% pochhammer_double
tff(fact_8357_pochhammer__double,axiom,
    ! [Z: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Z)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_real_real(times_times_real(aa_real_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))),Nb)))),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z),Nb))),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(plus_plus_real(Z),aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Nb)) ).

% pochhammer_double
tff(fact_8358_pochhammer__double,axiom,
    ! [Z: rat,Nb: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),Z)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = 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))),Nb)))),aa_nat_rat(comm_s4028243227959126397er_rat(Z),Nb))),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(plus_plus_rat(Z),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))),Nb)) ).

% pochhammer_double
tff(fact_8359_pochhammer__0,axiom,
    ! [A: complex] : aa_nat_complex(comm_s2602460028002588243omplex(A),zero_zero_nat) = one_one_complex ).

% pochhammer_0
tff(fact_8360_pochhammer__0,axiom,
    ! [A: real] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),zero_zero_nat) = one_one_real ).

% pochhammer_0
tff(fact_8361_pochhammer__0,axiom,
    ! [A: rat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),zero_zero_nat) = one_one_rat ).

% pochhammer_0
tff(fact_8362_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_8363_pochhammer__0,axiom,
    ! [A: int] : aa_nat_int(comm_s4660882817536571857er_int(A),zero_zero_nat) = one_one_int ).

% pochhammer_0
tff(fact_8364_pochhammer__pos,axiom,
    ! [Xb: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Xb),Nb)) ) ).

% pochhammer_pos
tff(fact_8365_pochhammer__pos,axiom,
    ! [Xb: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xb)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(comm_s4028243227959126397er_rat(Xb),Nb)) ) ).

% pochhammer_pos
tff(fact_8366_pochhammer__pos,axiom,
    ! [Xb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Xb)
     => 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,Xb),Nb)) ) ).

% pochhammer_pos
tff(fact_8367_pochhammer__pos,axiom,
    ! [Xb: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Xb)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(comm_s4660882817536571857er_int(Xb),Nb)) ) ).

% pochhammer_pos
tff(fact_8368_pochhammer__neq__0__mono,axiom,
    ! [A: complex,Mb: nat,Nb: nat] :
      ( ( aa_nat_complex(comm_s2602460028002588243omplex(A),Mb) != zero_zero_complex )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => ( aa_nat_complex(comm_s2602460028002588243omplex(A),Nb) != zero_zero_complex ) ) ) ).

% pochhammer_neq_0_mono
tff(fact_8369_pochhammer__neq__0__mono,axiom,
    ! [A: real,Mb: nat,Nb: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),Mb) != zero_zero_real )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),Nb) != zero_zero_real ) ) ) ).

% pochhammer_neq_0_mono
tff(fact_8370_pochhammer__neq__0__mono,axiom,
    ! [A: rat,Mb: nat,Nb: nat] :
      ( ( aa_nat_rat(comm_s4028243227959126397er_rat(A),Mb) != zero_zero_rat )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => ( aa_nat_rat(comm_s4028243227959126397er_rat(A),Nb) != zero_zero_rat ) ) ) ).

% pochhammer_neq_0_mono
tff(fact_8371_pochhammer__eq__0__mono,axiom,
    ! [A: complex,Nb: nat,Mb: nat] :
      ( ( aa_nat_complex(comm_s2602460028002588243omplex(A),Nb) = zero_zero_complex )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => ( aa_nat_complex(comm_s2602460028002588243omplex(A),Mb) = zero_zero_complex ) ) ) ).

% pochhammer_eq_0_mono
tff(fact_8372_pochhammer__eq__0__mono,axiom,
    ! [A: real,Nb: nat,Mb: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),Nb) = zero_zero_real )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),Mb) = zero_zero_real ) ) ) ).

% pochhammer_eq_0_mono
tff(fact_8373_pochhammer__eq__0__mono,axiom,
    ! [A: rat,Nb: nat,Mb: nat] :
      ( ( aa_nat_rat(comm_s4028243227959126397er_rat(A),Nb) = zero_zero_rat )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => ( aa_nat_rat(comm_s4028243227959126397er_rat(A),Mb) = zero_zero_rat ) ) ) ).

% pochhammer_eq_0_mono
tff(fact_8374_pochhammer__fact,axiom,
    ! [Nb: nat] : aa_nat_complex(semiri5044797733671781792omplex,Nb) = aa_nat_complex(comm_s2602460028002588243omplex(one_one_complex),Nb) ).

% pochhammer_fact
tff(fact_8375_pochhammer__fact,axiom,
    ! [Nb: nat] : aa_nat_rat(semiri773545260158071498ct_rat,Nb) = aa_nat_rat(comm_s4028243227959126397er_rat(one_one_rat),Nb) ).

% pochhammer_fact
tff(fact_8376_pochhammer__fact,axiom,
    ! [Nb: nat] : aa_nat_int(semiri1406184849735516958ct_int,Nb) = aa_nat_int(comm_s4660882817536571857er_int(one_one_int),Nb) ).

% pochhammer_fact
tff(fact_8377_pochhammer__fact,axiom,
    ! [Nb: nat] : aa_nat_real(semiri2265585572941072030t_real,Nb) = aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,one_one_real),Nb) ).

% pochhammer_fact
tff(fact_8378_pochhammer__fact,axiom,
    ! [Nb: nat] : aa_nat_nat(semiri1408675320244567234ct_nat,Nb) = aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,one_one_nat),Nb) ).

% pochhammer_fact
tff(fact_8379_pochhammer__nonneg,axiom,
    ! [Xb: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Xb),Nb)) ) ).

% pochhammer_nonneg
tff(fact_8380_pochhammer__nonneg,axiom,
    ! [Xb: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xb)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(comm_s4028243227959126397er_rat(Xb),Nb)) ) ).

% pochhammer_nonneg
tff(fact_8381_pochhammer__nonneg,axiom,
    ! [Xb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Xb)
     => 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,Xb),Nb)) ) ).

% pochhammer_nonneg
tff(fact_8382_pochhammer__nonneg,axiom,
    ! [Xb: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Xb)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(comm_s4660882817536571857er_int(Xb),Nb)) ) ).

% pochhammer_nonneg
tff(fact_8383_pochhammer__0__left,axiom,
    ! [Nb: nat] :
      aa_nat_complex(comm_s2602460028002588243omplex(zero_zero_complex),Nb) = $ite(Nb = zero_zero_nat,one_one_complex,zero_zero_complex) ).

% pochhammer_0_left
tff(fact_8384_pochhammer__0__left,axiom,
    ! [Nb: nat] :
      aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,zero_zero_real),Nb) = $ite(Nb = zero_zero_nat,one_one_real,zero_zero_real) ).

% pochhammer_0_left
tff(fact_8385_pochhammer__0__left,axiom,
    ! [Nb: nat] :
      aa_nat_rat(comm_s4028243227959126397er_rat(zero_zero_rat),Nb) = $ite(Nb = zero_zero_nat,one_one_rat,zero_zero_rat) ).

% pochhammer_0_left
tff(fact_8386_pochhammer__0__left,axiom,
    ! [Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,zero_zero_nat),Nb) = $ite(Nb = zero_zero_nat,one_one_nat,zero_zero_nat) ).

% pochhammer_0_left
tff(fact_8387_pochhammer__0__left,axiom,
    ! [Nb: nat] :
      aa_nat_int(comm_s4660882817536571857er_int(zero_zero_int),Nb) = $ite(Nb = zero_zero_nat,one_one_int,zero_zero_int) ).

% pochhammer_0_left
tff(fact_8388_pochhammer__rec,axiom,
    ! [A: complex,Nb: nat] : aa_nat_complex(comm_s2602460028002588243omplex(A),aa_nat_nat(suc,Nb)) = aa_complex_complex(times_times_complex(A),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(A),one_one_complex)),Nb)) ).

% pochhammer_rec
tff(fact_8389_pochhammer__rec,axiom,
    ! [A: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),aa_nat_nat(suc,Nb)) = aa_real_real(times_times_real(A),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(plus_plus_real(A),one_one_real)),Nb)) ).

% pochhammer_rec
tff(fact_8390_pochhammer__rec,axiom,
    ! [A: rat,Nb: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),aa_nat_nat(suc,Nb)) = aa_rat_rat(times_times_rat(A),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(plus_plus_rat(A),one_one_rat)),Nb)) ).

% pochhammer_rec
tff(fact_8391_pochhammer__rec,axiom,
    ! [A: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),aa_nat_nat(suc,Nb)) = 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)),Nb)) ).

% pochhammer_rec
tff(fact_8392_pochhammer__rec,axiom,
    ! [A: int,Nb: nat] : aa_nat_int(comm_s4660882817536571857er_int(A),aa_nat_nat(suc,Nb)) = 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)),Nb)) ).

% pochhammer_rec
tff(fact_8393_diffs__def,axiom,
    ! [C: fun_nat_int,X: nat] : aa_nat_int(diffs_int(C),X) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,X))),aa_nat_int(C,aa_nat_nat(suc,X))) ).

% diffs_def
tff(fact_8394_diffs__def,axiom,
    ! [C: fun_nat_real,X: nat] : aa_nat_real(diffs_real(C),X) = aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,X))),aa_nat_real(C,aa_nat_nat(suc,X))) ).

% diffs_def
tff(fact_8395_diffs__def,axiom,
    ! [C: fun_nat_rat,X: nat] : aa_nat_rat(diffs_rat(C),X) = aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,X))),aa_nat_rat(C,aa_nat_nat(suc,X))) ).

% diffs_def
tff(fact_8396_pochhammer__Suc,axiom,
    ! [A: int,Nb: nat] : aa_nat_int(comm_s4660882817536571857er_int(A),aa_nat_nat(suc,Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(comm_s4660882817536571857er_int(A),Nb)),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_nat_int(semiri1314217659103216013at_int,Nb))) ).

% pochhammer_Suc
tff(fact_8397_pochhammer__Suc,axiom,
    ! [A: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),aa_nat_nat(suc,Nb)) = aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),Nb)),aa_real_real(plus_plus_real(A),aa_nat_real(semiri5074537144036343181t_real,Nb))) ).

% pochhammer_Suc
tff(fact_8398_pochhammer__Suc,axiom,
    ! [A: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(semiri1316708129612266289at_nat,Nb))) ).

% pochhammer_Suc
tff(fact_8399_pochhammer__Suc,axiom,
    ! [A: rat,Nb: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),aa_nat_nat(suc,Nb)) = aa_rat_rat(times_times_rat(aa_nat_rat(comm_s4028243227959126397er_rat(A),Nb)),aa_rat_rat(plus_plus_rat(A),aa_nat_rat(semiri681578069525770553at_rat,Nb))) ).

% pochhammer_Suc
tff(fact_8400_pochhammer__rec_H,axiom,
    ! [Z: int,Nb: nat] : aa_nat_int(comm_s4660882817536571857er_int(Z),aa_nat_nat(suc,Nb)) = 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,Nb))),aa_nat_int(comm_s4660882817536571857er_int(Z),Nb)) ).

% pochhammer_rec'
tff(fact_8401_pochhammer__rec_H,axiom,
    ! [Z: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z),aa_nat_nat(suc,Nb)) = aa_real_real(times_times_real(aa_real_real(plus_plus_real(Z),aa_nat_real(semiri5074537144036343181t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z),Nb)) ).

% pochhammer_rec'
tff(fact_8402_pochhammer__rec_H,axiom,
    ! [Z: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),aa_nat_nat(suc,Nb)) = 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,Nb))),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),Nb)) ).

% pochhammer_rec'
tff(fact_8403_pochhammer__rec_H,axiom,
    ! [Z: rat,Nb: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(Z),aa_nat_nat(suc,Nb)) = aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(Z),aa_nat_rat(semiri681578069525770553at_rat,Nb))),aa_nat_rat(comm_s4028243227959126397er_rat(Z),Nb)) ).

% pochhammer_rec'
tff(fact_8404_pochhammer__eq__0__iff,axiom,
    ! [A: complex,Nb: nat] :
      ( ( aa_nat_complex(comm_s2602460028002588243omplex(A),Nb) = zero_zero_complex )
    <=> ? [K3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K3),Nb)
          & ( A = aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,K3)) ) ) ) ).

% pochhammer_eq_0_iff
tff(fact_8405_pochhammer__eq__0__iff,axiom,
    ! [A: real,Nb: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),Nb) = zero_zero_real )
    <=> ? [K3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K3),Nb)
          & ( A = aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,K3)) ) ) ) ).

% pochhammer_eq_0_iff
tff(fact_8406_pochhammer__eq__0__iff,axiom,
    ! [A: rat,Nb: nat] :
      ( ( aa_nat_rat(comm_s4028243227959126397er_rat(A),Nb) = zero_zero_rat )
    <=> ? [K3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K3),Nb)
          & ( A = aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,K3)) ) ) ) ).

% pochhammer_eq_0_iff
tff(fact_8407_pochhammer__of__nat__eq__0__iff,axiom,
    ! [Nb: nat,K: nat] :
      ( ( aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,Nb))),K) = zero_zero_complex )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),K) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_8408_pochhammer__of__nat__eq__0__iff,axiom,
    ! [Nb: nat,K: nat] :
      ( ( aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Nb))),K) = zero_z3403309356797280102nteger )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),K) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_8409_pochhammer__of__nat__eq__0__iff,axiom,
    ! [Nb: nat,K: nat] :
      ( ( aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,Nb))),K) = zero_zero_int )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),K) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_8410_pochhammer__of__nat__eq__0__iff,axiom,
    ! [Nb: nat,K: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,Nb))),K) = zero_zero_real )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),K) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_8411_pochhammer__of__nat__eq__0__iff,axiom,
    ! [Nb: nat,K: nat] :
      ( ( aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,Nb))),K) = zero_zero_rat )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),K) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_8412_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [Nb: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),K)
     => ( aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,Nb))),K) = zero_zero_complex ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_8413_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [Nb: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),K)
     => ( aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Nb))),K) = zero_z3403309356797280102nteger ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_8414_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [Nb: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),K)
     => ( aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,Nb))),K) = zero_zero_int ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_8415_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [Nb: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),K)
     => ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,Nb))),K) = zero_zero_real ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_8416_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [Nb: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),K)
     => ( aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,Nb))),K) = zero_zero_rat ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_8417_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => ( aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,Nb))),K) != zero_zero_complex ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_8418_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => ( aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Nb))),K) != zero_z3403309356797280102nteger ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_8419_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => ( aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,Nb))),K) != zero_zero_int ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_8420_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,Nb))),K) != zero_zero_real ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_8421_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => ( aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,Nb))),K) != zero_zero_rat ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_8422_pochhammer__product_H,axiom,
    ! [Z: int,Nb: nat,Mb: nat] : aa_nat_int(comm_s4660882817536571857er_int(Z),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(comm_s4660882817536571857er_int(Z),Nb)),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),aa_nat_int(semiri1314217659103216013at_int,Nb))),Mb)) ).

% pochhammer_product'
tff(fact_8423_pochhammer__product_H,axiom,
    ! [Z: real,Nb: nat,Mb: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)) = aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z),Nb)),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(plus_plus_real(Z),aa_nat_real(semiri5074537144036343181t_real,Nb))),Mb)) ).

% pochhammer_product'
tff(fact_8424_pochhammer__product_H,axiom,
    ! [Z: nat,Nb: nat,Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),Nb)),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,Nb))),Mb)) ).

% pochhammer_product'
tff(fact_8425_pochhammer__product_H,axiom,
    ! [Z: rat,Nb: nat,Mb: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(Z),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)) = aa_rat_rat(times_times_rat(aa_nat_rat(comm_s4028243227959126397er_rat(Z),Nb)),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(plus_plus_rat(Z),aa_nat_rat(semiri681578069525770553at_rat,Nb))),Mb)) ).

% pochhammer_product'
tff(fact_8426_termdiff__converges__all,axiom,
    ! [C: fun_nat_complex,Xb: complex] :
      ( ! [X3: complex] : summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_il(C),X3))
     => summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_jt(C),Xb)) ) ).

% termdiff_converges_all
tff(fact_8427_termdiff__converges__all,axiom,
    ! [C: fun_nat_real,Xb: real] :
      ( ! [X3: real] : summable_real(aa_real_fun_nat_real(aTP_Lamm_ij(C),X3))
     => summable_real(aa_real_fun_nat_real(aTP_Lamm_jv(C),Xb)) ) ).

% termdiff_converges_all
tff(fact_8428_pochhammer__product,axiom,
    ! [Mb: nat,Nb: nat,Z: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_int(comm_s4660882817536571857er_int(Z),Nb) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(comm_s4660882817536571857er_int(Z),Mb)),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),aa_nat_int(semiri1314217659103216013at_int,Mb))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb))) ) ) ).

% pochhammer_product
tff(fact_8429_pochhammer__product,axiom,
    ! [Mb: nat,Nb: nat,Z: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z),Nb) = aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z),Mb)),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(plus_plus_real(Z),aa_nat_real(semiri5074537144036343181t_real,Mb))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb))) ) ) ).

% pochhammer_product
tff(fact_8430_pochhammer__product,axiom,
    ! [Mb: nat,Nb: nat,Z: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),Mb)),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,Mb))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb))) ) ) ).

% pochhammer_product
tff(fact_8431_pochhammer__product,axiom,
    ! [Mb: nat,Nb: nat,Z: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_rat(comm_s4028243227959126397er_rat(Z),Nb) = aa_rat_rat(times_times_rat(aa_nat_rat(comm_s4028243227959126397er_rat(Z),Mb)),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(plus_plus_rat(Z),aa_nat_rat(semiri681578069525770553at_rat,Mb))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb))) ) ) ).

% pochhammer_product
tff(fact_8432_pochhammer__absorb__comp,axiom,
    ! [R2: complex,K: nat] : aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(R2),aa_nat_complex(semiri8010041392384452111omplex,K))),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,R2)),K)) = aa_complex_complex(times_times_complex(R2),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,R2)),one_one_complex)),K)) ).

% pochhammer_absorb_comp
tff(fact_8433_pochhammer__absorb__comp,axiom,
    ! [R2: code_integer,K: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,minus_8373710615458151222nteger(R2,aa_nat_Code_integer(semiri4939895301339042750nteger,K))),aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,R2)),K)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,R2),aa_nat_Code_integer(comm_s8582702949713902594nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,R2),one_one_Code_integer)),K)) ).

% pochhammer_absorb_comp
tff(fact_8434_pochhammer__absorb__comp,axiom,
    ! [R2: int,K: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,R2),aa_nat_int(semiri1314217659103216013at_int,K))),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,R2)),K)) = aa_int_int(aa_int_fun_int_int(times_times_int,R2),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,R2)),one_one_int)),K)) ).

% pochhammer_absorb_comp
tff(fact_8435_pochhammer__absorb__comp,axiom,
    ! [R2: real,K: nat] : aa_real_real(times_times_real(aa_real_real(minus_minus_real(R2),aa_nat_real(semiri5074537144036343181t_real,K))),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(uminus_uminus_real,R2)),K)) = aa_real_real(times_times_real(R2),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(plus_plus_real(aa_real_real(uminus_uminus_real,R2)),one_one_real)),K)) ).

% pochhammer_absorb_comp
tff(fact_8436_pochhammer__absorb__comp,axiom,
    ! [R2: rat,K: nat] : aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(R2),aa_nat_rat(semiri681578069525770553at_rat,K))),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,R2)),K)) = aa_rat_rat(times_times_rat(R2),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,R2)),one_one_rat)),K)) ).

% pochhammer_absorb_comp
tff(fact_8437_pochhammer__same,axiom,
    ! [Nb: nat] : aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,Nb))),Nb) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Nb)),aa_nat_complex(semiri5044797733671781792omplex,Nb)) ).

% pochhammer_same
tff(fact_8438_pochhammer__same,axiom,
    ! [Nb: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Nb))),Nb) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Nb)),aa_nat_Code_integer(semiri3624122377584611663nteger,Nb)) ).

% pochhammer_same
tff(fact_8439_pochhammer__same,axiom,
    ! [Nb: nat] : aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,Nb))),Nb) = 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)),Nb)),aa_nat_int(semiri1406184849735516958ct_int,Nb)) ).

% pochhammer_same
tff(fact_8440_pochhammer__same,axiom,
    ! [Nb: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,Nb))),Nb) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Nb)),aa_nat_rat(semiri773545260158071498ct_rat,Nb)) ).

% pochhammer_same
tff(fact_8441_pochhammer__same,axiom,
    ! [Nb: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,Nb))),Nb) = aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Nb)),aa_nat_real(semiri2265585572941072030t_real,Nb)) ).

% pochhammer_same
tff(fact_8442_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_8443_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_8444_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_8445_pochhammer__minus_H,axiom,
    ! [B: real,K: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(plus_plus_real(aa_real_real(minus_minus_real(B),aa_nat_real(semiri5074537144036343181t_real,K))),one_one_real)),K) = aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),K)),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(uminus_uminus_real,B)),K)) ).

% pochhammer_minus'
tff(fact_8446_pochhammer__minus_H,axiom,
    ! [B: rat,K: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(minus_minus_rat(B),aa_nat_rat(semiri681578069525770553at_rat,K))),one_one_rat)),K) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),K)),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,B)),K)) ).

% pochhammer_minus'
tff(fact_8447_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_8448_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_8449_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_8450_pochhammer__minus,axiom,
    ! [B: real,K: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(uminus_uminus_real,B)),K) = aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),K)),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(plus_plus_real(aa_real_real(minus_minus_real(B),aa_nat_real(semiri5074537144036343181t_real,K))),one_one_real)),K)) ).

% pochhammer_minus
tff(fact_8451_pochhammer__minus,axiom,
    ! [B: rat,K: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,B)),K) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),K)),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(minus_minus_rat(B),aa_nat_rat(semiri681578069525770553at_rat,K))),one_one_rat)),K)) ).

% pochhammer_minus
tff(fact_8452_termdiff__converges,axiom,
    ! [Xb: real,K5: real,C: fun_nat_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Xb)),K5)
     => ( ! [X3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,X3)),K5)
           => summable_real(aa_real_fun_nat_real(aTP_Lamm_ij(C),X3)) )
       => summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_jx(Xb),C)) ) ) ).

% termdiff_converges
tff(fact_8453_termdiff__converges,axiom,
    ! [Xb: complex,K5: real,C: fun_nat_complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Xb)),K5)
     => ( ! [X3: complex] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,X3)),K5)
           => summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_il(C),X3)) )
       => summable_complex(aa_fun2955221450537884980omplex(aTP_Lamm_jy(Xb),C)) ) ) ).

% termdiff_converges
tff(fact_8454_fact__double,axiom,
    ! [Nb: 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))),Nb)) = 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))),Nb))),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(divide1717551699836669952omplex(one_one_complex),numera6690914467698888265omplex(aa_num_num(bit0,one)))),Nb))),aa_nat_complex(semiri5044797733671781792omplex,Nb)) ).

% fact_double
tff(fact_8455_fact__double,axiom,
    ! [Nb: 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))),Nb)) = aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))),Nb))),aa_nat_rat(semiri773545260158071498ct_rat,Nb)) ).

% fact_double
tff(fact_8456_fact__double,axiom,
    ! [Nb: 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))),Nb)) = aa_real_real(times_times_real(aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),Nb))),aa_nat_real(semiri2265585572941072030t_real,Nb)) ).

% fact_double
tff(fact_8457_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_8458_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_8459_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_8460_monoseq__Suc,axiom,
    ! [X6: fun_nat_num] :
      ( topolo1459490580787246023eq_num(X6)
    <=> ( ! [N4: nat] : aa_num_o(aa_num_fun_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(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(X6,aa_nat_nat(suc,N4))),aa_nat_num(X6,N4)) ) ) ).

% monoseq_Suc
tff(fact_8461_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_8462_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_8463_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_8464_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_8465_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_8466_mono__SucI2,axiom,
    ! [X6: fun_nat_num] :
      ( ! [N: nat] : aa_num_o(aa_num_fun_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_8467_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_8468_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_8469_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_8470_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_8471_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_8472_mono__SucI1,axiom,
    ! [X6: fun_nat_num] :
      ( ! [N: nat] : aa_num_o(aa_num_fun_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_8473_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_8474_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_8475_pochhammer__times__pochhammer__half,axiom,
    ! [Z: complex,Nb: nat] : aa_complex_complex(times_times_complex(aa_nat_complex(comm_s2602460028002588243omplex(Z),aa_nat_nat(suc,Nb))),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,Nb))) = aa_set_nat_complex(groups6464643781859351333omplex(aTP_Lamm_jz(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))),Nb)),one_one_nat))) ).

% pochhammer_times_pochhammer_half
tff(fact_8476_pochhammer__times__pochhammer__half,axiom,
    ! [Z: real,Nb: nat] : aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z),aa_nat_nat(suc,Nb))),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(plus_plus_real(Z),aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_nat_nat(suc,Nb))) = aa_set_nat_real(groups129246275422532515t_real(aa_real_fun_nat_real(aTP_Lamm_ka,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))),Nb)),one_one_nat))) ).

% pochhammer_times_pochhammer_half
tff(fact_8477_pochhammer__times__pochhammer__half,axiom,
    ! [Z: rat,Nb: nat] : aa_rat_rat(times_times_rat(aa_nat_rat(comm_s4028243227959126397er_rat(Z),aa_nat_nat(suc,Nb))),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(plus_plus_rat(Z),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))),aa_nat_nat(suc,Nb))) = groups73079841787564623at_rat(aTP_Lamm_kb(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))),Nb)),one_one_nat))) ).

% pochhammer_times_pochhammer_half
tff(fact_8478_pochhammer__code,axiom,
    ! [A: complex,Nb: nat] :
      aa_nat_complex(comm_s2602460028002588243omplex(A),Nb) = $ite(Nb = zero_zero_nat,one_one_complex,aa_complex_complex(set_fo1517530859248394432omplex(aTP_Lamm_kc(A),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),one_one_complex)) ).

% pochhammer_code
tff(fact_8479_pochhammer__code,axiom,
    ! [A: int,Nb: nat] :
      aa_nat_int(comm_s4660882817536571857er_int(A),Nb) = $ite(Nb = zero_zero_nat,one_one_int,aa_int_int(set_fo2581907887559384638at_int(aTP_Lamm_kd(A),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),one_one_int)) ).

% pochhammer_code
tff(fact_8480_pochhammer__code,axiom,
    ! [A: real,Nb: nat] :
      aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),Nb) = $ite(Nb = zero_zero_nat,one_one_real,aa_real_real(aa_nat_fun_real_real(set_fo3111899725591712190t_real(aTP_Lamm_ke(A),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),one_one_real)) ).

% pochhammer_code
tff(fact_8481_pochhammer__code,axiom,
    ! [A: rat,Nb: nat] :
      aa_nat_rat(comm_s4028243227959126397er_rat(A),Nb) = $ite(Nb = zero_zero_nat,one_one_rat,aa_rat_rat(set_fo1949268297981939178at_rat(aTP_Lamm_kf(A),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),one_one_rat)) ).

% pochhammer_code
tff(fact_8482_pochhammer__code,axiom,
    ! [A: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),Nb) = $ite(Nb = zero_zero_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(aTP_Lamm_kg(A),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),one_one_nat)) ).

% pochhammer_code
tff(fact_8483_of__nat__code,axiom,
    ! [Nb: nat] : aa_nat_complex(semiri8010041392384452111omplex,Nb) = aa_complex_complex(semiri2816024913162550771omplex(aTP_Lamm_kh,Nb),zero_zero_complex) ).

% of_nat_code
tff(fact_8484_of__nat__code,axiom,
    ! [Nb: nat] : aa_nat_int(semiri1314217659103216013at_int,Nb) = aa_int_int(semiri8420488043553186161ux_int(aTP_Lamm_ki,Nb),zero_zero_int) ).

% of_nat_code
tff(fact_8485_of__nat__code,axiom,
    ! [Nb: nat] : aa_nat_real(semiri5074537144036343181t_real,Nb) = aa_real_real(aa_nat_fun_real_real(semiri7260567687927622513x_real(aTP_Lamm_kj),Nb),zero_zero_real) ).

% of_nat_code
tff(fact_8486_of__nat__code,axiom,
    ! [Nb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,Nb) = aa_nat_nat(aa_nat_fun_nat_nat(semiri8422978514062236437ux_nat(aTP_Lamm_kk),Nb),zero_zero_nat) ).

% of_nat_code
tff(fact_8487_of__nat__code,axiom,
    ! [Nb: nat] : aa_nat_rat(semiri681578069525770553at_rat,Nb) = aa_rat_rat(semiri7787848453975740701ux_rat(aTP_Lamm_kl,Nb),zero_zero_rat) ).

% of_nat_code
tff(fact_8488_gchoose__row__sum__weighted,axiom,
    ! [R2: complex,Mb: nat] : aa_set_nat_complex(groups2073611262835488442omplex(aTP_Lamm_km(R2)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Mb)) = aa_complex_complex(times_times_complex(aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,Mb))),numera6690914467698888265omplex(aa_num_num(bit0,one)))),aa_nat_complex(gbinomial_complex(R2),aa_nat_nat(suc,Mb))) ).

% gchoose_row_sum_weighted
tff(fact_8489_gchoose__row__sum__weighted,axiom,
    ! [R2: rat,Mb: nat] : groups2906978787729119204at_rat(aTP_Lamm_kn(R2),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Mb)) = aa_rat_rat(times_times_rat(aa_rat_rat(divide_divide_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,Mb))),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))),aa_nat_rat(gbinomial_rat(R2),aa_nat_nat(suc,Mb))) ).

% gchoose_row_sum_weighted
tff(fact_8490_gchoose__row__sum__weighted,axiom,
    ! [R2: real,Mb: nat] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ko,R2)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Mb)) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Mb))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,R2),aa_nat_nat(suc,Mb))) ).

% gchoose_row_sum_weighted
tff(fact_8491_central__binomial__lower__bound,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(divide_divide_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))),Nb)),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Nb)))),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))),Nb)),Nb))) ) ).

% central_binomial_lower_bound
tff(fact_8492_binomial__Suc__n,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,Nb)),Nb) = aa_nat_nat(suc,Nb) ).

% binomial_Suc_n
tff(fact_8493_binomial__n__n,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),Nb) = one_one_nat ).

% binomial_n_n
tff(fact_8494_prod_Oneutral__const,axiom,
    ! [A3: set_nat] : aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_kp),A3) = one_one_nat ).

% prod.neutral_const
tff(fact_8495_prod_Oneutral__const,axiom,
    ! [A3: set_nat] : groups705719431365010083at_int(aTP_Lamm_kq,A3) = one_one_int ).

% prod.neutral_const
tff(fact_8496_prod_Oneutral__const,axiom,
    ! [A3: set_int] : groups1705073143266064639nt_int(aTP_Lamm_kr,A3) = one_one_int ).

% prod.neutral_const
tff(fact_8497_of__nat__prod,axiom,
    ! [F: fun_int_nat,A3: set_int] : aa_nat_int(semiri1314217659103216013at_int,groups1707563613775114915nt_nat(F,A3)) = groups1705073143266064639nt_int(aTP_Lamm_fk(F),A3) ).

% of_nat_prod
tff(fact_8498_of__nat__prod,axiom,
    ! [F: fun_nat_nat,A3: set_nat] : aa_nat_real(semiri5074537144036343181t_real,aa_set_nat_nat(groups708209901874060359at_nat(F),A3)) = aa_set_nat_real(groups129246275422532515t_real(aTP_Lamm_fp(F)),A3) ).

% of_nat_prod
tff(fact_8499_of__nat__prod,axiom,
    ! [F: fun_nat_nat,A3: set_nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_set_nat_nat(groups708209901874060359at_nat(F),A3)) = groups73079841787564623at_rat(aTP_Lamm_fn(F),A3) ).

% of_nat_prod
tff(fact_8500_of__nat__prod,axiom,
    ! [F: fun_nat_nat,A3: set_nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_set_nat_nat(groups708209901874060359at_nat(F),A3)) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_fo(F)),A3) ).

% of_nat_prod
tff(fact_8501_of__nat__prod,axiom,
    ! [F: fun_nat_nat,A3: set_nat] : aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(groups708209901874060359at_nat(F),A3)) = groups705719431365010083at_int(aTP_Lamm_fm(F),A3) ).

% of_nat_prod
tff(fact_8502_of__int__prod,axiom,
    ! [F: fun_nat_int,A3: set_nat] : aa_int_real(ring_1_of_int_real,groups705719431365010083at_int(F,A3)) = aa_set_nat_real(groups129246275422532515t_real(aTP_Lamm_cm(F)),A3) ).

% of_int_prod
tff(fact_8503_of__int__prod,axiom,
    ! [F: fun_nat_int,A3: set_nat] : aa_int_rat(ring_1_of_int_rat,groups705719431365010083at_int(F,A3)) = groups73079841787564623at_rat(aTP_Lamm_ks(F),A3) ).

% of_int_prod
tff(fact_8504_of__int__prod,axiom,
    ! [F: fun_nat_int,A3: set_nat] : aa_int_int(ring_1_of_int_int,groups705719431365010083at_int(F,A3)) = groups705719431365010083at_int(aTP_Lamm_kt(F),A3) ).

% of_int_prod
tff(fact_8505_of__int__prod,axiom,
    ! [F: fun_int_int,A3: set_int] : aa_int_real(ring_1_of_int_real,groups1705073143266064639nt_int(F,A3)) = groups2316167850115554303t_real(aTP_Lamm_cn(F),A3) ).

% of_int_prod
tff(fact_8506_of__int__prod,axiom,
    ! [F: fun_int_int,A3: set_int] : aa_int_rat(ring_1_of_int_rat,groups1705073143266064639nt_int(F,A3)) = groups1072433553688619179nt_rat(aTP_Lamm_co(F),A3) ).

% of_int_prod
tff(fact_8507_of__int__prod,axiom,
    ! [F: fun_int_int,A3: set_int] : aa_int_int(ring_1_of_int_int,groups1705073143266064639nt_int(F,A3)) = groups1705073143266064639nt_int(aTP_Lamm_cp(F),A3) ).

% of_int_prod
tff(fact_8508_prod_Oempty,axiom,
    ! [G: fun_nat_complex] : aa_set_nat_complex(groups6464643781859351333omplex(G),bot_bot_set_nat) = one_one_complex ).

% prod.empty
tff(fact_8509_prod_Oempty,axiom,
    ! [G: fun_nat_real] : aa_set_nat_real(groups129246275422532515t_real(G),bot_bot_set_nat) = one_one_real ).

% prod.empty
tff(fact_8510_prod_Oempty,axiom,
    ! [G: fun_nat_rat] : groups73079841787564623at_rat(G,bot_bot_set_nat) = one_one_rat ).

% prod.empty
tff(fact_8511_prod_Oempty,axiom,
    ! [G: fun_int_complex] : groups7440179247065528705omplex(G,bot_bot_set_int) = one_one_complex ).

% prod.empty
tff(fact_8512_prod_Oempty,axiom,
    ! [G: fun_int_real] : groups2316167850115554303t_real(G,bot_bot_set_int) = one_one_real ).

% prod.empty
tff(fact_8513_prod_Oempty,axiom,
    ! [G: fun_int_rat] : groups1072433553688619179nt_rat(G,bot_bot_set_int) = one_one_rat ).

% prod.empty
tff(fact_8514_prod_Oempty,axiom,
    ! [G: fun_int_nat] : groups1707563613775114915nt_nat(G,bot_bot_set_int) = one_one_nat ).

% prod.empty
tff(fact_8515_prod_Oempty,axiom,
    ! [G: fun_real_complex] : groups713298508707869441omplex(G,bot_bot_set_real) = one_one_complex ).

% prod.empty
tff(fact_8516_prod_Oempty,axiom,
    ! [G: fun_real_real] : groups1681761925125756287l_real(G,bot_bot_set_real) = one_one_real ).

% prod.empty
tff(fact_8517_prod_Oempty,axiom,
    ! [G: fun_real_rat] : groups4061424788464935467al_rat(G,bot_bot_set_real) = one_one_rat ).

% prod.empty
tff(fact_8518_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_8519_gbinomial__0_I2_J,axiom,
    ! [K: nat] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,zero_zero_real),aa_nat_nat(suc,K)) = zero_zero_real ).

% gbinomial_0(2)
tff(fact_8520_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_8521_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_8522_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_8523_binomial__1,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),aa_nat_nat(suc,zero_zero_nat)) = Nb ).

% binomial_1
tff(fact_8524_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_8525_binomial__eq__0__iff,axiom,
    ! [Nb: nat,K: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K) = zero_zero_nat )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),K) ) ).

% binomial_eq_0_iff
tff(fact_8526_gbinomial__0_I1_J,axiom,
    ! [A: complex] : aa_nat_complex(gbinomial_complex(A),zero_zero_nat) = one_one_complex ).

% gbinomial_0(1)
tff(fact_8527_gbinomial__0_I1_J,axiom,
    ! [A: real] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),zero_zero_nat) = one_one_real ).

% gbinomial_0(1)
tff(fact_8528_gbinomial__0_I1_J,axiom,
    ! [A: rat] : aa_nat_rat(gbinomial_rat(A),zero_zero_nat) = one_one_rat ).

% gbinomial_0(1)
tff(fact_8529_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_8530_gbinomial__0_I1_J,axiom,
    ! [A: int] : aa_nat_int(gbinomial_int(A),zero_zero_nat) = one_one_int ).

% gbinomial_0(1)
tff(fact_8531_binomial__Suc__Suc,axiom,
    ! [Nb: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,Nb)),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,Nb),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),aa_nat_nat(suc,K))) ).

% binomial_Suc_Suc
tff(fact_8532_binomial__n__0,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),zero_zero_nat) = one_one_nat ).

% binomial_n_0
tff(fact_8533_zero__less__binomial__iff,axiom,
    ! [Nb: 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,Nb),K))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb) ) ).

% zero_less_binomial_iff
tff(fact_8534_prod_OlessThan__Suc,axiom,
    ! [G: fun_nat_real,Nb: nat] : aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = aa_real_real(times_times_real(aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_nat_real(G,Nb)) ).

% prod.lessThan_Suc
tff(fact_8535_prod_OlessThan__Suc,axiom,
    ! [G: fun_nat_rat,Nb: nat] : groups73079841787564623at_rat(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = aa_rat_rat(times_times_rat(groups73079841787564623at_rat(G,aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_nat_rat(G,Nb)) ).

% prod.lessThan_Suc
tff(fact_8536_prod_OlessThan__Suc,axiom,
    ! [G: fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = 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,Nb))),aa_nat_nat(G,Nb)) ).

% prod.lessThan_Suc
tff(fact_8537_prod_OlessThan__Suc,axiom,
    ! [G: fun_nat_int,Nb: nat] : groups705719431365010083at_int(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = aa_int_int(aa_int_fun_int_int(times_times_int,groups705719431365010083at_int(G,aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_nat_int(G,Nb)) ).

% prod.lessThan_Suc
tff(fact_8538_prod_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_complex,Mb: nat,Nb: nat] :
      aa_set_nat_complex(groups6464643781859351333omplex(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),Mb),one_one_complex,aa_complex_complex(times_times_complex(aa_set_nat_complex(groups6464643781859351333omplex(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))),aa_nat_complex(G,aa_nat_nat(suc,Nb)))) ).

% prod.cl_ivl_Suc
tff(fact_8539_prod_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_real,Mb: nat,Nb: nat] :
      aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),Mb),one_one_real,aa_real_real(times_times_real(aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))),aa_nat_real(G,aa_nat_nat(suc,Nb)))) ).

% prod.cl_ivl_Suc
tff(fact_8540_prod_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_rat,Mb: nat,Nb: nat] :
      groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),Mb),one_one_rat,aa_rat_rat(times_times_rat(groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))),aa_nat_rat(G,aa_nat_nat(suc,Nb)))) ).

% prod.cl_ivl_Suc
tff(fact_8541_prod_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_nat,Mb: nat,Nb: nat] :
      aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),Mb),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(Mb),Nb))),aa_nat_nat(G,aa_nat_nat(suc,Nb)))) ).

% prod.cl_ivl_Suc
tff(fact_8542_prod_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_int,Mb: nat,Nb: nat] :
      groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),Mb),one_one_int,aa_int_int(aa_int_fun_int_int(times_times_int,groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))),aa_nat_int(G,aa_nat_nat(suc,Nb)))) ).

% prod.cl_ivl_Suc
tff(fact_8543_choose__one,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),one_one_nat) = Nb ).

% choose_one
tff(fact_8544_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_complex_complex,A3: set_complex] :
      ( ( groups3708469109370488835omplex(G,A3) != one_one_complex )
     => ~ ! [A4: complex] :
            ( member_complex(A4,A3)
           => ( aa_complex_complex(G,A4) = one_one_complex ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_8545_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_real_complex,A3: set_real] :
      ( ( groups713298508707869441omplex(G,A3) != one_one_complex )
     => ~ ! [A4: real] :
            ( member_real(A4,A3)
           => ( aa_real_complex(G,A4) = one_one_complex ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_8546_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_nat_complex,A3: set_nat] :
      ( ( aa_set_nat_complex(groups6464643781859351333omplex(G),A3) != one_one_complex )
     => ~ ! [A4: nat] :
            ( aa_set_nat_o(member_nat(A4),A3)
           => ( aa_nat_complex(G,A4) = one_one_complex ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_8547_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_int_complex,A3: set_int] :
      ( ( groups7440179247065528705omplex(G,A3) != one_one_complex )
     => ~ ! [A4: int] :
            ( member_int(A4,A3)
           => ( aa_int_complex(G,A4) = one_one_complex ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_8548_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_complex_real,A3: set_complex] :
      ( ( groups766887009212190081x_real(G,A3) != one_one_real )
     => ~ ! [A4: complex] :
            ( member_complex(A4,A3)
           => ( aa_complex_real(G,A4) = one_one_real ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_8549_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_real_real,A3: set_real] :
      ( ( groups1681761925125756287l_real(G,A3) != one_one_real )
     => ~ ! [A4: real] :
            ( member_real(A4,A3)
           => ( aa_real_real(G,A4) = one_one_real ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_8550_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_nat_real,A3: set_nat] :
      ( ( aa_set_nat_real(groups129246275422532515t_real(G),A3) != one_one_real )
     => ~ ! [A4: nat] :
            ( aa_set_nat_o(member_nat(A4),A3)
           => ( aa_nat_real(G,A4) = one_one_real ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_8551_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_int_real,A3: set_int] :
      ( ( groups2316167850115554303t_real(G,A3) != one_one_real )
     => ~ ! [A4: int] :
            ( member_int(A4,A3)
           => ( aa_int_real(G,A4) = one_one_real ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_8552_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_complex_rat,A3: set_complex] :
      ( ( groups225925009352817453ex_rat(G,A3) != one_one_rat )
     => ~ ! [A4: complex] :
            ( member_complex(A4,A3)
           => ( aa_complex_rat(G,A4) = one_one_rat ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_8553_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_real_rat,A3: set_real] :
      ( ( groups4061424788464935467al_rat(G,A3) != one_one_rat )
     => ~ ! [A4: real] :
            ( member_real(A4,A3)
           => ( aa_real_rat(G,A4) = one_one_rat ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_8554_prod_Oneutral,axiom,
    ! [A3: set_nat,G: fun_nat_nat] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A3)
         => ( aa_nat_nat(G,X3) = one_one_nat ) )
     => ( aa_set_nat_nat(groups708209901874060359at_nat(G),A3) = one_one_nat ) ) ).

% prod.neutral
tff(fact_8555_prod_Oneutral,axiom,
    ! [A3: set_nat,G: fun_nat_int] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A3)
         => ( aa_nat_int(G,X3) = one_one_int ) )
     => ( groups705719431365010083at_int(G,A3) = one_one_int ) ) ).

% prod.neutral
tff(fact_8556_prod_Oneutral,axiom,
    ! [A3: set_int,G: fun_int_int] :
      ( ! [X3: int] :
          ( member_int(X3,A3)
         => ( aa_int_int(G,X3) = one_one_int ) )
     => ( groups1705073143266064639nt_int(G,A3) = one_one_int ) ) ).

% prod.neutral
tff(fact_8557_prod_Odistrib,axiom,
    ! [G: fun_nat_nat,H: fun_nat_nat,A3: set_nat] : aa_set_nat_nat(groups708209901874060359at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_ku(G),H)),A3) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups708209901874060359at_nat(G),A3)),aa_set_nat_nat(groups708209901874060359at_nat(H),A3)) ).

% prod.distrib
tff(fact_8558_prod_Odistrib,axiom,
    ! [G: fun_nat_int,H: fun_nat_int,A3: set_nat] : groups705719431365010083at_int(aa_fun5565043122120104496at_int(aTP_Lamm_kv(G),H),A3) = aa_int_int(aa_int_fun_int_int(times_times_int,groups705719431365010083at_int(G,A3)),groups705719431365010083at_int(H,A3)) ).

% prod.distrib
tff(fact_8559_prod_Odistrib,axiom,
    ! [G: fun_int_int,H: fun_int_int,A3: set_int] : groups1705073143266064639nt_int(aa_fun6280345759651049192nt_int(aTP_Lamm_kw(G),H),A3) = aa_int_int(aa_int_fun_int_int(times_times_int,groups1705073143266064639nt_int(G,A3)),groups1705073143266064639nt_int(H,A3)) ).

% prod.distrib
tff(fact_8560_prod__power__distrib,axiom,
    ! [F: fun_nat_nat,A3: set_nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_set_nat_nat(groups708209901874060359at_nat(F),A3)),Nb) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_kx(F),Nb)),A3) ).

% prod_power_distrib
tff(fact_8561_prod__power__distrib,axiom,
    ! [F: fun_nat_int,A3: set_nat,Nb: nat] : aa_nat_int(power_power_int(groups705719431365010083at_int(F,A3)),Nb) = groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_ky(F),Nb),A3) ).

% prod_power_distrib
tff(fact_8562_prod__power__distrib,axiom,
    ! [F: fun_int_int,A3: set_int,Nb: nat] : aa_nat_int(power_power_int(groups1705073143266064639nt_int(F,A3)),Nb) = groups1705073143266064639nt_int(aa_nat_fun_int_int(aTP_Lamm_kz(F),Nb),A3) ).

% prod_power_distrib
tff(fact_8563_mod__prod__eq,axiom,
    ! [F: fun_nat_nat,A: nat,A3: 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_cr(F),A)),A3)),A) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_set_nat_nat(groups708209901874060359at_nat(F),A3)),A) ).

% mod_prod_eq
tff(fact_8564_mod__prod__eq,axiom,
    ! [F: fun_nat_int,A: int,A3: set_nat] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,groups705719431365010083at_int(aa_int_fun_nat_int(aTP_Lamm_la(F),A),A3)),A) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,groups705719431365010083at_int(F,A3)),A) ).

% mod_prod_eq
tff(fact_8565_mod__prod__eq,axiom,
    ! [F: fun_int_int,A: int,A3: set_int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,groups1705073143266064639nt_int(aa_int_fun_int_int(aTP_Lamm_cq(F),A),A3)),A) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,groups1705073143266064639nt_int(F,A3)),A) ).

% mod_prod_eq
tff(fact_8566_prod__mono,axiom,
    ! [A3: set_complex,F: fun_complex_real,G: fun_complex_real] :
      ( ! [I2: complex] :
          ( member_complex(I2,A3)
         => ( 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,A3)),groups766887009212190081x_real(G,A3)) ) ).

% prod_mono
tff(fact_8567_prod__mono,axiom,
    ! [A3: set_real,F: fun_real_real,G: fun_real_real] :
      ( ! [I2: real] :
          ( member_real(I2,A3)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,I2))
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,I2)),aa_real_real(G,I2)) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups1681761925125756287l_real(F,A3)),groups1681761925125756287l_real(G,A3)) ) ).

% prod_mono
tff(fact_8568_prod__mono,axiom,
    ! [A3: set_nat,F: fun_nat_real,G: fun_nat_real] :
      ( ! [I2: nat] :
          ( aa_set_nat_o(member_nat(I2),A3)
         => ( 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,aa_nat_real(F,I2)),aa_nat_real(G,I2)) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_set_nat_real(groups129246275422532515t_real(F),A3)),aa_set_nat_real(groups129246275422532515t_real(G),A3)) ) ).

% prod_mono
tff(fact_8569_prod__mono,axiom,
    ! [A3: set_int,F: fun_int_real,G: fun_int_real] :
      ( ! [I2: int] :
          ( member_int(I2,A3)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(F,I2))
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(F,I2)),aa_int_real(G,I2)) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups2316167850115554303t_real(F,A3)),groups2316167850115554303t_real(G,A3)) ) ).

% prod_mono
tff(fact_8570_prod__mono,axiom,
    ! [A3: set_complex,F: fun_complex_rat,G: fun_complex_rat] :
      ( ! [I2: complex] :
          ( member_complex(I2,A3)
         => ( 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,A3)),groups225925009352817453ex_rat(G,A3)) ) ).

% prod_mono
tff(fact_8571_prod__mono,axiom,
    ! [A3: set_real,F: fun_real_rat,G: fun_real_rat] :
      ( ! [I2: real] :
          ( member_real(I2,A3)
         => ( 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,A3)),groups4061424788464935467al_rat(G,A3)) ) ).

% prod_mono
tff(fact_8572_prod__mono,axiom,
    ! [A3: set_nat,F: fun_nat_rat,G: fun_nat_rat] :
      ( ! [I2: nat] :
          ( aa_set_nat_o(member_nat(I2),A3)
         => ( 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,A3)),groups73079841787564623at_rat(G,A3)) ) ).

% prod_mono
tff(fact_8573_prod__mono,axiom,
    ! [A3: set_int,F: fun_int_rat,G: fun_int_rat] :
      ( ! [I2: int] :
          ( member_int(I2,A3)
         => ( 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,A3)),groups1072433553688619179nt_rat(G,A3)) ) ).

% prod_mono
tff(fact_8574_prod__mono,axiom,
    ! [A3: set_complex,F: fun_complex_nat,G: fun_complex_nat] :
      ( ! [I2: complex] :
          ( member_complex(I2,A3)
         => ( 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,A3)),groups861055069439313189ex_nat(G,A3)) ) ).

% prod_mono
tff(fact_8575_prod__mono,axiom,
    ! [A3: set_real,F: fun_real_nat,G: fun_real_nat] :
      ( ! [I2: real] :
          ( member_real(I2,A3)
         => ( 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,A3)),groups4696554848551431203al_nat(G,A3)) ) ).

% prod_mono
tff(fact_8576_prod__nonneg,axiom,
    ! [A3: set_nat,F: fun_nat_nat] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A3)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,X3)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_set_nat_nat(groups708209901874060359at_nat(F),A3)) ) ).

% prod_nonneg
tff(fact_8577_prod__nonneg,axiom,
    ! [A3: set_nat,F: fun_nat_int] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A3)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(F,X3)) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),groups705719431365010083at_int(F,A3)) ) ).

% prod_nonneg
tff(fact_8578_prod__nonneg,axiom,
    ! [A3: set_int,F: fun_int_int] :
      ( ! [X3: int] :
          ( member_int(X3,A3)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(F,X3)) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),groups1705073143266064639nt_int(F,A3)) ) ).

% prod_nonneg
tff(fact_8579_prod__pos,axiom,
    ! [A3: set_nat,F: fun_nat_nat] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A3)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(F,X3)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_set_nat_nat(groups708209901874060359at_nat(F),A3)) ) ).

% prod_pos
tff(fact_8580_prod__pos,axiom,
    ! [A3: set_nat,F: fun_nat_int] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A3)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(F,X3)) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),groups705719431365010083at_int(F,A3)) ) ).

% prod_pos
tff(fact_8581_prod__pos,axiom,
    ! [A3: set_int,F: fun_int_int] :
      ( ! [X3: int] :
          ( member_int(X3,A3)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(F,X3)) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),groups1705073143266064639nt_int(F,A3)) ) ).

% prod_pos
tff(fact_8582_prod__ge__1,axiom,
    ! [A3: set_complex,F: fun_complex_real] :
      ( ! [X3: complex] :
          ( member_complex(X3,A3)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_complex_real(F,X3)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),groups766887009212190081x_real(F,A3)) ) ).

% prod_ge_1
tff(fact_8583_prod__ge__1,axiom,
    ! [A3: set_real,F: fun_real_real] :
      ( ! [X3: real] :
          ( member_real(X3,A3)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(F,X3)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),groups1681761925125756287l_real(F,A3)) ) ).

% prod_ge_1
tff(fact_8584_prod__ge__1,axiom,
    ! [A3: set_nat,F: fun_nat_real] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A3)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_nat_real(F,X3)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_set_nat_real(groups129246275422532515t_real(F),A3)) ) ).

% prod_ge_1
tff(fact_8585_prod__ge__1,axiom,
    ! [A3: set_int,F: fun_int_real] :
      ( ! [X3: int] :
          ( member_int(X3,A3)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_int_real(F,X3)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),groups2316167850115554303t_real(F,A3)) ) ).

% prod_ge_1
tff(fact_8586_prod__ge__1,axiom,
    ! [A3: set_complex,F: fun_complex_rat] :
      ( ! [X3: complex] :
          ( member_complex(X3,A3)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_complex_rat(F,X3)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),groups225925009352817453ex_rat(F,A3)) ) ).

% prod_ge_1
tff(fact_8587_prod__ge__1,axiom,
    ! [A3: set_real,F: fun_real_rat] :
      ( ! [X3: real] :
          ( member_real(X3,A3)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_real_rat(F,X3)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),groups4061424788464935467al_rat(F,A3)) ) ).

% prod_ge_1
tff(fact_8588_prod__ge__1,axiom,
    ! [A3: set_nat,F: fun_nat_rat] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A3)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_nat_rat(F,X3)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),groups73079841787564623at_rat(F,A3)) ) ).

% prod_ge_1
tff(fact_8589_prod__ge__1,axiom,
    ! [A3: set_int,F: fun_int_rat] :
      ( ! [X3: int] :
          ( member_int(X3,A3)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_int_rat(F,X3)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),groups1072433553688619179nt_rat(F,A3)) ) ).

% prod_ge_1
tff(fact_8590_prod__ge__1,axiom,
    ! [A3: set_complex,F: fun_complex_nat] :
      ( ! [X3: complex] :
          ( member_complex(X3,A3)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),aa_complex_nat(F,X3)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),groups861055069439313189ex_nat(F,A3)) ) ).

% prod_ge_1
tff(fact_8591_prod__ge__1,axiom,
    ! [A3: set_real,F: fun_real_nat] :
      ( ! [X3: real] :
          ( member_real(X3,A3)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),aa_real_nat(F,X3)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),groups4696554848551431203al_nat(F,A3)) ) ).

% prod_ge_1
tff(fact_8592_binomial__eq__0,axiom,
    ! [Nb: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),K)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K) = zero_zero_nat ) ) ).

% binomial_eq_0
tff(fact_8593_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_lb(F),A,B),one_one_complex) ).

% prod_atLeastAtMost_code
tff(fact_8594_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_lc(F),A),B),one_one_real) ).

% prod_atLeastAtMost_code
tff(fact_8595_prod__atLeastAtMost__code,axiom,
    ! [F: fun_nat_rat,A: nat,B: nat] : groups73079841787564623at_rat(F,aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)) = aa_rat_rat(set_fo1949268297981939178at_rat(aTP_Lamm_ld(F),A,B),one_one_rat) ).

% prod_atLeastAtMost_code
tff(fact_8596_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_le(F),A),B),one_one_nat) ).

% prod_atLeastAtMost_code
tff(fact_8597_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_lf(F),A,B),one_one_int) ).

% prod_atLeastAtMost_code
tff(fact_8598_Suc__times__binomial__eq,axiom,
    ! [Nb: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),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,Nb)),aa_nat_nat(suc,K))),aa_nat_nat(suc,K)) ).

% Suc_times_binomial_eq
tff(fact_8599_Suc__times__binomial,axiom,
    ! [K: nat,Nb: 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,Nb)),aa_nat_nat(suc,K))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K)) ).

% Suc_times_binomial
tff(fact_8600_binomial__symmetric,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)) ) ) ).

% binomial_symmetric
tff(fact_8601_choose__mult__lemma,axiom,
    ! [Mb: nat,R2: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),R2)),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),K))),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),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,Mb),R2)),K)),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),R2)),Mb)) ).

% choose_mult_lemma
tff(fact_8602_binomial__le__pow,axiom,
    ! [R2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,R2),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),R2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Nb),R2)) ) ).

% binomial_le_pow
tff(fact_8603_prod_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: fun_nat_nat,Mb: nat,Nb: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb)),aa_nat_nat(suc,Nb))) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_dm(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) ).

% prod.shift_bounds_cl_Suc_ivl
tff(fact_8604_prod_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: fun_nat_int,Mb: nat,Nb: nat] : groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb)),aa_nat_nat(suc,Nb))) = groups705719431365010083at_int(aTP_Lamm_dx(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) ).

% prod.shift_bounds_cl_Suc_ivl
tff(fact_8605_power__sum,axiom,
    ! [C: real,F: fun_nat_nat,A3: set_nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,C),aa_set_nat_nat(groups3542108847815614940at_nat(F),A3)) = aa_set_nat_real(groups129246275422532515t_real(aa_fun1538629612289296340t_real(aTP_Lamm_lg(C),F)),A3) ).

% power_sum
tff(fact_8606_power__sum,axiom,
    ! [C: complex,F: fun_nat_nat,A3: set_nat] : aa_nat_complex(power_power_complex(C),aa_set_nat_nat(groups3542108847815614940at_nat(F),A3)) = aa_set_nat_complex(groups6464643781859351333omplex(aa_fun1958354139826840918omplex(aTP_Lamm_lh(C),F)),A3) ).

% power_sum
tff(fact_8607_power__sum,axiom,
    ! [C: nat,F: fun_nat_nat,A3: set_nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,C),aa_set_nat_nat(groups3542108847815614940at_nat(F),A3)) = aa_set_nat_nat(groups708209901874060359at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_li(C),F)),A3) ).

% power_sum
tff(fact_8608_power__sum,axiom,
    ! [C: int,F: fun_nat_nat,A3: set_nat] : aa_nat_int(power_power_int(C),aa_set_nat_nat(groups3542108847815614940at_nat(F),A3)) = groups705719431365010083at_int(aa_fun1613598493704395220at_int(aTP_Lamm_lj(C),F),A3) ).

% power_sum
tff(fact_8609_power__sum,axiom,
    ! [C: int,F: fun_int_nat,A3: set_int] : aa_nat_int(power_power_int(C),groups4541462559716669496nt_nat(F,A3)) = groups1705073143266064639nt_int(aa_fun2328901131235339916nt_int(aTP_Lamm_lk(C),F),A3) ).

% power_sum
tff(fact_8610_prod_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: fun_nat_nat,Mb: nat,K: nat,Nb: 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,Mb),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),K))) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_do(G),K)),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) ).

% prod.shift_bounds_cl_nat_ivl
tff(fact_8611_prod_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: fun_nat_int,Mb: nat,K: nat,Nb: nat] : groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),K))) = groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_ll(G),K),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) ).

% prod.shift_bounds_cl_nat_ivl
tff(fact_8612_prod__le__1,axiom,
    ! [A3: set_complex,F: fun_complex_real] :
      ( ! [X3: complex] :
          ( member_complex(X3,A3)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(F,X3))
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(F,X3)),one_one_real) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups766887009212190081x_real(F,A3)),one_one_real) ) ).

% prod_le_1
tff(fact_8613_prod__le__1,axiom,
    ! [A3: set_real,F: fun_real_real] :
      ( ! [X3: real] :
          ( member_real(X3,A3)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,X3))
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,X3)),one_one_real) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups1681761925125756287l_real(F,A3)),one_one_real) ) ).

% prod_le_1
tff(fact_8614_prod__le__1,axiom,
    ! [A3: set_nat,F: fun_nat_real] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A3)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,X3))
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,X3)),one_one_real) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_set_nat_real(groups129246275422532515t_real(F),A3)),one_one_real) ) ).

% prod_le_1
tff(fact_8615_prod__le__1,axiom,
    ! [A3: set_int,F: fun_int_real] :
      ( ! [X3: int] :
          ( member_int(X3,A3)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(F,X3))
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(F,X3)),one_one_real) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups2316167850115554303t_real(F,A3)),one_one_real) ) ).

% prod_le_1
tff(fact_8616_prod__le__1,axiom,
    ! [A3: set_complex,F: fun_complex_rat] :
      ( ! [X3: complex] :
          ( member_complex(X3,A3)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,X3))
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_complex_rat(F,X3)),one_one_rat) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups225925009352817453ex_rat(F,A3)),one_one_rat) ) ).

% prod_le_1
tff(fact_8617_prod__le__1,axiom,
    ! [A3: set_real,F: fun_real_rat] :
      ( ! [X3: real] :
          ( member_real(X3,A3)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,X3))
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_real_rat(F,X3)),one_one_rat) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups4061424788464935467al_rat(F,A3)),one_one_rat) ) ).

% prod_le_1
tff(fact_8618_prod__le__1,axiom,
    ! [A3: set_nat,F: fun_nat_rat] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A3)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,X3))
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X3)),one_one_rat) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups73079841787564623at_rat(F,A3)),one_one_rat) ) ).

% prod_le_1
tff(fact_8619_prod__le__1,axiom,
    ! [A3: set_int,F: fun_int_rat] :
      ( ! [X3: int] :
          ( member_int(X3,A3)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,X3))
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,X3)),one_one_rat) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups1072433553688619179nt_rat(F,A3)),one_one_rat) ) ).

% prod_le_1
tff(fact_8620_prod__le__1,axiom,
    ! [A3: set_complex,F: fun_complex_nat] :
      ( ! [X3: complex] :
          ( member_complex(X3,A3)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_complex_nat(F,X3))
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_complex_nat(F,X3)),one_one_nat) ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups861055069439313189ex_nat(F,A3)),one_one_nat) ) ).

% prod_le_1
tff(fact_8621_prod__le__1,axiom,
    ! [A3: set_real,F: fun_real_nat] :
      ( ! [X3: real] :
          ( member_real(X3,A3)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,X3))
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_real_nat(F,X3)),one_one_nat) ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups4696554848551431203al_nat(F,A3)),one_one_nat) ) ).

% prod_le_1
tff(fact_8622_zero__less__binomial,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K)) ) ).

% zero_less_binomial
tff(fact_8623_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_8624_choose__mult,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Mb),K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),K))) ) ) ) ).

% choose_mult
tff(fact_8625_binomial__Suc__Suc__eq__times,axiom,
    ! [Nb: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,Nb)),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,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K))),aa_nat_nat(suc,K)) ).

% binomial_Suc_Suc_eq_times
tff(fact_8626_binomial__absorb__comp,axiom,
    ! [Nb: 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,Nb),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),K)) ).

% binomial_absorb_comp
tff(fact_8627_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_8628_gbinomial__Suc__Suc,axiom,
    ! [A: real,K: nat] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(plus_plus_real(A),one_one_real)),aa_nat_nat(suc,K)) = aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),aa_nat_nat(suc,K))) ).

% gbinomial_Suc_Suc
tff(fact_8629_gbinomial__Suc__Suc,axiom,
    ! [A: rat,K: nat] : aa_nat_rat(gbinomial_rat(aa_rat_rat(plus_plus_rat(A),one_one_rat)),aa_nat_nat(suc,K)) = aa_rat_rat(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_8630_gbinomial__of__nat__symmetric,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => ( aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),K) = aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)) ) ) ).

% gbinomial_of_nat_symmetric
tff(fact_8631_gbinomial__of__nat__symmetric,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => ( aa_nat_rat(gbinomial_rat(aa_nat_rat(semiri681578069525770553at_rat,Nb)),K) = aa_nat_rat(gbinomial_rat(aa_nat_rat(semiri681578069525770553at_rat,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)) ) ) ).

% gbinomial_of_nat_symmetric
tff(fact_8632_prod_Onat__diff__reindex,axiom,
    ! [G: fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_gt(G),Nb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ).

% prod.nat_diff_reindex
tff(fact_8633_prod_Onat__diff__reindex,axiom,
    ! [G: fun_nat_int,Nb: nat] : groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_lm(G),Nb),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = groups705719431365010083at_int(G,aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ).

% prod.nat_diff_reindex
tff(fact_8634_prod_OatLeastAtMost__rev,axiom,
    ! [G: fun_nat_nat,Nb: nat,Mb: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Nb),Mb)) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_du(G),Nb),Mb)),aa_nat_set_nat(set_or1269000886237332187st_nat(Nb),Mb)) ).

% prod.atLeastAtMost_rev
tff(fact_8635_prod_OatLeastAtMost__rev,axiom,
    ! [G: fun_nat_int,Nb: nat,Mb: nat] : groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Nb),Mb)) = groups705719431365010083at_int(aa_nat_fun_nat_int(aa_nat1293065384339088724at_int(aTP_Lamm_ln(G),Nb),Mb),aa_nat_set_nat(set_or1269000886237332187st_nat(Nb),Mb)) ).

% prod.atLeastAtMost_rev
tff(fact_8636_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_lo(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_8637_gbinomial__Suc,axiom,
    ! [A: rat,K: nat] : aa_nat_rat(gbinomial_rat(A),aa_nat_nat(suc,K)) = aa_rat_rat(divide_divide_rat(groups73079841787564623at_rat(aTP_Lamm_lp(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_8638_gbinomial__Suc,axiom,
    ! [A: real,K: nat] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),aa_nat_nat(suc,K)) = aa_real_real(divide_divide_real(aa_set_nat_real(groups129246275422532515t_real(aa_real_fun_nat_real(aTP_Lamm_lq,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_8639_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_lr,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_8640_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_ls(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_8641_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_real,Nb: nat] : aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,Nb))) = aa_real_real(times_times_real(aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb))),aa_nat_real(G,aa_nat_nat(suc,Nb))) ).

% prod.atLeast0_atMost_Suc
tff(fact_8642_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_rat,Nb: nat] : groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,Nb))) = aa_rat_rat(times_times_rat(groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb))),aa_nat_rat(G,aa_nat_nat(suc,Nb))) ).

% prod.atLeast0_atMost_Suc
tff(fact_8643_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,Nb))) = 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),Nb))),aa_nat_nat(G,aa_nat_nat(suc,Nb))) ).

% prod.atLeast0_atMost_Suc
tff(fact_8644_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_int,Nb: nat] : groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,Nb))) = aa_int_int(aa_int_fun_int_int(times_times_int,groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb))),aa_nat_int(G,aa_nat_nat(suc,Nb))) ).

% prod.atLeast0_atMost_Suc
tff(fact_8645_prod_OatLeast__Suc__atMost,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) = aa_real_real(times_times_real(aa_nat_real(G,Mb)),aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb)),Nb))) ) ) ).

% prod.atLeast_Suc_atMost
tff(fact_8646_prod_OatLeast__Suc__atMost,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) = aa_rat_rat(times_times_rat(aa_nat_rat(G,Mb)),groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb)),Nb))) ) ) ).

% prod.atLeast_Suc_atMost
tff(fact_8647_prod_OatLeast__Suc__atMost,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(G,Mb)),aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb)),Nb))) ) ) ).

% prod.atLeast_Suc_atMost
tff(fact_8648_prod_OatLeast__Suc__atMost,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(G,Mb)),groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb)),Nb))) ) ) ).

% prod.atLeast_Suc_atMost
tff(fact_8649_prod_Onat__ivl__Suc_H,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb))
     => ( aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(suc,Nb))) = aa_real_real(times_times_real(aa_nat_real(G,aa_nat_nat(suc,Nb))),aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))) ) ) ).

% prod.nat_ivl_Suc'
tff(fact_8650_prod_Onat__ivl__Suc_H,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb))
     => ( groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(suc,Nb))) = aa_rat_rat(times_times_rat(aa_nat_rat(G,aa_nat_nat(suc,Nb))),groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))) ) ) ).

% prod.nat_ivl_Suc'
tff(fact_8651_prod_Onat__ivl__Suc_H,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb))
     => ( aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(G,aa_nat_nat(suc,Nb))),aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))) ) ) ).

% prod.nat_ivl_Suc'
tff(fact_8652_prod_Onat__ivl__Suc_H,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb))
     => ( groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(suc,Nb))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(G,aa_nat_nat(suc,Nb))),groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))) ) ) ).

% prod.nat_ivl_Suc'
tff(fact_8653_binomial__absorption,axiom,
    ! [K: nat,Nb: 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,Nb),aa_nat_nat(suc,K))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),K)) ).

% binomial_absorption
tff(fact_8654_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_8655_gbinomial__addition__formula,axiom,
    ! [A: real,K: nat] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),aa_nat_nat(suc,K)) = aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(minus_minus_real(A),one_one_real)),aa_nat_nat(suc,K))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(minus_minus_real(A),one_one_real)),K)) ).

% gbinomial_addition_formula
tff(fact_8656_gbinomial__addition__formula,axiom,
    ! [A: rat,K: nat] : aa_nat_rat(gbinomial_rat(A),aa_nat_nat(suc,K)) = aa_rat_rat(plus_plus_rat(aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(A),one_one_rat)),aa_nat_nat(suc,K))),aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(A),one_one_rat)),K)) ).

% gbinomial_addition_formula
tff(fact_8657_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_8658_gbinomial__absorb__comp,axiom,
    ! [A: real,K: nat] : aa_real_real(times_times_real(aa_real_real(minus_minus_real(A),aa_nat_real(semiri5074537144036343181t_real,K))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K)) = aa_real_real(times_times_real(A),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(minus_minus_real(A),one_one_real)),K)) ).

% gbinomial_absorb_comp
tff(fact_8659_gbinomial__absorb__comp,axiom,
    ! [A: rat,K: nat] : aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(A),aa_nat_rat(semiri681578069525770553at_rat,K))),aa_nat_rat(gbinomial_rat(A),K)) = aa_rat_rat(times_times_rat(A),aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(A),one_one_rat)),K)) ).

% gbinomial_absorb_comp
tff(fact_8660_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(aa_real_fun_nat_real(power_power_real,aa_real_real(divide_divide_real(A),aa_nat_real(semiri5074537144036343181t_real,K))),K)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K)) ) ).

% gbinomial_ge_n_over_k_pow_k
tff(fact_8661_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(aa_rat_rat(divide_divide_rat(A),aa_nat_rat(semiri681578069525770553at_rat,K))),K)),aa_nat_rat(gbinomial_rat(A),K)) ) ).

% gbinomial_ge_n_over_k_pow_k
tff(fact_8662_gbinomial__mult__1,axiom,
    ! [A: real,K: nat] : aa_real_real(times_times_real(A),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K)) = aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,K)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K))),aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,K))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),aa_nat_nat(suc,K)))) ).

% gbinomial_mult_1
tff(fact_8663_gbinomial__mult__1,axiom,
    ! [A: rat,K: nat] : aa_rat_rat(times_times_rat(A),aa_nat_rat(gbinomial_rat(A),K)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,K)),aa_nat_rat(gbinomial_rat(A),K))),aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K))),aa_nat_rat(gbinomial_rat(A),aa_nat_nat(suc,K)))) ).

% gbinomial_mult_1
tff(fact_8664_gbinomial__mult__1_H,axiom,
    ! [A: real,K: nat] : aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K)),A) = aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,K)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K))),aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,K))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),aa_nat_nat(suc,K)))) ).

% gbinomial_mult_1'
tff(fact_8665_gbinomial__mult__1_H,axiom,
    ! [A: rat,K: nat] : aa_rat_rat(times_times_rat(aa_nat_rat(gbinomial_rat(A),K)),A) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,K)),aa_nat_rat(gbinomial_rat(A),K))),aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K))),aa_nat_rat(gbinomial_rat(A),aa_nat_nat(suc,K)))) ).

% gbinomial_mult_1'
tff(fact_8666_binomial__fact__lemma,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => ( 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,Nb),K)))),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K)) = aa_nat_nat(semiri1408675320244567234ct_nat,Nb) ) ) ).

% binomial_fact_lemma
tff(fact_8667_prod_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_real,Nb: nat] : aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = aa_real_real(times_times_real(aa_nat_real(G,zero_zero_nat)),aa_set_nat_real(groups129246275422532515t_real(aTP_Lamm_dn(G)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% prod.lessThan_Suc_shift
tff(fact_8668_prod_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_rat,Nb: nat] : groups73079841787564623at_rat(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = aa_rat_rat(times_times_rat(aa_nat_rat(G,zero_zero_nat)),groups73079841787564623at_rat(aTP_Lamm_dw(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% prod.lessThan_Suc_shift
tff(fact_8669_prod_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = 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_dm(G)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% prod.lessThan_Suc_shift
tff(fact_8670_prod_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_int,Nb: nat] : groups705719431365010083at_int(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(G,zero_zero_nat)),groups705719431365010083at_int(aTP_Lamm_dx(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% prod.lessThan_Suc_shift
tff(fact_8671_prod_OSuc__reindex__ivl,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_real_real(times_times_real(aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))),aa_nat_real(G,aa_nat_nat(suc,Nb))) = aa_real_real(times_times_real(aa_nat_real(G,Mb)),aa_set_nat_real(groups129246275422532515t_real(aTP_Lamm_dn(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_8672_prod_OSuc__reindex__ivl,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_rat_rat(times_times_rat(groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))),aa_nat_rat(G,aa_nat_nat(suc,Nb))) = aa_rat_rat(times_times_rat(aa_nat_rat(G,Mb)),groups73079841787564623at_rat(aTP_Lamm_dw(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_8673_prod_OSuc__reindex__ivl,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( 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(Mb),Nb))),aa_nat_nat(G,aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(G,Mb)),aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_dm(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_8674_prod_OSuc__reindex__ivl,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))),aa_nat_int(G,aa_nat_nat(suc,Nb))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(G,Mb)),groups705719431365010083at_int(aTP_Lamm_dx(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_8675_prod_OatLeast1__atMost__eq,axiom,
    ! [G: fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),Nb)) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_dm(G)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ).

% prod.atLeast1_atMost_eq
tff(fact_8676_prod_OatLeast1__atMost__eq,axiom,
    ! [G: fun_nat_int,Nb: nat] : groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),Nb)) = groups705719431365010083at_int(aTP_Lamm_dx(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ).

% prod.atLeast1_atMost_eq
tff(fact_8677_fact__prod,axiom,
    ! [Nb: nat] : aa_nat_int(semiri1406184849735516958ct_int,Nb) = aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_ar),aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),Nb))) ).

% fact_prod
tff(fact_8678_fact__prod,axiom,
    ! [Nb: nat] : aa_nat_rat(semiri773545260158071498ct_rat,Nb) = aa_nat_rat(semiri681578069525770553at_rat,aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_ar),aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),Nb))) ).

% fact_prod
tff(fact_8679_fact__prod,axiom,
    ! [Nb: nat] : aa_nat_real(semiri2265585572941072030t_real,Nb) = aa_nat_real(semiri5074537144036343181t_real,aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_ar),aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),Nb))) ).

% fact_prod
tff(fact_8680_fact__prod,axiom,
    ! [Nb: nat] : aa_nat_nat(semiri1408675320244567234ct_nat,Nb) = aa_nat_nat(semiri1316708129612266289at_nat,aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_ar),aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),Nb))) ).

% fact_prod
tff(fact_8681_binomial__ge__n__over__k__pow__k,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(divide_divide_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(semiri5074537144036343181t_real,K))),K)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K))) ) ).

% binomial_ge_n_over_k_pow_k
tff(fact_8682_binomial__ge__n__over__k__pow__k,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_rat_rat(divide_divide_rat(aa_nat_rat(semiri681578069525770553at_rat,Nb)),aa_nat_rat(semiri681578069525770553at_rat,K))),K)),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K))) ) ).

% binomial_ge_n_over_k_pow_k
tff(fact_8683_binomial__mono,axiom,
    ! [K: nat,K6: nat,Nb: 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)),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K6)) ) ) ).

% binomial_mono
tff(fact_8684_binomial__maximum_H,axiom,
    ! [Nb: 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))),Nb)),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))),Nb)),Nb)) ).

% binomial_maximum'
tff(fact_8685_binomial__maximum,axiom,
    ! [Nb: nat,K: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% binomial_maximum
tff(fact_8686_binomial__antimono,axiom,
    ! [K: nat,K6: nat,Nb: 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,Nb),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),Nb)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K6)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K)) ) ) ) ).

% binomial_antimono
tff(fact_8687_binomial__le__pow2,axiom,
    ! [Nb: nat,K: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% binomial_le_pow2
tff(fact_8688_choose__reduce__nat,axiom,
    ! [Nb: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),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,Nb),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,Nb),one_one_nat)),K)) ) ) ) ).

% choose_reduce_nat
tff(fact_8689_times__binomial__minus1__eq,axiom,
    ! [K: nat,Nb: 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,Nb),K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))) ) ) ).

% times_binomial_minus1_eq
tff(fact_8690_prod_Oub__add__nat,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_real,P2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))
     => ( aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2))) = aa_real_real(times_times_real(aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))),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,Nb),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2)))) ) ) ).

% prod.ub_add_nat
tff(fact_8691_prod_Oub__add__nat,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_rat,P2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))
     => ( groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2))) = aa_rat_rat(times_times_rat(groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))),groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2)))) ) ) ).

% prod.ub_add_nat
tff(fact_8692_prod_Oub__add__nat,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_nat,P2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))
     => ( aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2))) = 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(Mb),Nb))),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,Nb),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2)))) ) ) ).

% prod.ub_add_nat
tff(fact_8693_prod_Oub__add__nat,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_int,P2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))
     => ( groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2))) = aa_int_int(aa_int_fun_int_int(times_times_int,groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))),groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2)))) ) ) ).

% prod.ub_add_nat
tff(fact_8694_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_8695_Suc__times__gbinomial,axiom,
    ! [K: nat,A: real] : aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,K))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(plus_plus_real(A),one_one_real)),aa_nat_nat(suc,K))) = aa_real_real(times_times_real(aa_real_real(plus_plus_real(A),one_one_real)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K)) ).

% Suc_times_gbinomial
tff(fact_8696_Suc__times__gbinomial,axiom,
    ! [K: nat,A: rat] : aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K))),aa_nat_rat(gbinomial_rat(aa_rat_rat(plus_plus_rat(A),one_one_rat)),aa_nat_nat(suc,K))) = aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(A),one_one_rat)),aa_nat_rat(gbinomial_rat(A),K)) ).

% Suc_times_gbinomial
tff(fact_8697_fold__atLeastAtMost__nat_Oelims,axiom,
    ! [Xb: fun_nat_fun_nat_nat,Xaa: nat,Xba: nat,Xc: nat,Yb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(Xb,Xaa),Xba),Xc) = Yb )
     => ( Yb = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xba),Xaa),Xc,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(Xb,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xaa),one_one_nat)),Xba),aa_nat_nat(aa_nat_fun_nat_nat(Xb,Xaa),Xc))) ) ) ).

% fold_atLeastAtMost_nat.elims
tff(fact_8698_fold__atLeastAtMost__nat_Osimps,axiom,
    ! [F: fun_nat_fun_nat_nat,A: nat,B: nat,Acc2: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(F,A),B),Acc2) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A),Acc2,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),Acc2))) ).

% fold_atLeastAtMost_nat.simps
tff(fact_8699_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_8700_gbinomial__absorption,axiom,
    ! [K: nat,A: real] : aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,K))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),aa_nat_nat(suc,K))) = aa_real_real(times_times_real(A),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(minus_minus_real(A),one_one_real)),K)) ).

% gbinomial_absorption
tff(fact_8701_gbinomial__absorption,axiom,
    ! [K: nat,A: rat] : aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K))),aa_nat_rat(gbinomial_rat(A),aa_nat_nat(suc,K))) = aa_rat_rat(times_times_rat(A),aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(A),one_one_rat)),K)) ).

% gbinomial_absorption
tff(fact_8702_binomial__altdef__nat,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Nb)),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,Nb),K)))) ) ) ).

% binomial_altdef_nat
tff(fact_8703_gbinomial__trinomial__revision,axiom,
    ! [K: nat,Mb: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Mb)
     => ( aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),Mb)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),K)) = aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(minus_minus_real(A),aa_nat_real(semiri5074537144036343181t_real,K))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),K))) ) ) ).

% gbinomial_trinomial_revision
tff(fact_8704_gbinomial__trinomial__revision,axiom,
    ! [K: nat,Mb: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Mb)
     => ( aa_rat_rat(times_times_rat(aa_nat_rat(gbinomial_rat(A),Mb)),aa_nat_rat(gbinomial_rat(aa_nat_rat(semiri681578069525770553at_rat,Mb)),K)) = aa_rat_rat(times_times_rat(aa_nat_rat(gbinomial_rat(A),K)),aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(A),aa_nat_rat(semiri681578069525770553at_rat,K))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),K))) ) ) ).

% gbinomial_trinomial_revision
tff(fact_8705_norm__prod__diff,axiom,
    ! [I6: set_complex,Z: fun_complex_real,W: fun_complex_real] :
      ( ! [I2: complex] :
          ( member_complex(I2,I6)
         => 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,I6)
           => 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(minus_minus_real(groups766887009212190081x_real(Z,I6)),groups766887009212190081x_real(W,I6)))),groups5808333547571424918x_real(aa_fun8933909628229591020x_real(aTP_Lamm_lt(Z),W),I6)) ) ) ).

% norm_prod_diff
tff(fact_8706_norm__prod__diff,axiom,
    ! [I6: set_real,Z: fun_real_real,W: fun_real_real] :
      ( ! [I2: real] :
          ( member_real(I2,I6)
         => 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,I6)
           => 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(minus_minus_real(groups1681761925125756287l_real(Z,I6)),groups1681761925125756287l_real(W,I6)))),groups8097168146408367636l_real(aa_fun6946558556880649704l_real(aTP_Lamm_lu(Z),W),I6)) ) ) ).

% norm_prod_diff
tff(fact_8707_norm__prod__diff,axiom,
    ! [I6: set_set_nat,Z: fun_set_nat_real,W: fun_set_nat_real] :
      ( ! [I2: set_nat] :
          ( member_set_nat(I2,I6)
         => 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,I6)
           => 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(minus_minus_real(groups3619160379726066777t_real(Z,I6)),groups3619160379726066777t_real(W,I6)))),groups5107569545109728110t_real(aa_fun3569254276315353500t_real(aTP_Lamm_lv(Z),W),I6)) ) ) ).

% norm_prod_diff
tff(fact_8708_norm__prod__diff,axiom,
    ! [I6: set_int,Z: fun_int_real,W: fun_int_real] :
      ( ! [I2: int] :
          ( member_int(I2,I6)
         => 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,I6)
           => 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(minus_minus_real(groups2316167850115554303t_real(Z,I6)),groups2316167850115554303t_real(W,I6)))),groups8778361861064173332t_real(aa_fun3252798400008884712t_real(aTP_Lamm_lw(Z),W),I6)) ) ) ).

% norm_prod_diff
tff(fact_8709_norm__prod__diff,axiom,
    ! [I6: set_complex,Z: fun_complex_complex,W: fun_complex_complex] :
      ( ! [I2: complex] :
          ( member_complex(I2,I6)
         => 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,I6)
           => 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,I6)),groups3708469109370488835omplex(W,I6)))),groups5808333547571424918x_real(aa_fun1953249243402654830x_real(aTP_Lamm_lx(Z),W),I6)) ) ) ).

% norm_prod_diff
tff(fact_8710_norm__prod__diff,axiom,
    ! [I6: set_real,Z: fun_real_complex,W: fun_real_complex] :
      ( ! [I2: real] :
          ( member_real(I2,I6)
         => 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,I6)
           => 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,I6)),groups713298508707869441omplex(W,I6)))),groups8097168146408367636l_real(aa_fun2786211375485122666l_real(aTP_Lamm_ly(Z),W),I6)) ) ) ).

% norm_prod_diff
tff(fact_8711_norm__prod__diff,axiom,
    ! [I6: set_set_nat,Z: fun_set_nat_complex,W: fun_set_nat_complex] :
      ( ! [I2: set_nat] :
          ( member_set_nat(I2,I6)
         => 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,I6)
           => 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,I6)),groups1092910753850256091omplex(W,I6)))),groups5107569545109728110t_real(aa_fun5767801665285160478t_real(aTP_Lamm_lz(Z),W),I6)) ) ) ).

% norm_prod_diff
tff(fact_8712_norm__prod__diff,axiom,
    ! [I6: set_int,Z: fun_int_complex,W: fun_int_complex] :
      ( ! [I2: int] :
          ( member_int(I2,I6)
         => 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,I6)
           => 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,I6)),groups7440179247065528705omplex(W,I6)))),groups8778361861064173332t_real(aa_fun1985759105522019690t_real(aTP_Lamm_ma(Z),W),I6)) ) ) ).

% norm_prod_diff
tff(fact_8713_norm__prod__diff,axiom,
    ! [I6: set_nat,Z: fun_nat_real,W: fun_nat_real] :
      ( ! [I2: nat] :
          ( aa_set_nat_o(member_nat(I2),I6)
         => 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),I6)
           => 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(minus_minus_real(aa_set_nat_real(groups129246275422532515t_real(Z),I6)),aa_set_nat_real(groups129246275422532515t_real(W),I6)))),aa_set_nat_real(groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_mb(Z),W)),I6)) ) ) ).

% norm_prod_diff
tff(fact_8714_norm__prod__diff,axiom,
    ! [I6: set_nat,Z: fun_nat_complex,W: fun_nat_complex] :
      ( ! [I2: nat] :
          ( aa_set_nat_o(member_nat(I2),I6)
         => 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),I6)
           => 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),I6)),aa_set_nat_complex(groups6464643781859351333omplex(W),I6)))),aa_set_nat_real(groups6591440286371151544t_real(aa_fun7054779078796324018t_real(aTP_Lamm_mc(Z),W)),I6)) ) ) ).

% norm_prod_diff
tff(fact_8715_fact__eq__fact__times,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => ( aa_nat_nat(semiri1408675320244567234ct_nat,Mb) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Nb)),aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_ar),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Nb)),Mb))) ) ) ).

% fact_eq_fact_times
tff(fact_8716_binomial__less__binomial__Suc,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),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,Nb),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),aa_nat_nat(suc,K))) ) ).

% binomial_less_binomial_Suc
tff(fact_8717_binomial__strict__mono,axiom,
    ! [K: nat,K6: nat,Nb: 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)),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K6)) ) ) ).

% binomial_strict_mono
tff(fact_8718_binomial__strict__antimono,axiom,
    ! [K: nat,K6: nat,Nb: 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,Nb),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),Nb)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K6)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K)) ) ) ) ).

% binomial_strict_antimono
tff(fact_8719_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_md(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_8720_gbinomial__code,axiom,
    ! [A: rat,K: nat] :
      aa_nat_rat(gbinomial_rat(A),K) = $ite(K = zero_zero_nat,one_one_rat,aa_rat_rat(divide_divide_rat(aa_rat_rat(set_fo1949268297981939178at_rat(aTP_Lamm_me(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_8721_gbinomial__code,axiom,
    ! [A: real,K: nat] :
      aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K) = $ite(K = zero_zero_nat,one_one_real,aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(set_fo3111899725591712190t_real(aTP_Lamm_mf(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_8722_central__binomial__odd,axiom,
    ! [Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ).

% central_binomial_odd
tff(fact_8723_binomial__addition__formula,axiom,
    ! [Nb: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),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,Nb),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,Nb),one_one_nat)),K)) ) ) ).

% binomial_addition_formula
tff(fact_8724_binomial__fact,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => ( aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K)) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(semiri5044797733671781792omplex,Nb)),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,Nb),K)))) ) ) ).

% binomial_fact
tff(fact_8725_binomial__fact,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => ( aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K)) = aa_rat_rat(divide_divide_rat(aa_nat_rat(semiri773545260158071498ct_rat,Nb)),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,Nb),K)))) ) ) ).

% binomial_fact
tff(fact_8726_binomial__fact,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => ( aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K)) = aa_real_real(divide_divide_real(aa_nat_real(semiri2265585572941072030t_real,Nb)),aa_real_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,Nb),K)))) ) ) ).

% binomial_fact
tff(fact_8727_fact__binomial,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => ( aa_complex_complex(times_times_complex(aa_nat_complex(semiri5044797733671781792omplex,K)),aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K))) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(semiri5044797733671781792omplex,Nb)),aa_nat_complex(semiri5044797733671781792omplex,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K))) ) ) ).

% fact_binomial
tff(fact_8728_fact__binomial,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => ( 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,Nb),K))) = aa_rat_rat(divide_divide_rat(aa_nat_rat(semiri773545260158071498ct_rat,Nb)),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K))) ) ) ).

% fact_binomial
tff(fact_8729_fact__binomial,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => ( aa_real_real(times_times_real(aa_nat_real(semiri2265585572941072030t_real,K)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K))) = aa_real_real(divide_divide_real(aa_nat_real(semiri2265585572941072030t_real,Nb)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K))) ) ) ).

% fact_binomial
tff(fact_8730_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_8731_gbinomial__factors,axiom,
    ! [A: real,K: nat] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(plus_plus_real(A),one_one_real)),aa_nat_nat(suc,K)) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(A),one_one_real)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,K)))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K)) ).

% gbinomial_factors
tff(fact_8732_gbinomial__factors,axiom,
    ! [A: rat,K: nat] : aa_nat_rat(gbinomial_rat(aa_rat_rat(plus_plus_rat(A),one_one_rat)),aa_nat_nat(suc,K)) = aa_rat_rat(times_times_rat(aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(A),one_one_rat)),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K)))),aa_nat_rat(gbinomial_rat(A),K)) ).

% gbinomial_factors
tff(fact_8733_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_8734_gbinomial__rec,axiom,
    ! [A: real,K: nat] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(plus_plus_real(A),one_one_real)),aa_nat_nat(suc,K)) = aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K)),aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(A),one_one_real)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,K)))) ).

% gbinomial_rec
tff(fact_8735_gbinomial__rec,axiom,
    ! [A: rat,K: nat] : aa_nat_rat(gbinomial_rat(aa_rat_rat(plus_plus_rat(A),one_one_rat)),aa_nat_nat(suc,K)) = aa_rat_rat(times_times_rat(aa_nat_rat(gbinomial_rat(A),K)),aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(A),one_one_rat)),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K)))) ).

% gbinomial_rec
tff(fact_8736_gbinomial__index__swap,axiom,
    ! [K: nat,Nb: 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,Nb))),one_one_complex)),K)) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Nb)),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,K))),one_one_complex)),Nb)) ).

% gbinomial_index_swap
tff(fact_8737_gbinomial__index__swap,axiom,
    ! [K: nat,Nb: nat] : aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),K)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,Nb))),one_one_real)),K)) = aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Nb)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(minus_minus_real(aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,K))),one_one_real)),Nb)) ).

% gbinomial_index_swap
tff(fact_8738_gbinomial__index__swap,axiom,
    ! [K: nat,Nb: nat] : aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),K)),aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,Nb))),one_one_rat)),K)) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Nb)),aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,K))),one_one_rat)),Nb)) ).

% gbinomial_index_swap
tff(fact_8739_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_8740_gbinomial__negated__upper,axiom,
    ! [A: real,K: nat] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K) = aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),K)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(minus_minus_real(aa_real_real(minus_minus_real(aa_nat_real(semiri5074537144036343181t_real,K)),A)),one_one_real)),K)) ).

% gbinomial_negated_upper
tff(fact_8741_gbinomial__negated__upper,axiom,
    ! [A: rat,K: nat] : aa_nat_rat(gbinomial_rat(A),K) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),K)),aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(minus_minus_rat(aa_nat_rat(semiri681578069525770553at_rat,K)),A)),one_one_rat)),K)) ).

% gbinomial_negated_upper
tff(fact_8742_pochhammer__Suc__prod,axiom,
    ! [A: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),aa_nat_nat(suc,Nb)) = aa_set_nat_real(groups129246275422532515t_real(aa_real_fun_nat_real(aTP_Lamm_mg,A)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb)) ).

% pochhammer_Suc_prod
tff(fact_8743_pochhammer__Suc__prod,axiom,
    ! [A: rat,Nb: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),aa_nat_nat(suc,Nb)) = groups73079841787564623at_rat(aTP_Lamm_mh(A),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb)) ).

% pochhammer_Suc_prod
tff(fact_8744_pochhammer__Suc__prod,axiom,
    ! [A: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),aa_nat_nat(suc,Nb)) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_mi,A)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb)) ).

% pochhammer_Suc_prod
tff(fact_8745_pochhammer__Suc__prod,axiom,
    ! [A: int,Nb: nat] : aa_nat_int(comm_s4660882817536571857er_int(A),aa_nat_nat(suc,Nb)) = groups705719431365010083at_int(aTP_Lamm_mj(A),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb)) ).

% pochhammer_Suc_prod
tff(fact_8746_pochhammer__prod__rev,axiom,
    ! [A: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),Nb) = aa_set_nat_real(groups129246275422532515t_real(aa_nat_fun_nat_real(aTP_Lamm_mk(A),Nb)),aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),Nb)) ).

% pochhammer_prod_rev
tff(fact_8747_pochhammer__prod__rev,axiom,
    ! [A: rat,Nb: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),Nb) = groups73079841787564623at_rat(aa_nat_fun_nat_rat(aTP_Lamm_ml(A),Nb),aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),Nb)) ).

% pochhammer_prod_rev
tff(fact_8748_pochhammer__prod__rev,axiom,
    ! [A: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),Nb) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_mm(A),Nb)),aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),Nb)) ).

% pochhammer_prod_rev
tff(fact_8749_pochhammer__prod__rev,axiom,
    ! [A: int,Nb: nat] : aa_nat_int(comm_s4660882817536571857er_int(A),Nb) = groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_mn(A),Nb),aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),Nb)) ).

% pochhammer_prod_rev
tff(fact_8750_binomial__code,axiom,
    ! [Nb: nat,K: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),K),
        zero_zero_nat,
        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),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,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),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,Nb),K)),one_one_nat)),Nb),one_one_nat)),aa_nat_nat(semiri1408675320244567234ct_nat,K))) ) ).

% binomial_code
tff(fact_8751_fact__div__fact,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Mb)),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_ar),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)),Mb)) ) ) ).

% fact_div_fact
tff(fact_8752_choose__two,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),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,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% choose_two
tff(fact_8753_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_8754_gbinomial__minus,axiom,
    ! [A: real,K: nat] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(uminus_uminus_real,A)),K) = aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),K)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(minus_minus_real(aa_real_real(plus_plus_real(A),aa_nat_real(semiri5074537144036343181t_real,K))),one_one_real)),K)) ).

% gbinomial_minus
tff(fact_8755_gbinomial__minus,axiom,
    ! [A: rat,K: nat] : aa_nat_rat(gbinomial_rat(aa_rat_rat(uminus_uminus_rat,A)),K) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),K)),aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(A),aa_nat_rat(semiri681578069525770553at_rat,K))),one_one_rat)),K)) ).

% gbinomial_minus
tff(fact_8756_prod_Oin__pairs,axiom,
    ! [G: fun_nat_real,Mb: nat,Nb: 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))),Mb)),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))),Nb)))) = aa_set_nat_real(groups129246275422532515t_real(aTP_Lamm_mo(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) ).

% prod.in_pairs
tff(fact_8757_prod_Oin__pairs,axiom,
    ! [G: fun_nat_rat,Mb: nat,Nb: 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))),Mb)),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))),Nb)))) = groups73079841787564623at_rat(aTP_Lamm_mp(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) ).

% prod.in_pairs
tff(fact_8758_prod_Oin__pairs,axiom,
    ! [G: fun_nat_nat,Mb: nat,Nb: 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))),Mb)),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))),Nb)))) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_mq(G)),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) ).

% prod.in_pairs
tff(fact_8759_prod_Oin__pairs,axiom,
    ! [G: fun_nat_int,Mb: nat,Nb: 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))),Mb)),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))),Nb)))) = groups705719431365010083at_int(aTP_Lamm_mr(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) ).

% prod.in_pairs
tff(fact_8760_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_8761_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(aa_real_fun_nat_real(gbinomial_real,A),K) = aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_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(aa_real_fun_nat_real(gbinomial_real,aa_real_real(minus_minus_real(A),one_one_real)),K)) ) ) ).

% gbinomial_reduce_nat
tff(fact_8762_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(plus_plus_rat(aa_nat_rat(gbinomial_rat(aa_rat_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(aa_rat_rat(minus_minus_rat(A),one_one_rat)),K)) ) ) ).

% gbinomial_reduce_nat
tff(fact_8763_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_ms(F),A,B),zero_zero_complex) ).

% sum_atLeastAtMost_code
tff(fact_8764_sum__atLeastAtMost__code,axiom,
    ! [F: fun_nat_rat,A: nat,B: nat] : groups2906978787729119204at_rat(F,aa_nat_set_nat(set_or1269000886237332187st_nat(A),B)) = aa_rat_rat(set_fo1949268297981939178at_rat(aTP_Lamm_mt(F),A,B),zero_zero_rat) ).

% sum_atLeastAtMost_code
tff(fact_8765_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_mu(F),A,B),zero_zero_int) ).

% sum_atLeastAtMost_code
tff(fact_8766_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_mv(F),A),B),zero_zero_nat) ).

% sum_atLeastAtMost_code
tff(fact_8767_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_mw(F),A),B),zero_zero_real) ).

% sum_atLeastAtMost_code
tff(fact_8768_pochhammer__Suc__prod__rev,axiom,
    ! [A: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),aa_nat_nat(suc,Nb)) = aa_set_nat_real(groups129246275422532515t_real(aa_nat_fun_nat_real(aTP_Lamm_mk(A),Nb)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb)) ).

% pochhammer_Suc_prod_rev
tff(fact_8769_pochhammer__Suc__prod__rev,axiom,
    ! [A: rat,Nb: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),aa_nat_nat(suc,Nb)) = groups73079841787564623at_rat(aa_nat_fun_nat_rat(aTP_Lamm_ml(A),Nb),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb)) ).

% pochhammer_Suc_prod_rev
tff(fact_8770_pochhammer__Suc__prod__rev,axiom,
    ! [A: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),aa_nat_nat(suc,Nb)) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_mm(A),Nb)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb)) ).

% pochhammer_Suc_prod_rev
tff(fact_8771_pochhammer__Suc__prod__rev,axiom,
    ! [A: int,Nb: nat] : aa_nat_int(comm_s4660882817536571857er_int(A),aa_nat_nat(suc,Nb)) = groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_mn(A),Nb),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb)) ).

% pochhammer_Suc_prod_rev
tff(fact_8772_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_8773_gbinomial__pochhammer,axiom,
    ! [A: rat,K: nat] : aa_nat_rat(gbinomial_rat(A),K) = aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),K)),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,A)),K))),aa_nat_rat(semiri773545260158071498ct_rat,K)) ).

% gbinomial_pochhammer
tff(fact_8774_gbinomial__pochhammer,axiom,
    ! [A: real,K: nat] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K) = aa_real_real(divide_divide_real(aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),K)),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(uminus_uminus_real,A)),K))),aa_nat_real(semiri2265585572941072030t_real,K)) ).

% gbinomial_pochhammer
tff(fact_8775_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_8776_gbinomial__pochhammer_H,axiom,
    ! [A: rat,K: nat] : aa_nat_rat(gbinomial_rat(A),K) = aa_rat_rat(divide_divide_rat(aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(plus_plus_rat(aa_rat_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_8777_gbinomial__pochhammer_H,axiom,
    ! [A: real,K: nat] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K) = aa_real_real(divide_divide_real(aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(plus_plus_real(aa_real_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_8778_gbinomial__sum__up__index,axiom,
    ! [K: nat,Nb: nat] : aa_set_nat_complex(groups2073611262835488442omplex(aTP_Lamm_mx(K)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb)) = aa_nat_complex(gbinomial_complex(aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,Nb)),one_one_complex)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),one_one_nat)) ).

% gbinomial_sum_up_index
tff(fact_8779_gbinomial__sum__up__index,axiom,
    ! [K: nat,Nb: nat] : groups2906978787729119204at_rat(aTP_Lamm_my(K),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb)) = aa_nat_rat(gbinomial_rat(aa_rat_rat(plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,Nb)),one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),one_one_nat)) ).

% gbinomial_sum_up_index
tff(fact_8780_gbinomial__sum__up__index,axiom,
    ! [K: nat,Nb: nat] : aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_mz,K)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb)) = aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(plus_plus_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),one_one_nat)) ).

% gbinomial_sum_up_index
tff(fact_8781_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_8782_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(aa_real_fun_nat_real(gbinomial_real,A),K) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(A),aa_nat_real(semiri5074537144036343181t_real,K))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_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_8783_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(aa_rat_rat(divide_divide_rat(A),aa_nat_rat(semiri681578069525770553at_rat,K))),aa_nat_rat(gbinomial_rat(aa_rat_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_8784_fact__code,axiom,
    ! [Nb: nat] : aa_nat_int(semiri1406184849735516958ct_int,Nb) = 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))),Nb),one_one_nat)) ).

% fact_code
tff(fact_8785_fact__code,axiom,
    ! [Nb: nat] : aa_nat_rat(semiri773545260158071498ct_rat,Nb) = 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))),Nb),one_one_nat)) ).

% fact_code
tff(fact_8786_fact__code,axiom,
    ! [Nb: nat] : aa_nat_real(semiri2265585572941072030t_real,Nb) = 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))),Nb),one_one_nat)) ).

% fact_code
tff(fact_8787_fact__code,axiom,
    ! [Nb: nat] : aa_nat_nat(semiri1408675320244567234ct_nat,Nb) = 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))),Nb),one_one_nat)) ).

% fact_code
tff(fact_8788_gbinomial__partial__row__sum,axiom,
    ! [A: complex,Mb: nat] : aa_set_nat_complex(groups2073611262835488442omplex(aTP_Lamm_km(A)),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = aa_complex_complex(times_times_complex(aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,Mb)),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,Mb),one_one_nat))) ).

% gbinomial_partial_row_sum
tff(fact_8789_gbinomial__partial__row__sum,axiom,
    ! [A: rat,Mb: nat] : groups2906978787729119204at_rat(aTP_Lamm_kn(A),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = aa_rat_rat(times_times_rat(aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,Mb)),one_one_rat)),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))),aa_nat_rat(gbinomial_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),one_one_nat))) ).

% gbinomial_partial_row_sum
tff(fact_8790_gbinomial__partial__row__sum,axiom,
    ! [A: real,Mb: nat] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ko,A)),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(aa_nat_real(semiri5074537144036343181t_real,Mb)),one_one_real)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),one_one_nat))) ).

% gbinomial_partial_row_sum
tff(fact_8791_choose__odd__sum,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_set_nat_complex(groups2073611262835488442omplex(aTP_Lamm_na(Nb)),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_nat_complex(power_power_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),Nb) ) ) ).

% choose_odd_sum
tff(fact_8792_choose__odd__sum,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( 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_nb,Nb),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb) ) ) ).

% choose_odd_sum
tff(fact_8793_choose__odd__sum,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),groups2906978787729119204at_rat(aTP_Lamm_nc(Nb),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),Nb) ) ) ).

% choose_odd_sum
tff(fact_8794_choose__odd__sum,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_nd,Nb)),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Nb) ) ) ).

% choose_odd_sum
tff(fact_8795_choose__even__sum,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_set_nat_complex(groups2073611262835488442omplex(aTP_Lamm_ne(Nb)),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_nat_complex(power_power_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),Nb) ) ) ).

% choose_even_sum
tff(fact_8796_choose__even__sum,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( 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_nf,Nb),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb) ) ) ).

% choose_even_sum
tff(fact_8797_choose__even__sum,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),groups2906978787729119204at_rat(aTP_Lamm_ng(Nb),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),Nb) ) ) ).

% choose_even_sum
tff(fact_8798_choose__even__sum,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_nh,Nb)),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Nb) ) ) ).

% choose_even_sum
tff(fact_8799_gbinomial__r__part__sum,axiom,
    ! [Mb: 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,Mb))),one_one_complex))),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = 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))),Mb)) ).

% gbinomial_r_part_sum
tff(fact_8800_gbinomial__r__part__sum,axiom,
    ! [Mb: nat] : groups2906978787729119204at_rat(gbinomial_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),aa_nat_rat(semiri681578069525770553at_rat,Mb))),one_one_rat)),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)) ).

% gbinomial_r_part_sum
tff(fact_8801_gbinomial__r__part__sum,axiom,
    ! [Mb: nat] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Mb))),one_one_real))),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)) ).

% gbinomial_r_part_sum
tff(fact_8802_Maclaurin__sin__bound,axiom,
    ! [Xb: real,Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(aa_real_real(sin_real,Xb)),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_jd,Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))))),aa_real_real(times_times_real(aa_real_real(inverse_inverse_real,aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,Xb)),Nb))) ).

% Maclaurin_sin_bound
tff(fact_8803_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_8804_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_8805_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_8806_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_8807_inverse__inverse__eq,axiom,
    ! [A: complex] : aa_complex_complex(invers8013647133539491842omplex,aa_complex_complex(invers8013647133539491842omplex,A)) = A ).

% inverse_inverse_eq
tff(fact_8808_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_8809_inverse__zero,axiom,
    aa_real_real(inverse_inverse_real,zero_zero_real) = zero_zero_real ).

% inverse_zero
tff(fact_8810_inverse__zero,axiom,
    aa_complex_complex(invers8013647133539491842omplex,zero_zero_complex) = zero_zero_complex ).

% inverse_zero
tff(fact_8811_inverse__zero,axiom,
    aa_rat_rat(inverse_inverse_rat,zero_zero_rat) = zero_zero_rat ).

% inverse_zero
tff(fact_8812_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_8813_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_8814_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_8815_inverse__mult__distrib,axiom,
    ! [A: real,B: real] : aa_real_real(inverse_inverse_real,aa_real_real(times_times_real(A),B)) = aa_real_real(times_times_real(aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B)) ).

% inverse_mult_distrib
tff(fact_8816_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_8817_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_8818_inverse__eq__1__iff,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(inverse_inverse_real,Xb) = one_one_real )
    <=> ( Xb = one_one_real ) ) ).

% inverse_eq_1_iff
tff(fact_8819_inverse__eq__1__iff,axiom,
    ! [Xb: complex] :
      ( ( aa_complex_complex(invers8013647133539491842omplex,Xb) = one_one_complex )
    <=> ( Xb = one_one_complex ) ) ).

% inverse_eq_1_iff
tff(fact_8820_inverse__eq__1__iff,axiom,
    ! [Xb: rat] :
      ( ( aa_rat_rat(inverse_inverse_rat,Xb) = one_one_rat )
    <=> ( Xb = one_one_rat ) ) ).

% inverse_eq_1_iff
tff(fact_8821_inverse__1,axiom,
    aa_real_real(inverse_inverse_real,one_one_real) = one_one_real ).

% inverse_1
tff(fact_8822_inverse__1,axiom,
    aa_complex_complex(invers8013647133539491842omplex,one_one_complex) = one_one_complex ).

% inverse_1
tff(fact_8823_inverse__1,axiom,
    aa_rat_rat(inverse_inverse_rat,one_one_rat) = one_one_rat ).

% inverse_1
tff(fact_8824_atMost__iff,axiom,
    ! [I3: real,K: real] :
      ( member_real(I3,set_ord_atMost_real(K))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,I3),K) ) ).

% atMost_iff
tff(fact_8825_atMost__iff,axiom,
    ! [I3: set_nat,K: set_nat] :
      ( member_set_nat(I3,set_or4236626031148496127et_nat(K))
    <=> aa_set_nat_o(ord_less_eq_set_nat(I3),K) ) ).

% atMost_iff
tff(fact_8826_atMost__iff,axiom,
    ! [I3: rat,K: rat] :
      ( member_rat(I3,set_ord_atMost_rat(K))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,I3),K) ) ).

% atMost_iff
tff(fact_8827_atMost__iff,axiom,
    ! [I3: num,K: num] :
      ( member_num(I3,set_ord_atMost_num(K))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,I3),K) ) ).

% atMost_iff
tff(fact_8828_atMost__iff,axiom,
    ! [I3: int,K: int] :
      ( member_int(I3,set_ord_atMost_int(K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I3),K) ) ).

% atMost_iff
tff(fact_8829_atMost__iff,axiom,
    ! [I3: nat,K: nat] :
      ( aa_set_nat_o(member_nat(I3),aa_nat_set_nat(set_ord_atMost_nat,K))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I3),K) ) ).

% atMost_iff
tff(fact_8830_inverse__divide,axiom,
    ! [A: real,B: real] : aa_real_real(inverse_inverse_real,aa_real_real(divide_divide_real(A),B)) = aa_real_real(divide_divide_real(B),A) ).

% inverse_divide
tff(fact_8831_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_8832_inverse__divide,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(inverse_inverse_rat,aa_rat_rat(divide_divide_rat(A),B)) = aa_rat_rat(divide_divide_rat(B),A) ).

% inverse_divide
tff(fact_8833_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_8834_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_8835_inverse__minus__eq,axiom,
    ! [A: rat] : aa_rat_rat(inverse_inverse_rat,aa_rat_rat(uminus_uminus_rat,A)) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(inverse_inverse_rat,A)) ).

% inverse_minus_eq
tff(fact_8836_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_8837_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_8838_abs__inverse,axiom,
    ! [A: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(inverse_inverse_rat,A)) = aa_rat_rat(inverse_inverse_rat,aa_rat_rat(abs_abs_rat,A)) ).

% abs_inverse
tff(fact_8839_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_8840_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_8841_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_8842_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_8843_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_8844_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_8845_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_8846_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_8847_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_8848_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_8849_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_8850_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_8851_atMost__subset__iff,axiom,
    ! [Xb: set_nat,Yb: set_nat] :
      ( ord_le6893508408891458716et_nat(set_or4236626031148496127et_nat(Xb),set_or4236626031148496127et_nat(Yb))
    <=> aa_set_nat_o(ord_less_eq_set_nat(Xb),Yb) ) ).

% atMost_subset_iff
tff(fact_8852_atMost__subset__iff,axiom,
    ! [Xb: rat,Yb: rat] :
      ( ord_less_eq_set_rat(set_ord_atMost_rat(Xb),set_ord_atMost_rat(Yb))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Yb) ) ).

% atMost_subset_iff
tff(fact_8853_atMost__subset__iff,axiom,
    ! [Xb: num,Yb: num] :
      ( ord_less_eq_set_num(set_ord_atMost_num(Xb),set_ord_atMost_num(Yb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Yb) ) ).

% atMost_subset_iff
tff(fact_8854_atMost__subset__iff,axiom,
    ! [Xb: int,Yb: int] :
      ( ord_less_eq_set_int(set_ord_atMost_int(Xb),set_ord_atMost_int(Yb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Yb) ) ).

% atMost_subset_iff
tff(fact_8855_atMost__subset__iff,axiom,
    ! [Xb: nat,Yb: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(set_ord_atMost_nat,Xb)),aa_nat_set_nat(set_ord_atMost_nat,Yb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Yb) ) ).

% atMost_subset_iff
tff(fact_8856_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_8857_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_8858_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_8859_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_8860_left__inverse,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(times_times_real(aa_real_real(inverse_inverse_real,A)),A) = one_one_real ) ) ).

% left_inverse
tff(fact_8861_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_8862_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_8863_right__inverse,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(times_times_real(A),aa_real_real(inverse_inverse_real,A)) = one_one_real ) ) ).

% right_inverse
tff(fact_8864_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_8865_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_8866_inverse__eq__divide__numeral,axiom,
    ! [W: num] : aa_real_real(inverse_inverse_real,aa_num_real(numeral_numeral_real,W)) = aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,W)) ).

% inverse_eq_divide_numeral
tff(fact_8867_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_8868_inverse__eq__divide__numeral,axiom,
    ! [W: num] : aa_rat_rat(inverse_inverse_rat,aa_num_rat(numeral_numeral_rat,W)) = aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,W)) ).

% inverse_eq_divide_numeral
tff(fact_8869_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_8870_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_8871_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(aa_num_fun_num_o(ord_less_eq_num,L),H)
        | aa_num_o(aa_num_fun_num_o(ord_less_eq_num,H),H2) ) ) ).

% Icc_subset_Iic_iff
tff(fact_8872_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_8873_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_8874_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_8875_sum_OatMost__Suc,axiom,
    ! [G: fun_nat_rat,Nb: nat] : groups2906978787729119204at_rat(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Nb))) = aa_rat_rat(plus_plus_rat(groups2906978787729119204at_rat(G,aa_nat_set_nat(set_ord_atMost_nat,Nb))),aa_nat_rat(G,aa_nat_nat(suc,Nb))) ).

% sum.atMost_Suc
tff(fact_8876_sum_OatMost__Suc,axiom,
    ! [G: fun_nat_int,Nb: nat] : groups3539618377306564664at_int(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Nb))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,aa_nat_set_nat(set_ord_atMost_nat,Nb))),aa_nat_int(G,aa_nat_nat(suc,Nb))) ).

% sum.atMost_Suc
tff(fact_8877_sum_OatMost__Suc,axiom,
    ! [G: fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Nb))) = 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,Nb))),aa_nat_nat(G,aa_nat_nat(suc,Nb))) ).

% sum.atMost_Suc
tff(fact_8878_sum_OatMost__Suc,axiom,
    ! [G: fun_nat_real,Nb: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Nb))) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_ord_atMost_nat,Nb))),aa_nat_real(G,aa_nat_nat(suc,Nb))) ).

% sum.atMost_Suc
tff(fact_8879_prod_OatMost__Suc,axiom,
    ! [G: fun_nat_real,Nb: nat] : aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Nb))) = aa_real_real(times_times_real(aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_ord_atMost_nat,Nb))),aa_nat_real(G,aa_nat_nat(suc,Nb))) ).

% prod.atMost_Suc
tff(fact_8880_prod_OatMost__Suc,axiom,
    ! [G: fun_nat_rat,Nb: nat] : groups73079841787564623at_rat(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Nb))) = aa_rat_rat(times_times_rat(groups73079841787564623at_rat(G,aa_nat_set_nat(set_ord_atMost_nat,Nb))),aa_nat_rat(G,aa_nat_nat(suc,Nb))) ).

% prod.atMost_Suc
tff(fact_8881_prod_OatMost__Suc,axiom,
    ! [G: fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Nb))) = 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,Nb))),aa_nat_nat(G,aa_nat_nat(suc,Nb))) ).

% prod.atMost_Suc
tff(fact_8882_prod_OatMost__Suc,axiom,
    ! [G: fun_nat_int,Nb: nat] : groups705719431365010083at_int(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Nb))) = aa_int_int(aa_int_fun_int_int(times_times_int,groups705719431365010083at_int(G,aa_nat_set_nat(set_ord_atMost_nat,Nb))),aa_nat_int(G,aa_nat_nat(suc,Nb))) ).

% prod.atMost_Suc
tff(fact_8883_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_8884_inverse__eq__divide__neg__numeral,axiom,
    ! [W: num] : aa_real_real(inverse_inverse_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))) = aa_real_real(divide_divide_real(one_one_real),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))) ).

% inverse_eq_divide_neg_numeral
tff(fact_8885_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_8886_inverse__eq__divide__neg__numeral,axiom,
    ! [W: num] : aa_rat_rat(inverse_inverse_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))) = aa_rat_rat(divide_divide_rat(one_one_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))) ).

% inverse_eq_divide_neg_numeral
tff(fact_8887_int__prod,axiom,
    ! [F: fun_int_nat,A3: set_int] : aa_nat_int(semiri1314217659103216013at_int,groups1707563613775114915nt_nat(F,A3)) = groups1705073143266064639nt_int(aTP_Lamm_fk(F),A3) ).

% int_prod
tff(fact_8888_int__prod,axiom,
    ! [F: fun_nat_nat,A3: set_nat] : aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(groups708209901874060359at_nat(F),A3)) = groups705719431365010083at_int(aTP_Lamm_fm(F),A3) ).

% int_prod
tff(fact_8889_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_8890_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_8891_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_8892_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_8893_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_8894_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_8895_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_8896_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_8897_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_8898_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_8899_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_8900_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_8901_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_8902_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_8903_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_8904_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_8905_field__class_Ofield__inverse__zero,axiom,
    aa_complex_complex(invers8013647133539491842omplex,zero_zero_complex) = zero_zero_complex ).

% field_class.field_inverse_zero
tff(fact_8906_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_8907_power__inverse,axiom,
    ! [A: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(inverse_inverse_real,A)),Nb) = aa_real_real(inverse_inverse_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ).

% power_inverse
tff(fact_8908_power__inverse,axiom,
    ! [A: complex,Nb: nat] : aa_nat_complex(power_power_complex(aa_complex_complex(invers8013647133539491842omplex,A)),Nb) = aa_complex_complex(invers8013647133539491842omplex,aa_nat_complex(power_power_complex(A),Nb)) ).

% power_inverse
tff(fact_8909_power__inverse,axiom,
    ! [A: rat,Nb: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(inverse_inverse_rat,A)),Nb) = aa_rat_rat(inverse_inverse_rat,aa_nat_rat(power_power_rat(A),Nb)) ).

% power_inverse
tff(fact_8910_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_8911_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_8912_mult__commute__imp__mult__inverse__commute,axiom,
    ! [Yb: real,Xb: real] :
      ( ( aa_real_real(times_times_real(Yb),Xb) = aa_real_real(times_times_real(Xb),Yb) )
     => ( aa_real_real(times_times_real(aa_real_real(inverse_inverse_real,Yb)),Xb) = aa_real_real(times_times_real(Xb),aa_real_real(inverse_inverse_real,Yb)) ) ) ).

% mult_commute_imp_mult_inverse_commute
tff(fact_8913_mult__commute__imp__mult__inverse__commute,axiom,
    ! [Yb: complex,Xb: complex] :
      ( ( aa_complex_complex(times_times_complex(Yb),Xb) = aa_complex_complex(times_times_complex(Xb),Yb) )
     => ( aa_complex_complex(times_times_complex(aa_complex_complex(invers8013647133539491842omplex,Yb)),Xb) = aa_complex_complex(times_times_complex(Xb),aa_complex_complex(invers8013647133539491842omplex,Yb)) ) ) ).

% mult_commute_imp_mult_inverse_commute
tff(fact_8914_mult__commute__imp__mult__inverse__commute,axiom,
    ! [Yb: rat,Xb: rat] :
      ( ( aa_rat_rat(times_times_rat(Yb),Xb) = aa_rat_rat(times_times_rat(Xb),Yb) )
     => ( aa_rat_rat(times_times_rat(aa_rat_rat(inverse_inverse_rat,Yb)),Xb) = aa_rat_rat(times_times_rat(Xb),aa_rat_rat(inverse_inverse_rat,Yb)) ) ) ).

% mult_commute_imp_mult_inverse_commute
tff(fact_8915_atMost__def,axiom,
    ! [U: real] : set_ord_atMost_real(U) = collect_real(aa_real_fun_real_o(aTP_Lamm_ni,U)) ).

% atMost_def
tff(fact_8916_atMost__def,axiom,
    ! [U: set_nat] : set_or4236626031148496127et_nat(U) = collect_set_nat(aTP_Lamm_nj(U)) ).

% atMost_def
tff(fact_8917_atMost__def,axiom,
    ! [U: rat] : set_ord_atMost_rat(U) = collect_rat(aa_rat_fun_rat_o(aTP_Lamm_nk,U)) ).

% atMost_def
tff(fact_8918_atMost__def,axiom,
    ! [U: num] : set_ord_atMost_num(U) = collect_num(aa_num_fun_num_o(aTP_Lamm_nl,U)) ).

% atMost_def
tff(fact_8919_atMost__def,axiom,
    ! [U: int] : set_ord_atMost_int(U) = collect_int(aa_int_fun_int_o(aTP_Lamm_nm,U)) ).

% atMost_def
tff(fact_8920_atMost__def,axiom,
    ! [U: nat] : aa_nat_set_nat(set_ord_atMost_nat,U) = collect_nat(aa_nat_fun_nat_o(aTP_Lamm_nn,U)) ).

% atMost_def
tff(fact_8921_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_8922_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_8923_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_8924_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_8925_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_8926_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_8927_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_8928_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_8929_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_8930_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_8931_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_8932_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_8933_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_8934_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_8935_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_8936_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_8937_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(times_times_real(A),B)) = aa_real_real(times_times_real(aa_real_real(inverse_inverse_real,B)),aa_real_real(inverse_inverse_real,A)) ) ) ) ).

% nonzero_inverse_mult_distrib
tff(fact_8938_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_8939_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_8940_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_8941_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_8942_nonzero__inverse__minus__eq,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(inverse_inverse_rat,aa_rat_rat(uminus_uminus_rat,A)) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(inverse_inverse_rat,A)) ) ) ).

% nonzero_inverse_minus_eq
tff(fact_8943_inverse__numeral__1,axiom,
    aa_real_real(inverse_inverse_real,aa_num_real(numeral_numeral_real,one)) = aa_num_real(numeral_numeral_real,one) ).

% inverse_numeral_1
tff(fact_8944_inverse__numeral__1,axiom,
    aa_complex_complex(invers8013647133539491842omplex,numera6690914467698888265omplex(one)) = numera6690914467698888265omplex(one) ).

% inverse_numeral_1
tff(fact_8945_inverse__numeral__1,axiom,
    aa_rat_rat(inverse_inverse_rat,aa_num_rat(numeral_numeral_rat,one)) = aa_num_rat(numeral_numeral_rat,one) ).

% inverse_numeral_1
tff(fact_8946_inverse__unique,axiom,
    ! [A: real,B: real] :
      ( ( aa_real_real(times_times_real(A),B) = one_one_real )
     => ( aa_real_real(inverse_inverse_real,A) = B ) ) ).

% inverse_unique
tff(fact_8947_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_8948_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_8949_field__class_Ofield__divide__inverse,axiom,
    ! [A: real,B: real] : aa_real_real(divide_divide_real(A),B) = aa_real_real(times_times_real(A),aa_real_real(inverse_inverse_real,B)) ).

% field_class.field_divide_inverse
tff(fact_8950_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_8951_field__class_Ofield__divide__inverse,axiom,
    ! [A: rat,B: rat] : aa_rat_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_8952_divide__inverse,axiom,
    ! [A: real,B: real] : aa_real_real(divide_divide_real(A),B) = aa_real_real(times_times_real(A),aa_real_real(inverse_inverse_real,B)) ).

% divide_inverse
tff(fact_8953_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_8954_divide__inverse,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(divide_divide_rat(A),B) = aa_rat_rat(times_times_rat(A),aa_rat_rat(inverse_inverse_rat,B)) ).

% divide_inverse
tff(fact_8955_divide__inverse__commute,axiom,
    ! [A: real,B: real] : aa_real_real(divide_divide_real(A),B) = aa_real_real(times_times_real(aa_real_real(inverse_inverse_real,B)),A) ).

% divide_inverse_commute
tff(fact_8956_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_8957_divide__inverse__commute,axiom,
    ! [A: rat,B: rat] : aa_rat_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_8958_inverse__eq__divide,axiom,
    ! [A: real] : aa_real_real(inverse_inverse_real,A) = aa_real_real(divide_divide_real(one_one_real),A) ).

% inverse_eq_divide
tff(fact_8959_inverse__eq__divide,axiom,
    ! [A: complex] : aa_complex_complex(invers8013647133539491842omplex,A) = aa_complex_complex(divide1717551699836669952omplex(one_one_complex),A) ).

% inverse_eq_divide
tff(fact_8960_inverse__eq__divide,axiom,
    ! [A: rat] : aa_rat_rat(inverse_inverse_rat,A) = aa_rat_rat(divide_divide_rat(one_one_rat),A) ).

% inverse_eq_divide
tff(fact_8961_power__mult__inverse__distrib,axiom,
    ! [Xb: real,Mb: nat] : aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Mb)),aa_real_real(inverse_inverse_real,Xb)) = aa_real_real(times_times_real(aa_real_real(inverse_inverse_real,Xb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Mb)) ).

% power_mult_inverse_distrib
tff(fact_8962_power__mult__inverse__distrib,axiom,
    ! [Xb: complex,Mb: nat] : aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(Xb),Mb)),aa_complex_complex(invers8013647133539491842omplex,Xb)) = aa_complex_complex(times_times_complex(aa_complex_complex(invers8013647133539491842omplex,Xb)),aa_nat_complex(power_power_complex(Xb),Mb)) ).

% power_mult_inverse_distrib
tff(fact_8963_power__mult__inverse__distrib,axiom,
    ! [Xb: rat,Mb: nat] : aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(Xb),Mb)),aa_rat_rat(inverse_inverse_rat,Xb)) = aa_rat_rat(times_times_rat(aa_rat_rat(inverse_inverse_rat,Xb)),aa_nat_rat(power_power_rat(Xb),Mb)) ).

% power_mult_inverse_distrib
tff(fact_8964_power__mult__power__inverse__commute,axiom,
    ! [Xb: real,Mb: nat,Nb: nat] : aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Mb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(inverse_inverse_real,Xb)),Nb)) = aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(inverse_inverse_real,Xb)),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Mb)) ).

% power_mult_power_inverse_commute
tff(fact_8965_power__mult__power__inverse__commute,axiom,
    ! [Xb: complex,Mb: nat,Nb: nat] : aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(Xb),Mb)),aa_nat_complex(power_power_complex(aa_complex_complex(invers8013647133539491842omplex,Xb)),Nb)) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(invers8013647133539491842omplex,Xb)),Nb)),aa_nat_complex(power_power_complex(Xb),Mb)) ).

% power_mult_power_inverse_commute
tff(fact_8966_power__mult__power__inverse__commute,axiom,
    ! [Xb: rat,Mb: nat,Nb: nat] : aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(Xb),Mb)),aa_nat_rat(power_power_rat(aa_rat_rat(inverse_inverse_rat,Xb)),Nb)) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(inverse_inverse_rat,Xb)),Nb)),aa_nat_rat(power_power_rat(Xb),Mb)) ).

% power_mult_power_inverse_commute
tff(fact_8967_mult__inverse__of__nat__commute,axiom,
    ! [Xaa: nat,Xb: real] : aa_real_real(times_times_real(aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,Xaa))),Xb) = aa_real_real(times_times_real(Xb),aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,Xaa))) ).

% mult_inverse_of_nat_commute
tff(fact_8968_mult__inverse__of__nat__commute,axiom,
    ! [Xaa: nat,Xb: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(invers8013647133539491842omplex,aa_nat_complex(semiri8010041392384452111omplex,Xaa))),Xb) = aa_complex_complex(times_times_complex(Xb),aa_complex_complex(invers8013647133539491842omplex,aa_nat_complex(semiri8010041392384452111omplex,Xaa))) ).

% mult_inverse_of_nat_commute
tff(fact_8969_mult__inverse__of__nat__commute,axiom,
    ! [Xaa: nat,Xb: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(inverse_inverse_rat,aa_nat_rat(semiri681578069525770553at_rat,Xaa))),Xb) = aa_rat_rat(times_times_rat(Xb),aa_rat_rat(inverse_inverse_rat,aa_nat_rat(semiri681578069525770553at_rat,Xaa))) ).

% mult_inverse_of_nat_commute
tff(fact_8970_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_8971_nonzero__abs__inverse,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(abs_abs_rat,aa_rat_rat(inverse_inverse_rat,A)) = aa_rat_rat(inverse_inverse_rat,aa_rat_rat(abs_abs_rat,A)) ) ) ).

% nonzero_abs_inverse
tff(fact_8972_mult__inverse__of__int__commute,axiom,
    ! [Xaa: int,Xb: real] : aa_real_real(times_times_real(aa_real_real(inverse_inverse_real,aa_int_real(ring_1_of_int_real,Xaa))),Xb) = aa_real_real(times_times_real(Xb),aa_real_real(inverse_inverse_real,aa_int_real(ring_1_of_int_real,Xaa))) ).

% mult_inverse_of_int_commute
tff(fact_8973_mult__inverse__of__int__commute,axiom,
    ! [Xaa: int,Xb: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(invers8013647133539491842omplex,aa_int_complex(ring_17405671764205052669omplex,Xaa))),Xb) = aa_complex_complex(times_times_complex(Xb),aa_complex_complex(invers8013647133539491842omplex,aa_int_complex(ring_17405671764205052669omplex,Xaa))) ).

% mult_inverse_of_int_commute
tff(fact_8974_mult__inverse__of__int__commute,axiom,
    ! [Xaa: int,Xb: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(inverse_inverse_rat,aa_int_rat(ring_1_of_int_rat,Xaa))),Xb) = aa_rat_rat(times_times_rat(Xb),aa_rat_rat(inverse_inverse_rat,aa_int_rat(ring_1_of_int_rat,Xaa))) ).

% mult_inverse_of_int_commute
tff(fact_8975_atMost__atLeast0,axiom,
    ! [Nb: nat] : aa_nat_set_nat(set_ord_atMost_nat,Nb) = aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb) ).

% atMost_atLeast0
tff(fact_8976_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_8977_divide__real__def,axiom,
    ! [Xb: real,Yb: real] : aa_real_real(divide_divide_real(Xb),Yb) = aa_real_real(times_times_real(Xb),aa_real_real(inverse_inverse_real,Yb)) ).

% divide_real_def
tff(fact_8978_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_8979_not__Iic__le__Icc,axiom,
    ! [H: int,L2: int,H2: int] : ~ ord_less_eq_set_int(set_ord_atMost_int(H),set_or1266510415728281911st_int(L2,H2)) ).

% not_Iic_le_Icc
tff(fact_8980_not__Iic__le__Icc,axiom,
    ! [H: real,L2: real,H2: real] : ~ ord_less_eq_set_real(set_ord_atMost_real(H),set_or1222579329274155063t_real(L2,H2)) ).

% not_Iic_le_Icc
tff(fact_8981_prod_Otriangle__reindex__eq,axiom,
    ! [G: fun_nat_fun_nat_nat,Nb: nat] : groups4077766827762148844at_nat(produc6842872674320459806at_nat(G),collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_no(Nb)))) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_nq(G)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% prod.triangle_reindex_eq
tff(fact_8982_prod_Otriangle__reindex__eq,axiom,
    ! [G: fun_nat_fun_nat_int,Nb: nat] : groups4075276357253098568at_int(produc6840382203811409530at_int(G),collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_no(Nb)))) = groups705719431365010083at_int(aTP_Lamm_ns(G),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% prod.triangle_reindex_eq
tff(fact_8983_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_8984_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_8985_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_8986_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_8987_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_8988_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_8989_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_8990_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_8991_inverse__le__1__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(inverse_inverse_real,Xb)),one_one_real)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real)
        | aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xb) ) ) ).

% inverse_le_1_iff
tff(fact_8992_inverse__le__1__iff,axiom,
    ! [Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(inverse_inverse_rat,Xb)),one_one_rat)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),zero_zero_rat)
        | aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),Xb) ) ) ).

% inverse_le_1_iff
tff(fact_8993_one__less__inverse__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(inverse_inverse_real,Xb))
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),one_one_real) ) ) ).

% one_less_inverse_iff
tff(fact_8994_one__less__inverse__iff,axiom,
    ! [Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_rat_rat(inverse_inverse_rat,Xb))
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xb)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),one_one_rat) ) ) ).

% one_less_inverse_iff
tff(fact_8995_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_8996_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_8997_field__class_Ofield__inverse,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(times_times_real(aa_real_real(inverse_inverse_real,A)),A) = one_one_real ) ) ).

% field_class.field_inverse
tff(fact_8998_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_8999_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_9000_division__ring__inverse__add,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( ( B != zero_zero_real )
       => ( aa_real_real(plus_plus_real(aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B)) = aa_real_real(times_times_real(aa_real_real(times_times_real(aa_real_real(inverse_inverse_real,A)),aa_real_real(plus_plus_real(A),B))),aa_real_real(inverse_inverse_real,B)) ) ) ) ).

% division_ring_inverse_add
tff(fact_9001_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_9002_division__ring__inverse__add,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( ( B != zero_zero_rat )
       => ( aa_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(plus_plus_rat(A),B))),aa_rat_rat(inverse_inverse_rat,B)) ) ) ) ).

% division_ring_inverse_add
tff(fact_9003_inverse__add,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( ( B != zero_zero_real )
       => ( aa_real_real(plus_plus_real(aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B)) = aa_real_real(times_times_real(aa_real_real(times_times_real(aa_real_real(plus_plus_real(A),B)),aa_real_real(inverse_inverse_real,A))),aa_real_real(inverse_inverse_real,B)) ) ) ) ).

% inverse_add
tff(fact_9004_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_9005_inverse__add,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( ( B != zero_zero_rat )
       => ( aa_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(plus_plus_rat(A),B)),aa_rat_rat(inverse_inverse_rat,A))),aa_rat_rat(inverse_inverse_rat,B)) ) ) ) ).

% inverse_add
tff(fact_9006_division__ring__inverse__diff,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( ( B != zero_zero_real )
       => ( aa_real_real(minus_minus_real(aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B)) = aa_real_real(times_times_real(aa_real_real(times_times_real(aa_real_real(inverse_inverse_real,A)),aa_real_real(minus_minus_real(B),A))),aa_real_real(inverse_inverse_real,B)) ) ) ) ).

% division_ring_inverse_diff
tff(fact_9007_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_9008_division__ring__inverse__diff,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( ( B != zero_zero_rat )
       => ( aa_rat_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)),aa_rat_rat(minus_minus_rat(B),A))),aa_rat_rat(inverse_inverse_rat,B)) ) ) ) ).

% division_ring_inverse_diff
tff(fact_9009_nonzero__inverse__eq__divide,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(inverse_inverse_real,A) = aa_real_real(divide_divide_real(one_one_real),A) ) ) ).

% nonzero_inverse_eq_divide
tff(fact_9010_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_9011_nonzero__inverse__eq__divide,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(inverse_inverse_rat,A) = aa_rat_rat(divide_divide_rat(one_one_rat),A) ) ) ).

% nonzero_inverse_eq_divide
tff(fact_9012_prod_Otriangle__reindex,axiom,
    ! [G: fun_nat_fun_nat_nat,Nb: nat] : groups4077766827762148844at_nat(produc6842872674320459806at_nat(G),collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_nt(Nb)))) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_nq(G)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ).

% prod.triangle_reindex
tff(fact_9013_prod_Otriangle__reindex,axiom,
    ! [G: fun_nat_fun_nat_int,Nb: nat] : groups4075276357253098568at_int(produc6840382203811409530at_int(G),collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_nt(Nb)))) = groups705719431365010083at_int(aTP_Lamm_ns(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ).

% prod.triangle_reindex
tff(fact_9014_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_9015_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_9016_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(aa_num_fun_num_o(ord_less_num,A),B) ) ).

% Iic_subset_Iio_iff
tff(fact_9017_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_9018_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_9019_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_9020_sum__choose__upper,axiom,
    ! [Mb: nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_nu,Mb)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,Nb)),aa_nat_nat(suc,Mb)) ).

% sum_choose_upper
tff(fact_9021_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(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(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_9022_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_9023_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(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(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_9024_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_9025_one__le__inverse__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(inverse_inverse_real,Xb))
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real) ) ) ).

% one_le_inverse_iff
tff(fact_9026_one__le__inverse__iff,axiom,
    ! [Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_rat_rat(inverse_inverse_rat,Xb))
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xb)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),one_one_rat) ) ) ).

% one_le_inverse_iff
tff(fact_9027_inverse__less__1__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(inverse_inverse_real,Xb)),one_one_real)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real)
        | aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb) ) ) ).

% inverse_less_1_iff
tff(fact_9028_inverse__less__1__iff,axiom,
    ! [Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(inverse_inverse_rat,Xb)),one_one_rat)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),zero_zero_rat)
        | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),Xb) ) ) ).

% inverse_less_1_iff
tff(fact_9029_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_9030_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_9031_inverse__diff__inverse,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( ( B != zero_zero_real )
       => ( aa_real_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(times_times_real(aa_real_real(times_times_real(aa_real_real(inverse_inverse_real,A)),aa_real_real(minus_minus_real(A),B))),aa_real_real(inverse_inverse_real,B))) ) ) ) ).

% inverse_diff_inverse
tff(fact_9032_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_9033_inverse__diff__inverse,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( ( B != zero_zero_rat )
       => ( aa_rat_rat(minus_minus_rat(aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B)) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(minus_minus_rat(A),B))),aa_rat_rat(inverse_inverse_rat,B))) ) ) ) ).

% inverse_diff_inverse
tff(fact_9034_reals__Archimedean,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ? [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)))),Xb) ) ).

% reals_Archimedean
tff(fact_9035_reals__Archimedean,axiom,
    ! [Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xb)
     => ? [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)))),Xb) ) ).

% reals_Archimedean
tff(fact_9036_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_9037_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_9038_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_9039_sum_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_rat,Nb: nat] : groups2906978787729119204at_rat(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Nb))) = aa_rat_rat(plus_plus_rat(aa_nat_rat(G,zero_zero_nat)),groups2906978787729119204at_rat(aTP_Lamm_dw(G),aa_nat_set_nat(set_ord_atMost_nat,Nb))) ).

% sum.atMost_Suc_shift
tff(fact_9040_sum_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_int,Nb: nat] : groups3539618377306564664at_int(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Nb))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,zero_zero_nat)),groups3539618377306564664at_int(aTP_Lamm_dx(G),aa_nat_set_nat(set_ord_atMost_nat,Nb))) ).

% sum.atMost_Suc_shift
tff(fact_9041_sum_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Nb))) = 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_dm(G)),aa_nat_set_nat(set_ord_atMost_nat,Nb))) ).

% sum.atMost_Suc_shift
tff(fact_9042_sum_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_real,Nb: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Nb))) = aa_real_real(plus_plus_real(aa_nat_real(G,zero_zero_nat)),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_dn(G)),aa_nat_set_nat(set_ord_atMost_nat,Nb))) ).

% sum.atMost_Suc_shift
tff(fact_9043_sum__telescope,axiom,
    ! [F: fun_nat_rat,I3: nat] : groups2906978787729119204at_rat(aTP_Lamm_gy(F),aa_nat_set_nat(set_ord_atMost_nat,I3)) = aa_rat_rat(minus_minus_rat(aa_nat_rat(F,zero_zero_nat)),aa_nat_rat(F,aa_nat_nat(suc,I3))) ).

% sum_telescope
tff(fact_9044_sum__telescope,axiom,
    ! [F: fun_nat_int,I3: nat] : groups3539618377306564664at_int(aTP_Lamm_gz(F),aa_nat_set_nat(set_ord_atMost_nat,I3)) = 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,I3))) ).

% sum_telescope
tff(fact_9045_sum__telescope,axiom,
    ! [F: fun_nat_real,I3: nat] : aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_ha(F)),aa_nat_set_nat(set_ord_atMost_nat,I3)) = aa_real_real(minus_minus_real(aa_nat_real(F,zero_zero_nat)),aa_nat_real(F,aa_nat_nat(suc,I3))) ).

% sum_telescope
tff(fact_9046_polyfun__eq__coeffs,axiom,
    ! [C: fun_nat_complex,Nb: nat,D: fun_nat_complex] :
      ( ! [X4: complex] : aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_il(C),X4)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_il(D),X4)),aa_nat_set_nat(set_ord_atMost_nat,Nb))
    <=> ! [I5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I5),Nb)
         => ( aa_nat_complex(C,I5) = aa_nat_complex(D,I5) ) ) ) ).

% polyfun_eq_coeffs
tff(fact_9047_polyfun__eq__coeffs,axiom,
    ! [C: fun_nat_real,Nb: nat,D: fun_nat_real] :
      ( ! [X4: real] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ij(C),X4)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ij(D),X4)),aa_nat_set_nat(set_ord_atMost_nat,Nb))
    <=> ! [I5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I5),Nb)
         => ( aa_nat_real(C,I5) = aa_nat_real(D,I5) ) ) ) ).

% polyfun_eq_coeffs
tff(fact_9048_bounded__imp__summable,axiom,
    ! [A: fun_nat_int,B4: 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))),B4)
       => summable_int(A) ) ) ).

% bounded_imp_summable
tff(fact_9049_bounded__imp__summable,axiom,
    ! [A: fun_nat_nat,B4: 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))),B4)
       => summable_nat(A) ) ) ).

% bounded_imp_summable
tff(fact_9050_bounded__imp__summable,axiom,
    ! [A: fun_nat_real,B4: 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))),B4)
       => summable_real(A) ) ) ).

% bounded_imp_summable
tff(fact_9051_prod_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_real,Nb: nat] : aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Nb))) = aa_real_real(times_times_real(aa_nat_real(G,zero_zero_nat)),aa_set_nat_real(groups129246275422532515t_real(aTP_Lamm_dn(G)),aa_nat_set_nat(set_ord_atMost_nat,Nb))) ).

% prod.atMost_Suc_shift
tff(fact_9052_prod_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_rat,Nb: nat] : groups73079841787564623at_rat(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Nb))) = aa_rat_rat(times_times_rat(aa_nat_rat(G,zero_zero_nat)),groups73079841787564623at_rat(aTP_Lamm_dw(G),aa_nat_set_nat(set_ord_atMost_nat,Nb))) ).

% prod.atMost_Suc_shift
tff(fact_9053_prod_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Nb))) = 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_dm(G)),aa_nat_set_nat(set_ord_atMost_nat,Nb))) ).

% prod.atMost_Suc_shift
tff(fact_9054_prod_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_int,Nb: nat] : groups705719431365010083at_int(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Nb))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(G,zero_zero_nat)),groups705719431365010083at_int(aTP_Lamm_dx(G),aa_nat_set_nat(set_ord_atMost_nat,Nb))) ).

% prod.atMost_Suc_shift
tff(fact_9055_prod__int__plus__eq,axiom,
    ! [I3: nat,J: nat] : groups705719431365010083at_int(semiri1314217659103216013at_int,aa_nat_set_nat(set_or1269000886237332187st_nat(I3),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I3),J))) = groups1705073143266064639nt_int(aTP_Lamm_as,set_or1266510415728281911st_int(aa_nat_int(semiri1314217659103216013at_int,I3),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I3),J)))) ).

% prod_int_plus_eq
tff(fact_9056_summable__exp,axiom,
    ! [Xb: complex] : summable_complex(aTP_Lamm_nv(Xb)) ).

% summable_exp
tff(fact_9057_summable__exp,axiom,
    ! [Xb: real] : summable_real(aa_real_fun_nat_real(aTP_Lamm_nw,Xb)) ).

% summable_exp
tff(fact_9058_sum_Onested__swap_H,axiom,
    ! [A: fun_nat_fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_nx(A)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_nz(A),Nb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ).

% sum.nested_swap'
tff(fact_9059_sum_Onested__swap_H,axiom,
    ! [A: fun_nat_fun_nat_real,Nb: nat] : aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_oa(A)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_oc(A),Nb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ).

% sum.nested_swap'
tff(fact_9060_prod_Onested__swap_H,axiom,
    ! [A: fun_nat_fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_od(A)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_oe(A),Nb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ).

% prod.nested_swap'
tff(fact_9061_prod_Onested__swap_H,axiom,
    ! [A: fun_nat_fun_nat_int,Nb: nat] : groups705719431365010083at_int(aTP_Lamm_of(A),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_oh(A),Nb),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ).

% prod.nested_swap'
tff(fact_9062_sum__choose__lower,axiom,
    ! [R2: nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_oi,R2)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,R2),Nb))),Nb) ).

% sum_choose_lower
tff(fact_9063_choose__rising__sum_I1_J,axiom,
    ! [Nb: nat,Mb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_oj,Nb)),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = 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,Nb),Mb)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)) ).

% choose_rising_sum(1)
tff(fact_9064_choose__rising__sum_I2_J,axiom,
    ! [Nb: nat,Mb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_oj,Nb)),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = 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,Nb),Mb)),one_one_nat)),Mb) ).

% choose_rising_sum(2)
tff(fact_9065_ex__inverse__of__nat__less,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ? [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))),Xb) ) ) ).

% ex_inverse_of_nat_less
tff(fact_9066_ex__inverse__of__nat__less,axiom,
    ! [Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xb)
     => ? [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))),Xb) ) ) ).

% ex_inverse_of_nat_less
tff(fact_9067_power__diff__conv__inverse,axiom,
    ! [Xb: real,Mb: nat,Nb: nat] :
      ( ( Xb != zero_zero_real )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)) = aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(inverse_inverse_real,Xb)),Mb)) ) ) ) ).

% power_diff_conv_inverse
tff(fact_9068_power__diff__conv__inverse,axiom,
    ! [Xb: complex,Mb: nat,Nb: nat] :
      ( ( Xb != zero_zero_complex )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
       => ( aa_nat_complex(power_power_complex(Xb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(Xb),Nb)),aa_nat_complex(power_power_complex(aa_complex_complex(invers8013647133539491842omplex,Xb)),Mb)) ) ) ) ).

% power_diff_conv_inverse
tff(fact_9069_power__diff__conv__inverse,axiom,
    ! [Xb: rat,Mb: nat,Nb: nat] :
      ( ( Xb != zero_zero_rat )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
       => ( aa_nat_rat(power_power_rat(Xb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(Xb),Nb)),aa_nat_rat(power_power_rat(aa_rat_rat(inverse_inverse_rat,Xb)),Mb)) ) ) ) ).

% power_diff_conv_inverse
tff(fact_9070_zero__polynom__imp__zero__coeffs,axiom,
    ! [C: fun_nat_complex,Nb: nat,K: nat] :
      ( ! [W2: complex] : aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_il(C),W2)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_zero_complex
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
       => ( aa_nat_complex(C,K) = zero_zero_complex ) ) ) ).

% zero_polynom_imp_zero_coeffs
tff(fact_9071_zero__polynom__imp__zero__coeffs,axiom,
    ! [C: fun_nat_real,Nb: nat,K: nat] :
      ( ! [W2: real] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ij(C),W2)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_zero_real
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
       => ( aa_nat_real(C,K) = zero_zero_real ) ) ) ).

% zero_polynom_imp_zero_coeffs
tff(fact_9072_polyfun__eq__0,axiom,
    ! [C: fun_nat_complex,Nb: nat] :
      ( ! [X4: complex] : aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_il(C),X4)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_zero_complex
    <=> ! [I5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I5),Nb)
         => ( aa_nat_complex(C,I5) = zero_zero_complex ) ) ) ).

% polyfun_eq_0
tff(fact_9073_polyfun__eq__0,axiom,
    ! [C: fun_nat_real,Nb: nat] :
      ( ! [X4: real] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ij(C),X4)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_zero_real
    <=> ! [I5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I5),Nb)
         => ( aa_nat_real(C,I5) = zero_zero_real ) ) ) ).

% polyfun_eq_0
tff(fact_9074_sum_OatMost__shift,axiom,
    ! [G: fun_nat_rat,Nb: nat] : groups2906978787729119204at_rat(G,aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_rat_rat(plus_plus_rat(aa_nat_rat(G,zero_zero_nat)),groups2906978787729119204at_rat(aTP_Lamm_dw(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% sum.atMost_shift
tff(fact_9075_sum_OatMost__shift,axiom,
    ! [G: fun_nat_int,Nb: nat] : groups3539618377306564664at_int(G,aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,zero_zero_nat)),groups3539618377306564664at_int(aTP_Lamm_dx(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% sum.atMost_shift
tff(fact_9076_sum_OatMost__shift,axiom,
    ! [G: fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = 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_dm(G)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% sum.atMost_shift
tff(fact_9077_sum_OatMost__shift,axiom,
    ! [G: fun_nat_real,Nb: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_real_real(plus_plus_real(aa_nat_real(G,zero_zero_nat)),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_dn(G)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% sum.atMost_shift
tff(fact_9078_sum__up__index__split,axiom,
    ! [F: fun_nat_rat,Mb: nat,Nb: nat] : groups2906978787729119204at_rat(F,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb))) = aa_rat_rat(plus_plus_rat(groups2906978787729119204at_rat(F,aa_nat_set_nat(set_ord_atMost_nat,Mb))),groups2906978787729119204at_rat(F,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)))) ).

% sum_up_index_split
tff(fact_9079_sum__up__index__split,axiom,
    ! [F: fun_nat_int,Mb: nat,Nb: nat] : groups3539618377306564664at_int(F,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(F,aa_nat_set_nat(set_ord_atMost_nat,Mb))),groups3539618377306564664at_int(F,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)))) ).

% sum_up_index_split
tff(fact_9080_sum__up__index__split,axiom,
    ! [F: fun_nat_nat,Mb: nat,Nb: 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,Mb),Nb))) = 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,Mb))),aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)))) ).

% sum_up_index_split
tff(fact_9081_sum__up__index__split,axiom,
    ! [F: fun_nat_real,Mb: nat,Nb: 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,Mb),Nb))) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_ord_atMost_nat,Mb))),aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)))) ).

% sum_up_index_split
tff(fact_9082_prod_OatMost__shift,axiom,
    ! [G: fun_nat_real,Nb: nat] : aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_real_real(times_times_real(aa_nat_real(G,zero_zero_nat)),aa_set_nat_real(groups129246275422532515t_real(aTP_Lamm_dn(G)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% prod.atMost_shift
tff(fact_9083_prod_OatMost__shift,axiom,
    ! [G: fun_nat_rat,Nb: nat] : groups73079841787564623at_rat(G,aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_rat_rat(times_times_rat(aa_nat_rat(G,zero_zero_nat)),groups73079841787564623at_rat(aTP_Lamm_dw(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% prod.atMost_shift
tff(fact_9084_prod_OatMost__shift,axiom,
    ! [G: fun_nat_nat,Nb: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = 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_dm(G)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% prod.atMost_shift
tff(fact_9085_prod_OatMost__shift,axiom,
    ! [G: fun_nat_int,Nb: nat] : groups705719431365010083at_int(G,aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(G,zero_zero_nat)),groups705719431365010083at_int(aTP_Lamm_dx(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% prod.atMost_shift
tff(fact_9086_atLeast1__atMost__eq__remove0,axiom,
    ! [Nb: nat] : aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),Nb) = minus_minus_set_nat(aa_nat_set_nat(set_ord_atMost_nat,Nb),insert_nat(zero_zero_nat,bot_bot_set_nat)) ).

% atLeast1_atMost_eq_remove0
tff(fact_9087_gbinomial__parallel__sum,axiom,
    ! [A: complex,Nb: nat] : aa_set_nat_complex(groups2073611262835488442omplex(aTP_Lamm_ok(A)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_nat_complex(gbinomial_complex(aa_complex_complex(plus_plus_complex(aa_complex_complex(plus_plus_complex(A),aa_nat_complex(semiri8010041392384452111omplex,Nb))),one_one_complex)),Nb) ).

% gbinomial_parallel_sum
tff(fact_9088_gbinomial__parallel__sum,axiom,
    ! [A: rat,Nb: nat] : groups2906978787729119204at_rat(aTP_Lamm_ol(A),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_nat_rat(gbinomial_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(A),aa_nat_rat(semiri681578069525770553at_rat,Nb))),one_one_rat)),Nb) ).

% gbinomial_parallel_sum
tff(fact_9089_gbinomial__parallel__sum,axiom,
    ! [A: real,Nb: nat] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_om,A)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(plus_plus_real(aa_real_real(plus_plus_real(A),aa_nat_real(semiri5074537144036343181t_real,Nb))),one_one_real)),Nb) ).

% gbinomial_parallel_sum
tff(fact_9090_sum_Otriangle__reindex__eq,axiom,
    ! [G: fun_nat_fun_nat_nat,Nb: nat] : groups977919841031483927at_nat(produc6842872674320459806at_nat(G),collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_no(Nb)))) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_on(G)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% sum.triangle_reindex_eq
tff(fact_9091_sum_Otriangle__reindex__eq,axiom,
    ! [G: fun_nat_fun_nat_real,Nb: nat] : groups4567486121110086003t_real(produc1703576794950452218t_real(G),collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_no(Nb)))) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_op(G)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% sum.triangle_reindex_eq
tff(fact_9092_sum__choose__diagonal,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_oq(Mb),Nb)),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,Nb)),Mb) ) ) ).

% sum_choose_diagonal
tff(fact_9093_vandermonde,axiom,
    ! [Mb: nat,Nb: nat,R2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_or(Mb),Nb),R2)),aa_nat_set_nat(set_ord_atMost_nat,R2)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)),R2) ).

% vandermonde
tff(fact_9094_sum__gp__basic,axiom,
    ! [Xb: complex,Nb: nat] : aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(one_one_complex),Xb)),aa_set_nat_complex(groups2073611262835488442omplex(power_power_complex(Xb)),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(power_power_complex(Xb),aa_nat_nat(suc,Nb))) ).

% sum_gp_basic
tff(fact_9095_sum__gp__basic,axiom,
    ! [Xb: rat,Nb: nat] : aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(one_one_rat),Xb)),groups2906978787729119204at_rat(power_power_rat(Xb),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_rat_rat(minus_minus_rat(one_one_rat),aa_nat_rat(power_power_rat(Xb),aa_nat_nat(suc,Nb))) ).

% sum_gp_basic
tff(fact_9096_sum__gp__basic,axiom,
    ! [Xb: int,Nb: 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),Xb)),groups3539618377306564664at_int(power_power_int(Xb),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),aa_nat_int(power_power_int(Xb),aa_nat_nat(suc,Nb))) ).

% sum_gp_basic
tff(fact_9097_sum__gp__basic,axiom,
    ! [Xb: real,Nb: nat] : aa_real_real(times_times_real(aa_real_real(minus_minus_real(one_one_real),Xb)),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,Xb)),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_real_real(minus_minus_real(one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_nat_nat(suc,Nb))) ).

% sum_gp_basic
tff(fact_9098_exp__plus__inverse__exp,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(plus_plus_real(aa_real_real(exp_real,Xb)),aa_real_real(inverse_inverse_real,aa_real_real(exp_real,Xb)))) ).

% exp_plus_inverse_exp
tff(fact_9099_polyfun__linear__factor__root,axiom,
    ! [C: fun_nat_complex,A: complex,Nb: nat] :
      ( ( aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_il(C),A)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_zero_complex )
     => ~ ! [B3: fun_nat_complex] :
            ~ ! [Z3: complex] : aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_il(C),Z3)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(Z3),A)),aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_il(B3),Z3)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ) ).

% polyfun_linear_factor_root
tff(fact_9100_polyfun__linear__factor__root,axiom,
    ! [C: fun_nat_rat,A: rat,Nb: nat] :
      ( ( groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_os(C),A),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_zero_rat )
     => ~ ! [B3: fun_nat_rat] :
            ~ ! [Z3: rat] : groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_os(C),Z3),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(Z3),A)),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_os(B3),Z3),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ) ).

% polyfun_linear_factor_root
tff(fact_9101_polyfun__linear__factor__root,axiom,
    ! [C: fun_nat_int,A: int,Nb: nat] :
      ( ( groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_ot(C),A),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_zero_int )
     => ~ ! [B3: fun_nat_int] :
            ~ ! [Z3: int] : groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_ot(C),Z3),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = 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_ot(B3),Z3),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ) ).

% polyfun_linear_factor_root
tff(fact_9102_polyfun__linear__factor__root,axiom,
    ! [C: fun_nat_real,A: real,Nb: nat] :
      ( ( aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ij(C),A)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = zero_zero_real )
     => ~ ! [B3: fun_nat_real] :
            ~ ! [Z3: real] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ij(C),Z3)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_real_real(times_times_real(aa_real_real(minus_minus_real(Z3),A)),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ij(B3),Z3)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ) ).

% polyfun_linear_factor_root
tff(fact_9103_polyfun__linear__factor,axiom,
    ! [C: fun_nat_complex,Nb: nat,A: complex] :
    ? [B3: fun_nat_complex] :
    ! [Z3: complex] : aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_il(C),Z3)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = 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_il(B3),Z3)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)))),aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_il(C),A)),aa_nat_set_nat(set_ord_atMost_nat,Nb))) ).

% polyfun_linear_factor
tff(fact_9104_polyfun__linear__factor,axiom,
    ! [C: fun_nat_rat,Nb: nat,A: rat] :
    ? [B3: fun_nat_rat] :
    ! [Z3: rat] : groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_os(C),Z3),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(Z3),A)),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_os(B3),Z3),aa_nat_set_nat(set_ord_lessThan_nat,Nb)))),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_os(C),A),aa_nat_set_nat(set_ord_atMost_nat,Nb))) ).

% polyfun_linear_factor
tff(fact_9105_polyfun__linear__factor,axiom,
    ! [C: fun_nat_int,Nb: nat,A: int] :
    ? [B3: fun_nat_int] :
    ! [Z3: int] : groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_ot(C),Z3),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = 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_ot(B3),Z3),aa_nat_set_nat(set_ord_lessThan_nat,Nb)))),groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_ot(C),A),aa_nat_set_nat(set_ord_atMost_nat,Nb))) ).

% polyfun_linear_factor
tff(fact_9106_polyfun__linear__factor,axiom,
    ! [C: fun_nat_real,Nb: nat,A: real] :
    ? [B3: fun_nat_real] :
    ! [Z3: real] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ij(C),Z3)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_real_real(minus_minus_real(Z3),A)),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ij(B3),Z3)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)))),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ij(C),A)),aa_nat_set_nat(set_ord_atMost_nat,Nb))) ).

% polyfun_linear_factor
tff(fact_9107_sum__power__shift,axiom,
    ! [Mb: nat,Nb: nat,Xb: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_set_nat_complex(groups2073611262835488442omplex(power_power_complex(Xb)),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(Xb),Mb)),aa_set_nat_complex(groups2073611262835488442omplex(power_power_complex(Xb)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)))) ) ) ).

% sum_power_shift
tff(fact_9108_sum__power__shift,axiom,
    ! [Mb: nat,Nb: nat,Xb: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( groups2906978787729119204at_rat(power_power_rat(Xb),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(Xb),Mb)),groups2906978787729119204at_rat(power_power_rat(Xb),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)))) ) ) ).

% sum_power_shift
tff(fact_9109_sum__power__shift,axiom,
    ! [Mb: nat,Nb: nat,Xb: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( groups3539618377306564664at_int(power_power_int(Xb),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(Xb),Mb)),groups3539618377306564664at_int(power_power_int(Xb),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)))) ) ) ).

% sum_power_shift
tff(fact_9110_sum__power__shift,axiom,
    ! [Mb: nat,Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,Xb)),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) = aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Mb)),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,Xb)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)))) ) ) ).

% sum_power_shift
tff(fact_9111_sum_Otriangle__reindex,axiom,
    ! [G: fun_nat_fun_nat_nat,Nb: nat] : groups977919841031483927at_nat(produc6842872674320459806at_nat(G),collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_nt(Nb)))) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_on(G)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ).

% sum.triangle_reindex
tff(fact_9112_sum_Otriangle__reindex,axiom,
    ! [G: fun_nat_fun_nat_real,Nb: nat] : groups4567486121110086003t_real(produc1703576794950452218t_real(G),collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_nt(Nb)))) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_op(G)),aa_nat_set_nat(set_ord_lessThan_nat,Nb)) ).

% sum.triangle_reindex
tff(fact_9113_summable__Cauchy__product,axiom,
    ! [A: fun_nat_complex,B: fun_nat_complex] :
      ( summable_real(aTP_Lamm_is(A))
     => ( summable_real(aTP_Lamm_is(B))
       => summable_complex(aa_fun2955221450537884980omplex(aTP_Lamm_ov(A),B)) ) ) ).

% summable_Cauchy_product
tff(fact_9114_summable__Cauchy__product,axiom,
    ! [A: fun_nat_real,B: fun_nat_real] :
      ( summable_real(aTP_Lamm_ow(A))
     => ( summable_real(aTP_Lamm_ow(B))
       => summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_oy(A),B)) ) ) ).

% summable_Cauchy_product
tff(fact_9115_choose__row__sum,axiom,
    ! [Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(binomial,Nb)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) ).

% choose_row_sum
tff(fact_9116_Cauchy__product,axiom,
    ! [A: fun_nat_complex,B: fun_nat_complex] :
      ( summable_real(aTP_Lamm_is(A))
     => ( summable_real(aTP_Lamm_is(B))
       => ( aa_complex_complex(times_times_complex(suminf_complex(A)),suminf_complex(B)) = suminf_complex(aa_fun2955221450537884980omplex(aTP_Lamm_ov(A),B)) ) ) ) ).

% Cauchy_product
tff(fact_9117_Cauchy__product,axiom,
    ! [A: fun_nat_real,B: fun_nat_real] :
      ( summable_real(aTP_Lamm_ow(A))
     => ( summable_real(aTP_Lamm_ow(B))
       => ( aa_real_real(times_times_real(suminf_real(A)),suminf_real(B)) = suminf_real(aa_fun1839206346889086768t_real(aTP_Lamm_oy(A),B)) ) ) ) ).

% Cauchy_product
tff(fact_9118_binomial,axiom,
    ! [A: nat,B: nat,Nb: 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)),Nb) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_oz(A),B),Nb)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% binomial
tff(fact_9119_plus__inverse__ge__2,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(plus_plus_real(Xb),aa_real_real(inverse_inverse_real,Xb))) ) ).

% plus_inverse_ge_2
tff(fact_9120_real__inv__sqrt__pow2,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(inverse_inverse_real,aa_real_real(sqrt,Xb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(inverse_inverse_real,Xb) ) ) ).

% real_inv_sqrt_pow2
tff(fact_9121_sum_Oin__pairs__0,axiom,
    ! [G: fun_nat_rat,Nb: 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))),Nb)))) = groups2906978787729119204at_rat(aTP_Lamm_ej(G),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% sum.in_pairs_0
tff(fact_9122_sum_Oin__pairs__0,axiom,
    ! [G: fun_nat_int,Nb: 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))),Nb)))) = groups3539618377306564664at_int(aTP_Lamm_ek(G),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% sum.in_pairs_0
tff(fact_9123_sum_Oin__pairs__0,axiom,
    ! [G: fun_nat_nat,Nb: 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))),Nb)))) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_el(G)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% sum.in_pairs_0
tff(fact_9124_sum_Oin__pairs__0,axiom,
    ! [G: fun_nat_real,Nb: 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))),Nb)))) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_em(G)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% sum.in_pairs_0
tff(fact_9125_polynomial__product,axiom,
    ! [Mb: nat,A: fun_nat_complex,Nb: nat,B: fun_nat_complex,Xb: complex] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),I2)
         => ( aa_nat_complex(A,I2) = zero_zero_complex ) )
     => ( ! [J2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),J2)
           => ( aa_nat_complex(B,J2) = zero_zero_complex ) )
       => ( aa_complex_complex(times_times_complex(aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_il(A),Xb)),aa_nat_set_nat(set_ord_atMost_nat,Mb))),aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_il(B),Xb)),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_fun3721018923249493343omplex(aTP_Lamm_pa(A),B),Xb)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb))) ) ) ) ).

% polynomial_product
tff(fact_9126_polynomial__product,axiom,
    ! [Mb: nat,A: fun_nat_rat,Nb: nat,B: fun_nat_rat,Xb: rat] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),I2)
         => ( aa_nat_rat(A,I2) = zero_zero_rat ) )
     => ( ! [J2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),J2)
           => ( aa_nat_rat(B,J2) = zero_zero_rat ) )
       => ( aa_rat_rat(times_times_rat(groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_os(A),Xb),aa_nat_set_nat(set_ord_atMost_nat,Mb))),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_os(B),Xb),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_fun5559679822142647773at_rat(aTP_Lamm_pc(A),B),Xb),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb))) ) ) ) ).

% polynomial_product
tff(fact_9127_polynomial__product,axiom,
    ! [Mb: nat,A: fun_nat_int,Nb: nat,B: fun_nat_int,Xb: int] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),I2)
         => ( aa_nat_int(A,I2) = zero_zero_int ) )
     => ( ! [J2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),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_ot(A),Xb),aa_nat_set_nat(set_ord_atMost_nat,Mb))),groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_ot(B),Xb),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = groups3539618377306564664at_int(aa_int_fun_nat_int(aa_fun2315716387381795033at_int(aTP_Lamm_pe(A),B),Xb),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb))) ) ) ) ).

% polynomial_product
tff(fact_9128_polynomial__product,axiom,
    ! [Mb: nat,A: fun_nat_real,Nb: nat,B: fun_nat_real,Xb: real] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),I2)
         => ( aa_nat_real(A,I2) = zero_zero_real ) )
     => ( ! [J2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),J2)
           => ( aa_nat_real(B,J2) = zero_zero_real ) )
       => ( aa_real_real(times_times_real(aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ij(A),Xb)),aa_nat_set_nat(set_ord_atMost_nat,Mb))),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ij(B),Xb)),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_fun4732485501138722137t_real(aTP_Lamm_pf(A),B),Xb)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb))) ) ) ) ).

% polynomial_product
tff(fact_9129_prod_Oin__pairs__0,axiom,
    ! [G: fun_nat_real,Nb: 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))),Nb)))) = aa_set_nat_real(groups129246275422532515t_real(aTP_Lamm_mo(G)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% prod.in_pairs_0
tff(fact_9130_prod_Oin__pairs__0,axiom,
    ! [G: fun_nat_rat,Nb: 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))),Nb)))) = groups73079841787564623at_rat(aTP_Lamm_mp(G),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% prod.in_pairs_0
tff(fact_9131_prod_Oin__pairs__0,axiom,
    ! [G: fun_nat_nat,Nb: 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))),Nb)))) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_mq(G)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% prod.in_pairs_0
tff(fact_9132_prod_Oin__pairs__0,axiom,
    ! [G: fun_nat_int,Nb: 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))),Nb)))) = groups705719431365010083at_int(aTP_Lamm_mr(G),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% prod.in_pairs_0
tff(fact_9133_polyfun__eq__const,axiom,
    ! [C: fun_nat_complex,Nb: nat,K: complex] :
      ( ! [X4: complex] : aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_il(C),X4)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = 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),Nb))
           => ( aa_nat_complex(C,X4) = zero_zero_complex ) ) ) ) ).

% polyfun_eq_const
tff(fact_9134_polyfun__eq__const,axiom,
    ! [C: fun_nat_real,Nb: nat,K: real] :
      ( ! [X4: real] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ij(C),X4)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = 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),Nb))
           => ( aa_nat_real(C,X4) = zero_zero_real ) ) ) ) ).

% polyfun_eq_const
tff(fact_9135_gbinomial__sum__lower__neg,axiom,
    ! [A: complex,Mb: nat] : aa_set_nat_complex(groups2073611262835488442omplex(aTP_Lamm_pg(A)),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Mb)),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(A),one_one_complex)),Mb)) ).

% gbinomial_sum_lower_neg
tff(fact_9136_gbinomial__sum__lower__neg,axiom,
    ! [A: rat,Mb: nat] : groups2906978787729119204at_rat(aTP_Lamm_ph(A),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Mb)),aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(A),one_one_rat)),Mb)) ).

% gbinomial_sum_lower_neg
tff(fact_9137_gbinomial__sum__lower__neg,axiom,
    ! [A: real,Mb: nat] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_pi,A)),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Mb)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(minus_minus_real(A),one_one_real)),Mb)) ).

% gbinomial_sum_lower_neg
tff(fact_9138_tan__cot,axiom,
    ! [Xb: real] : aa_real_real(tan_real,aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),Xb)) = aa_real_real(inverse_inverse_real,aa_real_real(tan_real,Xb)) ).

% tan_cot
tff(fact_9139_binomial__ring,axiom,
    ! [A: complex,B: complex,Nb: nat] : aa_nat_complex(power_power_complex(aa_complex_complex(plus_plus_complex(A),B)),Nb) = aa_set_nat_complex(groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_pj(A),B),Nb)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% binomial_ring
tff(fact_9140_binomial__ring,axiom,
    ! [A: int,B: int,Nb: nat] : aa_nat_int(power_power_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),Nb) = groups3539618377306564664at_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_pk(A),B),Nb),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% binomial_ring
tff(fact_9141_binomial__ring,axiom,
    ! [A: rat,B: rat,Nb: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(plus_plus_rat(A),B)),Nb) = groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_pl(A),B),Nb),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% binomial_ring
tff(fact_9142_binomial__ring,axiom,
    ! [A: nat,B: nat,Nb: 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)),Nb) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_oz(A),B),Nb)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% binomial_ring
tff(fact_9143_binomial__ring,axiom,
    ! [A: real,B: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(plus_plus_real(A),B)),Nb) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_pm(A),B),Nb)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% binomial_ring
tff(fact_9144_polynomial__product__nat,axiom,
    ! [Mb: nat,A: fun_nat_nat,Nb: nat,B: fun_nat_nat,Xb: nat] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),I2)
         => ( aa_nat_nat(A,I2) = zero_zero_nat ) )
     => ( ! [J2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),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_pn(A),Xb)),aa_nat_set_nat(set_ord_atMost_nat,Mb))),aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_pn(B),Xb)),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_fun6650658898577385413at_nat(aTP_Lamm_pp(A),B),Xb)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb))) ) ) ) ).

% polynomial_product_nat
tff(fact_9145_choose__square__sum,axiom,
    ! [Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_pq,Nb)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = 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))),Nb)),Nb) ).

% choose_square_sum
tff(fact_9146_real__le__x__sinh,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(exp_real,Xb)),aa_real_real(inverse_inverse_real,aa_real_real(exp_real,Xb)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ).

% real_le_x_sinh
tff(fact_9147_real__le__abs__sinh,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),aa_real_real(abs_abs_real,aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(exp_real,Xb)),aa_real_real(inverse_inverse_real,aa_real_real(exp_real,Xb)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% real_le_abs_sinh
tff(fact_9148_binomial__r__part__sum,axiom,
    ! [Mb: 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))),Mb)),one_one_nat))),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = 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))),Mb)) ).

% binomial_r_part_sum
tff(fact_9149_choose__linear__sum,axiom,
    ! [Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_pr,Nb)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),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,Nb),one_one_nat))) ).

% choose_linear_sum
tff(fact_9150_of__nat__id,axiom,
    ! [Nb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,Nb) = Nb ).

% of_nat_id
tff(fact_9151_divide__complex__def,axiom,
    ! [Xb: complex,Yb: complex] : aa_complex_complex(divide1717551699836669952omplex(Xb),Yb) = aa_complex_complex(times_times_complex(Xb),aa_complex_complex(invers8013647133539491842omplex,Yb)) ).

% divide_complex_def
tff(fact_9152_real__scaleR__def,axiom,
    ! [A: real,Xb: real] : aa_real_real(real_V1485227260804924795R_real(A),Xb) = aa_real_real(times_times_real(A),Xb) ).

% real_scaleR_def
tff(fact_9153_complex__scaleR,axiom,
    ! [R2: real,A: real,B: real] : aa_complex_complex(real_V2046097035970521341omplex(R2),aa_real_complex(complex2(A),B)) = aa_real_complex(complex2(aa_real_real(times_times_real(R2),A)),aa_real_real(times_times_real(R2),B)) ).

% complex_scaleR
tff(fact_9154_complex__inverse,axiom,
    ! [A: real,B: real] : aa_complex_complex(invers8013647133539491842omplex,aa_real_complex(complex2(A),B)) = aa_real_complex(complex2(aa_real_real(divide_divide_real(A),aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(divide_divide_real(aa_real_real(uminus_uminus_real,B)),aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% complex_inverse
tff(fact_9155_i__even__power,axiom,
    ! [Nb: nat] : aa_nat_complex(power_power_complex(imaginary_unit),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Nb) ).

% i_even_power
tff(fact_9156_log__base__10__eq1,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),Xb) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(ln_ln_real,aa_real_real(exp_real,one_one_real))),aa_real_real(ln_ln_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))))),aa_real_real(ln_ln_real,Xb)) ) ) ).

% log_base_10_eq1
tff(fact_9157_log__one,axiom,
    ! [A: real] : aa_real_real(log(A),one_one_real) = zero_zero_real ).

% log_one
tff(fact_9158_norm__ii,axiom,
    aa_complex_real(real_V1022390504157884413omplex,imaginary_unit) = one_one_real ).

% norm_ii
tff(fact_9159_complex__i__mult__minus,axiom,
    ! [Xb: complex] : aa_complex_complex(times_times_complex(imaginary_unit),aa_complex_complex(times_times_complex(imaginary_unit),Xb)) = aa_complex_complex(uminus1482373934393186551omplex,Xb) ).

% complex_i_mult_minus
tff(fact_9160_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(log(A),A) = one_one_real ) ) ) ).

% log_eq_one
tff(fact_9161_log__less__cancel__iff,axiom,
    ! [A: real,Xb: real,Yb: 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),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Yb)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(log(A),Xb)),aa_real_real(log(A),Yb))
          <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Yb) ) ) ) ) ).

% log_less_cancel_iff
tff(fact_9162_log__less__one__cancel__iff,axiom,
    ! [A: real,Xb: 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),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(log(A),Xb)),one_one_real)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),A) ) ) ) ).

% log_less_one_cancel_iff
tff(fact_9163_one__less__log__cancel__iff,axiom,
    ! [A: real,Xb: 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),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(log(A),Xb))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),Xb) ) ) ) ).

% one_less_log_cancel_iff
tff(fact_9164_log__less__zero__cancel__iff,axiom,
    ! [A: real,Xb: 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),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(log(A),Xb)),zero_zero_real)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),one_one_real) ) ) ) ).

% log_less_zero_cancel_iff
tff(fact_9165_zero__less__log__cancel__iff,axiom,
    ! [A: real,Xb: 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),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(log(A),Xb))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb) ) ) ) ).

% zero_less_log_cancel_iff
tff(fact_9166_divide__numeral__i,axiom,
    ! [Z: complex,Nb: num] : aa_complex_complex(divide1717551699836669952omplex(Z),aa_complex_complex(times_times_complex(numera6690914467698888265omplex(Nb)),imaginary_unit)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(times_times_complex(imaginary_unit),Z))),numera6690914467698888265omplex(Nb)) ).

% divide_numeral_i
tff(fact_9167_divide__i,axiom,
    ! [Xb: complex] : aa_complex_complex(divide1717551699836669952omplex(Xb),imaginary_unit) = aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,imaginary_unit)),Xb) ).

% divide_i
tff(fact_9168_i__squared,axiom,
    aa_complex_complex(times_times_complex(imaginary_unit),imaginary_unit) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% i_squared
tff(fact_9169_log__le__cancel__iff,axiom,
    ! [A: real,Xb: real,Yb: 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),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Yb)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(log(A),Xb)),aa_real_real(log(A),Yb))
          <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Yb) ) ) ) ) ).

% log_le_cancel_iff
tff(fact_9170_log__le__one__cancel__iff,axiom,
    ! [A: real,Xb: 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),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(log(A),Xb)),one_one_real)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),A) ) ) ) ).

% log_le_one_cancel_iff
tff(fact_9171_one__le__log__cancel__iff,axiom,
    ! [A: real,Xb: 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),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(log(A),Xb))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),Xb) ) ) ) ).

% one_le_log_cancel_iff
tff(fact_9172_log__le__zero__cancel__iff,axiom,
    ! [A: real,Xb: 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),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(log(A),Xb)),zero_zero_real)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real) ) ) ) ).

% log_le_zero_cancel_iff
tff(fact_9173_zero__le__log__cancel__iff,axiom,
    ! [A: real,Xb: 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),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(log(A),Xb))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xb) ) ) ) ).

% zero_le_log_cancel_iff
tff(fact_9174_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(log(A),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),B)) = aa_nat_real(semiri5074537144036343181t_real,B) ) ) ) ).

% log_pow_cancel
tff(fact_9175_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_9176_complex__i__not__one,axiom,
    imaginary_unit != one_one_complex ).

% complex_i_not_one
tff(fact_9177_cosh__real__ge__1,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(cosh_real,Xb)) ).

% cosh_real_ge_1
tff(fact_9178_i__times__eq__iff,axiom,
    ! [W: complex,Z: complex] :
      ( ( aa_complex_complex(times_times_complex(imaginary_unit),W) = Z )
    <=> ( W = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(times_times_complex(imaginary_unit),Z)) ) ) ).

% i_times_eq_iff
tff(fact_9179_log__ln,axiom,
    ! [Xb: real] : aa_real_real(ln_ln_real,Xb) = aa_real_real(log(aa_real_real(exp_real,one_one_real)),Xb) ).

% log_ln
tff(fact_9180_imaginary__unit_Ocode,axiom,
    imaginary_unit = aa_real_complex(complex2(zero_zero_real),one_one_real) ).

% imaginary_unit.code
tff(fact_9181_Complex__eq__i,axiom,
    ! [Xb: real,Yb: real] :
      ( ( aa_real_complex(complex2(Xb),Yb) = imaginary_unit )
    <=> ( ( Xb = zero_zero_real )
        & ( Yb = one_one_real ) ) ) ).

% Complex_eq_i
tff(fact_9182_i__mult__Complex,axiom,
    ! [A: real,B: real] : aa_complex_complex(times_times_complex(imaginary_unit),aa_real_complex(complex2(A),B)) = aa_real_complex(complex2(aa_real_real(uminus_uminus_real,B)),A) ).

% i_mult_Complex
tff(fact_9183_Complex__mult__i,axiom,
    ! [A: real,B: real] : aa_complex_complex(times_times_complex(aa_real_complex(complex2(A),B)),imaginary_unit) = aa_real_complex(complex2(aa_real_real(uminus_uminus_real,B)),A) ).

% Complex_mult_i
tff(fact_9184_log__base__change,axiom,
    ! [A: real,B: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( ( A != one_one_real )
       => ( aa_real_real(log(B),Xb) = aa_real_real(divide_divide_real(aa_real_real(log(A),Xb)),aa_real_real(log(A),B)) ) ) ) ).

% log_base_change
tff(fact_9185_log__of__power__eq,axiom,
    ! [Mb: nat,B: real,Nb: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,Mb) = aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
       => ( aa_nat_real(semiri5074537144036343181t_real,Nb) = aa_real_real(log(B),aa_nat_real(semiri5074537144036343181t_real,Mb)) ) ) ) ).

% log_of_power_eq
tff(fact_9186_less__log__of__power,axiom,
    ! [B: real,Nb: nat,Mb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb)),Mb)
     => ( 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,Nb)),aa_real_real(log(B),Mb)) ) ) ).

% less_log_of_power
tff(fact_9187_log__mult,axiom,
    ! [A: real,Xb: real,Yb: 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),Xb)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Yb)
           => ( aa_real_real(log(A),aa_real_real(times_times_real(Xb),Yb)) = aa_real_real(plus_plus_real(aa_real_real(log(A),Xb)),aa_real_real(log(A),Yb)) ) ) ) ) ) ).

% log_mult
tff(fact_9188_log__divide,axiom,
    ! [A: real,Xb: real,Yb: 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),Xb)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Yb)
           => ( aa_real_real(log(A),aa_real_real(divide_divide_real(Xb),Yb)) = aa_real_real(minus_minus_real(aa_real_real(log(A),Xb)),aa_real_real(log(A),Yb)) ) ) ) ) ) ).

% log_divide
tff(fact_9189_le__log__of__power,axiom,
    ! [B: real,Nb: nat,Mb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb)),Mb)
     => ( 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,Nb)),aa_real_real(log(B),Mb)) ) ) ).

% le_log_of_power
tff(fact_9190_log__nat__power,axiom,
    ! [Xb: real,B: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_real(log(B),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)) = aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(log(B),Xb)) ) ) ).

% log_nat_power
tff(fact_9191_log__inverse,axiom,
    ! [A: real,Xb: 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),Xb)
         => ( aa_real_real(log(A),aa_real_real(inverse_inverse_real,Xb)) = aa_real_real(uminus_uminus_real,aa_real_real(log(A),Xb)) ) ) ) ) ).

% log_inverse
tff(fact_9192_log2__of__power__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( Mb = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) )
     => ( aa_nat_real(semiri5074537144036343181t_real,Nb) = aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Mb)) ) ) ).

% log2_of_power_eq
tff(fact_9193_log__of__power__less,axiom,
    ! [Mb: nat,B: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb))
     => ( 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),Mb)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(log(B),aa_nat_real(semiri5074537144036343181t_real,Mb))),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ) ) ).

% log_of_power_less
tff(fact_9194_log__eq__div__ln__mult__log,axiom,
    ! [A: real,B: real,Xb: 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),Xb)
             => ( aa_real_real(log(A),Xb) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(ln_ln_real,B)),aa_real_real(ln_ln_real,A))),aa_real_real(log(B),Xb)) ) ) ) ) ) ) ).

% log_eq_div_ln_mult_log
tff(fact_9195_log__of__power__le,axiom,
    ! [Mb: nat,B: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B),Nb))
     => ( 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),Mb)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(log(B),aa_nat_real(semiri5074537144036343181t_real,Mb))),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ) ) ).

% log_of_power_le
tff(fact_9196_less__log2__of__power,axiom,
    ! [Nb: nat,Mb: 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))),Nb)),Mb)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Mb))) ) ).

% less_log2_of_power
tff(fact_9197_le__log2__of__power,axiom,
    ! [Nb: nat,Mb: 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))),Nb)),Mb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Mb))) ) ).

% le_log2_of_power
tff(fact_9198_log2__of__power__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Mb))),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ) ).

% log2_of_power_less
tff(fact_9199_cosh__ln__real,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_real(cosh_real,aa_real_real(ln_ln_real,Xb)) = aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(Xb),aa_real_real(inverse_inverse_real,Xb))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) ) ).

% cosh_ln_real
tff(fact_9200_log2__of__power__le,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Mb))),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ) ).

% log2_of_power_le
tff(fact_9201_log__base__10__eq2,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),Xb) = aa_real_real(times_times_real(aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),aa_real_real(exp_real,one_one_real))),aa_real_real(ln_ln_real,Xb)) ) ) ).

% log_base_10_eq2
tff(fact_9202_sinh__ln__real,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_real(sinh_real,aa_real_real(ln_ln_real,Xb)) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(Xb),aa_real_real(inverse_inverse_real,Xb))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) ) ).

% sinh_ln_real
tff(fact_9203_Arg__minus__ii,axiom,
    aa_complex_real(arg,aa_complex_complex(uminus1482373934393186551omplex,imaginary_unit)) = aa_real_real(divide_divide_real(aa_real_real(uminus_uminus_real,pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% Arg_minus_ii
tff(fact_9204_ceiling__log__nat__eq__powr__iff,axiom,
    ! [B: nat,K: nat,Nb: 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(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,Nb)),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),Nb)),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,Nb),one_one_nat))) ) ) ) ) ).

% ceiling_log_nat_eq_powr_iff
tff(fact_9205_Arg__ii,axiom,
    aa_complex_real(arg,imaginary_unit) = aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% Arg_ii
tff(fact_9206_ceiling__log2__div2,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_real_int(archim7802044766580827645g_real,aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Nb))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim7802044766580827645g_real,aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),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_9207_ceiling__divide__eq__div__numeral,axiom,
    ! [A: num,B: num] : aa_real_int(archim7802044766580827645g_real,aa_real_real(divide_divide_real(aa_num_real(numeral_numeral_real,A)),aa_num_real(numeral_numeral_real,B))) = aa_int_int(uminus_uminus_int,aa_int_int(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_9208_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(divide_divide_real(aa_num_real(numeral_numeral_real,A)),aa_num_real(numeral_numeral_real,B)))) = aa_int_int(uminus_uminus_int,aa_int_int(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_9209_ceiling__log__nat__eq__if,axiom,
    ! [B: nat,Nb: 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),Nb)),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,Nb),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(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,Nb)),one_one_int) ) ) ) ) ).

% ceiling_log_nat_eq_if
tff(fact_9210_cis__minus__pi__half,axiom,
    aa_real_complex(cis,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) = aa_complex_complex(uminus1482373934393186551omplex,imaginary_unit) ).

% cis_minus_pi_half
tff(fact_9211_ceiling__log__eq__powr__iff,axiom,
    ! [Xb: real,B: real,K: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
       => ( ( aa_real_int(archim7802044766580827645g_real,aa_real_real(log(B),Xb)) = 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(powr_real(B),aa_nat_real(semiri5074537144036343181t_real,K))),Xb)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_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_9212_floor__log__nat__eq__powr__iff,axiom,
    ! [B: nat,K: nat,Nb: 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(log(aa_nat_real(semiri5074537144036343181t_real,B)),aa_nat_real(semiri5074537144036343181t_real,K))) = aa_nat_int(semiri1314217659103216013at_int,Nb) )
        <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Nb)),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,Nb),one_one_nat))) ) ) ) ) ).

% floor_log_nat_eq_powr_iff
tff(fact_9213_floor__log__nat__eq__if,axiom,
    ! [B: nat,Nb: 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),Nb)),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,Nb),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(log(aa_nat_real(semiri5074537144036343181t_real,B)),aa_nat_real(semiri5074537144036343181t_real,K))) = aa_nat_int(semiri1314217659103216013at_int,Nb) ) ) ) ) ).

% floor_log_nat_eq_if
tff(fact_9214_powr__less__cancel__iff,axiom,
    ! [Xb: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(powr_real(Xb),A)),aa_real_real(powr_real(Xb),B))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ) ).

% powr_less_cancel_iff
tff(fact_9215_norm__cis,axiom,
    ! [A: real] : aa_complex_real(real_V1022390504157884413omplex,aa_real_complex(cis,A)) = one_one_real ).

% norm_cis
tff(fact_9216_cis__zero,axiom,
    aa_real_complex(cis,zero_zero_real) = one_one_complex ).

% cis_zero
tff(fact_9217_powr__eq__one__iff,axiom,
    ! [A: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( ( aa_real_real(powr_real(A),Xb) = one_one_real )
      <=> ( Xb = zero_zero_real ) ) ) ).

% powr_eq_one_iff
tff(fact_9218_powr__one,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_real(powr_real(Xb),one_one_real) = Xb ) ) ).

% powr_one
tff(fact_9219_powr__one__gt__zero__iff,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(powr_real(Xb),one_one_real) = Xb )
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb) ) ).

% powr_one_gt_zero_iff
tff(fact_9220_powr__le__cancel__iff,axiom,
    ! [Xb: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(powr_real(Xb),A)),aa_real_real(powr_real(Xb),B))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B) ) ) ).

% powr_le_cancel_iff
tff(fact_9221_numeral__powr__numeral__real,axiom,
    ! [Mb: num,Nb: num] : aa_real_real(powr_real(aa_num_real(numeral_numeral_real,Mb)),aa_num_real(numeral_numeral_real,Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,Mb)),aa_num_nat(numeral_numeral_nat,Nb)) ).

% numeral_powr_numeral_real
tff(fact_9222_cis__pi,axiom,
    aa_real_complex(cis,pi) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% cis_pi
tff(fact_9223_powr__log__cancel,axiom,
    ! [A: real,Xb: 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),Xb)
         => ( aa_real_real(powr_real(A),aa_real_real(log(A),Xb)) = Xb ) ) ) ) ).

% powr_log_cancel
tff(fact_9224_log__powr__cancel,axiom,
    ! [A: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( ( A != one_one_real )
       => ( aa_real_real(log(A),aa_real_real(powr_real(A),Yb)) = Yb ) ) ) ).

% log_powr_cancel
tff(fact_9225_floor__divide__eq__div__numeral,axiom,
    ! [A: num,B: num] : aa_real_int(archim6058952711729229775r_real,aa_real_real(divide_divide_real(aa_num_real(numeral_numeral_real,A)),aa_num_real(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_9226_powr__numeral,axiom,
    ! [Xb: real,Nb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_real(powr_real(Xb),aa_num_real(numeral_numeral_real,Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,Nb)) ) ) ).

% powr_numeral
tff(fact_9227_cis__pi__half,axiom,
    aa_real_complex(cis,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = imaginary_unit ).

% cis_pi_half
tff(fact_9228_floor__one__divide__eq__div__numeral,axiom,
    ! [B: num] : aa_real_int(archim6058952711729229775r_real,aa_real_real(divide_divide_real(one_one_real),aa_num_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_9229_cis__2pi,axiom,
    aa_real_complex(cis,aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)) = one_one_complex ).

% cis_2pi
tff(fact_9230_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(divide_divide_real(aa_num_real(numeral_numeral_real,A)),aa_num_real(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_9231_square__powr__half,axiom,
    ! [Xb: real] : aa_real_real(powr_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = aa_real_real(abs_abs_real,Xb) ).

% square_powr_half
tff(fact_9232_floor__minus__one__divide__eq__div__numeral,axiom,
    ! [B: num] : aa_real_int(archim6058952711729229775r_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(one_one_real),aa_num_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_9233_powr__powr,axiom,
    ! [Xb: real,A: real,B: real] : aa_real_real(powr_real(aa_real_real(powr_real(Xb),A)),B) = aa_real_real(powr_real(Xb),aa_real_real(times_times_real(A),B)) ).

% powr_powr
tff(fact_9234_powr__less__cancel,axiom,
    ! [Xb: real,A: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(powr_real(Xb),A)),aa_real_real(powr_real(Xb),B))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B) ) ) ).

% powr_less_cancel
tff(fact_9235_powr__less__mono,axiom,
    ! [A: real,B: real,Xb: 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),Xb)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(powr_real(Xb),A)),aa_real_real(powr_real(Xb),B)) ) ) ).

% powr_less_mono
tff(fact_9236_powr__mono,axiom,
    ! [A: real,B: real,Xb: 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),Xb)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(powr_real(Xb),A)),aa_real_real(powr_real(Xb),B)) ) ) ).

% powr_mono
tff(fact_9237_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(plus_plus_real(A),B)) ).

% cis_mult
tff(fact_9238_gr__one__powr,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Yb)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(powr_real(Xb),Yb)) ) ) ).

% gr_one_powr
tff(fact_9239_powr__inj,axiom,
    ! [A: real,Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( ( A != one_one_real )
       => ( ( aa_real_real(powr_real(A),Xb) = aa_real_real(powr_real(A),Yb) )
        <=> ( Xb = Yb ) ) ) ) ).

% powr_inj
tff(fact_9240_powr__le1,axiom,
    ! [A: real,Xb: 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),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(powr_real(Xb),A)),one_one_real) ) ) ) ).

% powr_le1
tff(fact_9241_powr__mono__both,axiom,
    ! [A: real,B: real,Xb: real,Yb: 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),Xb)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Yb)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(powr_real(Xb),A)),aa_real_real(powr_real(Yb),B)) ) ) ) ) ).

% powr_mono_both
tff(fact_9242_ge__one__powr__ge__zero,axiom,
    ! [Xb: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xb)
     => ( 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(powr_real(Xb),A)) ) ) ).

% ge_one_powr_ge_zero
tff(fact_9243_powr__mult,axiom,
    ! [Xb: real,Yb: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Yb)
       => ( aa_real_real(powr_real(aa_real_real(times_times_real(Xb),Yb)),A) = aa_real_real(times_times_real(aa_real_real(powr_real(Xb),A)),aa_real_real(powr_real(Yb),A)) ) ) ) ).

% powr_mult
tff(fact_9244_divide__powr__uminus,axiom,
    ! [A: real,B: real,C: real] : aa_real_real(divide_divide_real(A),aa_real_real(powr_real(B),C)) = aa_real_real(times_times_real(A),aa_real_real(powr_real(B),aa_real_real(uminus_uminus_real,C))) ).

% divide_powr_uminus
tff(fact_9245_ln__powr,axiom,
    ! [Xb: real,Yb: real] :
      ( ( Xb != zero_zero_real )
     => ( aa_real_real(ln_ln_real,aa_real_real(powr_real(Xb),Yb)) = aa_real_real(times_times_real(Yb),aa_real_real(ln_ln_real,Xb)) ) ) ).

% ln_powr
tff(fact_9246_log__powr,axiom,
    ! [Xb: real,B: real,Yb: real] :
      ( ( Xb != zero_zero_real )
     => ( aa_real_real(log(B),aa_real_real(powr_real(Xb),Yb)) = aa_real_real(times_times_real(Yb),aa_real_real(log(B),Xb)) ) ) ).

% log_powr
tff(fact_9247_floor__log__eq__powr__iff,axiom,
    ! [Xb: real,B: real,K: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
       => ( ( aa_real_int(archim6058952711729229775r_real,aa_real_real(log(B),Xb)) = K )
        <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(powr_real(B),aa_int_real(ring_1_of_int_real,K))),Xb)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_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_9248_less__log__iff,axiom,
    ! [B: real,Xb: real,Yb: 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),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Yb),aa_real_real(log(B),Xb))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(powr_real(B),Yb)),Xb) ) ) ) ).

% less_log_iff
tff(fact_9249_log__less__iff,axiom,
    ! [B: real,Xb: real,Yb: 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),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(log(B),Xb)),Yb)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(powr_real(B),Yb)) ) ) ) ).

% log_less_iff
tff(fact_9250_less__powr__iff,axiom,
    ! [B: real,Xb: real,Yb: 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),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(powr_real(B),Yb))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(log(B),Xb)),Yb) ) ) ) ).

% less_powr_iff
tff(fact_9251_powr__less__iff,axiom,
    ! [B: real,Xb: real,Yb: 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),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(powr_real(B),Yb)),Xb)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Yb),aa_real_real(log(B),Xb)) ) ) ) ).

% powr_less_iff
tff(fact_9252_real__of__int__floor__add__one__gt,axiom,
    ! [R2: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,R2),aa_real_real(plus_plus_real(aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,R2))),one_one_real)) ).

% real_of_int_floor_add_one_gt
tff(fact_9253_floor__eq,axiom,
    ! [Nb: int,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Nb)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(plus_plus_real(aa_int_real(ring_1_of_int_real,Nb)),one_one_real))
       => ( aa_real_int(archim6058952711729229775r_real,Xb) = Nb ) ) ) ).

% floor_eq
tff(fact_9254_real__of__int__floor__add__one__ge,axiom,
    ! [R2: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,R2),aa_real_real(plus_plus_real(aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,R2))),one_one_real)) ).

% real_of_int_floor_add_one_ge
tff(fact_9255_real__of__int__floor__gt__diff__one,axiom,
    ! [R2: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(minus_minus_real(R2),one_one_real)),aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,R2))) ).

% real_of_int_floor_gt_diff_one
tff(fact_9256_real__of__int__floor__ge__diff__one,axiom,
    ! [R2: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(minus_minus_real(R2),one_one_real)),aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,R2))) ).

% real_of_int_floor_ge_diff_one
tff(fact_9257_DeMoivre,axiom,
    ! [A: real,Nb: nat] : aa_nat_complex(power_power_complex(aa_real_complex(cis,A)),Nb) = aa_real_complex(cis,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),A)) ).

% DeMoivre
tff(fact_9258_powr__neg__one,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_real(powr_real(Xb),aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(divide_divide_real(one_one_real),Xb) ) ) ).

% powr_neg_one
tff(fact_9259_powr__mult__base,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_real(times_times_real(Xb),aa_real_real(powr_real(Xb),Yb)) = aa_real_real(powr_real(Xb),aa_real_real(plus_plus_real(one_one_real),Yb)) ) ) ).

% powr_mult_base
tff(fact_9260_le__log__iff,axiom,
    ! [B: real,Xb: real,Yb: 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),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Yb),aa_real_real(log(B),Xb))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(powr_real(B),Yb)),Xb) ) ) ) ).

% le_log_iff
tff(fact_9261_log__le__iff,axiom,
    ! [B: real,Xb: real,Yb: 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),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(log(B),Xb)),Yb)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(powr_real(B),Yb)) ) ) ) ).

% log_le_iff
tff(fact_9262_le__powr__iff,axiom,
    ! [B: real,Xb: real,Yb: 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),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(powr_real(B),Yb))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(log(B),Xb)),Yb) ) ) ) ).

% le_powr_iff
tff(fact_9263_powr__le__iff,axiom,
    ! [B: real,Xb: real,Yb: 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),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(powr_real(B),Yb)),Xb)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Yb),aa_real_real(log(B),Xb)) ) ) ) ).

% powr_le_iff
tff(fact_9264_floor__eq2,axiom,
    ! [Nb: int,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Nb)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(plus_plus_real(aa_int_real(ring_1_of_int_real,Nb)),one_one_real))
       => ( aa_real_int(archim6058952711729229775r_real,Xb) = Nb ) ) ) ).

% floor_eq2
tff(fact_9265_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(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_9266_ln__powr__bound,axiom,
    ! [Xb: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xb)
     => ( 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,Xb)),aa_real_real(divide_divide_real(aa_real_real(powr_real(Xb),A)),A)) ) ) ).

% ln_powr_bound
tff(fact_9267_ln__powr__bound2,axiom,
    ! [Xb: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb)
     => ( 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(powr_real(aa_real_real(ln_ln_real,Xb)),A)),aa_real_real(times_times_real(aa_real_real(powr_real(A),A)),Xb)) ) ) ).

% ln_powr_bound2
tff(fact_9268_log__add__eq__powr,axiom,
    ! [B: real,Xb: real,Yb: 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),Xb)
         => ( aa_real_real(plus_plus_real(aa_real_real(log(B),Xb)),Yb) = aa_real_real(log(B),aa_real_real(times_times_real(Xb),aa_real_real(powr_real(B),Yb))) ) ) ) ) ).

% log_add_eq_powr
tff(fact_9269_add__log__eq__powr,axiom,
    ! [B: real,Xb: real,Yb: 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),Xb)
         => ( aa_real_real(plus_plus_real(Yb),aa_real_real(log(B),Xb)) = aa_real_real(log(B),aa_real_real(times_times_real(aa_real_real(powr_real(B),Yb)),Xb)) ) ) ) ) ).

% add_log_eq_powr
tff(fact_9270_minus__log__eq__powr,axiom,
    ! [B: real,Xb: real,Yb: 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),Xb)
         => ( aa_real_real(minus_minus_real(Yb),aa_real_real(log(B),Xb)) = aa_real_real(log(B),aa_real_real(divide_divide_real(aa_real_real(powr_real(B),Yb)),Xb)) ) ) ) ) ).

% minus_log_eq_powr
tff(fact_9271_log__minus__eq__powr,axiom,
    ! [B: real,Xb: real,Yb: 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),Xb)
         => ( aa_real_real(minus_minus_real(aa_real_real(log(B),Xb)),Yb) = aa_real_real(log(B),aa_real_real(times_times_real(Xb),aa_real_real(powr_real(B),aa_real_real(uminus_uminus_real,Yb)))) ) ) ) ) ).

% log_minus_eq_powr
tff(fact_9272_powr__half__sqrt,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_real(powr_real(Xb),aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = aa_real_real(sqrt,Xb) ) ) ).

% powr_half_sqrt
tff(fact_9273_powr__neg__numeral,axiom,
    ! [Xb: real,Nb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_real(powr_real(Xb),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))) = aa_real_real(divide_divide_real(one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,Nb))) ) ) ).

% powr_neg_numeral
tff(fact_9274_floor__log2__div2,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_real_int(archim6058952711729229775r_real,aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Nb))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim6058952711729229775r_real,aa_real_real(log(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),one_one_int) ) ) ).

% floor_log2_div2
tff(fact_9275_bij__betw__roots__unity,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => bij_betw_nat_complex(aTP_Lamm_ps(Nb),aa_nat_set_nat(set_ord_lessThan_nat,Nb),collect_complex(aTP_Lamm_dl(Nb))) ) ).

% bij_betw_roots_unity
tff(fact_9276_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_9277_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_9278_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_9279_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_9280_complex__exp__exists,axiom,
    ! [Z: complex] :
    ? [A4: complex,R3: real] : Z = aa_complex_complex(times_times_complex(aa_real_complex(real_V4546457046886955230omplex,R3)),aa_complex_complex(exp_complex,A4)) ).

% complex_exp_exists
tff(fact_9281_Complex__mult__complex__of__real,axiom,
    ! [Xb: real,Yb: real,R2: real] : aa_complex_complex(times_times_complex(aa_real_complex(complex2(Xb),Yb)),aa_real_complex(real_V4546457046886955230omplex,R2)) = aa_real_complex(complex2(aa_real_real(times_times_real(Xb),R2)),aa_real_real(times_times_real(Yb),R2)) ).

% Complex_mult_complex_of_real
tff(fact_9282_complex__of__real__mult__Complex,axiom,
    ! [R2: real,Xb: real,Yb: real] : aa_complex_complex(times_times_complex(aa_real_complex(real_V4546457046886955230omplex,R2)),aa_real_complex(complex2(Xb),Yb)) = aa_real_complex(complex2(aa_real_real(times_times_real(R2),Xb)),aa_real_real(times_times_real(R2),Yb)) ).

% complex_of_real_mult_Complex
tff(fact_9283_Complex__add__complex__of__real,axiom,
    ! [Xb: real,Yb: real,R2: real] : aa_complex_complex(plus_plus_complex(aa_real_complex(complex2(Xb),Yb)),aa_real_complex(real_V4546457046886955230omplex,R2)) = aa_real_complex(complex2(aa_real_real(plus_plus_real(Xb),R2)),Yb) ).

% Complex_add_complex_of_real
tff(fact_9284_complex__of__real__add__Complex,axiom,
    ! [R2: real,Xb: real,Yb: real] : aa_complex_complex(plus_plus_complex(aa_real_complex(real_V4546457046886955230omplex,R2)),aa_real_complex(complex2(Xb),Yb)) = aa_real_complex(complex2(aa_real_real(plus_plus_real(R2),Xb)),Yb) ).

% complex_of_real_add_Complex
tff(fact_9285_cis__conv__exp,axiom,
    ! [B: real] : aa_real_complex(cis,B) = aa_complex_complex(exp_complex,aa_complex_complex(times_times_complex(imaginary_unit),aa_real_complex(real_V4546457046886955230omplex,B))) ).

% cis_conv_exp
tff(fact_9286_i__complex__of__real,axiom,
    ! [R2: real] : aa_complex_complex(times_times_complex(imaginary_unit),aa_real_complex(real_V4546457046886955230omplex,R2)) = aa_real_complex(complex2(zero_zero_real),R2) ).

% i_complex_of_real
tff(fact_9287_complex__of__real__i,axiom,
    ! [R2: real] : aa_complex_complex(times_times_complex(aa_real_complex(real_V4546457046886955230omplex,R2)),imaginary_unit) = aa_real_complex(complex2(zero_zero_real),R2) ).

% complex_of_real_i
tff(fact_9288_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_9289_complex__split__polar,axiom,
    ! [Z: complex] :
    ? [R3: real,A4: real] : Z = aa_complex_complex(times_times_complex(aa_real_complex(real_V4546457046886955230omplex,R3)),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_9290_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_9291_cmod__complex__polar,axiom,
    ! [R2: real,A: real] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(times_times_complex(aa_real_complex(real_V4546457046886955230omplex,R2)),aa_complex_complex(plus_plus_complex(aa_real_complex(real_V4546457046886955230omplex,aa_real_real(cos_real,A))),aa_complex_complex(times_times_complex(imaginary_unit),aa_real_complex(real_V4546457046886955230omplex,aa_real_real(sin_real,A)))))) = aa_real_real(abs_abs_real,R2) ).

% cmod_complex_polar
tff(fact_9292_csqrt__ii,axiom,
    aa_complex_complex(csqrt,imaginary_unit) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(one_one_complex),imaginary_unit)),aa_real_complex(real_V4546457046886955230omplex,aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% csqrt_ii
tff(fact_9293_arctan__def,axiom,
    ! [Yb: real] : aa_real_real(arctan,Yb) = the_real(aa_real_fun_real_o(aTP_Lamm_pt,Yb)) ).

% arctan_def
tff(fact_9294_arcsin__def,axiom,
    ! [Yb: real] : aa_real_real(arcsin,Yb) = the_real(aa_real_fun_real_o(aTP_Lamm_pu,Yb)) ).

% arcsin_def
tff(fact_9295_modulo__int__unfold,axiom,
    ! [K: int,Mb: nat,L: int,Nb: nat] :
      aa_int_int(aa_int_fun_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,Mb))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,Nb))) = $ite(
        ( ( aa_int_int(sgn_sgn_int,L) = zero_zero_int )
        | ( aa_int_int(sgn_sgn_int,K) = zero_zero_int )
        | ( Nb = 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,Mb)),
        $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,Mb),Nb))),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,Nb),zero_n2687167440665602831ol_nat(~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Nb),Mb))))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb))))) ) ).

% modulo_int_unfold
tff(fact_9296_csqrt__eq__1,axiom,
    ! [Z: complex] :
      ( ( aa_complex_complex(csqrt,Z) = one_one_complex )
    <=> ( Z = one_one_complex ) ) ).

% csqrt_eq_1
tff(fact_9297_csqrt__1,axiom,
    aa_complex_complex(csqrt,one_one_complex) = one_one_complex ).

% csqrt_1
tff(fact_9298_dvd__mult__sgn__iff,axiom,
    ! [L: int,K: int,R2: 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,R2)))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K)
        | ( R2 = zero_zero_int ) ) ) ).

% dvd_mult_sgn_iff
tff(fact_9299_dvd__sgn__mult__iff,axiom,
    ! [L: int,R2: 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,R2)),K))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K)
        | ( R2 = zero_zero_int ) ) ) ).

% dvd_sgn_mult_iff
tff(fact_9300_mult__sgn__dvd__iff,axiom,
    ! [L: int,R2: 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,R2))),K)
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K)
        & ( ( R2 = zero_zero_int )
         => ( K = zero_zero_int ) ) ) ) ).

% mult_sgn_dvd_iff
tff(fact_9301_sgn__mult__dvd__iff,axiom,
    ! [R2: 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,R2)),L)),K)
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K)
        & ( ( R2 = zero_zero_int )
         => ( K = zero_zero_int ) ) ) ) ).

% sgn_mult_dvd_iff
tff(fact_9302_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_9303_int__sgnE,axiom,
    ! [K: int] :
      ~ ! [N: nat,L3: int] : K != aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L3)),aa_nat_int(semiri1314217659103216013at_int,N)) ).

% int_sgnE
tff(fact_9304_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_9305_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(aa_int_fun_int_int(modulo_modulo_int,K),L)) = aa_int_int(sgn_sgn_int,L) ) ) ) ).

% sgn_mod
tff(fact_9306_zsgn__def,axiom,
    ! [I3: int] :
      aa_int_int(sgn_sgn_int,I3) = $ite(
        I3 = zero_zero_int,
        zero_zero_int,
        $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),I3),one_one_int,aa_int_int(uminus_uminus_int,one_one_int)) ) ).

% zsgn_def
tff(fact_9307_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_9308_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_9309_eucl__rel__int__remainderI,axiom,
    ! [R2: int,L: int,K: int,Q2: int] :
      ( ( aa_int_int(sgn_sgn_int,R2) = aa_int_int(sgn_sgn_int,L) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,R2)),aa_int_int(abs_abs_int,L))
       => ( ( K = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Q2),L)),R2) )
         => aa_Pro4592068669816069595_int_o(eucl_rel_int(K,L),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q2),R2)) ) ) ) ).

% eucl_rel_int_remainderI
tff(fact_9310_eucl__rel__int_Osimps,axiom,
    ! [A1: int,A22: int,A32: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(eucl_rel_int(A1,A22),A32)
    <=> ( ? [K3: int] :
            ( ( A1 = K3 )
            & ( A22 = zero_zero_int )
            & ( A32 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),K3) ) )
        | ? [L4: int,K3: int,Q4: int] :
            ( ( A1 = K3 )
            & ( A22 = L4 )
            & ( A32 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q4),zero_zero_int) )
            & ( L4 != zero_zero_int )
            & ( K3 = aa_int_int(aa_int_fun_int_int(times_times_int,Q4),L4) ) )
        | ? [R5: int,L4: int,K3: int,Q4: int] :
            ( ( A1 = K3 )
            & ( A22 = L4 )
            & ( A32 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q4),R5) )
            & ( aa_int_int(sgn_sgn_int,R5) = aa_int_int(sgn_sgn_int,L4) )
            & aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,R5)),aa_int_int(abs_abs_int,L4))
            & ( K3 = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Q4),L4)),R5) ) ) ) ) ).

% eucl_rel_int.simps
tff(fact_9311_eucl__rel__int_Ocases,axiom,
    ! [A1: int,A22: int,A32: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(eucl_rel_int(A1,A22),A32)
     => ( ( ( A22 = zero_zero_int )
         => ( A32 != aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),A1) ) )
       => ( ! [Q3: int] :
              ( ( A32 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q3),zero_zero_int) )
             => ( ( A22 != zero_zero_int )
               => ( A1 != aa_int_int(aa_int_fun_int_int(times_times_int,Q3),A22) ) ) )
         => ~ ! [R3: int,Q3: int] :
                ( ( A32 = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Q3),R3) )
               => ( ( aa_int_int(sgn_sgn_int,R3) = aa_int_int(sgn_sgn_int,A22) )
                 => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,R3)),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,Q3),A22)),R3) ) ) ) ) ) ) ) ).

% eucl_rel_int.cases
tff(fact_9312_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_9313_pi__half,axiom,
    aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) = the_real(aTP_Lamm_pv) ).

% pi_half
tff(fact_9314_pi__def,axiom,
    pi = aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),the_real(aTP_Lamm_pv)) ).

% pi_def
tff(fact_9315_divide__int__unfold,axiom,
    ! [K: int,Mb: nat,L: int,Nb: 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,Mb))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,Nb))) = $ite(
        ( ( aa_int_int(sgn_sgn_int,L) = zero_zero_int )
        | ( aa_int_int(sgn_sgn_int,K) = zero_zero_int )
        | ( Nb = 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,Mb),Nb)),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,Mb),Nb)),zero_n2687167440665602831ol_nat(~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Nb),Mb)))))) ) ).

% divide_int_unfold
tff(fact_9316_modulo__int__def,axiom,
    ! [K: int,L: int] :
      aa_int_int(aa_int_fun_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_9317_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)))),zero_n2687167440665602831ol_nat(~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K)))))) ) ).

% divide_int_def
tff(fact_9318_powr__int,axiom,
    ! [Xb: real,I3: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_real(powr_real(Xb),aa_int_real(ring_1_of_int_real,I3)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),I3),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_int_nat(nat2,I3)),aa_real_real(divide_divide_real(one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_int_nat(nat2,aa_int_int(uminus_uminus_int,I3))))) ) ) ).

% powr_int
tff(fact_9319_nat__int,axiom,
    ! [Nb: nat] : aa_int_nat(nat2,aa_nat_int(semiri1314217659103216013at_int,Nb)) = Nb ).

% nat_int
tff(fact_9320_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_9321_nat__of__bool,axiom,
    ! [P: $o] : aa_int_nat(nat2,aa_o_int(zero_n2684676970156552555ol_int,(P))) = zero_n2687167440665602831ol_nat((P)) ).

% nat_of_bool
tff(fact_9322_nat__1,axiom,
    aa_int_nat(nat2,one_one_int) = aa_nat_nat(suc,zero_zero_nat) ).

% nat_1
tff(fact_9323_nat__0__iff,axiom,
    ! [I3: int] :
      ( ( aa_int_nat(nat2,I3) = zero_zero_nat )
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I3),zero_zero_int) ) ).

% nat_0_iff
tff(fact_9324_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_9325_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_9326_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_9327_nat__zminus__int,axiom,
    ! [Nb: nat] : aa_int_nat(nat2,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,Nb))) = zero_zero_nat ).

% nat_zminus_int
tff(fact_9328_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_9329_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_9330_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_9331_nat__eq__numeral__power__cancel__iff,axiom,
    ! [Yb: int,Xb: num,Nb: nat] :
      ( ( aa_int_nat(nat2,Yb) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb) )
    <=> ( Yb = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb) ) ) ).

% nat_eq_numeral_power_cancel_iff
tff(fact_9332_numeral__power__eq__nat__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,Yb: int] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb) = aa_int_nat(nat2,Yb) )
    <=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb) = Yb ) ) ).

% numeral_power_eq_nat_cancel_iff
tff(fact_9333_dvd__nat__abs__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Nb),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,Nb)),K) ) ).

% dvd_nat_abs_iff
tff(fact_9334_nat__abs__dvd__iff,axiom,
    ! [K: int,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),Nb)
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,K),aa_nat_int(semiri1314217659103216013at_int,Nb)) ) ).

% nat_abs_dvd_iff
tff(fact_9335_nat__ceiling__le__eq,axiom,
    ! [Xb: real,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,aa_real_int(archim7802044766580827645g_real,Xb))),A)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_nat_real(semiri5074537144036343181t_real,A)) ) ).

% nat_ceiling_le_eq
tff(fact_9336_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_9337_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_9338_nat__less__numeral__power__cancel__iff,axiom,
    ! [A: int,Xb: num,Nb: 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,Xb)),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb)) ) ).

% nat_less_numeral_power_cancel_iff
tff(fact_9339_numeral__power__less__nat__cancel__iff,axiom,
    ! [Xb: num,Nb: 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,Xb)),Nb)),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,Xb)),Nb)),A) ) ).

% numeral_power_less_nat_cancel_iff
tff(fact_9340_nat__le__numeral__power__cancel__iff,axiom,
    ! [A: int,Xb: num,Nb: 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,Xb)),Nb))
    <=> 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,Xb)),Nb)) ) ).

% nat_le_numeral_power_cancel_iff
tff(fact_9341_numeral__power__le__nat__cancel__iff,axiom,
    ! [Xb: num,Nb: 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,Xb)),Nb)),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,Xb)),Nb)),A) ) ).

% numeral_power_le_nat_cancel_iff
tff(fact_9342_nat__zero__as__int,axiom,
    zero_zero_nat = aa_int_nat(nat2,zero_zero_int) ).

% nat_zero_as_int
tff(fact_9343_nat__numeral__as__int,axiom,
    ! [X: num] : aa_num_nat(numeral_numeral_nat,X) = aa_int_nat(nat2,aa_num_int(numeral_numeral_int,X)) ).

% nat_numeral_as_int
tff(fact_9344_nat__mono,axiom,
    ! [Xb: int,Yb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Yb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,Xb)),aa_int_nat(nat2,Yb)) ) ).

% nat_mono
tff(fact_9345_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_9346_all__nat,axiom,
    ! [P: fun_nat_o] :
      ( ! [X_13: nat] : aa_nat_o(P,X_13)
    <=> ! [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_9347_ex__nat,axiom,
    ! [P: fun_nat_o] :
      ( ? [X_13: nat] : aa_nat_o(P,X_13)
    <=> ? [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_9348_nat__one__as__int,axiom,
    one_one_nat = aa_int_nat(nat2,one_one_int) ).

% nat_one_as_int
tff(fact_9349_unset__bit__nat__def,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,Mb),Nb) = aa_int_nat(nat2,aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,Mb),aa_nat_int(semiri1314217659103216013at_int,Nb))) ).

% unset_bit_nat_def
tff(fact_9350_nat__mask__eq,axiom,
    ! [Nb: nat] : aa_int_nat(nat2,aa_nat_int(bit_se2000444600071755411sk_int,Nb)) = aa_nat_nat(bit_se2002935070580805687sk_nat,Nb) ).

% nat_mask_eq
tff(fact_9351_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_9352_zless__nat__eq__int__zless,axiom,
    ! [Mb: nat,Z: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_int_nat(nat2,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),Z) ) ).

% zless_nat_eq_int_zless
tff(fact_9353_nat__le__iff,axiom,
    ! [Xb: int,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,Xb)),Nb)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),aa_nat_int(semiri1314217659103216013at_int,Nb)) ) ).

% nat_le_iff
tff(fact_9354_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_9355_int__eq__iff,axiom,
    ! [Mb: nat,Z: int] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,Mb) = Z )
    <=> ( ( Mb = 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_9356_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_9357_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_9358_and__nat__def,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,Mb),Nb) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb))) ).

% and_nat_def
tff(fact_9359_nat__plus__as__int,axiom,
    ! [X: nat,Xa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Xa) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(semiri1314217659103216013at_int,Xa))) ).

% nat_plus_as_int
tff(fact_9360_or__nat__def,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,Mb),Nb) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb))) ).

% or_nat_def
tff(fact_9361_nat__times__as__int,axiom,
    ! [X: nat,Xa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X),Xa) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(semiri1314217659103216013at_int,Xa))) ).

% nat_times_as_int
tff(fact_9362_real__nat__ceiling__ge,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_nat_real(semiri5074537144036343181t_real,aa_int_nat(nat2,aa_real_int(archim7802044766580827645g_real,Xb)))) ).

% real_nat_ceiling_ge
tff(fact_9363_nat__div__as__int,axiom,
    ! [X: nat,Xa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Xa) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(semiri1314217659103216013at_int,X)),aa_nat_int(semiri1314217659103216013at_int,Xa))) ).

% nat_div_as_int
tff(fact_9364_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_9365_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_9366_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_9367_nat__eq__iff2,axiom,
    ! [Mb: nat,W: int] :
      ( ( Mb = 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,Mb),Mb = zero_zero_nat) ) ).

% nat_eq_iff2
tff(fact_9368_nat__eq__iff,axiom,
    ! [W: int,Mb: nat] :
      ( ( aa_int_nat(nat2,W) = Mb )
    <=> $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),W),W = aa_nat_int(semiri1314217659103216013at_int,Mb),Mb = zero_zero_nat) ) ).

% nat_eq_iff
tff(fact_9369_split__nat,axiom,
    ! [P: fun_nat_o,I3: int] :
      ( aa_nat_o(P,aa_int_nat(nat2,I3))
    <=> ( ! [N4: nat] :
            ( ( I3 = aa_nat_int(semiri1314217659103216013at_int,N4) )
           => aa_nat_o(P,N4) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,I3),zero_zero_int)
         => aa_nat_o(P,zero_zero_nat) ) ) ) ).

% split_nat
tff(fact_9370_le__nat__iff,axiom,
    ! [K: int,Nb: 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,Nb),aa_int_nat(nat2,K))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),K) ) ) ).

% le_nat_iff
tff(fact_9371_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_9372_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_9373_Suc__as__int,axiom,
    ! [X: nat] : aa_nat_nat(suc,X) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,X)),one_one_int)) ).

% Suc_as_int
tff(fact_9374_nat__diff__distrib_H,axiom,
    ! [Xb: int,Yb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Yb)
       => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),Yb)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(nat2,Xb)),aa_int_nat(nat2,Yb)) ) ) ) ).

% nat_diff_distrib'
tff(fact_9375_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_9376_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_9377_nat__div__distrib,axiom,
    ! [Xb: int,Yb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
     => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(divide_divide_int,Xb),Yb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(nat2,Xb)),aa_int_nat(nat2,Yb)) ) ) ).

% nat_div_distrib
tff(fact_9378_nat__div__distrib_H,axiom,
    ! [Yb: int,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Yb)
     => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(divide_divide_int,Xb),Yb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(nat2,Xb)),aa_int_nat(nat2,Yb)) ) ) ).

% nat_div_distrib'
tff(fact_9379_nat__power__eq,axiom,
    ! [Z: int,Nb: 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),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_int_nat(nat2,Z)),Nb) ) ) ).

% nat_power_eq
tff(fact_9380_nat__floor__neg,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real)
     => ( aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,Xb)) = zero_zero_nat ) ) ).

% nat_floor_neg
tff(fact_9381_nat__mod__distrib,axiom,
    ! [Xb: int,Yb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Yb)
       => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Xb),Yb)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_int_nat(nat2,Xb)),aa_int_nat(nat2,Yb)) ) ) ) ).

% nat_mod_distrib
tff(fact_9382_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_9383_floor__eq3,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Nb)))
       => ( aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,Xb)) = Nb ) ) ) ).

% floor_eq3
tff(fact_9384_le__nat__floor,axiom,
    ! [Xb: nat,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Xb)),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,A))) ) ).

% le_nat_floor
tff(fact_9385_mod__abs__eq__div__nat,axiom,
    ! [K: int,L: int] : aa_int_int(aa_int_fun_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_9386_take__bit__nat__eq,axiom,
    ! [K: int,Nb: 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,Nb),aa_int_nat(nat2,K)) = aa_int_nat(nat2,aa_int_int(bit_se2923211474154528505it_int(Nb),K)) ) ) ).

% take_bit_nat_eq
tff(fact_9387_nat__take__bit__eq,axiom,
    ! [K: int,Nb: 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(Nb),K)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),aa_int_nat(nat2,K)) ) ) ).

% nat_take_bit_eq
tff(fact_9388_bit__nat__iff,axiom,
    ! [K: int,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_int_nat(nat2,K)),Nb)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
        & aa_nat_o(bit_se1146084159140164899it_int(K),Nb) ) ) ).

% bit_nat_iff
tff(fact_9389_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_9390_sgn__power__injE,axiom,
    ! [A: real,Nb: nat,Xb: real,B: real] :
      ( ( aa_real_real(times_times_real(aa_real_real(sgn_sgn_real,A)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,A)),Nb)) = Xb )
     => ( ( Xb = aa_real_real(times_times_real(aa_real_real(sgn_sgn_real,B)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,B)),Nb)) )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
         => ( A = B ) ) ) ) ).

% sgn_power_injE
tff(fact_9391_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_9392_nat__less__iff,axiom,
    ! [W: int,Mb: 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)),Mb)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W),aa_nat_int(semiri1314217659103216013at_int,Mb)) ) ) ).

% nat_less_iff
tff(fact_9393_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_9394_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_9395_floor__eq4,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Nb)))
       => ( aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,Xb)) = Nb ) ) ) ).

% floor_eq4
tff(fact_9396_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_9397_nat__dvd__iff,axiom,
    ! [Z: int,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_int_nat(nat2,Z)),Mb)
    <=> $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,Mb)),Mb = zero_zero_nat) ) ).

% nat_dvd_iff
tff(fact_9398_floor__real__def,axiom,
    ! [Xb: real] : aa_real_int(archim6058952711729229775r_real,Xb) = the_int(aTP_Lamm_pw(Xb)) ).

% floor_real_def
tff(fact_9399_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_9400_powr__real__of__int,axiom,
    ! [Xb: real,Nb: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_real(powr_real(Xb),aa_int_real(ring_1_of_int_real,Nb)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Nb),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_int_nat(nat2,Nb)),aa_real_real(inverse_inverse_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_int_nat(nat2,aa_int_int(uminus_uminus_int,Nb))))) ) ) ).

% powr_real_of_int
tff(fact_9401_arctan__inverse,axiom,
    ! [Xb: real] :
      ( ( Xb != zero_zero_real )
     => ( aa_real_real(arctan,aa_real_real(divide_divide_real(one_one_real),Xb)) = aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(aa_real_real(times_times_real(aa_real_real(sgn_sgn_real,Xb)),pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_real_real(arctan,Xb)) ) ) ).

% arctan_inverse
tff(fact_9402_cis__multiple__2pi,axiom,
    ! [Nb: real] :
      ( member_real(Nb,ring_1_Ints_real)
     => ( aa_real_complex(cis,aa_real_real(times_times_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)),Nb)) = one_one_complex ) ) ).

% cis_multiple_2pi
tff(fact_9403_Suc__0__xor__eq,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(suc,zero_zero_nat)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),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))),Nb)))),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))),Nb))) ).

% Suc_0_xor_eq
tff(fact_9404_xor__Suc__0__eq,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,Nb),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,Nb),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))),Nb)))),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))),Nb))) ).

% xor_Suc_0_eq
tff(fact_9405_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_9406_xor__nat__numerals_I1_J,axiom,
    ! [Yb: 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,Yb))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Yb)) ).

% xor_nat_numerals(1)
tff(fact_9407_xor__nat__numerals_I2_J,axiom,
    ! [Yb: 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,Yb))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Yb)) ).

% xor_nat_numerals(2)
tff(fact_9408_xor__nat__numerals_I3_J,axiom,
    ! [Xb: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Xb))),aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xb)) ).

% xor_nat_numerals(3)
tff(fact_9409_xor__nat__numerals_I4_J,axiom,
    ! [Xb: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xb))),aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Xb)) ).

% xor_nat_numerals(4)
tff(fact_9410_sin__times__pi__eq__0,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(sin_real,aa_real_real(times_times_real(Xb),pi)) = zero_zero_real )
    <=> member_real(Xb,ring_1_Ints_real) ) ).

% sin_times_pi_eq_0
tff(fact_9411_sin__integer__2pi,axiom,
    ! [Nb: real] :
      ( member_real(Nb,ring_1_Ints_real)
     => ( aa_real_real(sin_real,aa_real_real(times_times_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)),Nb)) = zero_zero_real ) ) ).

% sin_integer_2pi
tff(fact_9412_cos__integer__2pi,axiom,
    ! [Nb: real] :
      ( member_real(Nb,ring_1_Ints_real)
     => ( aa_real_real(cos_real,aa_real_real(times_times_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)),Nb)) = one_one_real ) ) ).

% cos_integer_2pi
tff(fact_9413_xor__nat__unfold,axiom,
    ! [Mb: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,Mb),Nb) = $ite(
        Mb = zero_zero_nat,
        Nb,
        $ite(Nb = zero_zero_nat,Mb,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,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),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,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ) ).

% xor_nat_unfold
tff(fact_9414_xor__nat__rec,axiom,
    ! [Mb: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,Mb),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_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))),Mb) != ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))),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,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% xor_nat_rec
tff(fact_9415_floor__rat__def,axiom,
    ! [Xb: rat] : aa_rat_int(archim3151403230148437115or_rat,Xb) = the_int(aTP_Lamm_px(Xb)) ).

% floor_rat_def
tff(fact_9416_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_9417_push__bit__nonnegative__int__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se545348938243370406it_int(Nb),K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K) ) ).

% push_bit_nonnegative_int_iff
tff(fact_9418_push__bit__negative__int__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se545348938243370406it_int(Nb),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_9419_concat__bit__of__zero__1,axiom,
    ! [Nb: nat,L: int] : aa_int_int(aa_int_fun_int_int(bit_concat_bit(Nb),zero_zero_int),L) = aa_int_int(bit_se545348938243370406it_int(Nb),L) ).

% concat_bit_of_zero_1
tff(fact_9420_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(aa_int_fun_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_9421_xor__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_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_9422_set__encode__empty,axiom,
    aa_set_nat_nat(nat_set_encode,bot_bot_set_nat) = zero_zero_nat ).

% set_encode_empty
tff(fact_9423_push__bit__of__Suc__0,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Nb),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))),Nb) ).

% push_bit_of_Suc_0
tff(fact_9424_sgn__rat__def,axiom,
    ! [A: rat] :
      aa_rat_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,aa_rat_rat(uminus_uminus_rat,one_one_rat)) ) ).

% sgn_rat_def
tff(fact_9425_bit__xor__int__iff,axiom,
    ! [K: int,L: int,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,K),L)),Nb)
    <=> ~ ( aa_nat_o(bit_se1146084159140164899it_int(K),Nb)
        <=> aa_nat_o(bit_se1146084159140164899it_int(L),Nb) ) ) ).

% bit_xor_int_iff
tff(fact_9426_flip__bit__int__def,axiom,
    ! [Nb: nat,K: int] : aa_int_int(bit_se2159334234014336723it_int(Nb),K) = aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,K),aa_int_int(bit_se545348938243370406it_int(Nb),one_one_int)) ).

% flip_bit_int_def
tff(fact_9427_obtain__pos__sum,axiom,
    ! [R2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),R2)
     => ~ ! [S: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),S)
           => ! [T2: rat] :
                ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),T2)
               => ( R2 != aa_rat_rat(plus_plus_rat(S),T2) ) ) ) ) ).

% obtain_pos_sum
tff(fact_9428_push__bit__nat__eq,axiom,
    ! [Nb: nat,K: int] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Nb),aa_int_nat(nat2,K)) = aa_int_nat(nat2,aa_int_int(bit_se545348938243370406it_int(Nb),K)) ).

% push_bit_nat_eq
tff(fact_9429_XOR__lower,axiom,
    ! [Xb: int,Yb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Yb)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,Xb),Yb)) ) ) ).

% XOR_lower
tff(fact_9430_set__bit__nat__def,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,Mb),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Mb),one_one_nat)) ).

% set_bit_nat_def
tff(fact_9431_flip__bit__nat__def,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,Mb),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Mb),one_one_nat)) ).

% flip_bit_nat_def
tff(fact_9432_bit__push__bit__iff__int,axiom,
    ! [Mb: nat,K: int,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(bit_se545348938243370406it_int(Mb),K)),Nb)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
        & aa_nat_o(bit_se1146084159140164899it_int(K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)) ) ) ).

% bit_push_bit_iff_int
tff(fact_9433_xor__nat__def,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,Mb),Nb) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb))) ).

% xor_nat_def
tff(fact_9434_bit__push__bit__iff__nat,axiom,
    ! [Mb: nat,Q2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Mb),Q2)),Nb)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
        & aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,Q2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)) ) ) ).

% bit_push_bit_iff_nat
tff(fact_9435_concat__bit__eq,axiom,
    ! [Nb: nat,K: int,L: int] : aa_int_int(aa_int_fun_int_int(bit_concat_bit(Nb),K),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(bit_se2923211474154528505it_int(Nb),K)),aa_int_int(bit_se545348938243370406it_int(Nb),L)) ).

% concat_bit_eq
tff(fact_9436_concat__bit__def,axiom,
    ! [Nb: nat,K: int,L: int] : aa_int_int(aa_int_fun_int_int(bit_concat_bit(Nb),K),L) = aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(bit_se2923211474154528505it_int(Nb),K)),aa_int_int(bit_se545348938243370406it_int(Nb),L)) ).

% concat_bit_def
tff(fact_9437_set__bit__int__def,axiom,
    ! [Nb: nat,K: int] : aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,Nb),K) = aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,K),aa_int_int(bit_se545348938243370406it_int(Nb),one_one_int)) ).

% set_bit_int_def
tff(fact_9438_push__bit__int__def,axiom,
    ! [Nb: nat,K: int] : aa_int_int(bit_se545348938243370406it_int(Nb),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))),Nb)) ).

% push_bit_int_def
tff(fact_9439_push__bit__nat__def,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Nb),Mb) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% push_bit_nat_def
tff(fact_9440_push__bit__minus__one,axiom,
    ! [Nb: nat] : aa_int_int(bit_se545348938243370406it_int(Nb),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))),Nb)) ).

% push_bit_minus_one
tff(fact_9441_XOR__upper,axiom,
    ! [Xb: int,Nb: nat,Yb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Yb),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,Xb),Yb)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ) ) ) ).

% XOR_upper
tff(fact_9442_xor__int__rec,axiom,
    ! [K: int,L: int] :
      aa_int_int(aa_int_fun_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(aa_int_fun_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_9443_xor__int__unfold,axiom,
    ! [K: int,L: int] :
      aa_int_int(aa_int_fun_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(aa_int_fun_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(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(aa_int_fun_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_9444_set__encode__insert,axiom,
    ! [A3: set_nat,Nb: nat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( ~ aa_set_nat_o(member_nat(Nb),A3)
       => ( aa_set_nat_nat(nat_set_encode,insert_nat(Nb,A3)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),aa_set_nat_nat(nat_set_encode,A3)) ) ) ) ).

% set_encode_insert
tff(fact_9445_rat__inverse__code,axiom,
    ! [P2: rat] : quotient_of(aa_rat_rat(inverse_inverse_rat,P2)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_py),quotient_of(P2)) ).

% rat_inverse_code
tff(fact_9446_normalize__negative,axiom,
    ! [Q2: int,P2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Q2),zero_zero_int)
     => ( aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,P2),Q2)) = aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(uminus_uminus_int,P2)),aa_int_int(uminus_uminus_int,Q2))) ) ) ).

% normalize_negative
tff(fact_9447_set__vebt__finite,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => aa_set_nat_o(finite_finite_nat,vEBT_VEBT_set_vebt(Ta)) ) ).

% set_vebt_finite
tff(fact_9448_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_9449_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_9450_quotient__of__number_I3_J,axiom,
    ! [K: num] : quotient_of(aa_num_rat(numeral_numeral_rat,K)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_num_int(numeral_numeral_int,K)),one_one_int) ).

% quotient_of_number(3)
tff(fact_9451_normalize__denom__zero,axiom,
    ! [P2: int] : aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,P2),zero_zero_int)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),one_one_int) ).

% normalize_denom_zero
tff(fact_9452_rat__one__code,axiom,
    quotient_of(one_one_rat) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,one_one_int),one_one_int) ).

% rat_one_code
tff(fact_9453_rat__zero__code,axiom,
    quotient_of(zero_zero_rat) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),one_one_int) ).

% rat_zero_code
tff(fact_9454_quotient__of__number_I5_J,axiom,
    ! [K: num] : quotient_of(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,K))) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(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_9455_quotient__of__number_I4_J,axiom,
    quotient_of(aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(uminus_uminus_int,one_one_int)),one_one_int) ).

% quotient_of_number(4)
tff(fact_9456_or__minus__minus__numerals,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,Mb)),one_one_int)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,Nb)),one_one_int))) ).

% or_minus_minus_numerals
tff(fact_9457_and__minus__minus__numerals,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,Mb)),one_one_int)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,Nb)),one_one_int))) ).

% and_minus_minus_numerals
tff(fact_9458_finite__M__bounded__by__nat,axiom,
    ! [P: fun_nat_o,I3: nat] : aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_pz(P),I3))) ).

% finite_M_bounded_by_nat
tff(fact_9459_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_qa(F),U))) ) ).

% finite_less_ub
tff(fact_9460_divide__rat__def,axiom,
    ! [Q2: rat,R2: rat] : aa_rat_rat(divide_divide_rat(Q2),R2) = aa_rat_rat(times_times_rat(Q2),aa_rat_rat(inverse_inverse_rat,R2)) ).

% divide_rat_def
tff(fact_9461_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_9462_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_9463_bounded__nat__set__is__finite,axiom,
    ! [N2: set_nat,Nb: nat] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),N2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Nb) )
     => aa_set_nat_o(finite_finite_nat,N2) ) ).

% bounded_nat_set_is_finite
tff(fact_9464_diff__rat__def,axiom,
    ! [Q2: rat,R2: rat] : aa_rat_rat(minus_minus_rat(Q2),R2) = aa_rat_rat(plus_plus_rat(Q2),aa_rat_rat(uminus_uminus_rat,R2)) ).

% diff_rat_def
tff(fact_9465_bit__not__int__iff,axiom,
    ! [K: int,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(bit_ri7919022796975470100ot_int,K)),Nb)
    <=> ~ aa_nat_o(bit_se1146084159140164899it_int(K),Nb) ) ).

% bit_not_int_iff
tff(fact_9466_or__int__def,axiom,
    ! [K: int,L: int] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,K),L) = aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(bit_ri7919022796975470100ot_int,K)),aa_int_int(bit_ri7919022796975470100ot_int,L))) ).

% or_int_def
tff(fact_9467_rat__divide__code,axiom,
    ! [P2: rat,Q2: rat] : quotient_of(aa_rat_rat(divide_divide_rat(P2),Q2)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_qc(Q2)),quotient_of(P2)) ).

% rat_divide_code
tff(fact_9468_rat__times__code,axiom,
    ! [P2: rat,Q2: rat] : quotient_of(aa_rat_rat(times_times_rat(P2),Q2)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_qe(Q2)),quotient_of(P2)) ).

% rat_times_code
tff(fact_9469_set__encode__inf,axiom,
    ! [A3: set_nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_nat_nat(nat_set_encode,A3) = zero_zero_nat ) ) ).

% set_encode_inf
tff(fact_9470_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_9471_and__not__numerals_I1_J,axiom,
    aa_int_int(aa_int_fun_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_9472_or__not__numerals_I1_J,axiom,
    aa_int_int(aa_int_fun_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_9473_rat__plus__code,axiom,
    ! [P2: rat,Q2: rat] : quotient_of(aa_rat_rat(plus_plus_rat(P2),Q2)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_qg(Q2)),quotient_of(P2)) ).

% rat_plus_code
tff(fact_9474_rat__minus__code,axiom,
    ! [P2: rat,Q2: rat] : quotient_of(aa_rat_rat(minus_minus_rat(P2),Q2)) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aTP_Lamm_qi(Q2)),quotient_of(P2)) ).

% rat_minus_code
tff(fact_9475_unset__bit__int__def,axiom,
    ! [Nb: nat,K: int] : aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,Nb),K) = aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K),aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(bit_se545348938243370406it_int(Nb),one_one_int))) ).

% unset_bit_int_def
tff(fact_9476_xor__int__def,axiom,
    ! [K: int,L: int] : aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,K),L) = aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K),aa_int_int(bit_ri7919022796975470100ot_int,L))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(bit_ri7919022796975470100ot_int,K)),L)) ).

% xor_int_def
tff(fact_9477_finite__divisors__nat,axiom,
    ! [Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
     => aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_ai,Mb))) ) ).

% finite_divisors_nat
tff(fact_9478_quotient__of__denom__pos,axiom,
    ! [R2: rat,P2: int,Q2: int] :
      ( ( quotient_of(R2) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,P2),Q2) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Q2) ) ).

% quotient_of_denom_pos
tff(fact_9479_subset__eq__atLeast0__atMost__finite,axiom,
    ! [N2: set_nat,Nb: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(N2),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb))
     => aa_set_nat_o(finite_finite_nat,N2) ) ).

% subset_eq_atLeast0_atMost_finite
tff(fact_9480_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_9481_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_9482_and__not__numerals_I2_J,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,one_one_int),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))) = one_one_int ).

% and_not_numerals(2)
tff(fact_9483_and__not__numerals_I4_J,axiom,
    ! [Mb: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Mb))),aa_int_int(bit_ri7919022796975470100ot_int,one_one_int)) = aa_num_int(numeral_numeral_int,aa_num_num(bit0,Mb)) ).

% and_not_numerals(4)
tff(fact_9484_or__not__numerals_I2_J,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,one_one_int),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))) = aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb))) ).

% or_not_numerals(2)
tff(fact_9485_or__not__numerals_I4_J,axiom,
    ! [Mb: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Mb))),aa_int_int(bit_ri7919022796975470100ot_int,one_one_int)) = aa_int_int(bit_ri7919022796975470100ot_int,one_one_int) ).

% or_not_numerals(4)
tff(fact_9486_bit__minus__int__iff,axiom,
    ! [K: int,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(uminus_uminus_int,K)),Nb)
    <=> 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))),Nb) ) ).

% bit_minus_int_iff
tff(fact_9487_int__numeral__or__not__num__neg,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit_or_not_num_neg(Mb),Nb))) ).

% int_numeral_or_not_num_neg
tff(fact_9488_int__numeral__not__or__num__neg,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Mb))),aa_num_int(numeral_numeral_int,Nb)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit_or_not_num_neg(Nb),Mb))) ).

% int_numeral_not_or_num_neg
tff(fact_9489_numeral__or__not__num__eq,axiom,
    ! [Mb: num,Nb: num] : aa_num_int(numeral_numeral_int,aa_num_num(bit_or_not_num_neg(Mb),Nb)) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb)))) ).

% numeral_or_not_num_eq
tff(fact_9490_and__not__numerals_I5_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Mb))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))) = 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(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb)))) ).

% and_not_numerals(5)
tff(fact_9491_and__not__numerals_I7_J,axiom,
    ! [Mb: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Mb))),aa_int_int(bit_ri7919022796975470100ot_int,one_one_int)) = aa_num_int(numeral_numeral_int,aa_num_num(bit0,Mb)) ).

% and_not_numerals(7)
tff(fact_9492_or__not__numerals_I3_J,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,one_one_int),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))) = aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb))) ).

% or_not_numerals(3)
tff(fact_9493_and__not__numerals_I3_J,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,one_one_int),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))) = zero_zero_int ).

% and_not_numerals(3)
tff(fact_9494_or__not__numerals_I7_J,axiom,
    ! [Mb: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Mb))),aa_int_int(bit_ri7919022796975470100ot_int,one_one_int)) = aa_int_int(bit_ri7919022796975470100ot_int,zero_zero_int) ).

% or_not_numerals(7)
tff(fact_9495_normalize__denom__pos,axiom,
    ! [R2: product_prod_int_int,P2: int,Q2: int] :
      ( ( aa_Pro6970504127699319548nt_int(normalize,R2) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,P2),Q2) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Q2) ) ).

% normalize_denom_pos
tff(fact_9496_normalize__crossproduct,axiom,
    ! [Q2: int,S2: int,P2: int,R2: int] :
      ( ( Q2 != zero_zero_int )
     => ( ( S2 != zero_zero_int )
       => ( ( aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,P2),Q2)) = aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,R2),S2)) )
         => ( aa_int_int(aa_int_fun_int_int(times_times_int,P2),S2) = aa_int_int(aa_int_fun_int_int(times_times_int,R2),Q2) ) ) ) ) ).

% normalize_crossproduct
tff(fact_9497_and__not__numerals_I6_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Mb))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))) = 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(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb)))) ).

% and_not_numerals(6)
tff(fact_9498_and__not__numerals_I9_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Mb))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))) = 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(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb)))) ).

% and_not_numerals(9)
tff(fact_9499_or__not__numerals_I6_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Mb))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))) = 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(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb)))) ).

% or_not_numerals(6)
tff(fact_9500_rat__less__code,axiom,
    ! [P2: rat,Q2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,P2),Q2)
    <=> aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(aTP_Lamm_qk(Q2)),quotient_of(P2)) ) ).

% rat_less_code
tff(fact_9501_rat__floor__code,axiom,
    ! [P2: rat] : aa_rat_int(archim3151403230148437115or_rat,P2) = aa_Pro6799258434465214121nt_int(produc8211389475949308722nt_int(divide_divide_int),quotient_of(P2)) ).

% rat_floor_code
tff(fact_9502_rat__less__eq__code,axiom,
    ! [P2: rat,Q2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,P2),Q2)
    <=> aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(aTP_Lamm_qm(Q2)),quotient_of(P2)) ) ).

% rat_less_eq_code
tff(fact_9503_even__set__encode__iff,axiom,
    ! [A3: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( 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,A3))
      <=> ~ aa_set_nat_o(member_nat(zero_zero_nat),A3) ) ) ).

% even_set_encode_iff
tff(fact_9504_or__not__numerals_I5_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Mb))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))) = 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(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb))))) ).

% or_not_numerals(5)
tff(fact_9505_and__not__numerals_I8_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Mb))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))) = 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(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb))))) ).

% and_not_numerals(8)
tff(fact_9506_or__not__numerals_I9_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Mb))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))) = 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(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb))))) ).

% or_not_numerals(9)
tff(fact_9507_or__not__numerals_I8_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Mb))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))) = 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(bit_se1409905431419307370or_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb))))) ).

% or_not_numerals(8)
tff(fact_9508_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_9509_finite__Collect__le__nat,axiom,
    ! [K: nat] : aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_nn,K))) ).

% finite_Collect_le_nat
tff(fact_9510_finite__Collect__less__nat,axiom,
    ! [K: nat] : aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_ei,K))) ).

% finite_Collect_less_nat
tff(fact_9511_finite__interval__int1,axiom,
    ! [A: int,B: int] : finite_finite_int(collect_int(aa_int_fun_int_o(aTP_Lamm_qn(A),B))) ).

% finite_interval_int1
tff(fact_9512_finite__interval__int4,axiom,
    ! [A: int,B: int] : finite_finite_int(collect_int(aa_int_fun_int_o(aTP_Lamm_qo(A),B))) ).

% finite_interval_int4
tff(fact_9513_finite__interval__int3,axiom,
    ! [A: int,B: int] : finite_finite_int(collect_int(aa_int_fun_int_o(aTP_Lamm_qp(A),B))) ).

% finite_interval_int3
tff(fact_9514_finite__interval__int2,axiom,
    ! [A: int,B: int] : finite_finite_int(collect_int(aa_int_fun_int_o(aTP_Lamm_qq(A),B))) ).

% finite_interval_int2
tff(fact_9515_finite__nth__roots,axiom,
    ! [Nb: nat,C: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => finite3207457112153483333omplex(collect_complex(aa_com5808801942499186519plex_o(aTP_Lamm_dk(Nb),C))) ) ).

% finite_nth_roots
tff(fact_9516_finite__divisors__int,axiom,
    ! [I3: int] :
      ( ( I3 != zero_zero_int )
     => finite_finite_int(collect_int(aa_int_fun_int_o(aTP_Lamm_ag,I3))) ) ).

% finite_divisors_int
tff(fact_9517_quotient__of__int,axiom,
    ! [A: int] : quotient_of(aa_int_rat(of_int,A)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),one_one_int) ).

% quotient_of_int
tff(fact_9518_bij__betw__nth__root__unity,axiom,
    ! [C: complex,Nb: nat] :
      ( ( C != zero_zero_complex )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => 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,Nb),aa_complex_real(real_V1022390504157884413omplex,C)))),aa_real_complex(cis,aa_real_real(divide_divide_real(aa_complex_real(arg,C)),aa_nat_real(semiri5074537144036343181t_real,Nb))))),collect_complex(aTP_Lamm_dl(Nb)),collect_complex(aa_nat_fun_complex_o(aTP_Lamm_qr(C),Nb))) ) ) ).

% bij_betw_nth_root_unity
tff(fact_9519_Frct__code__post_I5_J,axiom,
    ! [K: num] : frct(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,one_one_int),aa_num_int(numeral_numeral_int,K))) = aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,K)) ).

% Frct_code_post(5)
tff(fact_9520_real__root__Suc__0,axiom,
    ! [Xb: real] : aa_real_real(aa_nat_fun_real_real(root,aa_nat_nat(suc,zero_zero_nat)),Xb) = Xb ).

% real_root_Suc_0
tff(fact_9521_root__0,axiom,
    ! [Xb: real] : aa_real_real(aa_nat_fun_real_real(root,zero_zero_nat),Xb) = zero_zero_real ).

% root_0
tff(fact_9522_real__root__eq__iff,axiom,
    ! [Nb: nat,Xb: real,Yb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( ( aa_real_real(aa_nat_fun_real_real(root,Nb),Xb) = aa_real_real(aa_nat_fun_real_real(root,Nb),Yb) )
      <=> ( Xb = Yb ) ) ) ).

% real_root_eq_iff
tff(fact_9523_real__root__eq__0__iff,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( ( aa_real_real(aa_nat_fun_real_real(root,Nb),Xb) = zero_zero_real )
      <=> ( Xb = zero_zero_real ) ) ) ).

% real_root_eq_0_iff
tff(fact_9524_real__root__less__iff,axiom,
    ! [Nb: nat,Xb: real,Yb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),aa_real_real(aa_nat_fun_real_real(root,Nb),Yb))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Yb) ) ) ).

% real_root_less_iff
tff(fact_9525_real__root__le__iff,axiom,
    ! [Nb: nat,Xb: real,Yb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),aa_real_real(aa_nat_fun_real_real(root,Nb),Yb))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Yb) ) ) ).

% real_root_le_iff
tff(fact_9526_real__root__eq__1__iff,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( ( aa_real_real(aa_nat_fun_real_real(root,Nb),Xb) = one_one_real )
      <=> ( Xb = one_one_real ) ) ) ).

% real_root_eq_1_iff
tff(fact_9527_real__root__one,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_real(aa_nat_fun_real_real(root,Nb),one_one_real) = one_one_real ) ) ).

% real_root_one
tff(fact_9528_real__root__lt__0__iff,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),zero_zero_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real) ) ) ).

% real_root_lt_0_iff
tff(fact_9529_real__root__gt__0__iff,axiom,
    ! [Nb: nat,Yb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,Nb),Yb))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Yb) ) ) ).

% real_root_gt_0_iff
tff(fact_9530_real__root__le__0__iff,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),zero_zero_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real) ) ) ).

% real_root_le_0_iff
tff(fact_9531_real__root__ge__0__iff,axiom,
    ! [Nb: nat,Yb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,Nb),Yb))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Yb) ) ) ).

% real_root_ge_0_iff
tff(fact_9532_real__root__lt__1__iff,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),one_one_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),one_one_real) ) ) ).

% real_root_lt_1_iff
tff(fact_9533_real__root__gt__1__iff,axiom,
    ! [Nb: nat,Yb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(aa_nat_fun_real_real(root,Nb),Yb))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Yb) ) ) ).

% real_root_gt_1_iff
tff(fact_9534_real__root__le__1__iff,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),one_one_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real) ) ) ).

% real_root_le_1_iff
tff(fact_9535_real__root__ge__1__iff,axiom,
    ! [Nb: nat,Yb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(aa_nat_fun_real_real(root,Nb),Yb))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Yb) ) ) ).

% real_root_ge_1_iff
tff(fact_9536_real__root__pow__pos2,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),Nb) = Xb ) ) ) ).

% real_root_pow_pos2
tff(fact_9537_real__root__mult__exp,axiom,
    ! [Mb: nat,Nb: nat,Xb: real] : aa_real_real(aa_nat_fun_real_real(root,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)),Xb) = aa_real_real(aa_nat_fun_real_real(root,Mb),aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)) ).

% real_root_mult_exp
tff(fact_9538_real__root__mult,axiom,
    ! [Nb: nat,Xb: real,Yb: real] : aa_real_real(aa_nat_fun_real_real(root,Nb),aa_real_real(times_times_real(Xb),Yb)) = aa_real_real(times_times_real(aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),aa_real_real(aa_nat_fun_real_real(root,Nb),Yb)) ).

% real_root_mult
tff(fact_9539_real__root__less__mono,axiom,
    ! [Nb: nat,Xb: real,Yb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Yb)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),aa_real_real(aa_nat_fun_real_real(root,Nb),Yb)) ) ) ).

% real_root_less_mono
tff(fact_9540_real__root__le__mono,axiom,
    ! [Nb: nat,Xb: real,Yb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Yb)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),aa_real_real(aa_nat_fun_real_real(root,Nb),Yb)) ) ) ).

% real_root_le_mono
tff(fact_9541_real__root__power,axiom,
    ! [Nb: nat,Xb: real,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_real(aa_nat_fun_real_real(root,Nb),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),K)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),K) ) ) ).

% real_root_power
tff(fact_9542_real__root__abs,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_real(aa_nat_fun_real_real(root,Nb),aa_real_real(abs_abs_real,Xb)) = aa_real_real(abs_abs_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)) ) ) ).

% real_root_abs
tff(fact_9543_sgn__root,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_real(sgn_sgn_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)) = aa_real_real(sgn_sgn_real,Xb) ) ) ).

% sgn_root
tff(fact_9544_real__root__gt__zero,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)) ) ) ).

% real_root_gt_zero
tff(fact_9545_real__root__strict__decreasing,axiom,
    ! [Nb: nat,N2: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N2)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,N2),Xb)),aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)) ) ) ) ).

% real_root_strict_decreasing
tff(fact_9546_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_9547_root__abs__power,axiom,
    ! [Nb: nat,Yb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_real(abs_abs_real,aa_real_real(aa_nat_fun_real_real(root,Nb),aa_nat_real(aa_real_fun_nat_real(power_power_real,Yb),Nb))) = aa_real_real(abs_abs_real,Yb) ) ) ).

% root_abs_power
tff(fact_9548_real__root__pos__pos,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)) ) ) ).

% real_root_pos_pos
tff(fact_9549_real__root__strict__increasing,axiom,
    ! [Nb: nat,N2: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N2)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),one_one_real)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),aa_real_real(aa_nat_fun_real_real(root,N2),Xb)) ) ) ) ) ).

% real_root_strict_increasing
tff(fact_9550_real__root__decreasing,axiom,
    ! [Nb: nat,N2: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N2)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xb)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,N2),Xb)),aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)) ) ) ) ).

% real_root_decreasing
tff(fact_9551_real__root__pow__pos,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),Nb) = Xb ) ) ) ).

% real_root_pow_pos
tff(fact_9552_real__root__pos__unique,axiom,
    ! [Nb: nat,Yb: real,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Yb)
       => ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Yb),Nb) = Xb )
         => ( aa_real_real(aa_nat_fun_real_real(root,Nb),Xb) = Yb ) ) ) ) ).

% real_root_pos_unique
tff(fact_9553_real__root__power__cancel,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
       => ( aa_real_real(aa_nat_fun_real_real(root,Nb),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)) = Xb ) ) ) ).

% real_root_power_cancel
tff(fact_9554_odd__real__root__power__cancel,axiom,
    ! [Nb: nat,Xb: real] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_real_real(aa_nat_fun_real_real(root,Nb),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)) = Xb ) ) ).

% odd_real_root_power_cancel
tff(fact_9555_odd__real__root__unique,axiom,
    ! [Nb: nat,Yb: real,Xb: real] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Yb),Nb) = Xb )
       => ( aa_real_real(aa_nat_fun_real_real(root,Nb),Xb) = Yb ) ) ) ).

% odd_real_root_unique
tff(fact_9556_odd__real__root__pow,axiom,
    ! [Nb: nat,Xb: real] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),Nb) = Xb ) ) ).

% odd_real_root_pow
tff(fact_9557_Frct__code__post_I1_J,axiom,
    ! [A: int] : frct(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),A)) = zero_zero_rat ).

% Frct_code_post(1)
tff(fact_9558_Frct__code__post_I2_J,axiom,
    ! [A: int] : frct(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,A),zero_zero_int)) = zero_zero_rat ).

% Frct_code_post(2)
tff(fact_9559_real__root__increasing,axiom,
    ! [Nb: nat,N2: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N2)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),aa_real_real(aa_nat_fun_real_real(root,N2),Xb)) ) ) ) ) ).

% real_root_increasing
tff(fact_9560_Frct__code__post_I3_J,axiom,
    frct(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,one_one_int),one_one_int)) = one_one_rat ).

% Frct_code_post(3)
tff(fact_9561_root__sgn__power,axiom,
    ! [Nb: nat,Yb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_real(aa_nat_fun_real_real(root,Nb),aa_real_real(times_times_real(aa_real_real(sgn_sgn_real,Yb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,Yb)),Nb))) = Yb ) ) ).

% root_sgn_power
tff(fact_9562_sgn__power__root,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_real(times_times_real(aa_real_real(sgn_sgn_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb))),Nb)) = Xb ) ) ).

% sgn_power_root
tff(fact_9563_ln__root,axiom,
    ! [Nb: nat,B: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( 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,Nb),B)) = aa_real_real(divide_divide_real(aa_real_real(ln_ln_real,B)),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ) ) ).

% ln_root
tff(fact_9564_log__root,axiom,
    ! [Nb: nat,A: real,B: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => ( aa_real_real(log(B),aa_real_real(aa_nat_fun_real_real(root,Nb),A)) = aa_real_real(divide_divide_real(aa_real_real(log(B),A)),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ) ) ).

% log_root
tff(fact_9565_log__base__root,axiom,
    ! [Nb: nat,B: real,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B)
       => ( aa_real_real(log(aa_real_real(aa_nat_fun_real_real(root,Nb),B)),Xb) = aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(log(B),Xb)) ) ) ) ).

% log_base_root
tff(fact_9566_split__root,axiom,
    ! [P: fun_real_o,Nb: nat,Xb: real] :
      ( aa_real_o(P,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb))
    <=> ( ( ( Nb = zero_zero_nat )
         => aa_real_o(P,zero_zero_real) )
        & ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
         => ! [Y4: real] :
              ( ( aa_real_real(times_times_real(aa_real_real(sgn_sgn_real,Y4)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,Y4)),Nb)) = Xb )
             => aa_real_o(P,Y4) ) ) ) ) ).

% split_root
tff(fact_9567_Frct__code__post_I4_J,axiom,
    ! [K: num] : frct(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_num_int(numeral_numeral_int,K)),one_one_int)) = aa_num_rat(numeral_numeral_rat,K) ).

% Frct_code_post(4)
tff(fact_9568_root__powr__inverse,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_real_real(aa_nat_fun_real_real(root,Nb),Xb) = aa_real_real(powr_real(Xb),aa_real_real(divide_divide_real(one_one_real),aa_nat_real(semiri5074537144036343181t_real,Nb))) ) ) ) ).

% root_powr_inverse
tff(fact_9569_Frct__code__post_I6_J,axiom,
    ! [K: num,L: num] : frct(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_num_int(numeral_numeral_int,K)),aa_num_int(numeral_numeral_int,L))) = aa_rat_rat(divide_divide_rat(aa_num_rat(numeral_numeral_rat,K)),aa_num_rat(numeral_numeral_rat,L)) ).

% Frct_code_post(6)
tff(fact_9570_infinite__int__iff__unbounded,axiom,
    ! [S3: set_int] :
      ( ~ finite_finite_int(S3)
    <=> ! [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,S3) ) ) ).

% infinite_int_iff_unbounded
tff(fact_9571_infinite__int__iff__unbounded__le,axiom,
    ! [S3: set_int] :
      ( ~ finite_finite_int(S3)
    <=> ! [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,S3) ) ) ).

% infinite_int_iff_unbounded_le
tff(fact_9572_infinite__nat__iff__unbounded,axiom,
    ! [S3: set_nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,S3)
    <=> ! [M5: nat] :
        ? [N4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M5),N4)
          & aa_set_nat_o(member_nat(N4),S3) ) ) ).

% infinite_nat_iff_unbounded
tff(fact_9573_unbounded__k__infinite,axiom,
    ! [K: nat,S3: set_nat] :
      ( ! [M2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),M2)
         => ? [N7: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),N7)
              & aa_set_nat_o(member_nat(N7),S3) ) )
     => ~ aa_set_nat_o(finite_finite_nat,S3) ) ).

% unbounded_k_infinite
tff(fact_9574_infinite__nat__iff__unbounded__le,axiom,
    ! [S3: set_nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,S3)
    <=> ! [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),S3) ) ) ).

% infinite_nat_iff_unbounded_le
tff(fact_9575_valid__eq2,axiom,
    ! [Ta: vEBT_VEBT,D: nat] :
      ( aa_nat_o(vEBT_VEBT_valid(Ta),D)
     => aa_nat_o(vEBT_invar_vebt(Ta),D) ) ).

% valid_eq2
tff(fact_9576_valid__eq,axiom,
    ! [Ta: vEBT_VEBT,D: nat] :
      ( aa_nat_o(vEBT_VEBT_valid(Ta),D)
    <=> aa_nat_o(vEBT_invar_vebt(Ta),D) ) ).

% valid_eq
tff(fact_9577_valid__eq1,axiom,
    ! [Ta: vEBT_VEBT,D: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),D)
     => aa_nat_o(vEBT_VEBT_valid(Ta),D) ) ).

% valid_eq1
tff(fact_9578_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_9579_Sum__Ico__nat,axiom,
    ! [Mb: nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_ar),aa_nat_set_nat(set_or4665077453230672383an_nat(Mb),Nb)) = 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,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% Sum_Ico_nat
tff(fact_9580_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_9581_Cauchy__iff2,axiom,
    ! [X6: fun_nat_real] :
      ( topolo4055970368930404560y_real(X6)
    <=> ! [J3: nat] :
        ? [M7: nat] :
        ! [M5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M7),M5)
         => ! [N4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M7),N4)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_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_9582_atLeastLessThan__singleton,axiom,
    ! [Mb: nat] : aa_nat_set_nat(set_or4665077453230672383an_nat(Mb),aa_nat_nat(suc,Mb)) = insert_nat(Mb,bot_bot_set_nat) ).

% atLeastLessThan_singleton
tff(fact_9583_ex__nat__less__eq,axiom,
    ! [Nb: nat,P: fun_nat_o] :
      ( ? [M5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M5),Nb)
          & aa_nat_o(P,M5) )
    <=> ? [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Nb))
          & aa_nat_o(P,X4) ) ) ).

% ex_nat_less_eq
tff(fact_9584_all__nat__less__eq,axiom,
    ! [Nb: nat,P: fun_nat_o] :
      ( ! [M5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M5),Nb)
         => aa_nat_o(P,M5) )
    <=> ! [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Nb))
         => aa_nat_o(P,X4) ) ) ).

% all_nat_less_eq
tff(fact_9585_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_9586_lessThan__atLeast0,axiom,
    ! [Nb: nat] : aa_nat_set_nat(set_ord_lessThan_nat,Nb) = aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Nb) ).

% lessThan_atLeast0
tff(fact_9587_atLeastLessThan0,axiom,
    ! [Mb: nat] : aa_nat_set_nat(set_or4665077453230672383an_nat(Mb),zero_zero_nat) = bot_bot_set_nat ).

% atLeastLessThan0
tff(fact_9588_atLeast0__lessThan__Suc,axiom,
    ! [Nb: nat] : aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),aa_nat_nat(suc,Nb)) = insert_nat(Nb,aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Nb)) ).

% atLeast0_lessThan_Suc
tff(fact_9589_subset__eq__atLeast0__lessThan__finite,axiom,
    ! [N2: set_nat,Nb: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(N2),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Nb))
     => aa_set_nat_o(finite_finite_nat,N2) ) ).

% subset_eq_atLeast0_lessThan_finite
tff(fact_9590_atLeastLessThanSuc,axiom,
    ! [Mb: nat,Nb: nat] :
      aa_nat_set_nat(set_or4665077453230672383an_nat(Mb),aa_nat_nat(suc,Nb)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb),insert_nat(Nb,aa_nat_set_nat(set_or4665077453230672383an_nat(Mb),Nb)),bot_bot_set_nat) ).

% atLeastLessThanSuc
tff(fact_9591_prod__Suc__fact,axiom,
    ! [Nb: nat] : aa_set_nat_nat(groups708209901874060359at_nat(suc),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Nb)) = aa_nat_nat(semiri1408675320244567234ct_nat,Nb) ).

% prod_Suc_fact
tff(fact_9592_prod__Suc__Suc__fact,axiom,
    ! [Nb: nat] : aa_set_nat_nat(groups708209901874060359at_nat(suc),aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(suc,zero_zero_nat)),Nb)) = aa_nat_nat(semiri1408675320244567234ct_nat,Nb) ).

% prod_Suc_Suc_fact
tff(fact_9593_atLeastLessThan__nat__numeral,axiom,
    ! [Mb: nat,K: num] :
      aa_nat_set_nat(set_or4665077453230672383an_nat(Mb),aa_num_nat(numeral_numeral_nat,K)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_num_nat(pred_numeral,K)),insert_nat(aa_num_nat(pred_numeral,K),aa_nat_set_nat(set_or4665077453230672383an_nat(Mb),aa_num_nat(pred_numeral,K))),bot_bot_set_nat) ).

% atLeastLessThan_nat_numeral
tff(fact_9594_atLeast1__lessThan__eq__remove0,axiom,
    ! [Nb: nat] : aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(suc,zero_zero_nat)),Nb) = minus_minus_set_nat(aa_nat_set_nat(set_ord_lessThan_nat,Nb),insert_nat(zero_zero_nat,bot_bot_set_nat)) ).

% atLeast1_lessThan_eq_remove0
tff(fact_9595_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_9596_Chebyshev__sum__upper__nat,axiom,
    ! [Nb: 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),Nb)
           => 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),Nb)
             => 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,Nb),aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_ku(A),B)),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Nb)))),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),Nb))),aa_set_nat_nat(groups3542108847815614940at_nat(B),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Nb)))) ) ) ).

% Chebyshev_sum_upper_nat
tff(fact_9597_finite__atLeastZeroLessThan__int,axiom,
    ! [U: int] : finite_finite_int(set_or4662586982721622107an_int(zero_zero_int,U)) ).

% finite_atLeastZeroLessThan_int
tff(fact_9598_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_9599_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_9600_Code__Target__Int_Opositive__def,axiom,
    code_Target_positive = numeral_numeral_int ).

% Code_Target_Int.positive_def
tff(fact_9601_VEBT_Osize__gen_I2_J,axiom,
    ! [X21: $o,X22: $o] : aa_VEBT_VEBT_nat(vEBT_size_VEBT,vEBT_Leaf((X21),(X22))) = zero_zero_nat ).

% VEBT.size_gen(2)
tff(fact_9602_divmod__step__integer__def,axiom,
    ! [L: num,Qr: produc8923325533196201883nteger] : unique4921790084139445826nteger(L,Qr) = produc6916734918728496179nteger(aTP_Lamm_bg(L),Qr) ).

% divmod_step_integer_def
tff(fact_9603_csqrt_Osimps_I1_J,axiom,
    ! [Z: complex] : aa_complex_real(re,aa_complex_complex(csqrt,Z)) = aa_real_real(sqrt,aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(aa_complex_real(real_V1022390504157884413omplex,Z)),aa_complex_real(re,Z))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ).

% csqrt.simps(1)
tff(fact_9604_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_9605_times__integer__code_I1_J,axiom,
    ! [K: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,K),zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% times_integer_code(1)
tff(fact_9606_times__integer__code_I2_J,axiom,
    ! [L: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,zero_z3403309356797280102nteger),L) = zero_z3403309356797280102nteger ).

% times_integer_code(2)
tff(fact_9607_plus__integer__code_I1_J,axiom,
    ! [K: code_integer] : plus_p5714425477246183910nteger(K,zero_z3403309356797280102nteger) = K ).

% plus_integer_code(1)
tff(fact_9608_plus__integer__code_I2_J,axiom,
    ! [L: code_integer] : plus_p5714425477246183910nteger(zero_z3403309356797280102nteger,L) = L ).

% plus_integer_code(2)
tff(fact_9609_one__complex_Osimps_I1_J,axiom,
    aa_complex_real(re,one_one_complex) = one_one_real ).

% one_complex.simps(1)
tff(fact_9610_plus__complex_Osimps_I1_J,axiom,
    ! [Xb: complex,Yb: complex] : aa_complex_real(re,aa_complex_complex(plus_plus_complex(Xb),Yb)) = aa_real_real(plus_plus_real(aa_complex_real(re,Xb)),aa_complex_real(re,Yb)) ).

% plus_complex.simps(1)
tff(fact_9611_scaleR__complex_Osimps_I1_J,axiom,
    ! [R2: real,Xb: complex] : aa_complex_real(re,aa_complex_complex(real_V2046097035970521341omplex(R2),Xb)) = aa_real_real(times_times_real(R2),aa_complex_real(re,Xb)) ).

% scaleR_complex.simps(1)
tff(fact_9612_zero__natural_Orsp,axiom,
    zero_zero_nat = zero_zero_nat ).

% zero_natural.rsp
tff(fact_9613_zero__integer_Orsp,axiom,
    zero_zero_int = zero_zero_int ).

% zero_integer.rsp
tff(fact_9614_one__integer_Orsp,axiom,
    one_one_int = one_one_int ).

% one_integer.rsp
tff(fact_9615_one__natural_Orsp,axiom,
    one_one_nat = one_one_nat ).

% one_natural.rsp
tff(fact_9616_cmod__plus__Re__le__0__iff,axiom,
    ! [Z: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_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_9617_cos__n__Re__cis__pow__n,axiom,
    ! [Nb: nat,A: real] : aa_real_real(cos_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),A)) = aa_complex_real(re,aa_nat_complex(power_power_complex(aa_real_complex(cis,A)),Nb)) ).

% cos_n_Re_cis_pow_n
tff(fact_9618_csqrt_Ocode,axiom,
    ! [Z: complex] :
      aa_complex_complex(csqrt,Z) = aa_real_complex(complex2(aa_real_real(sqrt,aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(aa_complex_real(real_V1022390504157884413omplex,Z)),aa_complex_real(re,Z))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),
        aa_real_real(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(divide_divide_real(aa_real_real(minus_minus_real(aa_complex_real(real_V1022390504157884413omplex,Z)),aa_complex_real(re,Z))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))) ).

% csqrt.code
tff(fact_9619_csqrt_Osimps_I2_J,axiom,
    ! [Z: complex] :
      aa_complex_real(im,aa_complex_complex(csqrt,Z)) = aa_real_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(divide_divide_real(aa_real_real(minus_minus_real(aa_complex_real(real_V1022390504157884413omplex,Z)),aa_complex_real(re,Z))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% csqrt.simps(2)
tff(fact_9620_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(aa_int_fun_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_9621_Im__i__times,axiom,
    ! [Z: complex] : aa_complex_real(im,aa_complex_complex(times_times_complex(imaginary_unit),Z)) = aa_complex_real(re,Z) ).

% Im_i_times
tff(fact_9622_Re__i__times,axiom,
    ! [Z: complex] : aa_complex_real(re,aa_complex_complex(times_times_complex(imaginary_unit),Z)) = aa_real_real(uminus_uminus_real,aa_complex_real(im,Z)) ).

% Re_i_times
tff(fact_9623_csqrt__minus,axiom,
    ! [Xb: complex] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(im,Xb)),zero_zero_real)
        | ( ( aa_complex_real(im,Xb) = zero_zero_real )
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(re,Xb)) ) )
     => ( aa_complex_complex(csqrt,aa_complex_complex(uminus1482373934393186551omplex,Xb)) = aa_complex_complex(times_times_complex(imaginary_unit),aa_complex_complex(csqrt,Xb)) ) ) ).

% csqrt_minus
tff(fact_9624_csqrt__of__real__nonpos,axiom,
    ! [Xb: complex] :
      ( ( aa_complex_real(im,Xb) = zero_zero_real )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(re,Xb)),zero_zero_real)
       => ( aa_complex_complex(csqrt,Xb) = aa_complex_complex(times_times_complex(imaginary_unit),aa_real_complex(real_V4546457046886955230omplex,aa_real_real(sqrt,aa_real_real(abs_abs_real,aa_complex_real(re,Xb))))) ) ) ) ).

% csqrt_of_real_nonpos
tff(fact_9625_divide__integer_Oabs__eq,axiom,
    ! [Xaa: int,Xb: int] : divide6298287555418463151nteger(aa_int_Code_integer(code_integer_of_int,Xaa),aa_int_Code_integer(code_integer_of_int,Xb)) = aa_int_Code_integer(code_integer_of_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,Xaa),Xb)) ).

% divide_integer.abs_eq
tff(fact_9626_less__integer_Oabs__eq,axiom,
    ! [Xaa: int,Xb: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_int_Code_integer(code_integer_of_int,Xaa)),aa_int_Code_integer(code_integer_of_int,Xb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Xaa),Xb) ) ).

% less_integer.abs_eq
tff(fact_9627_zero__integer__def,axiom,
    zero_z3403309356797280102nteger = aa_int_Code_integer(code_integer_of_int,zero_zero_int) ).

% zero_integer_def
tff(fact_9628_imaginary__unit_Osimps_I2_J,axiom,
    aa_complex_real(im,imaginary_unit) = one_one_real ).

% imaginary_unit.simps(2)
tff(fact_9629_one__complex_Osimps_I2_J,axiom,
    aa_complex_real(im,one_one_complex) = zero_zero_real ).

% one_complex.simps(2)
tff(fact_9630_plus__integer_Oabs__eq,axiom,
    ! [Xaa: int,Xb: int] : plus_p5714425477246183910nteger(aa_int_Code_integer(code_integer_of_int,Xaa),aa_int_Code_integer(code_integer_of_int,Xb)) = aa_int_Code_integer(code_integer_of_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xaa),Xb)) ).

% plus_integer.abs_eq
tff(fact_9631_times__integer_Oabs__eq,axiom,
    ! [Xaa: int,Xb: int] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_int_Code_integer(code_integer_of_int,Xaa)),aa_int_Code_integer(code_integer_of_int,Xb)) = aa_int_Code_integer(code_integer_of_int,aa_int_int(aa_int_fun_int_int(times_times_int,Xaa),Xb)) ).

% times_integer.abs_eq
tff(fact_9632_one__integer__def,axiom,
    one_one_Code_integer = aa_int_Code_integer(code_integer_of_int,one_one_int) ).

% one_integer_def
tff(fact_9633_less__eq__integer_Oabs__eq,axiom,
    ! [Xaa: int,Xb: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_int_Code_integer(code_integer_of_int,Xaa)),aa_int_Code_integer(code_integer_of_int,Xb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xaa),Xb) ) ).

% less_eq_integer.abs_eq
tff(fact_9634_plus__complex_Osimps_I2_J,axiom,
    ! [Xb: complex,Yb: complex] : aa_complex_real(im,aa_complex_complex(plus_plus_complex(Xb),Yb)) = aa_real_real(plus_plus_real(aa_complex_real(im,Xb)),aa_complex_real(im,Yb)) ).

% plus_complex.simps(2)
tff(fact_9635_scaleR__complex_Osimps_I2_J,axiom,
    ! [R2: real,Xb: complex] : aa_complex_real(im,aa_complex_complex(real_V2046097035970521341omplex(R2),Xb)) = aa_real_real(times_times_real(R2),aa_complex_real(im,Xb)) ).

% scaleR_complex.simps(2)
tff(fact_9636_times__complex_Osimps_I2_J,axiom,
    ! [Xb: complex,Yb: complex] : aa_complex_real(im,aa_complex_complex(times_times_complex(Xb),Yb)) = aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_complex_real(re,Xb)),aa_complex_real(im,Yb))),aa_real_real(times_times_real(aa_complex_real(im,Xb)),aa_complex_real(re,Yb))) ).

% times_complex.simps(2)
tff(fact_9637_times__complex_Osimps_I1_J,axiom,
    ! [Xb: complex,Yb: complex] : aa_complex_real(re,aa_complex_complex(times_times_complex(Xb),Yb)) = aa_real_real(minus_minus_real(aa_real_real(times_times_real(aa_complex_real(re,Xb)),aa_complex_real(re,Yb))),aa_real_real(times_times_real(aa_complex_real(im,Xb)),aa_complex_real(im,Yb))) ).

% times_complex.simps(1)
tff(fact_9638_plus__complex_Ocode,axiom,
    ! [Xb: complex,Yb: complex] : aa_complex_complex(plus_plus_complex(Xb),Yb) = aa_real_complex(complex2(aa_real_real(plus_plus_real(aa_complex_real(re,Xb)),aa_complex_real(re,Yb))),aa_real_real(plus_plus_real(aa_complex_real(im,Xb)),aa_complex_real(im,Yb))) ).

% plus_complex.code
tff(fact_9639_scaleR__complex_Ocode,axiom,
    ! [R2: real,Xb: complex] : aa_complex_complex(real_V2046097035970521341omplex(R2),Xb) = aa_real_complex(complex2(aa_real_real(times_times_real(R2),aa_complex_real(re,Xb))),aa_real_real(times_times_real(R2),aa_complex_real(im,Xb))) ).

% scaleR_complex.code
tff(fact_9640_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(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_9641_sin__n__Im__cis__pow__n,axiom,
    ! [Nb: nat,A: real] : aa_real_real(sin_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),A)) = aa_complex_real(im,aa_nat_complex(power_power_complex(aa_real_complex(cis,A)),Nb)) ).

% sin_n_Im_cis_pow_n
tff(fact_9642_Re__exp,axiom,
    ! [Z: complex] : aa_complex_real(re,aa_complex_complex(exp_complex,Z)) = aa_real_real(times_times_real(aa_real_real(exp_real,aa_complex_real(re,Z))),aa_real_real(cos_real,aa_complex_real(im,Z))) ).

% Re_exp
tff(fact_9643_Im__exp,axiom,
    ! [Z: complex] : aa_complex_real(im,aa_complex_complex(exp_complex,Z)) = aa_real_real(times_times_real(aa_real_real(exp_real,aa_complex_real(re,Z))),aa_real_real(sin_real,aa_complex_real(im,Z))) ).

% Im_exp
tff(fact_9644_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_9645_times__complex_Ocode,axiom,
    ! [Xb: complex,Yb: complex] : aa_complex_complex(times_times_complex(Xb),Yb) = aa_real_complex(complex2(aa_real_real(minus_minus_real(aa_real_real(times_times_real(aa_complex_real(re,Xb)),aa_complex_real(re,Yb))),aa_real_real(times_times_real(aa_complex_real(im,Xb)),aa_complex_real(im,Yb)))),aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_complex_real(re,Xb)),aa_complex_real(im,Yb))),aa_real_real(times_times_real(aa_complex_real(im,Xb)),aa_complex_real(re,Yb)))) ).

% times_complex.code
tff(fact_9646_exp__eq__polar,axiom,
    ! [Z: complex] : aa_complex_complex(exp_complex,Z) = aa_complex_complex(times_times_complex(aa_real_complex(real_V4546457046886955230omplex,aa_real_real(exp_real,aa_complex_real(re,Z)))),aa_real_complex(cis,aa_complex_real(im,Z))) ).

% exp_eq_polar
tff(fact_9647_cmod__power2,axiom,
    ! [Z: complex] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% cmod_power2
tff(fact_9648_Im__power2,axiom,
    ! [Xb: complex] : aa_complex_real(im,aa_nat_complex(power_power_complex(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_real_real(times_times_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_complex_real(re,Xb))),aa_complex_real(im,Xb)) ).

% Im_power2
tff(fact_9649_Re__power2,axiom,
    ! [Xb: complex] : aa_complex_real(re,aa_nat_complex(power_power_complex(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_real_real(minus_minus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% Re_power2
tff(fact_9650_complex__eq__0,axiom,
    ! [Z: complex] :
      ( ( Z = zero_zero_complex )
    <=> ( aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = zero_zero_real ) ) ).

% complex_eq_0
tff(fact_9651_norm__complex__def,axiom,
    ! [Z: complex] : aa_complex_real(real_V1022390504157884413omplex,Z) = aa_real_real(sqrt,aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% norm_complex_def
tff(fact_9652_inverse__complex_Osimps_I1_J,axiom,
    ! [Xb: complex] : aa_complex_real(re,aa_complex_complex(invers8013647133539491842omplex,Xb)) = aa_real_real(divide_divide_real(aa_complex_real(re,Xb)),aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% inverse_complex.simps(1)
tff(fact_9653_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(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% complex_neq_0
tff(fact_9654_Re__divide,axiom,
    ! [Xb: complex,Yb: complex] : aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(Xb),Yb)) = aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_complex_real(re,Xb)),aa_complex_real(re,Yb))),aa_real_real(times_times_real(aa_complex_real(im,Xb)),aa_complex_real(im,Yb)))),aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Yb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Yb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% Re_divide
tff(fact_9655_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_9656_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_9657_inverse__complex_Osimps_I2_J,axiom,
    ! [Xb: complex] : aa_complex_real(im,aa_complex_complex(invers8013647133539491842omplex,Xb)) = aa_real_real(divide_divide_real(aa_real_real(uminus_uminus_real,aa_complex_real(im,Xb))),aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% inverse_complex.simps(2)
tff(fact_9658_Im__divide,axiom,
    ! [Xb: complex,Yb: complex] : aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(Xb),Yb)) = aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(times_times_real(aa_complex_real(im,Xb)),aa_complex_real(re,Yb))),aa_real_real(times_times_real(aa_complex_real(re,Xb)),aa_complex_real(im,Yb)))),aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Yb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Yb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% Im_divide
tff(fact_9659_complex__abs__le__norm,axiom,
    ! [Z: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_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(times_times_real(aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_complex_real(real_V1022390504157884413omplex,Z))) ).

% complex_abs_le_norm
tff(fact_9660_complex__unit__circle,axiom,
    ! [Z: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(divide_divide_real(aa_complex_real(re,Z)),aa_complex_real(real_V1022390504157884413omplex,Z))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(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_9661_inverse__complex_Ocode,axiom,
    ! [Xb: complex] : aa_complex_complex(invers8013647133539491842omplex,Xb) = aa_real_complex(complex2(aa_real_real(divide_divide_real(aa_complex_real(re,Xb)),aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(divide_divide_real(aa_real_real(uminus_uminus_real,aa_complex_real(im,Xb))),aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% inverse_complex.code
tff(fact_9662_Complex__divide,axiom,
    ! [Xb: complex,Yb: complex] : aa_complex_complex(divide1717551699836669952omplex(Xb),Yb) = aa_real_complex(complex2(aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(aa_real_real(times_times_real(aa_complex_real(re,Xb)),aa_complex_real(re,Yb))),aa_real_real(times_times_real(aa_complex_real(im,Xb)),aa_complex_real(im,Yb)))),aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Yb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Yb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(divide_divide_real(aa_real_real(minus_minus_real(aa_real_real(times_times_real(aa_complex_real(im,Xb)),aa_complex_real(re,Yb))),aa_real_real(times_times_real(aa_complex_real(re,Xb)),aa_complex_real(im,Yb)))),aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Yb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Yb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% Complex_divide
tff(fact_9663_Im__Reals__divide,axiom,
    ! [R2: complex,Z: complex] :
      ( member_complex(R2,real_V2521375963428798218omplex)
     => ( aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(R2),Z)) = aa_real_real(divide_divide_real(aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,aa_complex_real(re,R2))),aa_complex_real(im,Z))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ).

% Im_Reals_divide
tff(fact_9664_Re__Reals__divide,axiom,
    ! [R2: complex,Z: complex] :
      ( member_complex(R2,real_V2521375963428798218omplex)
     => ( aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(R2),Z)) = aa_real_real(divide_divide_real(aa_real_real(times_times_real(aa_complex_real(re,R2)),aa_complex_real(re,Z))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ).

% Re_Reals_divide
tff(fact_9665_imaginary__eq__real__iff,axiom,
    ! [Yb: complex,Xb: complex] :
      ( member_complex(Yb,real_V2521375963428798218omplex)
     => ( member_complex(Xb,real_V2521375963428798218omplex)
       => ( ( aa_complex_complex(times_times_complex(imaginary_unit),Yb) = Xb )
        <=> ( ( Xb = zero_zero_complex )
            & ( Yb = zero_zero_complex ) ) ) ) ) ).

% imaginary_eq_real_iff
tff(fact_9666_real__eq__imaginary__iff,axiom,
    ! [Yb: complex,Xb: complex] :
      ( member_complex(Yb,real_V2521375963428798218omplex)
     => ( member_complex(Xb,real_V2521375963428798218omplex)
       => ( ( Xb = aa_complex_complex(times_times_complex(imaginary_unit),Yb) )
        <=> ( ( Xb = zero_zero_complex )
            & ( Yb = zero_zero_complex ) ) ) ) ) ).

% real_eq_imaginary_iff
tff(fact_9667_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(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% complex_mult_cnj
tff(fact_9668_integer__of__num_I3_J,axiom,
    ! [Nb: num] :
      code_integer_of_num(aa_num_num(bit1,Nb)) = $let(
        k: code_integer,
        k:= code_integer_of_num(Nb),
        plus_p5714425477246183910nteger(plus_p5714425477246183910nteger(k,k),one_one_Code_integer) ) ).

% integer_of_num(3)
tff(fact_9669_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_qs,code_divmod_integer(K,numera6620942414471956472nteger(aa_num_num(bit0,one))))) ) ).

% int_of_integer_code
tff(fact_9670_complex__cnj__mult,axiom,
    ! [Xb: complex,Yb: complex] : aa_complex_complex(cnj,aa_complex_complex(times_times_complex(Xb),Yb)) = aa_complex_complex(times_times_complex(aa_complex_complex(cnj,Xb)),aa_complex_complex(cnj,Yb)) ).

% complex_cnj_mult
tff(fact_9671_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_9672_complex__cnj__one,axiom,
    aa_complex_complex(cnj,one_one_complex) = one_one_complex ).

% complex_cnj_one
tff(fact_9673_complex__cnj__add,axiom,
    ! [Xb: complex,Yb: complex] : aa_complex_complex(cnj,aa_complex_complex(plus_plus_complex(Xb),Yb)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(cnj,Xb)),aa_complex_complex(cnj,Yb)) ).

% complex_cnj_add
tff(fact_9674_zero__integer_Orep__eq,axiom,
    code_int_of_integer(zero_z3403309356797280102nteger) = zero_zero_int ).

% zero_integer.rep_eq
tff(fact_9675_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_9676_plus__integer_Orep__eq,axiom,
    ! [Xb: code_integer,Xaa: code_integer] : code_int_of_integer(plus_p5714425477246183910nteger(Xb,Xaa)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,code_int_of_integer(Xb)),code_int_of_integer(Xaa)) ).

% plus_integer.rep_eq
tff(fact_9677_times__integer_Orep__eq,axiom,
    ! [Xb: code_integer,Xaa: code_integer] : code_int_of_integer(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Xb),Xaa)) = aa_int_int(aa_int_fun_int_int(times_times_int,code_int_of_integer(Xb)),code_int_of_integer(Xaa)) ).

% times_integer.rep_eq
tff(fact_9678_one__integer_Orep__eq,axiom,
    code_int_of_integer(one_one_Code_integer) = one_one_int ).

% one_integer.rep_eq
tff(fact_9679_divide__integer_Orep__eq,axiom,
    ! [Xb: code_integer,Xaa: code_integer] : code_int_of_integer(divide6298287555418463151nteger(Xb,Xaa)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,code_int_of_integer(Xb)),code_int_of_integer(Xaa)) ).

% divide_integer.rep_eq
tff(fact_9680_complex__In__mult__cnj__zero,axiom,
    ! [Z: complex] : aa_complex_real(im,aa_complex_complex(times_times_complex(Z),aa_complex_complex(cnj,Z))) = zero_zero_real ).

% complex_In_mult_cnj_zero
tff(fact_9681_less__integer_Orep__eq,axiom,
    ! [Xb: code_integer,Xaa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Xb),Xaa)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,code_int_of_integer(Xb)),code_int_of_integer(Xaa)) ) ).

% less_integer.rep_eq
tff(fact_9682_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_9683_less__eq__integer_Orep__eq,axiom,
    ! [Xb: code_integer,Xaa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Xb),Xaa)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,code_int_of_integer(Xb)),code_int_of_integer(Xaa)) ) ).

% less_eq_integer.rep_eq
tff(fact_9684_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_9685_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_9686_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_9687_complex__mod__sqrt__Re__mult__cnj,axiom,
    ! [Z: complex] : aa_complex_real(real_V1022390504157884413omplex,Z) = aa_real_real(sqrt,aa_complex_real(re,aa_complex_complex(times_times_complex(Z),aa_complex_complex(cnj,Z)))) ).

% complex_mod_sqrt_Re_mult_cnj
tff(fact_9688_integer__of__num__triv_I1_J,axiom,
    code_integer_of_num(one) = one_one_Code_integer ).

% integer_of_num_triv(1)
tff(fact_9689_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_9690_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_9691_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_9692_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_9693_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_9694_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_9695_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_9696_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_9697_integer__of__num_I2_J,axiom,
    ! [Nb: num] :
      code_integer_of_num(aa_num_num(bit0,Nb)) = $let(
        k: code_integer,
        k:= code_integer_of_num(Nb),
        plus_p5714425477246183910nteger(k,k) ) ).

% integer_of_num(2)
tff(fact_9698_complex__mod__mult__cnj,axiom,
    ! [Z: complex] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(times_times_complex(Z),aa_complex_complex(cnj,Z))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% complex_mod_mult_cnj
tff(fact_9699_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_9700_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_9701_complex__norm__square,axiom,
    ! [Z: complex] : aa_real_complex(real_V4546457046886955230omplex,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_complex_complex(times_times_complex(Z),aa_complex_complex(cnj,Z)) ).

% complex_norm_square
tff(fact_9702_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(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_complex_real(re,Z))) ).

% complex_add_cnj
tff(fact_9703_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(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_complex_real(im,Z)))),imaginary_unit) ).

% complex_diff_cnj
tff(fact_9704_complex__div__cnj,axiom,
    ! [A: complex,B: complex] : aa_complex_complex(divide1717551699836669952omplex(A),B) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B))),aa_real_complex(real_V4546457046886955230omplex,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(real_V1022390504157884413omplex,B)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% complex_div_cnj
tff(fact_9705_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(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_complex_real(re,aa_complex_complex(times_times_complex(Z),aa_complex_complex(cnj,W))))) ).

% cnj_add_mult_eq_Re
tff(fact_9706_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_qt,code_divmod_integer(K,numera6620942414471956472nteger(aa_num_num(bit0,one))))) ).

% num_of_integer_code
tff(fact_9707_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_qu,code_divmod_integer(K,numera6620942414471956472nteger(aa_num_num(bit0,one))))) ).

% nat_of_integer_code
tff(fact_9708_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_9709_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_9710_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_9711_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_9712_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_9713_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_qv(K),code_divmod_abs(K,numera6620942414471956472nteger(aa_num_num(bit0,one))))) ).

% bit_cut_integer_code
tff(fact_9714_card__Collect__less__nat,axiom,
    ! [Nb: nat] : aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_ei,Nb))) = Nb ).

% card_Collect_less_nat
tff(fact_9715_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_9716_card__Collect__le__nat,axiom,
    ! [Nb: nat] : aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_nn,Nb))) = aa_nat_nat(suc,Nb) ).

% card_Collect_le_nat
tff(fact_9717_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_9718_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_9719_card__less,axiom,
    ! [M6: set_nat,I3: nat] :
      ( aa_set_nat_o(member_nat(zero_zero_nat),M6)
     => ( aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_qw(M6),I3))) != zero_zero_nat ) ) ).

% card_less
tff(fact_9720_card__less__Suc,axiom,
    ! [M6: set_nat,I3: nat] :
      ( aa_set_nat_o(member_nat(zero_zero_nat),M6)
     => ( aa_nat_nat(suc,aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_qx(M6),I3)))) = aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_qw(M6),I3))) ) ) ).

% card_less_Suc
tff(fact_9721_card__less__Suc2,axiom,
    ! [M6: set_nat,I3: nat] :
      ( ~ aa_set_nat_o(member_nat(zero_zero_nat),M6)
     => ( aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_qx(M6),I3))) = aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_qw(M6),I3))) ) ) ).

% card_less_Suc2
tff(fact_9722_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_9723_subset__card__intvl__is__intvl,axiom,
    ! [A3: set_nat,K: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A3),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,A3))))
     => ( A3 = 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,A3))) ) ) ).

% subset_card_intvl_is_intvl
tff(fact_9724_subset__eq__atLeast0__lessThan__card,axiom,
    ! [N2: set_nat,Nb: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(N2),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Nb))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,N2)),Nb) ) ).

% subset_eq_atLeast0_lessThan_card
tff(fact_9725_card__sum__le__nat__sum,axiom,
    ! [S3: set_nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_ar),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),aa_set_nat_nat(finite_card_nat,S3)))),aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_ar),S3)) ).

% card_sum_le_nat_sum
tff(fact_9726_card__nth__roots,axiom,
    ! [C: complex,Nb: nat] :
      ( ( C != zero_zero_complex )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( finite_card_complex(collect_complex(aa_nat_fun_complex_o(aTP_Lamm_qr(C),Nb))) = Nb ) ) ) ).

% card_nth_roots
tff(fact_9727_card__roots__unity__eq,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( finite_card_complex(collect_complex(aTP_Lamm_dl(Nb))) = Nb ) ) ).

% card_roots_unity_eq
tff(fact_9728_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_qy(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_qz(L),code_divmod_abs(K,L)))) ) ) ) ).

% divmod_integer_code
tff(fact_9729_bezw__0,axiom,
    ! [Xb: nat] : bezw(Xb,zero_zero_nat) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,one_one_int),zero_zero_int) ).

% bezw_0
tff(fact_9730_nat_Odisc__eq__case_I1_J,axiom,
    ! [Nat: nat] :
      ( ( Nat = zero_zero_nat )
    <=> aa_nat_o(case_nat_o($true,aTP_Lamm_ra),Nat) ) ).

% nat.disc_eq_case(1)
tff(fact_9731_nat_Odisc__eq__case_I2_J,axiom,
    ! [Nat: nat] :
      ( ( Nat != zero_zero_nat )
    <=> aa_nat_o(case_nat_o($false,aTP_Lamm_rb),Nat) ) ).

% nat.disc_eq_case(2)
tff(fact_9732_less__eq__nat_Osimps_I2_J,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Mb)),Nb)
    <=> aa_nat_o(case_nat_o($false,aa_nat_fun_nat_o(ord_less_eq_nat,Mb)),Nb) ) ).

% less_eq_nat.simps(2)
tff(fact_9733_max__Suc1,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(suc,Nb)),Mb) = aa_nat_nat(case_nat_nat(aa_nat_nat(suc,Nb),aa_nat_fun_nat_nat(aTP_Lamm_rc,Nb)),Mb) ).

% max_Suc1
tff(fact_9734_max__Suc2,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Mb),aa_nat_nat(suc,Nb)) = aa_nat_nat(case_nat_nat(aa_nat_nat(suc,Nb),aa_nat_fun_nat_nat(aTP_Lamm_rd,Nb)),Mb) ).

% max_Suc2
tff(fact_9735_diff__Suc,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),aa_nat_nat(suc,Nb)) = aa_nat_nat(case_nat_nat(zero_zero_nat,aTP_Lamm_ar),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) ).

% diff_Suc
tff(fact_9736_binomial__def,axiom,
    ! [Nb: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K) = finite_card_set_nat(collect_set_nat(aa_nat_fun_set_nat_o(aTP_Lamm_re(Nb),K))) ).

% binomial_def
tff(fact_9737_prod__decode__aux_Osimps,axiom,
    ! [K: nat,Mb: nat] :
      nat_prod_decode_aux(K,Mb) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),K),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),Mb)),nat_prod_decode_aux(aa_nat_nat(suc,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),aa_nat_nat(suc,K)))) ).

% prod_decode_aux.simps
tff(fact_9738_pred__def,axiom,
    ! [Nat: nat] : aa_nat_nat(pred,Nat) = aa_nat_nat(case_nat_nat(zero_zero_nat,aTP_Lamm_ar),Nat) ).

% pred_def
tff(fact_9739_prod__decode__aux_Oelims,axiom,
    ! [Xb: nat,Xaa: nat,Yb: product_prod_nat_nat] :
      ( ( nat_prod_decode_aux(Xb,Xaa) = Yb )
     => ( Yb = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xaa),Xb),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xaa),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xb),Xaa)),nat_prod_decode_aux(aa_nat_nat(suc,Xb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xaa),aa_nat_nat(suc,Xb)))) ) ) ).

% prod_decode_aux.elims
tff(fact_9740_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_9741_drop__bit__numeral__minus__bit1,axiom,
    ! [L: num,K: num] : aa_int_int(bit_se8568078237143864401it_int(aa_num_nat(numeral_numeral_nat,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K)))) = aa_int_int(bit_se8568078237143864401it_int(aa_num_nat(pred_numeral,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(inc,K)))) ).

% drop_bit_numeral_minus_bit1
tff(fact_9742_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_9743_drop__bit__nonnegative__int__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se8568078237143864401it_int(Nb),K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K) ) ).

% drop_bit_nonnegative_int_iff
tff(fact_9744_drop__bit__negative__int__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se8568078237143864401it_int(Nb),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_9745_drop__bit__minus__one,axiom,
    ! [Nb: nat] : aa_int_int(bit_se8568078237143864401it_int(Nb),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,one_one_int) ).

% drop_bit_minus_one
tff(fact_9746_fst__divmod__nat,axiom,
    ! [Mb: nat,Nb: nat] : aa_Pro4243173770044397973at_nat(product_fst_nat_nat,divmod_nat(Mb,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb) ).

% fst_divmod_nat
tff(fact_9747_snd__divmod__nat,axiom,
    ! [Mb: nat,Nb: nat] : aa_Pro4243173770044397973at_nat(product_snd_nat_nat,divmod_nat(Mb,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb) ).

% snd_divmod_nat
tff(fact_9748_drop__bit__Suc__minus__bit0,axiom,
    ! [Nb: nat,K: num] : aa_int_int(bit_se8568078237143864401it_int(aa_nat_nat(suc,Nb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,K)))) = aa_int_int(bit_se8568078237143864401it_int(Nb),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) ).

% drop_bit_Suc_minus_bit0
tff(fact_9749_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_9750_drop__bit__Suc__minus__bit1,axiom,
    ! [Nb: nat,K: num] : aa_int_int(bit_se8568078237143864401it_int(aa_nat_nat(suc,Nb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K)))) = aa_int_int(bit_se8568078237143864401it_int(Nb),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(inc,K)))) ).

% drop_bit_Suc_minus_bit1
tff(fact_9751_drop__bit__push__bit__int,axiom,
    ! [Mb: nat,Nb: nat,K: int] : aa_int_int(bit_se8568078237143864401it_int(Mb),aa_int_int(bit_se545348938243370406it_int(Nb),K)) = aa_int_int(bit_se8568078237143864401it_int(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),aa_int_int(bit_se545348938243370406it_int(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)),K)) ).

% drop_bit_push_bit_int
tff(fact_9752_drop__bit__int__def,axiom,
    ! [Nb: nat,K: int] : aa_int_int(bit_se8568078237143864401it_int(Nb),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))),Nb)) ).

% drop_bit_int_def
tff(fact_9753_drop__bit__of__Suc__0,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se8570568707652914677it_nat,Nb),aa_nat_nat(suc,zero_zero_nat)) = zero_n2687167440665602831ol_nat(Nb = zero_zero_nat) ).

% drop_bit_of_Suc_0
tff(fact_9754_drop__bit__nat__eq,axiom,
    ! [Nb: nat,K: int] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se8570568707652914677it_nat,Nb),aa_int_nat(nat2,K)) = aa_int_nat(nat2,aa_int_int(bit_se8568078237143864401it_int(Nb),K)) ).

% drop_bit_nat_eq
tff(fact_9755_quotient__of__denom__pos_H,axiom,
    ! [R2: rat] : aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_Pro6799258434465214121nt_int(product_snd_int_int,quotient_of(R2))) ).

% quotient_of_denom_pos'
tff(fact_9756_bezw__non__0,axiom,
    ! [Yb: nat,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Yb)
     => ( bezw(Xb,Yb) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,bezw(Yb,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Yb)))),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,bezw(Yb,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Yb)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,bezw(Yb,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Yb)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xb),Yb))))) ) ) ).

% bezw_non_0
tff(fact_9757_bezw_Oelims,axiom,
    ! [Xb: nat,Xaa: nat,Yb: product_prod_int_int] :
      ( ( bezw(Xb,Xaa) = Yb )
     => ( Yb = $ite(Xaa = zero_zero_nat,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,one_one_int),zero_zero_int),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,bezw(Xaa,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Xaa)))),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,bezw(Xaa,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Xaa)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,bezw(Xaa,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Xaa)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xb),Xaa)))))) ) ) ).

% bezw.elims
tff(fact_9758_bezw_Osimps,axiom,
    ! [Xb: nat,Yb: nat] :
      bezw(Xb,Yb) = $ite(Yb = zero_zero_nat,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,one_one_int),zero_zero_int),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,bezw(Yb,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Yb)))),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,bezw(Yb,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Yb)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,bezw(Yb,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Yb)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xb),Yb)))))) ).

% bezw.simps
tff(fact_9759_drop__bit__nat__def,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se8570568707652914677it_nat,Nb),Mb) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% drop_bit_nat_def
tff(fact_9760_rat__sgn__code,axiom,
    ! [P2: rat] : quotient_of(aa_rat_rat(sgn_sgn_rat,P2)) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(sgn_sgn_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,quotient_of(P2)))),one_one_int) ).

% rat_sgn_code
tff(fact_9761_minus__one__mod__numeral,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,Nb)) = aa_int_int(aa_int_fun_int_int(adjust_mod,aa_num_int(numeral_numeral_int,Nb)),aa_Pro6799258434465214121nt_int(product_snd_int_int,unique5052692396658037445od_int(one,Nb))) ).

% minus_one_mod_numeral
tff(fact_9762_one__mod__minus__numeral,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(adjust_mod,aa_num_int(numeral_numeral_int,Nb)),aa_Pro6799258434465214121nt_int(product_snd_int_int,unique5052692396658037445od_int(one,Nb)))) ).

% one_mod_minus_numeral
tff(fact_9763_minus__numeral__mod__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),aa_num_int(numeral_numeral_int,Nb)) = aa_int_int(aa_int_fun_int_int(adjust_mod,aa_num_int(numeral_numeral_int,Nb)),aa_Pro6799258434465214121nt_int(product_snd_int_int,unique5052692396658037445od_int(Mb,Nb))) ).

% minus_numeral_mod_numeral
tff(fact_9764_numeral__mod__minus__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(adjust_mod,aa_num_int(numeral_numeral_int,Nb)),aa_Pro6799258434465214121nt_int(product_snd_int_int,unique5052692396658037445od_int(Mb,Nb)))) ).

% numeral_mod_minus_numeral
tff(fact_9765_Divides_Oadjust__mod__def,axiom,
    ! [L: int,R2: int] :
      aa_int_int(aa_int_fun_int_int(adjust_mod,L),R2) = $ite(R2 = zero_zero_int,zero_zero_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,L),R2)) ).

% Divides.adjust_mod_def
tff(fact_9766_bezw_Opelims,axiom,
    ! [Xb: nat,Xaa: nat,Yb: product_prod_int_int] :
      ( ( bezw(Xb,Xaa) = Yb )
     => ( aa_Pro6150637604445964947_nat_o(accp_P4275260045618599050at_nat(bezw_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xb),Xaa))
       => ~ ( ( Yb = $ite(Xaa = zero_zero_nat,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,one_one_int),zero_zero_int),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,bezw(Xaa,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Xaa)))),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,bezw(Xaa,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Xaa)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,bezw(Xaa,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Xaa)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xb),Xaa)))))) )
           => ~ aa_Pro6150637604445964947_nat_o(accp_P4275260045618599050at_nat(bezw_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xb),Xaa)) ) ) ) ).

% bezw.pelims
tff(fact_9767_normalize__def,axiom,
    ! [P2: product_prod_int_int] :
      aa_Pro6970504127699319548nt_int(normalize,P2) = $ite(
        aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_Pro6799258434465214121nt_int(product_snd_int_int,P2)),
        $let(
          a: int,
          a:= aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,P2)),aa_Pro6799258434465214121nt_int(product_snd_int_int,P2)),
          aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,P2)),a)),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,P2)),a)) ),
        $ite(
          aa_Pro6799258434465214121nt_int(product_snd_int_int,P2) = zero_zero_int,
          aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_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,aa_Pro6799258434465214121nt_int(product_fst_int_int,P2)),aa_Pro6799258434465214121nt_int(product_snd_int_int,P2))),
            aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,P2)),a)),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,P2)),a)) ) ) ) ).

% normalize_def
tff(fact_9768_gcd__1__int,axiom,
    ! [Mb: int] : aa_int_int(aa_int_fun_int_int(gcd_gcd_int,Mb),one_one_int) = one_one_int ).

% gcd_1_int
tff(fact_9769_gcd__pos__int,axiom,
    ! [Mb: int,Nb: 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,Mb),Nb))
    <=> ( ( Mb != zero_zero_int )
        | ( Nb != zero_zero_int ) ) ) ).

% gcd_pos_int
tff(fact_9770_gcd__neg__numeral__2__int,axiom,
    ! [Xb: int,Nb: num] : aa_int_int(aa_int_fun_int_int(gcd_gcd_int,Xb),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,Xb),aa_num_int(numeral_numeral_int,Nb)) ).

% gcd_neg_numeral_2_int
tff(fact_9771_gcd__neg__numeral__1__int,axiom,
    ! [Nb: num,Xb: int] : aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))),Xb) = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_num_int(numeral_numeral_int,Nb)),Xb) ).

% gcd_neg_numeral_1_int
tff(fact_9772_gcd__0__int,axiom,
    ! [Xb: int] : aa_int_int(aa_int_fun_int_int(gcd_gcd_int,Xb),zero_zero_int) = aa_int_int(abs_abs_int,Xb) ).

% gcd_0_int
tff(fact_9773_gcd__0__left__int,axiom,
    ! [Xb: int] : aa_int_int(aa_int_fun_int_int(gcd_gcd_int,zero_zero_int),Xb) = aa_int_int(abs_abs_int,Xb) ).

% gcd_0_left_int
tff(fact_9774_gcd__ge__0__int,axiom,
    ! [Xb: int,Yb: 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,Xb),Yb)) ).

% gcd_ge_0_int
tff(fact_9775_bezout__int,axiom,
    ! [Xb: int,Yb: 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),Xb)),aa_int_int(aa_int_fun_int_int(times_times_int,V2),Yb)) = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,Xb),Yb) ).

% bezout_int
tff(fact_9776_gcd__mult__distrib__int,axiom,
    ! [K: int,Mb: int,Nb: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,K)),aa_int_int(aa_int_fun_int_int(gcd_gcd_int,Mb),Nb)) = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),Mb)),aa_int_int(aa_int_fun_int_int(times_times_int,K),Nb)) ).

% gcd_mult_distrib_int
tff(fact_9777_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_9778_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_9779_gcd__cases__int,axiom,
    ! [Xb: int,Yb: int,P: fun_int_o] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Yb)
         => aa_int_o(P,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,Xb),Yb)) ) )
     => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Yb),zero_zero_int)
           => aa_int_o(P,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,Xb),aa_int_int(uminus_uminus_int,Yb))) ) )
       => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),zero_zero_int)
           => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Yb)
             => aa_int_o(P,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_int_int(uminus_uminus_int,Xb)),Yb)) ) )
         => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),zero_zero_int)
             => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Yb),zero_zero_int)
               => aa_int_o(P,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_int_int(uminus_uminus_int,Xb)),aa_int_int(uminus_uminus_int,Yb))) ) )
           => aa_int_o(P,aa_int_int(aa_int_fun_int_int(gcd_gcd_int,Xb),Yb)) ) ) ) ) ).

% gcd_cases_int
tff(fact_9780_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_9781_gcd__non__0__int,axiom,
    ! [Yb: int,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Yb)
     => ( aa_int_int(aa_int_fun_int_int(gcd_gcd_int,Xb),Yb) = aa_int_int(aa_int_fun_int_int(gcd_gcd_int,Yb),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Xb),Yb)) ) ) ).

% gcd_non_0_int
tff(fact_9782_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(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(abs_abs_int,K)),aa_int_int(abs_abs_int,L))))) ).

% gcd_code_int
tff(fact_9783_nat__descend__induct,axiom,
    ! [Nb: nat,P: fun_nat_o,Mb: nat] :
      ( ! [K2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),K2)
         => aa_nat_o(P,K2) )
     => ( ! [K2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
           => ( ! [I: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K2),I)
                 => aa_nat_o(P,I) )
             => aa_nat_o(P,K2) ) )
       => aa_nat_o(P,Mb) ) ) ).

% nat_descend_induct
tff(fact_9784_prod__decode__aux_Opelims,axiom,
    ! [Xb: nat,Xaa: nat,Yb: product_prod_nat_nat] :
      ( ( nat_prod_decode_aux(Xb,Xaa) = Yb )
     => ( aa_Pro6150637604445964947_nat_o(accp_P4275260045618599050at_nat(nat_pr5047031295181774490ux_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xb),Xaa))
       => ~ ( ( Yb = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xaa),Xb),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xaa),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xb),Xaa)),nat_prod_decode_aux(aa_nat_nat(suc,Xb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xaa),aa_nat_nat(suc,Xb)))) )
           => ~ aa_Pro6150637604445964947_nat_o(accp_P4275260045618599050at_nat(nat_pr5047031295181774490ux_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xb),Xaa)) ) ) ) ).

% prod_decode_aux.pelims
tff(fact_9785_finite__enumerate,axiom,
    ! [S3: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,S3)
     => ? [R3: fun_nat_nat] :
          ( aa_set_nat_o(strict1292158309912662752at_nat(R3),aa_nat_set_nat(set_ord_lessThan_nat,aa_set_nat_nat(finite_card_nat,S3)))
          & ! [N7: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N7),aa_set_nat_nat(finite_card_nat,S3))
             => aa_set_nat_o(member_nat(aa_nat_nat(R3,N7)),S3) ) ) ) ).

% finite_enumerate
tff(fact_9786_gcd__0__left__nat,axiom,
    ! [Xb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,zero_zero_nat),Xb) = Xb ).

% gcd_0_left_nat
tff(fact_9787_gcd__0__nat,axiom,
    ! [Xb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Xb),zero_zero_nat) = Xb ).

% gcd_0_nat
tff(fact_9788_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_9789_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_9790_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_9791_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_9792_gcd__1__nat,axiom,
    ! [Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Mb),one_one_nat) = one_one_nat ).

% gcd_1_nat
tff(fact_9793_gcd__Suc__0,axiom,
    ! [Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Mb),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(suc,zero_zero_nat) ).

% gcd_Suc_0
tff(fact_9794_gcd__pos__nat,axiom,
    ! [Mb: nat,Nb: 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,Mb),Nb))
    <=> ( ( Mb != zero_zero_nat )
        | ( Nb != zero_zero_nat ) ) ) ).

% gcd_pos_nat
tff(fact_9795_gcd__mult__distrib__nat,axiom,
    ! [K: nat,Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb)) ).

% gcd_mult_distrib_nat
tff(fact_9796_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_9797_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_9798_gcd__diff1__nat,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Mb),Nb) ) ) ).

% gcd_diff1_nat
tff(fact_9799_gcd__diff2__nat,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Mb),Nb) ) ) ).

% gcd_diff2_nat
tff(fact_9800_gcd__nat_Oelims,axiom,
    ! [Xb: nat,Xaa: nat,Yb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Xb),Xaa) = Yb )
     => ( Yb = $ite(Xaa = zero_zero_nat,Xb,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Xaa))) ) ) ).

% gcd_nat.elims
tff(fact_9801_gcd__nat_Osimps,axiom,
    ! [Xb: nat,Yb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Xb),Yb) = $ite(Yb = zero_zero_nat,Xb,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Yb),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Yb))) ).

% gcd_nat.simps
tff(fact_9802_gcd__non__0__nat,axiom,
    ! [Yb: nat,Xb: nat] :
      ( ( Yb != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Xb),Yb) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Yb),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Yb)) ) ) ).

% gcd_non_0_nat
tff(fact_9803_bezout__nat,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ? [X3: nat,Y: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X3) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y)),aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B)) ) ).

% bezout_nat
tff(fact_9804_bezout__gcd__nat_H,axiom,
    ! [B: nat,A: nat] :
    ? [X3: nat,Y: 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),Y)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X3))
        & ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),X3)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),Y)) = 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),Y)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),X3))
        & ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),X3)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B) ) ) ) ).

% bezout_gcd_nat'
tff(fact_9805_bezw__aux,axiom,
    ! [Xb: nat,Yb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Xb),Yb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,bezw(Xb,Yb))),aa_nat_int(semiri1314217659103216013at_int,Xb))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,bezw(Xb,Yb))),aa_nat_int(semiri1314217659103216013at_int,Yb))) ).

% bezw_aux
tff(fact_9806_gcd__nat_Opelims,axiom,
    ! [Xb: nat,Xaa: nat,Yb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Xb),Xaa) = Yb )
     => ( aa_Pro6150637604445964947_nat_o(accp_P4275260045618599050at_nat(gcd_nat_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xb),Xaa))
       => ~ ( ( Yb = $ite(Xaa = zero_zero_nat,Xb,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Xaa))) )
           => ~ aa_Pro6150637604445964947_nat_o(accp_P4275260045618599050at_nat(gcd_nat_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xb),Xaa)) ) ) ) ).

% gcd_nat.pelims
tff(fact_9807_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_9808_xor__minus__numerals_I1_J,axiom,
    ! [Nb: num,K: int] : aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))),K) = aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,aa_num_int(neg_numeral_sub_int(Nb),one)),K)) ).

% xor_minus_numerals(1)
tff(fact_9809_xor__minus__numerals_I2_J,axiom,
    ! [K: int,Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,K),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,K),aa_num_int(neg_numeral_sub_int(Nb),one))) ).

% xor_minus_numerals(2)
tff(fact_9810_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_9811_sub__BitM__One__eq,axiom,
    ! [Nb: num] : aa_num_int(neg_numeral_sub_int(aa_num_num(bitM,Nb)),one) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_num_int(neg_numeral_sub_int(Nb),one)) ).

% sub_BitM_One_eq
tff(fact_9812_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_9813_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_9814_tanh__real__bounds,axiom,
    ! [Xb: real] : member_real(aa_real_real(tanh_real,Xb),set_or1633881224788618240n_real(aa_real_real(uminus_uminus_real,one_one_real),one_one_real)) ).

% tanh_real_bounds
tff(fact_9815_Suc__funpow,axiom,
    ! [Nb: nat] : compow_fun_nat_nat(Nb,suc) = aa_nat_fun_nat_nat(plus_plus_nat,Nb) ).

% Suc_funpow
tff(fact_9816_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_rf(L),code_divmod_abs(K,L)))) ) ) ).

% divmod_integer_eq_cases
tff(fact_9817_max__nat_Osemilattice__neutr__order__axioms,axiom,
    semila1623282765462674594er_nat(ord_max_nat,zero_zero_nat,aTP_Lamm_nn,aTP_Lamm_ei) ).

% max_nat.semilattice_neutr_order_axioms
tff(fact_9818_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_9819_gcd__nat_Osemilattice__neutr__order__axioms,axiom,
    semila1623282765462674594er_nat(gcd_gcd_nat,zero_zero_nat,dvd_dvd_nat,aTP_Lamm_rg) ).

% gcd_nat.semilattice_neutr_order_axioms
tff(fact_9820_Sup__nat__empty,axiom,
    aa_set_nat_nat(complete_Sup_Sup_nat,bot_bot_set_nat) = zero_zero_nat ).

% Sup_nat_empty
tff(fact_9821_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_9822_times__int_Oabs__eq,axiom,
    ! [Xaa: product_prod_nat_nat,Xb: product_prod_nat_nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro4240683299535347697at_int(abs_Integ,Xaa)),aa_Pro4240683299535347697at_int(abs_Integ,Xb)) = aa_Pro4240683299535347697at_int(abs_Integ,aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(aTP_Lamm_ri),Xaa),Xb)) ).

% times_int.abs_eq
tff(fact_9823_int_Oabs__induct,axiom,
    ! [P: fun_int_o,Xb: int] :
      ( ! [Y: product_prod_nat_nat] : aa_int_o(P,aa_Pro4240683299535347697at_int(abs_Integ,Y))
     => aa_int_o(P,Xb) ) ).

% int.abs_induct
tff(fact_9824_eq__Abs__Integ,axiom,
    ! [Z: int] :
      ~ ! [X3: nat,Y: nat] : Z != aa_Pro4240683299535347697at_int(abs_Integ,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),Y)) ).

% eq_Abs_Integ
tff(fact_9825_nat_Oabs__eq,axiom,
    ! [Xb: product_prod_nat_nat] : aa_int_nat(nat2,aa_Pro4240683299535347697at_int(abs_Integ,Xb)) = aa_Pro4243173770044397973at_nat(produc6842872674320459806at_nat(minus_minus_nat),Xb) ).

% nat.abs_eq
tff(fact_9826_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_9827_int__def,axiom,
    ! [Nb: nat] : aa_nat_int(semiri1314217659103216013at_int,Nb) = aa_Pro4240683299535347697at_int(abs_Integ,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Nb),zero_zero_nat)) ).

% int_def
tff(fact_9828_uminus__int_Oabs__eq,axiom,
    ! [Xb: product_prod_nat_nat] : aa_int_int(uminus_uminus_int,aa_Pro4240683299535347697at_int(abs_Integ,Xb)) = aa_Pro4240683299535347697at_int(abs_Integ,aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_rj),Xb)) ).

% uminus_int.abs_eq
tff(fact_9829_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_9830_less__int_Oabs__eq,axiom,
    ! [Xaa: product_prod_nat_nat,Xb: product_prod_nat_nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_Pro4240683299535347697at_int(abs_Integ,Xaa)),aa_Pro4240683299535347697at_int(abs_Integ,Xb))
    <=> aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(aTP_Lamm_rl),Xaa),Xb) ) ).

% less_int.abs_eq
tff(fact_9831_less__eq__int_Oabs__eq,axiom,
    ! [Xaa: product_prod_nat_nat,Xb: product_prod_nat_nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_Pro4240683299535347697at_int(abs_Integ,Xaa)),aa_Pro4240683299535347697at_int(abs_Integ,Xb))
    <=> aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(aTP_Lamm_rn),Xaa),Xb) ) ).

% less_eq_int.abs_eq
tff(fact_9832_plus__int_Oabs__eq,axiom,
    ! [Xaa: product_prod_nat_nat,Xb: product_prod_nat_nat] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_Pro4240683299535347697at_int(abs_Integ,Xaa)),aa_Pro4240683299535347697at_int(abs_Integ,Xb)) = aa_Pro4240683299535347697at_int(abs_Integ,aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(aTP_Lamm_rp),Xaa),Xb)) ).

% plus_int.abs_eq
tff(fact_9833_minus__int_Oabs__eq,axiom,
    ! [Xaa: product_prod_nat_nat,Xb: product_prod_nat_nat] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_Pro4240683299535347697at_int(abs_Integ,Xaa)),aa_Pro4240683299535347697at_int(abs_Integ,Xb)) = aa_Pro4240683299535347697at_int(abs_Integ,aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(aTP_Lamm_rr),Xaa),Xb)) ).

% minus_int.abs_eq
tff(fact_9834_Gcd__remove0__nat,axiom,
    ! [M6: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,M6)
     => ( aa_set_nat_nat(gcd_Gcd_nat,M6) = aa_set_nat_nat(gcd_Gcd_nat,minus_minus_set_nat(M6,insert_nat(zero_zero_nat,bot_bot_set_nat))) ) ) ).

% Gcd_remove0_nat
tff(fact_9835_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_9836_less__eq__int_Orep__eq,axiom,
    ! [Xb: int,Xaa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Xaa)
    <=> aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(aTP_Lamm_rn),aa_int1417691210696471063at_nat(rep_Integ,Xb)),aa_int1417691210696471063at_nat(rep_Integ,Xaa)) ) ).

% less_eq_int.rep_eq
tff(fact_9837_less__int_Orep__eq,axiom,
    ! [Xb: int,Xaa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Xaa)
    <=> aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(aTP_Lamm_rl),aa_int1417691210696471063at_nat(rep_Integ,Xb)),aa_int1417691210696471063at_nat(rep_Integ,Xaa)) ) ).

% less_int.rep_eq
tff(fact_9838_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_9839_nat_Orep__eq,axiom,
    ! [Xb: int] : aa_int_nat(nat2,Xb) = aa_Pro4243173770044397973at_nat(produc6842872674320459806at_nat(minus_minus_nat),aa_int1417691210696471063at_nat(rep_Integ,Xb)) ).

% nat.rep_eq
tff(fact_9840_prod__encode__def,axiom,
    nat_prod_encode = produc6842872674320459806at_nat(aTP_Lamm_rs) ).

% prod_encode_def
tff(fact_9841_uminus__int__def,axiom,
    uminus_uminus_int = aa_fun8399641135911395862nt_int(map_fu3667384564859982768at_int(rep_Integ,abs_Integ),produc2626176000494625587at_nat(aTP_Lamm_rj)) ).

% uminus_int_def
tff(fact_9842_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_9843_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_9844_prod__encode__prod__decode__aux,axiom,
    ! [K: nat,Mb: nat] : aa_Pro4243173770044397973at_nat(nat_prod_encode,nat_prod_decode_aux(K,Mb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(nat_triangle,K)),Mb) ).

% prod_encode_prod_decode_aux
tff(fact_9845_times__int__def,axiom,
    times_times_int = map_fu8294830438409148311nt_int(rep_Integ,map_fu3667384564859982768at_int(rep_Integ,abs_Integ),produc4528391901023682385at_nat(aTP_Lamm_ri)) ).

% times_int_def
tff(fact_9846_minus__int__def,axiom,
    minus_minus_int = map_fu8294830438409148311nt_int(rep_Integ,map_fu3667384564859982768at_int(rep_Integ,abs_Integ),produc4528391901023682385at_nat(aTP_Lamm_rr)) ).

% minus_int_def
tff(fact_9847_plus__int__def,axiom,
    plus_plus_int = map_fu8294830438409148311nt_int(rep_Integ,map_fu3667384564859982768at_int(rep_Integ,abs_Integ),produc4528391901023682385at_nat(aTP_Lamm_rp)) ).

% plus_int_def
tff(fact_9848_num__of__nat_Osimps_I2_J,axiom,
    ! [Nb: nat] :
      aa_nat_num(num_of_nat,aa_nat_nat(suc,Nb)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb),aa_num_num(inc,aa_nat_num(num_of_nat,Nb)),one) ).

% num_of_nat.simps(2)
tff(fact_9849_num__of__nat__numeral__eq,axiom,
    ! [Q2: num] : aa_nat_num(num_of_nat,aa_num_nat(numeral_numeral_nat,Q2)) = Q2 ).

% num_of_nat_numeral_eq
tff(fact_9850_num__of__nat_Osimps_I1_J,axiom,
    aa_nat_num(num_of_nat,zero_zero_nat) = one ).

% num_of_nat.simps(1)
tff(fact_9851_numeral__num__of__nat,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_num_nat(numeral_numeral_nat,aa_nat_num(num_of_nat,Nb)) = Nb ) ) ).

% numeral_num_of_nat
tff(fact_9852_num__of__nat__One,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),one_one_nat)
     => ( aa_nat_num(num_of_nat,Nb) = one ) ) ).

% num_of_nat_One
tff(fact_9853_num__of__nat__double,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_num(num_of_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Nb)) = aa_num_num(bit0,aa_nat_num(num_of_nat,Nb)) ) ) ).

% num_of_nat_double
tff(fact_9854_num__of__nat__plus__distrib,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( aa_nat_num(num_of_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) = aa_num_num(plus_plus_num(aa_nat_num(num_of_nat,Mb)),aa_nat_num(num_of_nat,Nb)) ) ) ) ).

% num_of_nat_plus_distrib
tff(fact_9855_nth__sorted__list__of__set__greaterThanLessThan,axiom,
    ! [Nb: nat,J: nat,I3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),aa_nat_nat(suc,I3)))
     => ( aa_nat_nat(nth_nat(linord2614967742042102400et_nat(aa_nat_set_nat(set_or5834768355832116004an_nat(I3),J))),Nb) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I3),Nb)) ) ) ).

% nth_sorted_list_of_set_greaterThanLessThan
tff(fact_9856_pow_Osimps_I3_J,axiom,
    ! [Xb: num,Yb: num] : aa_num_num(pow(Xb),aa_num_num(bit1,Yb)) = aa_num_num(times_times_num(aa_num_num(sqr,aa_num_num(pow(Xb),Yb))),Xb) ).

% pow.simps(3)
tff(fact_9857_sqr_Osimps_I1_J,axiom,
    aa_num_num(sqr,one) = one ).

% sqr.simps(1)
tff(fact_9858_sqr_Osimps_I2_J,axiom,
    ! [Nb: num] : aa_num_num(sqr,aa_num_num(bit0,Nb)) = aa_num_num(bit0,aa_num_num(bit0,aa_num_num(sqr,Nb))) ).

% sqr.simps(2)
tff(fact_9859_sqr__conv__mult,axiom,
    ! [Xb: num] : aa_num_num(sqr,Xb) = aa_num_num(times_times_num(Xb),Xb) ).

% sqr_conv_mult
tff(fact_9860_pow_Osimps_I2_J,axiom,
    ! [Xb: num,Yb: num] : aa_num_num(pow(Xb),aa_num_num(bit0,Yb)) = aa_num_num(sqr,aa_num_num(pow(Xb),Yb)) ).

% pow.simps(2)
tff(fact_9861_sqr_Osimps_I3_J,axiom,
    ! [Nb: num] : aa_num_num(sqr,aa_num_num(bit1,Nb)) = aa_num_num(bit1,aa_num_num(bit0,aa_num_num(plus_plus_num(aa_num_num(sqr,Nb)),Nb))) ).

% sqr.simps(3)
tff(fact_9862_nth__sorted__list__of__set__greaterThanAtMost,axiom,
    ! [Nb: nat,J: nat,I3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I3))
     => ( aa_nat_nat(nth_nat(linord2614967742042102400et_nat(aa_nat_set_nat(set_or6659071591806873216st_nat(I3),J))),Nb) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I3),Nb)) ) ) ).

% nth_sorted_list_of_set_greaterThanAtMost
tff(fact_9863_image__minus__const__atLeastLessThan__nat,axiom,
    ! [C: nat,Xb: nat,Yb: nat] :
      image_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_rt,C),aa_nat_set_nat(set_or4665077453230672383an_nat(Xb),Yb)) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),Yb),
        aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xb),C)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Yb),C)),
        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Yb),insert_nat(zero_zero_nat,bot_bot_set_nat),bot_bot_set_nat) ) ).

% image_minus_const_atLeastLessThan_nat
tff(fact_9864_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(fract(A),B))
      & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_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_9865_bij__betw__Suc,axiom,
    ! [M6: set_nat,N2: set_nat] :
      ( aa_set_nat_o(bij_betw_nat_nat(suc,M6),N2)
    <=> ( image_nat_nat(suc,M6) = N2 ) ) ).

% bij_betw_Suc
tff(fact_9866_image__Suc__atLeastAtMost,axiom,
    ! [I3: nat,J: nat] : image_nat_nat(suc,aa_nat_set_nat(set_or1269000886237332187st_nat(I3),J)) = aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,I3)),aa_nat_nat(suc,J)) ).

% image_Suc_atLeastAtMost
tff(fact_9867_image__Suc__atLeastLessThan,axiom,
    ! [I3: nat,J: nat] : image_nat_nat(suc,aa_nat_set_nat(set_or4665077453230672383an_nat(I3),J)) = aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(suc,I3)),aa_nat_nat(suc,J)) ).

% image_Suc_atLeastLessThan
tff(fact_9868_mult__rat,axiom,
    ! [A: int,B: int,C: int,D: int] : aa_rat_rat(times_times_rat(aa_int_rat(fract(A),B)),aa_int_rat(fract(C),D)) = aa_int_rat(fract(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_rat
tff(fact_9869_divide__rat,axiom,
    ! [A: int,B: int,C: int,D: int] : aa_rat_rat(divide_divide_rat(aa_int_rat(fract(A),B)),aa_int_rat(fract(C),D)) = aa_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_9870_floor__Fract,axiom,
    ! [A: int,B: int] : aa_rat_int(archim3151403230148437115or_rat,aa_int_rat(fract(A),B)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) ).

% floor_Fract
tff(fact_9871_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(fract(A),B)),aa_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_9872_add__rat,axiom,
    ! [B: int,D: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( ( D != zero_zero_int )
       => ( aa_rat_rat(plus_plus_rat(aa_int_rat(fract(A),B)),aa_int_rat(fract(C),D)) = aa_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_9873_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(fract(A),B)),aa_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_9874_diff__rat,axiom,
    ! [B: int,D: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( ( D != zero_zero_int )
       => ( aa_rat_rat(minus_minus_rat(aa_int_rat(fract(A),B)),aa_int_rat(fract(C),D)) = aa_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_9875_sgn__rat,axiom,
    ! [A: int,B: int] : aa_rat_rat(sgn_sgn_rat,aa_int_rat(fract(A),B)) = aa_int_rat(ring_1_of_int_rat,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_rat
tff(fact_9876_rat__number__collapse_I6_J,axiom,
    ! [K: int] : aa_int_rat(fract(K),zero_zero_int) = zero_zero_rat ).

% rat_number_collapse(6)
tff(fact_9877_rat__number__collapse_I1_J,axiom,
    ! [K: int] : aa_int_rat(fract(zero_zero_int),K) = zero_zero_rat ).

% rat_number_collapse(1)
tff(fact_9878_eq__rat_I1_J,axiom,
    ! [B: int,D: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( ( D != zero_zero_int )
       => ( ( aa_int_rat(fract(A),B) = aa_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_9879_mult__rat__cancel,axiom,
    ! [C: int,A: int,B: int] :
      ( ( C != zero_zero_int )
     => ( aa_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(fract(A),B) ) ) ).

% mult_rat_cancel
tff(fact_9880_Fract__of__nat__eq,axiom,
    ! [K: nat] : aa_int_rat(fract(aa_nat_int(semiri1314217659103216013at_int,K)),one_one_int) = aa_nat_rat(semiri681578069525770553at_rat,K) ).

% Fract_of_nat_eq
tff(fact_9881_zero__notin__Suc__image,axiom,
    ! [A3: set_nat] : ~ aa_set_nat_o(member_nat(zero_zero_nat),image_nat_nat(suc,A3)) ).

% zero_notin_Suc_image
tff(fact_9882_eq__rat_I3_J,axiom,
    ! [A: int,C: int] : aa_int_rat(fract(zero_zero_int),A) = aa_int_rat(fract(zero_zero_int),C) ).

% eq_rat(3)
tff(fact_9883_Rat__induct__pos,axiom,
    ! [P: fun_rat_o,Q2: rat] :
      ( ! [A4: int,B3: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B3)
         => aa_rat_o(P,aa_int_rat(fract(A4),B3)) )
     => aa_rat_o(P,Q2) ) ).

% Rat_induct_pos
tff(fact_9884_eq__rat_I2_J,axiom,
    ! [A: int] : aa_int_rat(fract(A),zero_zero_int) = aa_int_rat(fract(zero_zero_int),one_one_int) ).

% eq_rat(2)
tff(fact_9885_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_9886_Fract__coprime,axiom,
    ! [A: int,B: int] : aa_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(fract(A),B) ).

% Fract_coprime
tff(fact_9887_One__rat__def,axiom,
    one_one_rat = aa_int_rat(fract(one_one_int),one_one_int) ).

% One_rat_def
tff(fact_9888_Fract__of__int__eq,axiom,
    ! [K: int] : aa_int_rat(fract(K),one_one_int) = aa_int_rat(ring_1_of_int_rat,K) ).

% Fract_of_int_eq
tff(fact_9889_Zero__rat__def,axiom,
    zero_zero_rat = aa_int_rat(fract(zero_zero_int),one_one_int) ).

% Zero_rat_def
tff(fact_9890_rat__number__expand_I3_J,axiom,
    ! [K: num] : aa_num_rat(numeral_numeral_rat,K) = aa_int_rat(fract(aa_num_int(numeral_numeral_int,K)),one_one_int) ).

% rat_number_expand(3)
tff(fact_9891_rat__number__collapse_I3_J,axiom,
    ! [W: num] : aa_int_rat(fract(aa_num_int(numeral_numeral_int,W)),one_one_int) = aa_num_rat(numeral_numeral_rat,W) ).

% rat_number_collapse(3)
tff(fact_9892_image__Suc__lessThan,axiom,
    ! [Nb: nat] : image_nat_nat(suc,aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),Nb) ).

% image_Suc_lessThan
tff(fact_9893_image__Suc__atMost,axiom,
    ! [Nb: nat] : image_nat_nat(suc,aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),aa_nat_nat(suc,Nb)) ).

% image_Suc_atMost
tff(fact_9894_atLeast0__atMost__Suc__eq__insert__0,axiom,
    ! [Nb: nat] : aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,Nb)) = insert_nat(zero_zero_nat,image_nat_nat(suc,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb))) ).

% atLeast0_atMost_Suc_eq_insert_0
tff(fact_9895_atLeast0__lessThan__Suc__eq__insert__0,axiom,
    ! [Nb: nat] : aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),aa_nat_nat(suc,Nb)) = insert_nat(zero_zero_nat,image_nat_nat(suc,aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Nb))) ).

% atLeast0_lessThan_Suc_eq_insert_0
tff(fact_9896_lessThan__Suc__eq__insert__0,axiom,
    ! [Nb: nat] : aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb)) = insert_nat(zero_zero_nat,image_nat_nat(suc,aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% lessThan_Suc_eq_insert_0
tff(fact_9897_atMost__Suc__eq__insert__0,axiom,
    ! [Nb: nat] : aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Nb)) = insert_nat(zero_zero_nat,image_nat_nat(suc,aa_nat_set_nat(set_ord_atMost_nat,Nb))) ).

% atMost_Suc_eq_insert_0
tff(fact_9898_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(fract(A),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A) ) ) ).

% zero_less_Fract_iff
tff(fact_9899_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(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_9900_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(fract(A),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ) ).

% one_less_Fract_iff
tff(fact_9901_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(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_9902_rat__number__collapse_I5_J,axiom,
    aa_int_rat(fract(aa_int_int(uminus_uminus_int,one_one_int)),one_one_int) = aa_rat_rat(uminus_uminus_rat,one_one_rat) ).

% rat_number_collapse(5)
tff(fact_9903_Fract__add__one,axiom,
    ! [Nb: int,Mb: int] :
      ( ( Nb != zero_zero_int )
     => ( aa_int_rat(fract(aa_int_int(aa_int_fun_int_int(plus_plus_int,Mb),Nb)),Nb) = aa_rat_rat(plus_plus_rat(aa_int_rat(fract(Mb),Nb)),one_one_rat) ) ) ).

% Fract_add_one
tff(fact_9904_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(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_9905_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(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_9906_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(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_9907_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(fract(A),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ) ).

% one_le_Fract_iff
tff(fact_9908_rat__number__expand_I5_J,axiom,
    ! [K: num] : aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,K)) = aa_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_9909_rat__number__collapse_I4_J,axiom,
    ! [W: num] : aa_int_rat(fract(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))),one_one_int) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)) ).

% rat_number_collapse(4)
tff(fact_9910_of__nat__eq__id,axiom,
    semiri1316708129612266289at_nat = id_nat ).

% of_nat_eq_id
tff(fact_9911_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_9912_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_rl)) ).

% less_int_def
tff(fact_9913_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_rn)) ).

% less_eq_int_def
tff(fact_9914_nat__def,axiom,
    nat2 = map_fu2345160673673942751at_nat(rep_Integ,id_nat,produc6842872674320459806at_nat(minus_minus_nat)) ).

% nat_def
tff(fact_9915_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_9916_image__add__int__atLeastLessThan,axiom,
    ! [L: int,U: int] : image_int_int(aa_int_fun_int_int(aTP_Lamm_ru,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_9917_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_9918_range__mod,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( image_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_rv,Nb),top_top_set_nat) = aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Nb) ) ) ).

% range_mod
tff(fact_9919_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_9920_card__UNIV__unit,axiom,
    finite410649719033368117t_unit(top_to1996260823553986621t_unit) = one_one_nat ).

% card_UNIV_unit
tff(fact_9921_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_9922_range__mult,axiom,
    ! [A: real] :
      image_real_real(times_times_real(A),top_top_set_real) = $ite(A = zero_zero_real,insert_real(zero_zero_real,bot_bot_set_real),top_top_set_real) ).

% range_mult
tff(fact_9923_infinite__UNIV__int,axiom,
    ~ finite_finite_int(top_top_set_int) ).

% infinite_UNIV_int
tff(fact_9924_int__in__range__abs,axiom,
    ! [Nb: nat] : member_int(aa_nat_int(semiri1314217659103216013at_int,Nb),image_int_int(abs_abs_int,top_top_set_int)) ).

% int_in_range_abs
tff(fact_9925_root__def,axiom,
    ! [Nb: nat,Xb: real] :
      aa_real_real(aa_nat_fun_real_real(root,Nb),Xb) = $ite(Nb = zero_zero_nat,zero_zero_real,aa_real_real(the_in5290026491893676941l_real(top_top_set_real,aa_nat_fun_real_real(aTP_Lamm_rw,Nb)),Xb)) ).

% root_def
tff(fact_9926_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_9927_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_9928_char_Osize_I2_J,axiom,
    ! [X1: $o,X2: $o,X32: $o,X42: $o,X52: $o,X62: $o,X7: $o,X8: $o] : size_size_char(char2((X1),(X2),(X32),(X42),(X52),(X62),(X7),(X8))) = zero_zero_nat ).

% char.size(2)
tff(fact_9929_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_9930_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_9931_integer__of__char__code,axiom,
    ! [B0: $o,B1: $o,B22: $o,B32: $o,B42: $o,B52: $o,B62: $o,B72: $o] : integer_of_char(char2((B0),(B1),(B22),(B32),(B42),(B52),(B62),(B72))) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,zero_n356916108424825756nteger((B72))),numera6620942414471956472nteger(aa_num_num(bit0,one))),zero_n356916108424825756nteger((B62)))),numera6620942414471956472nteger(aa_num_num(bit0,one))),zero_n356916108424825756nteger((B52)))),numera6620942414471956472nteger(aa_num_num(bit0,one))),zero_n356916108424825756nteger((B42)))),numera6620942414471956472nteger(aa_num_num(bit0,one))),zero_n356916108424825756nteger((B32)))),numera6620942414471956472nteger(aa_num_num(bit0,one))),zero_n356916108424825756nteger((B22)))),numera6620942414471956472nteger(aa_num_num(bit0,one))),zero_n356916108424825756nteger((B1)))),numera6620942414471956472nteger(aa_num_num(bit0,one))),zero_n356916108424825756nteger((B0))) ).

% integer_of_char_code
tff(fact_9932_char_Osize__gen,axiom,
    ! [X1: $o,X2: $o,X32: $o,X42: $o,X52: $o,X62: $o,X7: $o,X8: $o] : size_char(char2((X1),(X2),(X32),(X42),(X52),(X62),(X7),(X8))) = zero_zero_nat ).

% char.size_gen
tff(fact_9933_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_9934_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_9935_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_9936_sorted__list__of__set__greaterThanAtMost,axiom,
    ! [I3: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,I3)),J)
     => ( linord2614967742042102400et_nat(aa_nat_set_nat(set_or6659071591806873216st_nat(I3),J)) = cons_nat(aa_nat_nat(suc,I3),linord2614967742042102400et_nat(aa_nat_set_nat(set_or6659071591806873216st_nat(aa_nat_nat(suc,I3)),J))) ) ) ).

% sorted_list_of_set_greaterThanAtMost
tff(fact_9937_sorted__list__of__set__greaterThanLessThan,axiom,
    ! [I3: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,I3)),J)
     => ( linord2614967742042102400et_nat(aa_nat_set_nat(set_or5834768355832116004an_nat(I3),J)) = cons_nat(aa_nat_nat(suc,I3),linord2614967742042102400et_nat(aa_nat_set_nat(set_or5834768355832116004an_nat(aa_nat_nat(suc,I3)),J))) ) ) ).

% sorted_list_of_set_greaterThanLessThan
tff(fact_9938_upto__aux__rec,axiom,
    ! [I3: int,J: int,Js: list_int] :
      upto_aux(I3,J,Js) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,J),I3),Js,upto_aux(I3,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int),cons_int(J,Js))) ).

% upto_aux_rec
tff(fact_9939_list__encode_Oelims,axiom,
    ! [Xb: list_nat,Yb: nat] :
      ( ( nat_list_encode(Xb) = Yb )
     => ( ( ( Xb = nil_nat )
         => ( Yb != zero_zero_nat ) )
       => ~ ! [X3: nat,Xs2: list_nat] :
              ( ( Xb = cons_nat(X3,Xs2) )
             => ( Yb != aa_nat_nat(suc,aa_Pro4243173770044397973at_nat(nat_prod_encode,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),nat_list_encode(Xs2)))) ) ) ) ) ).

% list_encode.elims
tff(fact_9940_upto_Opsimps,axiom,
    ! [I3: int,J: int] :
      ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(upto_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,I3),J))
     => ( upto(I3,J) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I3),J),cons_int(I3,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,I3),one_one_int),J)),nil_int) ) ) ).

% upto.psimps
tff(fact_9941_upto__empty,axiom,
    ! [J: int,I3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,J),I3)
     => ( upto(I3,J) = nil_int ) ) ).

% upto_empty
tff(fact_9942_upto__Nil2,axiom,
    ! [I3: int,J: int] :
      ( ( nil_int = upto(I3,J) )
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,J),I3) ) ).

% upto_Nil2
tff(fact_9943_upto__Nil,axiom,
    ! [I3: int,J: int] :
      ( ( upto(I3,J) = nil_int )
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,J),I3) ) ).

% upto_Nil
tff(fact_9944_upto__single,axiom,
    ! [I3: int] : upto(I3,I3) = cons_int(I3,nil_int) ).

% upto_single
tff(fact_9945_nth__upto,axiom,
    ! [I3: 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,I3),aa_nat_int(semiri1314217659103216013at_int,K))),J)
     => ( aa_nat_int(nth_int(upto(I3,J)),K) = aa_int_int(aa_int_fun_int_int(plus_plus_int,I3),aa_nat_int(semiri1314217659103216013at_int,K)) ) ) ).

% nth_upto
tff(fact_9946_length__upto,axiom,
    ! [I3: int,J: int] : size_size_list_int(upto(I3,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),I3)),one_one_int)) ).

% length_upto
tff(fact_9947_upto__rec__numeral_I1_J,axiom,
    ! [Mb: num,Nb: num] :
      upto(aa_num_int(numeral_numeral_int,Mb),aa_num_int(numeral_numeral_int,Nb)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,Nb)),cons_int(aa_num_int(numeral_numeral_int,Mb),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,Mb)),one_one_int),aa_num_int(numeral_numeral_int,Nb))),nil_int) ).

% upto_rec_numeral(1)
tff(fact_9948_upto__rec__numeral_I4_J,axiom,
    ! [Mb: num,Nb: num] :
      upto(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = $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,Mb))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))),cons_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb)),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)))),nil_int) ).

% upto_rec_numeral(4)
tff(fact_9949_upto__rec__numeral_I3_J,axiom,
    ! [Mb: num,Nb: num] :
      upto(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,Nb)) = $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,Mb))),aa_num_int(numeral_numeral_int,Nb)),cons_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb)),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),one_one_int),aa_num_int(numeral_numeral_int,Nb))),nil_int) ).

% upto_rec_numeral(3)
tff(fact_9950_upto__rec__numeral_I2_J,axiom,
    ! [Mb: num,Nb: num] :
      upto(aa_num_int(numeral_numeral_int,Mb),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))),cons_int(aa_num_int(numeral_numeral_int,Mb),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,Mb)),one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)))),nil_int) ).

% upto_rec_numeral(2)
tff(fact_9951_upto__aux__def,axiom,
    ! [I3: int,J: int,Js: list_int] : upto_aux(I3,J,Js) = append_int(upto(I3,J),Js) ).

% upto_aux_def
tff(fact_9952_upto__code,axiom,
    ! [I3: int,J: int] : upto(I3,J) = upto_aux(I3,J,nil_int) ).

% upto_code
tff(fact_9953_distinct__upto,axiom,
    ! [I3: int,J: int] : aa_list_int_o(distinct_int,upto(I3,J)) ).

% distinct_upto
tff(fact_9954_atLeastAtMost__upto,axiom,
    ! [I3: int,J: int] : set_or1266510415728281911st_int(I3,J) = set_int2(upto(I3,J)) ).

% atLeastAtMost_upto
tff(fact_9955_upto__split2,axiom,
    ! [I3: int,J: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I3),J)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J),K)
       => ( upto(I3,K) = append_int(upto(I3,J),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,J),one_one_int),K)) ) ) ) ).

% upto_split2
tff(fact_9956_upto__split1,axiom,
    ! [I3: int,J: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I3),J)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J),K)
       => ( upto(I3,K) = append_int(upto(I3,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int)),upto(J,K)) ) ) ) ).

% upto_split1
tff(fact_9957_atLeastLessThan__upto,axiom,
    ! [I3: int,J: int] : set_or4662586982721622107an_int(I3,J) = set_int2(upto(I3,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int))) ).

% atLeastLessThan_upto
tff(fact_9958_list__encode_Osimps_I1_J,axiom,
    nat_list_encode(nil_nat) = zero_zero_nat ).

% list_encode.simps(1)
tff(fact_9959_greaterThanAtMost__upto,axiom,
    ! [I3: int,J: int] : set_or6656581121297822940st_int(I3,J) = set_int2(upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,I3),one_one_int),J)) ).

% greaterThanAtMost_upto
tff(fact_9960_upto_Oelims,axiom,
    ! [Xb: int,Xaa: int,Yb: list_int] :
      ( ( upto(Xb,Xaa) = Yb )
     => ( Yb = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Xaa),cons_int(Xb,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),one_one_int),Xaa)),nil_int) ) ) ).

% upto.elims
tff(fact_9961_upto_Osimps,axiom,
    ! [I3: int,J: int] :
      upto(I3,J) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I3),J),cons_int(I3,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,I3),one_one_int),J)),nil_int) ).

% upto.simps
tff(fact_9962_upto__rec1,axiom,
    ! [I3: int,J: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I3),J)
     => ( upto(I3,J) = cons_int(I3,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,I3),one_one_int),J)) ) ) ).

% upto_rec1
tff(fact_9963_upto__rec2,axiom,
    ! [I3: int,J: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I3),J)
     => ( upto(I3,J) = append_int(upto(I3,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int)),cons_int(J,nil_int)) ) ) ).

% upto_rec2
tff(fact_9964_greaterThanLessThan__upto,axiom,
    ! [I3: int,J: int] : set_or5832277885323065728an_int(I3,J) = set_int2(upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,I3),one_one_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int))) ).

% greaterThanLessThan_upto
tff(fact_9965_upto__split3,axiom,
    ! [I3: int,J: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I3),J)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J),K)
       => ( upto(I3,K) = append_int(upto(I3,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_9966_list__encode_Osimps_I2_J,axiom,
    ! [Xb: nat,Xs: list_nat] : nat_list_encode(cons_nat(Xb,Xs)) = aa_nat_nat(suc,aa_Pro4243173770044397973at_nat(nat_prod_encode,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xb),nat_list_encode(Xs)))) ).

% list_encode.simps(2)
tff(fact_9967_upto_Opelims,axiom,
    ! [Xb: int,Xaa: int,Yb: list_int] :
      ( ( upto(Xb,Xaa) = Yb )
     => ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(upto_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Xb),Xaa))
       => ~ ( ( Yb = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Xaa),cons_int(Xb,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),one_one_int),Xaa)),nil_int) )
           => ~ aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(upto_rel),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Xb),Xaa)) ) ) ) ).

% upto.pelims
tff(fact_9968_DERIV__even__real__root,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real)
         => has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,Nb),aa_real_real(inverse_inverse_real,aa_real_real(times_times_real(aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ) ) ).

% DERIV_even_real_root
tff(fact_9969_DERIV__real__root__generic,axiom,
    ! [Nb: nat,Xb: real,D4: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( ( Xb != 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))),Nb)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
             => ( D4 = aa_real_real(inverse_inverse_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),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))),Nb)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real)
               => ( D4 = aa_real_real(uminus_uminus_real,aa_real_real(inverse_inverse_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),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))),Nb)
               => ( D4 = aa_real_real(inverse_inverse_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))) ) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,Nb),D4,topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ) ) ) ) ).

% DERIV_real_root_generic
tff(fact_9970_has__real__derivative__pos__inc__right,axiom,
    ! [F: fun_real_real,L: real,Xb: real,S3: set_real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xb,S3))
     => ( 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(plus_plus_real(Xb),H3),S3)
                 => ( 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,Xb)),aa_real_real(F,aa_real_real(plus_plus_real(Xb),H3))) ) ) ) ) ) ) ).

% has_real_derivative_pos_inc_right
tff(fact_9971_has__real__derivative__neg__dec__right,axiom,
    ! [F: fun_real_real,L: real,Xb: real,S3: set_real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xb,S3))
     => ( 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(plus_plus_real(Xb),H3),S3)
                 => ( 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(plus_plus_real(Xb),H3))),aa_real_real(F,Xb)) ) ) ) ) ) ) ).

% has_real_derivative_neg_dec_right
tff(fact_9972_DERIV__pos__inc__right,axiom,
    ! [F: fun_real_real,L: real,Xb: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xb,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,Xb)),aa_real_real(F,aa_real_real(plus_plus_real(Xb),H3))) ) ) ) ) ) ).

% DERIV_pos_inc_right
tff(fact_9973_DERIV__neg__dec__right,axiom,
    ! [F: fun_real_real,L: real,Xb: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xb,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(plus_plus_real(Xb),H3))),aa_real_real(F,Xb)) ) ) ) ) ) ).

% DERIV_neg_dec_right
tff(fact_9974_DERIV__const__ratio__const,axiom,
    ! [A: real,B: real,F: fun_real_real,K: real] :
      ( ( A != B )
     => ( ! [X3: real] : has_fi5821293074295781190e_real(F,K,topolo2177554685111907308n_real(X3,top_top_set_real))
       => ( aa_real_real(minus_minus_real(aa_real_real(F,B)),aa_real_real(F,A)) = aa_real_real(times_times_real(aa_real_real(minus_minus_real(B),A)),K) ) ) ) ).

% DERIV_const_ratio_const
tff(fact_9975_MVT2,axiom,
    ! [A: real,B: real,F: fun_real_real,F3: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B)
     => ( ! [X3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X3)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),B)
             => has_fi5821293074295781190e_real(F,aa_real_real(F3,X3),topolo2177554685111907308n_real(X3,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(minus_minus_real(aa_real_real(F,B)),aa_real_real(F,A)) = aa_real_real(times_times_real(aa_real_real(minus_minus_real(B),A)),aa_real_real(F3,Z2)) ) ) ) ) ).

% MVT2
tff(fact_9976_DERIV__const__average,axiom,
    ! [A: real,B: real,V: fun_real_real,K: real] :
      ( ( A != B )
     => ( ! [X3: real] : has_fi5821293074295781190e_real(V,K,topolo2177554685111907308n_real(X3,top_top_set_real))
       => ( aa_real_real(V,aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(A),B)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = aa_real_real(divide_divide_real(aa_real_real(plus_plus_real(aa_real_real(V,A)),aa_real_real(V,B))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) ) ) ).

% DERIV_const_average
tff(fact_9977_DERIV__ln__divide,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => has_fi5821293074295781190e_real(ln_ln_real,aa_real_real(divide_divide_real(one_one_real),Xb),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ).

% DERIV_ln_divide
tff(fact_9978_DERIV__pow,axiom,
    ! [Nb: nat,Xb: real,S2: set_real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(aTP_Lamm_rx,Nb),aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat)))),topolo2177554685111907308n_real(Xb,S2)) ).

% DERIV_pow
tff(fact_9979_DERIV__fun__pow,axiom,
    ! [G: fun_real_real,Mb: real,Xb: real,Nb: nat] :
      ( has_fi5821293074295781190e_real(G,Mb,topolo2177554685111907308n_real(Xb,top_top_set_real))
     => has_fi5821293074295781190e_real(aa_nat_fun_real_real(aTP_Lamm_ry(G),Nb),aa_real_real(times_times_real(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(G,Xb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)))),Mb),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ).

% DERIV_fun_pow
tff(fact_9980_has__real__derivative__powr,axiom,
    ! [Z: real,R2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Z)
     => has_fi5821293074295781190e_real(aTP_Lamm_rz(R2),aa_real_real(times_times_real(R2),aa_real_real(powr_real(Z),aa_real_real(minus_minus_real(R2),one_one_real))),topolo2177554685111907308n_real(Z,top_top_set_real)) ) ).

% has_real_derivative_powr
tff(fact_9981_DERIV__log,axiom,
    ! [Xb: real,B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => has_fi5821293074295781190e_real(log(B),aa_real_real(divide_divide_real(one_one_real),aa_real_real(times_times_real(aa_real_real(ln_ln_real,B)),Xb)),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ).

% DERIV_log
tff(fact_9982_DERIV__fun__powr,axiom,
    ! [G: fun_real_real,Mb: real,Xb: real,R2: real] :
      ( has_fi5821293074295781190e_real(G,Mb,topolo2177554685111907308n_real(Xb,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(G,Xb))
       => has_fi5821293074295781190e_real(aa_rea7340605660300375871l_real(aTP_Lamm_sa(G),R2),aa_real_real(times_times_real(aa_real_real(times_times_real(R2),aa_real_real(powr_real(aa_real_real(G,Xb)),aa_real_real(minus_minus_real(R2),aa_nat_real(semiri5074537144036343181t_real,one_one_nat))))),Mb),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ) ).

% DERIV_fun_powr
tff(fact_9983_DERIV__powr,axiom,
    ! [G: fun_real_real,Mb: real,Xb: real,F: fun_real_real,R2: real] :
      ( has_fi5821293074295781190e_real(G,Mb,topolo2177554685111907308n_real(Xb,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(G,Xb))
       => ( has_fi5821293074295781190e_real(F,R2,topolo2177554685111907308n_real(Xb,top_top_set_real))
         => has_fi5821293074295781190e_real(aa_fun6946558556880649704l_real(aTP_Lamm_sb(G),F),aa_real_real(times_times_real(aa_real_real(powr_real(aa_real_real(G,Xb)),aa_real_real(F,Xb))),aa_real_real(plus_plus_real(aa_real_real(times_times_real(R2),aa_real_real(ln_ln_real,aa_real_real(G,Xb)))),aa_real_real(divide_divide_real(aa_real_real(times_times_real(Mb),aa_real_real(F,Xb))),aa_real_real(G,Xb)))),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ) ) ).

% DERIV_powr
tff(fact_9984_DERIV__real__sqrt,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => has_fi5821293074295781190e_real(sqrt,aa_real_real(divide_divide_real(aa_real_real(inverse_inverse_real,aa_real_real(sqrt,Xb))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ).

% DERIV_real_sqrt
tff(fact_9985_DERIV__series_H,axiom,
    ! [F: fun_re4238438383274954274t_real,F3: fun_re4238438383274954274t_real,X0: real,A: real,B: real,L5: fun_nat_real] :
      ( ! [N: nat] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(aTP_Lamm_sc(F),N),aa_nat_real(aa_real_fun_nat_real(F3,X0),N),topolo2177554685111907308n_real(X0,top_top_set_real))
     => ( ! [X3: real] :
            ( member_real(X3,set_or1633881224788618240n_real(A,B))
           => summable_real(aa_real_fun_nat_real(F,X3)) )
       => ( member_real(X0,set_or1633881224788618240n_real(A,B))
         => ( summable_real(aa_real_fun_nat_real(F3,X0))
           => ( summable_real(L5)
             => ( ! [N: nat,X3: real,Y: real] :
                    ( member_real(X3,set_or1633881224788618240n_real(A,B))
                   => ( member_real(Y,set_or1633881224788618240n_real(A,B))
                     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(aa_nat_real(aa_real_fun_nat_real(F,X3),N)),aa_nat_real(aa_real_fun_nat_real(F,Y),N)))),aa_real_real(times_times_real(aa_nat_real(L5,N)),aa_real_real(abs_abs_real,aa_real_real(minus_minus_real(X3),Y)))) ) )
               => has_fi5821293074295781190e_real(aTP_Lamm_sd(F),suminf_real(aa_real_fun_nat_real(F3,X0)),topolo2177554685111907308n_real(X0,top_top_set_real)) ) ) ) ) ) ) ).

% DERIV_series'
tff(fact_9986_DERIV__arctan,axiom,
    ! [Xb: real] : has_fi5821293074295781190e_real(arctan,aa_real_real(inverse_inverse_real,aa_real_real(plus_plus_real(one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),topolo2177554685111907308n_real(Xb,top_top_set_real)) ).

% DERIV_arctan
tff(fact_9987_arsinh__real__has__field__derivative,axiom,
    ! [Xb: real,A3: set_real] : has_fi5821293074295781190e_real(arsinh_real,aa_real_real(divide_divide_real(one_one_real),aa_real_real(sqrt,aa_real_real(plus_plus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real))),topolo2177554685111907308n_real(Xb,A3)) ).

% arsinh_real_has_field_derivative
tff(fact_9988_DERIV__real__sqrt__generic,axiom,
    ! [Xb: real,D4: real] :
      ( ( Xb != zero_zero_real )
     => ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
         => ( D4 = aa_real_real(divide_divide_real(aa_real_real(inverse_inverse_real,aa_real_real(sqrt,Xb))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) )
       => ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real)
           => ( D4 = aa_real_real(divide_divide_real(aa_real_real(uminus_uminus_real,aa_real_real(inverse_inverse_real,aa_real_real(sqrt,Xb)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) )
         => has_fi5821293074295781190e_real(sqrt,D4,topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ) ) ).

% DERIV_real_sqrt_generic
tff(fact_9989_arcosh__real__has__field__derivative,axiom,
    ! [Xb: real,A3: set_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb)
     => has_fi5821293074295781190e_real(arcosh_real,aa_real_real(divide_divide_real(one_one_real),aa_real_real(sqrt,aa_real_real(minus_minus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real))),topolo2177554685111907308n_real(Xb,A3)) ) ).

% arcosh_real_has_field_derivative
tff(fact_9990_artanh__real__has__field__derivative,axiom,
    ! [Xb: real,A3: set_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xb)),one_one_real)
     => has_fi5821293074295781190e_real(artanh_real,aa_real_real(divide_divide_real(one_one_real),aa_real_real(minus_minus_real(one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),topolo2177554685111907308n_real(Xb,A3)) ) ).

% artanh_real_has_field_derivative
tff(fact_9991_DERIV__power__series_H,axiom,
    ! [R: real,F: fun_nat_real,X0: real] :
      ( ! [X3: real] :
          ( member_real(X3,set_or1633881224788618240n_real(aa_real_real(uminus_uminus_real,R),R))
         => summable_real(aa_real_fun_nat_real(aTP_Lamm_se(F),X3)) )
     => ( 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_sg(F),suminf_real(aa_real_fun_nat_real(aTP_Lamm_se(F),X0)),topolo2177554685111907308n_real(X0,top_top_set_real)) ) ) ) ).

% DERIV_power_series'
tff(fact_9992_DERIV__real__root,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,Nb),aa_real_real(inverse_inverse_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ) ).

% DERIV_real_root
tff(fact_9993_DERIV__arccos,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),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(minus_minus_real(one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ) ).

% DERIV_arccos
tff(fact_9994_DERIV__arcsin,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),one_one_real)
       => has_fi5821293074295781190e_real(arcsin,aa_real_real(inverse_inverse_real,aa_real_real(sqrt,aa_real_real(minus_minus_real(one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ) ).

% DERIV_arcsin
tff(fact_9995_Maclaurin__all__le__objl,axiom,
    ! [Diff: fun_na6724076748692969250l_real,F: fun_real_real,Xb: real,Nb: nat] :
      ( ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
        & ! [M2: nat,X3: real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M2),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M2)),X3),topolo2177554685111907308n_real(X3,top_top_set_real)) )
     => ? [T2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T2)),aa_real_real(abs_abs_real,Xb))
          & ( aa_real_real(F,Xb) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_sh(Diff),Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,Nb),T2)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb))) ) ) ) ).

% Maclaurin_all_le_objl
tff(fact_9996_Maclaurin__all__le,axiom,
    ! [Diff: fun_na6724076748692969250l_real,F: fun_real_real,Xb: real,Nb: nat] :
      ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
     => ( ! [M2: nat,X3: real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M2),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M2)),X3),topolo2177554685111907308n_real(X3,top_top_set_real))
       => ? [T2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T2)),aa_real_real(abs_abs_real,Xb))
            & ( aa_real_real(F,Xb) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_sh(Diff),Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,Nb),T2)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb))) ) ) ) ) ).

% Maclaurin_all_le
tff(fact_9997_DERIV__odd__real__root,axiom,
    ! [Nb: nat,Xb: real] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( ( Xb != zero_zero_real )
       => has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,Nb),aa_real_real(inverse_inverse_real,aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ) ).

% DERIV_odd_real_root
tff(fact_9998_Maclaurin,axiom,
    ! [H: real,Nb: 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),Nb)
       => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
         => ( ! [M2: nat,T2: real] :
                ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Nb)
                  & 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(Diff,M2),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M2)),T2),topolo2177554685111907308n_real(T2,top_top_set_real)) )
           => ? [T2: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),T2)
                & aa_real_o(aa_real_fun_real_o(ord_less_real,T2),H)
                & ( aa_real_real(F,H) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_fun1113820866546176217t_real(aTP_Lamm_iy(H),Diff)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,Nb),T2)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,H),Nb))) ) ) ) ) ) ) ).

% Maclaurin
tff(fact_9999_Maclaurin2,axiom,
    ! [H: real,Diff: fun_na6724076748692969250l_real,F: fun_real_real,Nb: 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 )
       => ( ! [M2: nat,T2: real] :
              ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Nb)
                & 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(Diff,M2),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M2)),T2),topolo2177554685111907308n_real(T2,top_top_set_real)) )
         => ? [T2: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),T2)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T2),H)
              & ( aa_real_real(F,H) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_fun1113820866546176217t_real(aTP_Lamm_iy(H),Diff)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,Nb),T2)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,H),Nb))) ) ) ) ) ) ).

% Maclaurin2
tff(fact_10000_Maclaurin__minus,axiom,
    ! [H: real,Nb: 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),Nb)
       => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
         => ( ! [M2: nat,T2: real] :
                ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Nb)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,H),T2)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T2),zero_zero_real) )
               => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M2),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M2)),T2),topolo2177554685111907308n_real(T2,top_top_set_real)) )
           => ? [T2: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,H),T2)
                & aa_real_o(aa_real_fun_real_o(ord_less_real,T2),zero_zero_real)
                & ( aa_real_real(F,H) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_fun1113820866546176217t_real(aTP_Lamm_iy(H),Diff)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,Nb),T2)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,H),Nb))) ) ) ) ) ) ) ).

% Maclaurin_minus
tff(fact_10001_Maclaurin__all__lt,axiom,
    ! [Diff: fun_na6724076748692969250l_real,F: fun_real_real,Nb: nat,Xb: 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),Nb)
       => ( ( Xb != zero_zero_real )
         => ( ! [M2: nat,X3: real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M2),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M2)),X3),topolo2177554685111907308n_real(X3,top_top_set_real))
           => ? [T2: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(abs_abs_real,T2))
                & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,T2)),aa_real_real(abs_abs_real,Xb))
                & ( aa_real_real(F,Xb) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_sh(Diff),Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,Nb),T2)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb))) ) ) ) ) ) ) ).

% Maclaurin_all_lt
tff(fact_10002_Maclaurin__bi__le,axiom,
    ! [Diff: fun_na6724076748692969250l_real,F: fun_real_real,Nb: nat,Xb: real] :
      ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
     => ( ! [M2: nat,T2: real] :
            ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Nb)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T2)),aa_real_real(abs_abs_real,Xb)) )
           => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M2),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M2)),T2),topolo2177554685111907308n_real(T2,top_top_set_real)) )
       => ? [T2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T2)),aa_real_real(abs_abs_real,Xb))
            & ( aa_real_real(F,Xb) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_sh(Diff),Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,Nb),T2)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb))) ) ) ) ) ).

% Maclaurin_bi_le
tff(fact_10003_Taylor,axiom,
    ! [Nb: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real,A: real,B: real,C: real,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
       => ( ! [M2: nat,T2: real] :
              ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Nb)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),T2)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T2),B) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M2),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M2)),T2),topolo2177554685111907308n_real(T2,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),Xb)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),B)
                 => ( ( Xb != C )
                   => ? [T2: real] :
                        ( $ite(
                            aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),C),
                            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),T2)
                            & aa_real_o(aa_real_fun_real_o(ord_less_real,T2),C) ),
                            ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),T2)
                            & aa_real_o(aa_real_fun_real_o(ord_less_real,T2),Xb) ) )
                        & ( aa_real_real(F,Xb) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_si(Diff),C),Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,Nb),T2)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(minus_minus_real(Xb),C)),Nb))) ) ) ) ) ) ) ) ) ) ) ).

% Taylor
tff(fact_10004_Taylor__up,axiom,
    ! [Nb: 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),Nb)
     => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
       => ( ! [M2: nat,T2: real] :
              ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Nb)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),T2)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T2),B) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M2),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M2)),T2),topolo2177554685111907308n_real(T2,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)
             => ? [T2: real] :
                  ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),T2)
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,T2),B)
                  & ( aa_real_real(F,B) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_sj(Diff),B),C)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,Nb),T2)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(minus_minus_real(B),C)),Nb))) ) ) ) ) ) ) ) ).

% Taylor_up
tff(fact_10005_Taylor__down,axiom,
    ! [Nb: 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),Nb)
     => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
       => ( ! [M2: nat,T2: real] :
              ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Nb)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),T2)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T2),B) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M2),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M2)),T2),topolo2177554685111907308n_real(T2,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)
             => ? [T2: real] :
                  ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),T2)
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,T2),C)
                  & ( aa_real_real(F,A) = aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_sj(Diff),A),C)),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Diff,Nb),T2)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(minus_minus_real(A),C)),Nb))) ) ) ) ) ) ) ) ).

% Taylor_down
tff(fact_10006_Maclaurin__lemma2,axiom,
    ! [Nb: nat,H: real,Diff: fun_na6724076748692969250l_real,K: nat,B4: real] :
      ( ! [M2: nat,T2: real] :
          ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Nb)
            & 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(Diff,M2),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M2)),T2),topolo2177554685111907308n_real(T2,top_top_set_real)) )
     => ( ( Nb = aa_nat_nat(suc,K) )
       => ! [M: nat,T3: real] :
            ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T3)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T3),H) )
           => has_fi5821293074295781190e_real(aa_nat_fun_real_real(aa_rea1103652080130114956l_real(aa_fun2567993205388704171l_real(aTP_Lamm_sl(Nb),Diff),B4),M),aa_real_real(minus_minus_real(aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M)),T3)),aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_sm(Diff),M),T3)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,M))))),aa_real_real(times_times_real(B4),aa_real_real(divide_divide_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,T3),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,M)))),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,M))))))),topolo2177554685111907308n_real(T3,top_top_set_real)) ) ) ) ).

% Maclaurin_lemma2
tff(fact_10007_DERIV__arctan__series,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xb)),one_one_real)
     => has_fi5821293074295781190e_real(aTP_Lamm_sn,suminf_real(aa_real_fun_nat_real(aTP_Lamm_so,Xb)),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ).

% DERIV_arctan_series
tff(fact_10008_take__bit__numeral__minus__numeral__int,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(bit_se2923211474154528505it_int(aa_num_nat(numeral_numeral_nat,Mb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = case_option_int_num(zero_zero_int,aTP_Lamm_sp(Mb),bit_take_bit_num(aa_num_nat(numeral_numeral_nat,Mb),Nb)) ).

% take_bit_numeral_minus_numeral_int
tff(fact_10009_take__bit__num__simps_I1_J,axiom,
    ! [Mb: num] : bit_take_bit_num(zero_zero_nat,Mb) = none_num ).

% take_bit_num_simps(1)
tff(fact_10010_take__bit__num__simps_I2_J,axiom,
    ! [Nb: nat] : bit_take_bit_num(aa_nat_nat(suc,Nb),one) = some_num(one) ).

% take_bit_num_simps(2)
tff(fact_10011_take__bit__num__simps_I5_J,axiom,
    ! [R2: num] : bit_take_bit_num(aa_num_nat(numeral_numeral_nat,R2),one) = some_num(one) ).

% take_bit_num_simps(5)
tff(fact_10012_take__bit__num__simps_I3_J,axiom,
    ! [Nb: nat,Mb: num] : bit_take_bit_num(aa_nat_nat(suc,Nb),aa_num_num(bit0,Mb)) = case_o6005452278849405969um_num(none_num,aTP_Lamm_sq,bit_take_bit_num(Nb,Mb)) ).

% take_bit_num_simps(3)
tff(fact_10013_take__bit__num__simps_I4_J,axiom,
    ! [Nb: nat,Mb: num] : bit_take_bit_num(aa_nat_nat(suc,Nb),aa_num_num(bit1,Mb)) = some_num(case_option_num_num(one,bit1,bit_take_bit_num(Nb,Mb))) ).

% take_bit_num_simps(4)
tff(fact_10014_take__bit__num__simps_I6_J,axiom,
    ! [R2: num,Mb: num] : bit_take_bit_num(aa_num_nat(numeral_numeral_nat,R2),aa_num_num(bit0,Mb)) = case_o6005452278849405969um_num(none_num,aTP_Lamm_sq,bit_take_bit_num(aa_num_nat(pred_numeral,R2),Mb)) ).

% take_bit_num_simps(6)
tff(fact_10015_take__bit__num__simps_I7_J,axiom,
    ! [R2: num,Mb: num] : bit_take_bit_num(aa_num_nat(numeral_numeral_nat,R2),aa_num_num(bit1,Mb)) = some_num(case_option_num_num(one,bit1,bit_take_bit_num(aa_num_nat(pred_numeral,R2),Mb))) ).

% take_bit_num_simps(7)
tff(fact_10016_Code__Abstract__Nat_Otake__bit__num__code_I2_J,axiom,
    ! [Nb: nat,Mb: num] : bit_take_bit_num(Nb,aa_num_num(bit0,Mb)) = case_nat_option_num(none_num,aTP_Lamm_sr(Mb),Nb) ).

% Code_Abstract_Nat.take_bit_num_code(2)
tff(fact_10017_Code__Abstract__Nat_Otake__bit__num__code_I1_J,axiom,
    ! [Nb: nat] : bit_take_bit_num(Nb,one) = case_nat_option_num(none_num,aTP_Lamm_ss,Nb) ).

% Code_Abstract_Nat.take_bit_num_code(1)
tff(fact_10018_Code__Abstract__Nat_Otake__bit__num__code_I3_J,axiom,
    ! [Nb: nat,Mb: num] : bit_take_bit_num(Nb,aa_num_num(bit1,Mb)) = case_nat_option_num(none_num,aTP_Lamm_st(Mb),Nb) ).

% Code_Abstract_Nat.take_bit_num_code(3)
tff(fact_10019_take__bit__num__def,axiom,
    ! [Nb: nat,Mb: num] :
      bit_take_bit_num(Nb,Mb) = $ite(aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),aa_num_nat(numeral_numeral_nat,Mb)) = zero_zero_nat,none_num,some_num(aa_nat_num(num_of_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),aa_num_nat(numeral_numeral_nat,Mb))))) ).

% take_bit_num_def
tff(fact_10020_and__minus__numerals_I7_J,axiom,
    ! [Nb: num,Mb: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))),aa_num_int(numeral_numeral_int,Mb)) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(Mb,aa_num_num(bitM,Nb))) ).

% and_minus_numerals(7)
tff(fact_10021_and__minus__numerals_I3_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(Mb,aa_num_num(bitM,Nb))) ).

% and_minus_numerals(3)
tff(fact_10022_and__minus__numerals_I4_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(Mb,aa_num_num(bit0,Nb))) ).

% and_minus_numerals(4)
tff(fact_10023_and__minus__numerals_I8_J,axiom,
    ! [Nb: num,Mb: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))),aa_num_int(numeral_numeral_int,Mb)) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(Mb,aa_num_num(bit0,Nb))) ).

% and_minus_numerals(8)
tff(fact_10024_and__not__num_Osimps_I1_J,axiom,
    bit_and_not_num(one,one) = none_num ).

% and_not_num.simps(1)
tff(fact_10025_and__not__num_Osimps_I4_J,axiom,
    ! [Mb: num] : bit_and_not_num(aa_num_num(bit0,Mb),one) = some_num(aa_num_num(bit0,Mb)) ).

% and_not_num.simps(4)
tff(fact_10026_and__not__num_Osimps_I2_J,axiom,
    ! [Nb: num] : bit_and_not_num(one,aa_num_num(bit0,Nb)) = some_num(one) ).

% and_not_num.simps(2)
tff(fact_10027_and__not__num_Osimps_I3_J,axiom,
    ! [Nb: num] : bit_and_not_num(one,aa_num_num(bit1,Nb)) = none_num ).

% and_not_num.simps(3)
tff(fact_10028_and__not__num_Osimps_I7_J,axiom,
    ! [Mb: num] : bit_and_not_num(aa_num_num(bit1,Mb),one) = some_num(aa_num_num(bit0,Mb)) ).

% and_not_num.simps(7)
tff(fact_10029_and__not__num__eq__Some__iff,axiom,
    ! [Mb: num,Nb: num,Q2: num] :
      ( ( bit_and_not_num(Mb,Nb) = some_num(Q2) )
    <=> ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb))) = aa_num_int(numeral_numeral_int,Q2) ) ) ).

% and_not_num_eq_Some_iff
tff(fact_10030_and__not__num_Osimps_I8_J,axiom,
    ! [Mb: num,Nb: num] : bit_and_not_num(aa_num_num(bit1,Mb),aa_num_num(bit0,Nb)) = case_o6005452278849405969um_num(some_num(one),aTP_Lamm_su,bit_and_not_num(Mb,Nb)) ).

% and_not_num.simps(8)
tff(fact_10031_and__not__num__eq__None__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( ( bit_and_not_num(Mb,Nb) = none_num )
    <=> ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb))) = zero_zero_int ) ) ).

% and_not_num_eq_None_iff
tff(fact_10032_int__numeral__not__and__num,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Mb))),aa_num_int(numeral_numeral_int,Nb)) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(Nb,Mb)) ).

% int_numeral_not_and_num
tff(fact_10033_int__numeral__and__not__num,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb))) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(Mb,Nb)) ).

% int_numeral_and_not_num
tff(fact_10034_Bit__Operations_Otake__bit__num__code,axiom,
    ! [Nb: nat,Mb: num] : bit_take_bit_num(Nb,Mb) = produc478579273971653890on_num(aTP_Lamm_sy,product_Pair_nat_num(Nb,Mb)) ).

% Bit_Operations.take_bit_num_code
tff(fact_10035_isCont__arcosh,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb)
     => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Xb,top_top_set_real),arcosh_real) ) ).

% isCont_arcosh
tff(fact_10036_isCont__arccos,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),one_one_real)
       => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Xb,top_top_set_real),arccos) ) ) ).

% isCont_arccos
tff(fact_10037_isCont__arcsin,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),one_one_real)
       => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Xb,top_top_set_real),arcsin) ) ) ).

% isCont_arcsin
tff(fact_10038_isCont__artanh,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),one_one_real)
       => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Xb,top_top_set_real),artanh_real) ) ) ).

% isCont_artanh
tff(fact_10039_GMVT_H,axiom,
    ! [A: real,B: real,F: fun_real_real,G: fun_real_real,G2: fun_real_real,F3: 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(F3,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(times_times_real(aa_real_real(minus_minus_real(aa_real_real(F,B)),aa_real_real(F,A))),aa_real_real(G2,C3)) = aa_real_real(times_times_real(aa_real_real(minus_minus_real(aa_real_real(G,B)),aa_real_real(G,A))),aa_real_real(F3,C3)) ) ) ) ) ) ) ) ).

% GMVT'
tff(fact_10040_LIM__cos__div__sin,axiom,
    filterlim_real_real(aTP_Lamm_sz,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),top_top_set_real)) ).

% LIM_cos_div_sin
tff(fact_10041_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_ta(A)),set_or1222579329274155063t_real(aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_ta(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_ta(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_10042_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_ta(A)),set_or1222579329274155063t_real(aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_ta(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_ta(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_10043_filterlim__Suc,axiom,
    filterlim_nat_nat(suc,at_top_nat,at_top_nat) ).

% filterlim_Suc
tff(fact_10044_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_10045_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_bb,C),at_top_nat,at_top_nat) ) ).

% mult_nat_right_at_top
tff(fact_10046_LIMSEQ__root,axiom,
    filterlim_nat_real(aTP_Lamm_tb,topolo2815343760600316023s_real(one_one_real),at_top_nat) ).

% LIMSEQ_root
tff(fact_10047_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_tc(F),G),topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
           => ? [L3: real] :
                ( ! [N7: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,N7)),L3)
                & filterlim_nat_real(F,topolo2815343760600316023s_real(L3),at_top_nat)
                & ! [N7: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L3),aa_nat_real(G,N7))
                & filterlim_nat_real(G,topolo2815343760600316023s_real(L3),at_top_nat) ) ) ) ) ) ).

% nested_sequence_unique
tff(fact_10048_LIMSEQ__inverse__zero,axiom,
    ! [X6: fun_nat_real] :
      ( ! [R3: 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,R3),aa_nat_real(X6,N)) )
     => filterlim_nat_real(aTP_Lamm_td(X6),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_inverse_zero
tff(fact_10049_lim__inverse__n_H,axiom,
    filterlim_nat_real(aTP_Lamm_te,topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ).

% lim_inverse_n'
tff(fact_10050_LIMSEQ__root__const,axiom,
    ! [C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_tf,C),topolo2815343760600316023s_real(one_one_real),at_top_nat) ) ).

% LIMSEQ_root_const
tff(fact_10051_LIMSEQ__inverse__real__of__nat,axiom,
    filterlim_nat_real(aTP_Lamm_tg,topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ).

% LIMSEQ_inverse_real_of_nat
tff(fact_10052_LIMSEQ__inverse__real__of__nat__add,axiom,
    ! [R2: real] : filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_th,R2),topolo2815343760600316023s_real(R2),at_top_nat) ).

% LIMSEQ_inverse_real_of_nat_add
tff(fact_10053_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(plus_plus_real(aa_nat_real(F,N7)),E)) )
         => filterlim_nat_real(F,topolo2815343760600316023s_real(L),at_top_nat) ) ) ) ).

% increasing_LIMSEQ
tff(fact_10054_LIMSEQ__realpow__zero,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),one_one_real)
       => filterlim_nat_real(aa_real_fun_nat_real(power_power_real,Xb),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ) ).

% LIMSEQ_realpow_zero
tff(fact_10055_LIMSEQ__divide__realpow__zero,axiom,
    ! [Xb: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb)
     => filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_ti(Xb),A),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_divide_realpow_zero
tff(fact_10056_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(aa_real_fun_nat_real(power_power_real,C),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_abs_realpow_zero2
tff(fact_10057_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(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,C)),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_abs_realpow_zero
tff(fact_10058_LIMSEQ__inverse__realpow__zero,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb)
     => filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_tj,Xb),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_inverse_realpow_zero
tff(fact_10059_LIMSEQ__inverse__real__of__nat__add__minus,axiom,
    ! [R2: real] : filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_tk,R2),topolo2815343760600316023s_real(R2),at_top_nat) ).

% LIMSEQ_inverse_real_of_nat_add_minus
tff(fact_10060_tendsto__exp__limit__sequentially,axiom,
    ! [Xb: real] : filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_tl,Xb),topolo2815343760600316023s_real(aa_real_real(exp_real,Xb)),at_top_nat) ).

% tendsto_exp_limit_sequentially
tff(fact_10061_LIMSEQ__inverse__real__of__nat__add__minus__mult,axiom,
    ! [R2: real] : filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_tm,R2),topolo2815343760600316023s_real(R2),at_top_nat) ).

% LIMSEQ_inverse_real_of_nat_add_minus_mult
tff(fact_10062_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_ta(A)) ) ) ).

% summable_Leibniz(1)
tff(fact_10063_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_ta(A)) ) ) ) ).

% summable
tff(fact_10064_cos__diff__limit__1,axiom,
    ! [Theta: fun_nat_real,Theta2: real] :
      ( filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_tn(Theta),Theta2),topolo2815343760600316023s_real(one_one_real),at_top_nat)
     => ~ ! [K2: fun_nat_int] : ~ filterlim_nat_real(aa_fun4551936527957760560t_real(aTP_Lamm_to(Theta),K2),topolo2815343760600316023s_real(Theta2),at_top_nat) ) ).

% cos_diff_limit_1
tff(fact_10065_cos__limit__1,axiom,
    ! [Theta: fun_nat_real] :
      ( filterlim_nat_real(aTP_Lamm_tp(Theta),topolo2815343760600316023s_real(one_one_real),at_top_nat)
     => ? [K2: fun_nat_int] : filterlim_nat_real(aa_fun4551936527957760560t_real(aTP_Lamm_to(Theta),K2),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% cos_limit_1
tff(fact_10066_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_tq(A),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_ta(A))),at_top_nat) ) ) ).

% summable_Leibniz(4)
tff(fact_10067_zeroseq__arctan__series,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),one_one_real)
     => filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_ga,Xb),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% zeroseq_arctan_series
tff(fact_10068_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_tq(A),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_ta(A))),at_top_nat) ) ) ) ).

% summable_Leibniz'(3)
tff(fact_10069_summable__Leibniz_H_I2_J,axiom,
    ! [A: fun_nat_real,Nb: 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_ta(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))),Nb)))),suminf_real(aTP_Lamm_ta(A))) ) ) ) ).

% summable_Leibniz'(2)
tff(fact_10070_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)
         => ? [L3: real] :
              ( ! [N7: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_ta(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)))),L3)
              & filterlim_nat_real(aTP_Lamm_tq(A),topolo2815343760600316023s_real(L3),at_top_nat)
              & ! [N7: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L3),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_ta(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_tr(A),topolo2815343760600316023s_real(L3),at_top_nat) ) ) ) ) ).

% sums_alternating_upper_lower
tff(fact_10071_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_tr(A),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_ta(A))),at_top_nat) ) ) ).

% summable_Leibniz(5)
tff(fact_10072_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_tr(A),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_ta(A))),at_top_nat) ) ) ) ).

% summable_Leibniz'(5)
tff(fact_10073_summable__Leibniz_H_I4_J,axiom,
    ! [A: fun_nat_real,Nb: 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_ta(A))),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_ta(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))),Nb)),one_one_nat)))) ) ) ) ).

% summable_Leibniz'(4)
tff(fact_10074_real__bounded__linear,axiom,
    ! [F: fun_real_real] :
      ( real_V5970128139526366754l_real(F)
    <=> ? [C2: real] :
        ! [X4: real] : aa_real_real(F,X4) = aa_real_real(times_times_real(X4),C2) ) ).

% real_bounded_linear
tff(fact_10075_tendsto__exp__limit__at__right,axiom,
    ! [Xb: real] : filterlim_real_real(aTP_Lamm_ts(Xb),topolo2815343760600316023s_real(aa_real_real(exp_real,Xb)),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real))) ).

% tendsto_exp_limit_at_right
tff(fact_10076_tendsto__arctan__at__bot,axiom,
    filterlim_real_real(arctan,topolo2815343760600316023s_real(aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),at_bot_real) ).

% tendsto_arctan_at_bot
tff(fact_10077_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_10078_filterlim__tan__at__right,axiom,
    filterlim_real_real(tan_real,at_bot_real,topolo2177554685111907308n_real(aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),set_or5849166863359141190n_real(aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))))) ).

% filterlim_tan_at_right
tff(fact_10079_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_10080_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_10081_filterlim__pow__at__bot__odd,axiom,
    ! [Nb: nat,F: fun_real_real,F4: filter_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( filterlim_real_real(F,at_bot_real,F4)
       => ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
         => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_tt(Nb),F),at_bot_real,F4) ) ) ) ).

% filterlim_pow_at_bot_odd
tff(fact_10082_filterlim__pow__at__bot__even,axiom,
    ! [Nb: nat,F: fun_real_real,F4: filter_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( filterlim_real_real(F,at_bot_real,F4)
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
         => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_tt(Nb),F),at_top_real,F4) ) ) ) ).

% filterlim_pow_at_bot_even
tff(fact_10083_atLeast__0,axiom,
    aa_nat_set_nat(set_ord_atLeast_nat,zero_zero_nat) = top_top_set_nat ).

% atLeast_0
tff(fact_10084_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_10085_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_10086_tanh__real__at__top,axiom,
    filterlim_real_real(tanh_real,topolo2815343760600316023s_real(one_one_real),at_top_real) ).

% tanh_real_at_top
tff(fact_10087_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_10088_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_10089_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_10090_tendsto__exp__limit__at__top,axiom,
    ! [Xb: real] : filterlim_real_real(aTP_Lamm_tu(Xb),topolo2815343760600316023s_real(aa_real_real(exp_real,Xb)),at_top_real) ).

% tendsto_exp_limit_at_top
tff(fact_10091_filterlim__tan__at__left,axiom,
    filterlim_real_real(tan_real,at_top_real,topolo2177554685111907308n_real(aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),set_or5984915006950818249n_real(aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))) ).

% filterlim_tan_at_left
tff(fact_10092_tendsto__arctan__at__top,axiom,
    filterlim_real_real(arctan,topolo2815343760600316023s_real(aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),at_top_real) ).

% tendsto_arctan_at_top
tff(fact_10093_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)
     => ( ! [X3: real] :
            ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X3)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),B) )
           => topolo4422821103128117721l_real(topolo2177554685111907308n_real(X3,top_top_set_real),F) )
       => ( ! [X3: real] :
              ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),X3)
                & aa_real_o(aa_real_fun_real_o(ord_less_real,X3),B) )
             => differ6690327859849518006l_real(F,topolo2177554685111907308n_real(X3,top_top_set_real)) )
         => ( ! [X3: real] :
                ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X3)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),B) )
               => topolo4422821103128117721l_real(topolo2177554685111907308n_real(X3,top_top_set_real),G) )
           => ( ! [X3: real] :
                  ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),X3)
                    & aa_real_o(aa_real_fun_real_o(ord_less_real,X3),B) )
                 => differ6690327859849518006l_real(G,topolo2177554685111907308n_real(X3,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(times_times_real(aa_real_real(minus_minus_real(aa_real_real(F,B)),aa_real_real(F,A))),G_c) = aa_real_real(times_times_real(aa_real_real(minus_minus_real(aa_real_real(G,B)),aa_real_real(G,A))),F_c) ) ) ) ) ) ) ) ).

% GMVT
tff(fact_10094_eventually__sequentially__Suc,axiom,
    ! [P: fun_nat_o] :
      ( eventually_nat(aTP_Lamm_tv(P),at_top_nat)
    <=> eventually_nat(P,at_top_nat) ) ).

% eventually_sequentially_Suc
tff(fact_10095_eventually__sequentially__seg,axiom,
    ! [P: fun_nat_o,K: nat] :
      ( eventually_nat(aa_nat_fun_nat_o(aTP_Lamm_tw(P),K),at_top_nat)
    <=> eventually_nat(P,at_top_nat) ) ).

% eventually_sequentially_seg
tff(fact_10096_sequentially__offset,axiom,
    ! [P: fun_nat_o,K: nat] :
      ( eventually_nat(P,at_top_nat)
     => eventually_nat(aa_nat_fun_nat_o(aTP_Lamm_tw(P),K),at_top_nat) ) ).

% sequentially_offset
tff(fact_10097_eventually__sequentiallyI,axiom,
    ! [C: nat,P: fun_nat_o] :
      ( ! [X3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),X3)
         => aa_nat_o(P,X3) )
     => eventually_nat(P,at_top_nat) ) ).

% eventually_sequentiallyI
tff(fact_10098_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_10099_le__sequentially,axiom,
    ! [F4: filter_nat] :
      ( ord_le2510731241096832064er_nat(F4,at_top_nat)
    <=> ! [N8: nat] : eventually_nat(aa_nat_fun_nat_o(ord_less_eq_nat,N8),F4) ) ).

% le_sequentially
tff(fact_10100_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_tx(P),A),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real))) ) ).

% eventually_at_right_to_0
tff(fact_10101_VEBT__internal_Ovalid_H_Oelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(vEBT_VEBT_valid(Xb),Xaa)
     => ( ( ? [Uu2: $o,Uv2: $o] : Xb = vEBT_Leaf((Uu2),(Uv2))
         => ( Xaa = one_one_nat ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
              ( ( Xb = vEBT_Node(Mima,Deg,TreeList,Summary) )
             => ( ( Deg = Xaa )
                & $let(
                    n: nat,
                    n:= 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),n),
                      ( ! [X4: vEBT_VEBT] :
                          ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList))
                         => aa_nat_o(vEBT_VEBT_valid(X4),n) )
                      & 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(( ~ ? [X9: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary),X9)
                            & ! [X4: vEBT_VEBT] :
                                ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList))
                               => ~ ? [X9: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X4),X9) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_ty(Deg),TreeList),Summary),n),m2))),Mima) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(3)
tff(fact_10102_GreatestI__ex__nat,axiom,
    ! [P: fun_nat_o,B: nat] :
      ( ? [X_1: nat] : aa_nat_o(P,X_1)
     => ( ! [Y: nat] :
            ( aa_nat_o(P,Y)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),B) )
       => aa_nat_o(P,order_Greatest_nat(P)) ) ) ).

% GreatestI_ex_nat
tff(fact_10103_Greatest__le__nat,axiom,
    ! [P: fun_nat_o,K: nat,B: nat] :
      ( aa_nat_o(P,K)
     => ( ! [Y: nat] :
            ( aa_nat_o(P,Y)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),B) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),order_Greatest_nat(P)) ) ) ).

% Greatest_le_nat
tff(fact_10104_GreatestI__nat,axiom,
    ! [P: fun_nat_o,K: nat,B: nat] :
      ( aa_nat_o(P,K)
     => ( ! [Y: nat] :
            ( aa_nat_o(P,Y)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),B) )
       => aa_nat_o(P,order_Greatest_nat(P)) ) ) ).

% GreatestI_nat
tff(fact_10105_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(
            n: nat,
            n:= 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),n),
              ( ! [X4: vEBT_VEBT] :
                  ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeLista))
                 => aa_nat_o(vEBT_VEBT_valid(X4),n) )
              & 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(( ~ ? [X9: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summarya),X9)
                    & ! [X4: vEBT_VEBT] :
                        ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeLista))
                       => ~ ? [X9: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X4),X9) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_ty(Dega),TreeLista),Summarya),n),m2))),Mima2) ) ) ) ) ) ).

% VEBT_internal.valid'.simps(2)
tff(fact_10106_VEBT__internal_Ovalid_H_Oelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Yb: $o] :
      ( ( aa_nat_o(vEBT_VEBT_valid(Xb),Xaa)
      <=> (Yb) )
     => ( ( ? [Uu2: $o,Uv2: $o] : Xb = vEBT_Leaf((Uu2),(Uv2))
         => ( (Yb)
          <=> ( Xaa != one_one_nat ) ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
              ( ( Xb = vEBT_Node(Mima,Deg,TreeList,Summary) )
             => ( (Yb)
              <=> ~ ( ( Deg = Xaa )
                    & $let(
                        n: nat,
                        n:= 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),n),
                          ( ! [X4: vEBT_VEBT] :
                              ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList))
                             => aa_nat_o(vEBT_VEBT_valid(X4),n) )
                          & 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(( ~ ? [X9: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary),X9)
                                & ! [X4: vEBT_VEBT] :
                                    ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList))
                                   => ~ ? [X9: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X4),X9) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_ty(Deg),TreeList),Summary),n),m2))),Mima) ) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(1)
tff(fact_10107_VEBT__internal_Ovalid_H_Oelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(vEBT_VEBT_valid(Xb),Xaa)
     => ( ( ? [Uu2: $o,Uv2: $o] : Xb = vEBT_Leaf((Uu2),(Uv2))
         => ( Xaa != one_one_nat ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
              ( ( Xb = vEBT_Node(Mima,Deg,TreeList,Summary) )
             => ~ ( ( Deg = Xaa )
                  & $let(
                      n: nat,
                      n:= 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),n),
                        ( ! [X4: vEBT_VEBT] :
                            ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList))
                           => aa_nat_o(vEBT_VEBT_valid(X4),n) )
                        & 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(( ~ ? [X9: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary),X9)
                              & ! [X4: vEBT_VEBT] :
                                  ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList))
                                 => ~ ? [X9: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X4),X9) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_ty(Deg),TreeList),Summary),n),m2))),Mima) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(2)
tff(fact_10108_VEBT__internal_Ovalid_H_Opelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Yb: $o] :
      ( ( aa_nat_o(vEBT_VEBT_valid(Xb),Xaa)
      <=> (Yb) )
     => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(Xb,Xaa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( Xb = vEBT_Leaf((Uu2),(Uv2)) )
             => ( ( (Yb)
                <=> ( Xaa = one_one_nat ) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(Uv2)),Xaa)) ) )
         => ~ ! [Mima: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Mima,Deg,TreeList,Summary) )
               => ( ( (Yb)
                  <=> ( ( Deg = Xaa )
                      & $let(
                          n: nat,
                          n:= 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),n),
                            ( ! [X4: vEBT_VEBT] :
                                ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList))
                               => aa_nat_o(vEBT_VEBT_valid(X4),n) )
                            & 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(( ~ ? [X9: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary),X9)
                                  & ! [X4: vEBT_VEBT] :
                                      ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList))
                                     => ~ ? [X9: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X4),X9) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_ty(Deg),TreeList),Summary),n),m2))),Mima) ) ) ) ) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Node(Mima,Deg,TreeList,Summary),Xaa)) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(1)
tff(fact_10109_VEBT__internal_Ovalid_H_Opelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(vEBT_VEBT_valid(Xb),Xaa)
     => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(Xb,Xaa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( Xb = vEBT_Leaf((Uu2),(Uv2)) )
             => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(Uv2)),Xaa))
               => ( Xaa != one_one_nat ) ) )
         => ~ ! [Mima: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Mima,Deg,TreeList,Summary) )
               => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Node(Mima,Deg,TreeList,Summary),Xaa))
                 => ~ ( ( Deg = Xaa )
                      & $let(
                          n: nat,
                          n:= 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),n),
                            ( ! [X4: vEBT_VEBT] :
                                ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList))
                               => aa_nat_o(vEBT_VEBT_valid(X4),n) )
                            & 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(( ~ ? [X9: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary),X9)
                                  & ! [X4: vEBT_VEBT] :
                                      ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList))
                                     => ~ ? [X9: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X4),X9) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_ty(Deg),TreeList),Summary),n),m2))),Mima) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(2)
tff(fact_10110_Sup__int__def,axiom,
    ! [X6: set_int] : complete_Sup_Sup_int(X6) = the_int(aTP_Lamm_tz(X6)) ).

% Sup_int_def
tff(fact_10111_VEBT__internal_Ovalid_H_Opelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(vEBT_VEBT_valid(Xb),Xaa)
     => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(Xb,Xaa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( Xb = vEBT_Leaf((Uu2),(Uv2)) )
             => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(Uv2)),Xaa))
               => ( Xaa = one_one_nat ) ) )
         => ~ ! [Mima: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Mima,Deg,TreeList,Summary) )
               => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Node(Mima,Deg,TreeList,Summary),Xaa))
                 => ( ( Deg = Xaa )
                    & $let(
                        n: nat,
                        n:= 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),n),
                          ( ! [X4: vEBT_VEBT] :
                              ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList))
                             => aa_nat_o(vEBT_VEBT_valid(X4),n) )
                          & 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(( ~ ? [X9: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary),X9)
                                & ! [X4: vEBT_VEBT] :
                                    ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList))
                                   => ~ ? [X9: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X4),X9) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_ty(Deg),TreeList),Summary),n),m2))),Mima) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(3)
tff(fact_10112_Bseq__realpow,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real)
       => bfun_nat_real(aa_real_fun_nat_real(power_power_real,Xb),at_top_nat) ) ) ).

% Bseq_realpow
tff(fact_10113_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)
       => ( ! [X3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),X3)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),B)
               => differ6690327859849518006l_real(F,topolo2177554685111907308n_real(X3,top_top_set_real)) ) )
         => ? [L3: 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,L3,topolo2177554685111907308n_real(Z2,top_top_set_real))
              & ( aa_real_real(minus_minus_real(aa_real_real(F,B)),aa_real_real(F,A)) = aa_real_real(times_times_real(aa_real_real(minus_minus_real(B),A)),L3) ) ) ) ) ) ).

% MVT
tff(fact_10114_continuous__on__arcosh,axiom,
    ! [A3: set_real] :
      ( ord_less_eq_set_real(A3,set_ord_atLeast_real(one_one_real))
     => topolo5044208981011980120l_real(A3,arcosh_real) ) ).

% continuous_on_arcosh
tff(fact_10115_continuous__on__arcosh_H,axiom,
    ! [A3: set_real,F: fun_real_real] :
      ( topolo5044208981011980120l_real(A3,F)
     => ( ! [X3: real] :
            ( member_real(X3,A3)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(F,X3)) )
       => topolo5044208981011980120l_real(A3,aTP_Lamm_ua(F)) ) ) ).

% continuous_on_arcosh'
tff(fact_10116_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_10117_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_10118_continuous__on__artanh,axiom,
    ! [A3: set_real] :
      ( ord_less_eq_set_real(A3,set_or1633881224788618240n_real(aa_real_real(uminus_uminus_real,one_one_real),one_one_real))
     => topolo5044208981011980120l_real(A3,artanh_real) ) ).

% continuous_on_artanh
tff(fact_10119_continuous__on__artanh_H,axiom,
    ! [A3: set_real,F: fun_real_real] :
      ( topolo5044208981011980120l_real(A3,F)
     => ( ! [X3: real] :
            ( member_real(X3,A3)
           => member_real(aa_real_real(F,X3),set_or1633881224788618240n_real(aa_real_real(uminus_uminus_real,one_one_real),one_one_real)) )
       => topolo5044208981011980120l_real(A3,aTP_Lamm_ub(F)) ) ) ).

% continuous_on_artanh'
tff(fact_10120_mono__Suc,axiom,
    order_mono_nat_nat(suc) ).

% mono_Suc
tff(fact_10121_mono__times__nat,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => order_mono_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb)) ) ).

% mono_times_nat
tff(fact_10122_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_uc,K)) ) ).

% mono_ge2_power_minus_self
tff(fact_10123_and__not__num_Oelims,axiom,
    ! [Xb: num,Xaa: num,Yb: option_num] :
      ( ( bit_and_not_num(Xb,Xaa) = Yb )
     => ( ( ( Xb = one )
         => ( ( Xaa = one )
           => ( Yb != none_num ) ) )
       => ( ( ( Xb = one )
           => ( ? [N: num] : Xaa = aa_num_num(bit0,N)
             => ( Yb != some_num(one) ) ) )
         => ( ( ( Xb = one )
             => ( ? [N: num] : Xaa = aa_num_num(bit1,N)
               => ( Yb != none_num ) ) )
           => ( ! [M2: num] :
                  ( ( Xb = aa_num_num(bit0,M2) )
                 => ( ( Xaa = one )
                   => ( Yb != some_num(aa_num_num(bit0,M2)) ) ) )
             => ( ! [M2: num] :
                    ( ( Xb = aa_num_num(bit0,M2) )
                   => ! [N: num] :
                        ( ( Xaa = aa_num_num(bit0,N) )
                       => ( Yb != map_option_num_num(bit0,bit_and_not_num(M2,N)) ) ) )
               => ( ! [M2: num] :
                      ( ( Xb = aa_num_num(bit0,M2) )
                     => ! [N: num] :
                          ( ( Xaa = aa_num_num(bit1,N) )
                         => ( Yb != map_option_num_num(bit0,bit_and_not_num(M2,N)) ) ) )
                 => ( ! [M2: num] :
                        ( ( Xb = aa_num_num(bit1,M2) )
                       => ( ( Xaa = one )
                         => ( Yb != some_num(aa_num_num(bit0,M2)) ) ) )
                   => ( ! [M2: num] :
                          ( ( Xb = aa_num_num(bit1,M2) )
                         => ! [N: num] :
                              ( ( Xaa = aa_num_num(bit0,N) )
                             => ( Yb != case_o6005452278849405969um_num(some_num(one),aTP_Lamm_su,bit_and_not_num(M2,N)) ) ) )
                     => ~ ! [M2: num] :
                            ( ( Xb = aa_num_num(bit1,M2) )
                           => ! [N: num] :
                                ( ( Xaa = aa_num_num(bit1,N) )
                               => ( Yb != map_option_num_num(bit0,bit_and_not_num(M2,N)) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_not_num.elims
tff(fact_10124_and__not__num_Osimps_I5_J,axiom,
    ! [Mb: num,Nb: num] : bit_and_not_num(aa_num_num(bit0,Mb),aa_num_num(bit0,Nb)) = map_option_num_num(bit0,bit_and_not_num(Mb,Nb)) ).

% and_not_num.simps(5)
tff(fact_10125_and__not__num_Osimps_I6_J,axiom,
    ! [Mb: num,Nb: num] : bit_and_not_num(aa_num_num(bit0,Mb),aa_num_num(bit1,Nb)) = map_option_num_num(bit0,bit_and_not_num(Mb,Nb)) ).

% and_not_num.simps(6)
tff(fact_10126_and__not__num_Osimps_I9_J,axiom,
    ! [Mb: num,Nb: num] : bit_and_not_num(aa_num_num(bit1,Mb),aa_num_num(bit1,Nb)) = map_option_num_num(bit0,bit_and_not_num(Mb,Nb)) ).

% and_not_num.simps(9)
tff(fact_10127_and__num_Oelims,axiom,
    ! [Xb: num,Xaa: num,Yb: option_num] :
      ( ( aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,Xb),Xaa) = Yb )
     => ( ( ( Xb = one )
         => ( ( Xaa = one )
           => ( Yb != some_num(one) ) ) )
       => ( ( ( Xb = one )
           => ( ? [N: num] : Xaa = aa_num_num(bit0,N)
             => ( Yb != none_num ) ) )
         => ( ( ( Xb = one )
             => ( ? [N: num] : Xaa = aa_num_num(bit1,N)
               => ( Yb != some_num(one) ) ) )
           => ( ( ? [M2: num] : Xb = aa_num_num(bit0,M2)
               => ( ( Xaa = one )
                 => ( Yb != none_num ) ) )
             => ( ! [M2: num] :
                    ( ( Xb = aa_num_num(bit0,M2) )
                   => ! [N: num] :
                        ( ( Xaa = aa_num_num(bit0,N) )
                       => ( Yb != map_option_num_num(bit0,aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,M2),N)) ) ) )
               => ( ! [M2: num] :
                      ( ( Xb = aa_num_num(bit0,M2) )
                     => ! [N: num] :
                          ( ( Xaa = aa_num_num(bit1,N) )
                         => ( Yb != map_option_num_num(bit0,aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,M2),N)) ) ) )
                 => ( ( ? [M2: num] : Xb = aa_num_num(bit1,M2)
                     => ( ( Xaa = one )
                       => ( Yb != some_num(one) ) ) )
                   => ( ! [M2: num] :
                          ( ( Xb = aa_num_num(bit1,M2) )
                         => ! [N: num] :
                              ( ( Xaa = aa_num_num(bit0,N) )
                             => ( Yb != map_option_num_num(bit0,aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,M2),N)) ) ) )
                     => ~ ! [M2: num] :
                            ( ( Xb = aa_num_num(bit1,M2) )
                           => ! [N: num] :
                                ( ( Xaa = aa_num_num(bit1,N) )
                               => ( Yb != case_o6005452278849405969um_num(some_num(one),aTP_Lamm_su,aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,M2),N)) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_num.elims
tff(fact_10128_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_10129_and__num_Osimps_I5_J,axiom,
    ! [Mb: num,Nb: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,aa_num_num(bit0,Mb)),aa_num_num(bit0,Nb)) = map_option_num_num(bit0,aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,Mb),Nb)) ).

% and_num.simps(5)
tff(fact_10130_and__num_Osimps_I3_J,axiom,
    ! [Nb: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,one),aa_num_num(bit1,Nb)) = some_num(one) ).

% and_num.simps(3)
tff(fact_10131_and__num_Osimps_I7_J,axiom,
    ! [Mb: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,aa_num_num(bit1,Mb)),one) = some_num(one) ).

% and_num.simps(7)
tff(fact_10132_and__num_Osimps_I2_J,axiom,
    ! [Nb: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,one),aa_num_num(bit0,Nb)) = none_num ).

% and_num.simps(2)
tff(fact_10133_and__num_Osimps_I4_J,axiom,
    ! [Mb: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,aa_num_num(bit0,Mb)),one) = none_num ).

% and_num.simps(4)
tff(fact_10134_and__num_Osimps_I8_J,axiom,
    ! [Mb: num,Nb: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,aa_num_num(bit1,Mb)),aa_num_num(bit0,Nb)) = map_option_num_num(bit0,aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,Mb),Nb)) ).

% and_num.simps(8)
tff(fact_10135_and__num_Osimps_I6_J,axiom,
    ! [Mb: num,Nb: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,aa_num_num(bit0,Mb)),aa_num_num(bit1,Nb)) = map_option_num_num(bit0,aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,Mb),Nb)) ).

% and_num.simps(6)
tff(fact_10136_and__num_Osimps_I9_J,axiom,
    ! [Mb: num,Nb: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,aa_num_num(bit1,Mb)),aa_num_num(bit1,Nb)) = case_o6005452278849405969um_num(some_num(one),aTP_Lamm_su,aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,Mb),Nb)) ).

% and_num.simps(9)
tff(fact_10137_xor__num_Oelims,axiom,
    ! [Xb: num,Xaa: num,Yb: option_num] :
      ( ( aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,Xb),Xaa) = Yb )
     => ( ( ( Xb = one )
         => ( ( Xaa = one )
           => ( Yb != none_num ) ) )
       => ( ( ( Xb = one )
           => ! [N: num] :
                ( ( Xaa = aa_num_num(bit0,N) )
               => ( Yb != some_num(aa_num_num(bit1,N)) ) ) )
         => ( ( ( Xb = one )
             => ! [N: num] :
                  ( ( Xaa = aa_num_num(bit1,N) )
                 => ( Yb != some_num(aa_num_num(bit0,N)) ) ) )
           => ( ! [M2: num] :
                  ( ( Xb = aa_num_num(bit0,M2) )
                 => ( ( Xaa = one )
                   => ( Yb != some_num(aa_num_num(bit1,M2)) ) ) )
             => ( ! [M2: num] :
                    ( ( Xb = aa_num_num(bit0,M2) )
                   => ! [N: num] :
                        ( ( Xaa = aa_num_num(bit0,N) )
                       => ( Yb != map_option_num_num(bit0,aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,M2),N)) ) ) )
               => ( ! [M2: num] :
                      ( ( Xb = aa_num_num(bit0,M2) )
                     => ! [N: num] :
                          ( ( Xaa = aa_num_num(bit1,N) )
                         => ( Yb != some_num(case_option_num_num(one,bit1,aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,M2),N))) ) ) )
                 => ( ! [M2: num] :
                        ( ( Xb = aa_num_num(bit1,M2) )
                       => ( ( Xaa = one )
                         => ( Yb != some_num(aa_num_num(bit0,M2)) ) ) )
                   => ( ! [M2: num] :
                          ( ( Xb = aa_num_num(bit1,M2) )
                         => ! [N: num] :
                              ( ( Xaa = aa_num_num(bit0,N) )
                             => ( Yb != some_num(case_option_num_num(one,bit1,aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,M2),N))) ) ) )
                     => ~ ! [M2: num] :
                            ( ( Xb = aa_num_num(bit1,M2) )
                           => ! [N: num] :
                                ( ( Xaa = aa_num_num(bit1,N) )
                               => ( Yb != map_option_num_num(bit0,aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,M2),N)) ) ) ) ) ) ) ) ) ) ) ) ) ).

% xor_num.elims
tff(fact_10138_and__num__dict,axiom,
    bit_un7362597486090784418nd_num = bit_un1837492267222099188nd_num ).

% and_num_dict
tff(fact_10139_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_10140_xor__num_Osimps_I5_J,axiom,
    ! [Mb: num,Nb: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,aa_num_num(bit0,Mb)),aa_num_num(bit0,Nb)) = map_option_num_num(bit0,aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,Mb),Nb)) ).

% xor_num.simps(5)
tff(fact_10141_xor__num_Osimps_I9_J,axiom,
    ! [Mb: num,Nb: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,aa_num_num(bit1,Mb)),aa_num_num(bit1,Nb)) = map_option_num_num(bit0,aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,Mb),Nb)) ).

% xor_num.simps(9)
tff(fact_10142_xor__num_Osimps_I7_J,axiom,
    ! [Mb: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,aa_num_num(bit1,Mb)),one) = some_num(aa_num_num(bit0,Mb)) ).

% xor_num.simps(7)
tff(fact_10143_xor__num_Osimps_I4_J,axiom,
    ! [Mb: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,aa_num_num(bit0,Mb)),one) = some_num(aa_num_num(bit1,Mb)) ).

% xor_num.simps(4)
tff(fact_10144_xor__num_Osimps_I3_J,axiom,
    ! [Nb: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,one),aa_num_num(bit1,Nb)) = some_num(aa_num_num(bit0,Nb)) ).

% xor_num.simps(3)
tff(fact_10145_xor__num_Osimps_I2_J,axiom,
    ! [Nb: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,one),aa_num_num(bit0,Nb)) = some_num(aa_num_num(bit1,Nb)) ).

% xor_num.simps(2)
tff(fact_10146_xor__num_Osimps_I8_J,axiom,
    ! [Mb: num,Nb: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,aa_num_num(bit1,Mb)),aa_num_num(bit0,Nb)) = some_num(case_option_num_num(one,bit1,aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,Mb),Nb))) ).

% xor_num.simps(8)
tff(fact_10147_xor__num_Osimps_I6_J,axiom,
    ! [Mb: num,Nb: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,aa_num_num(bit0,Mb)),aa_num_num(bit1,Nb)) = some_num(case_option_num_num(one,bit1,aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,Mb),Nb))) ).

% xor_num.simps(6)
tff(fact_10148_xor__num__dict,axiom,
    bit_un2480387367778600638or_num = bit_un6178654185764691216or_num ).

% xor_num_dict
tff(fact_10149_range__abs__Nats,axiom,
    image_int_int(abs_abs_int,top_top_set_int) = semiring_1_Nats_int ).

% range_abs_Nats
tff(fact_10150_inj__sgn__power,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => inj_on_real_real(aa_nat_fun_real_real(aTP_Lamm_rw,Nb),top_top_set_real) ) ).

% inj_sgn_power
tff(fact_10151_log__inj,axiom,
    ! [B: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B)
     => inj_on_real_real(log(B),set_or5849166863359141190n_real(zero_zero_real)) ) ).

% log_inj
tff(fact_10152_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_rt,K)),N2) ) ).

% inj_on_diff_nat
tff(fact_10153_inj__Suc,axiom,
    ! [N2: set_nat] : aa_set_nat_o(inj_on_nat_nat(suc),N2) ).

% inj_Suc
tff(fact_10154_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_10155_Rats__eq__int__div__nat,axiom,
    field_5140801741446780682s_real = collect_real(aTP_Lamm_ud) ).

% Rats_eq_int_div_nat
tff(fact_10156_Rats__abs__iff,axiom,
    ! [Xb: real] :
      ( member_real(aa_real_real(abs_abs_real,Xb),field_5140801741446780682s_real)
    <=> member_real(Xb,field_5140801741446780682s_real) ) ).

% Rats_abs_iff
tff(fact_10157_Rats__no__top__le,axiom,
    ! [Xb: real] :
    ? [X3: real] :
      ( member_real(X3,field_5140801741446780682s_real)
      & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),X3) ) ).

% Rats_no_top_le
tff(fact_10158_Rats__no__bot__less,axiom,
    ! [Xb: real] :
    ? [X3: real] :
      ( member_real(X3,field_5140801741446780682s_real)
      & aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Xb) ) ).

% Rats_no_bot_less
tff(fact_10159_Rats__dense__in__real,axiom,
    ! [Xb: real,Yb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Yb)
     => ? [X3: real] :
          ( member_real(X3,field_5140801741446780682s_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),X3)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Yb) ) ) ).

% Rats_dense_in_real
tff(fact_10160_Rats__eq__int__div__int,axiom,
    field_5140801741446780682s_real = collect_real(aTP_Lamm_ue) ).

% Rats_eq_int_div_int
tff(fact_10161_sup__int__def,axiom,
    sup_sup_int = ord_max_int ).

% sup_int_def
tff(fact_10162_sup__nat__def,axiom,
    sup_sup_nat = ord_max_nat ).

% sup_nat_def
tff(fact_10163_atLeastLessThan__add__Un,axiom,
    ! [I3: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I3),J)
     => ( aa_nat_set_nat(set_or4665077453230672383an_nat(I3),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K)) = sup_sup_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(I3),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_10164_powr__real__of__int_H,axiom,
    ! [Xb: real,Nb: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( ( ( Xb != zero_zero_real )
          | aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Nb) )
       => ( aa_real_real(powr_real(Xb),aa_int_real(ring_1_of_int_real,Nb)) = aa_int_real(power_int_real(Xb),Nb) ) ) ) ).

% powr_real_of_int'
tff(fact_10165_pairs__le__eq__Sigma,axiom,
    ! [Mb: nat] : collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_no(Mb))) = produc457027306803732586at_nat(aa_nat_set_nat(set_ord_atMost_nat,Mb),aTP_Lamm_uf(Mb)) ).

% pairs_le_eq_Sigma
tff(fact_10166_remdups__upt,axiom,
    ! [Mb: nat,Nb: nat] : remdups_nat(upt(Mb,Nb)) = upt(Mb,Nb) ).

% remdups_upt
tff(fact_10167_length__upt,axiom,
    ! [I3: nat,J: nat] : size_size_list_nat(upt(I3,J)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I3) ).

% length_upt
tff(fact_10168_upt__conv__Nil,axiom,
    ! [J: nat,I3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),I3)
     => ( upt(I3,J) = nil_nat ) ) ).

% upt_conv_Nil
tff(fact_10169_sorted__list__of__set__range,axiom,
    ! [Mb: nat,Nb: nat] : linord2614967742042102400et_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(Mb),Nb)) = upt(Mb,Nb) ).

% sorted_list_of_set_range
tff(fact_10170_upt__eq__Nil__conv,axiom,
    ! [I3: nat,J: nat] :
      ( ( upt(I3,J) = nil_nat )
    <=> ( ( J = zero_zero_nat )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),I3) ) ) ).

% upt_eq_Nil_conv
tff(fact_10171_nth__upt,axiom,
    ! [I3: 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,I3),K)),J)
     => ( aa_nat_nat(nth_nat(upt(I3,J)),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I3),K) ) ) ).

% nth_upt
tff(fact_10172_upt__rec__numeral,axiom,
    ! [Mb: num,Nb: num] :
      upt(aa_num_nat(numeral_numeral_nat,Mb),aa_num_nat(numeral_numeral_nat,Nb)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Nb)),cons_nat(aa_num_nat(numeral_numeral_nat,Mb),upt(aa_nat_nat(suc,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Nb))),nil_nat) ).

% upt_rec_numeral
tff(fact_10173_map__Suc__upt,axiom,
    ! [Mb: nat,Nb: nat] : map_nat_nat(suc,upt(Mb,Nb)) = upt(aa_nat_nat(suc,Mb),aa_nat_nat(suc,Nb)) ).

% map_Suc_upt
tff(fact_10174_map__add__upt,axiom,
    ! [Nb: nat,Mb: nat] : map_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ug,Nb),upt(zero_zero_nat,Mb)) = upt(Nb,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) ).

% map_add_upt
tff(fact_10175_distinct__upt,axiom,
    ! [I3: nat,J: nat] : aa_list_nat_o(distinct_nat,upt(I3,J)) ).

% distinct_upt
tff(fact_10176_upt__conv__Cons__Cons,axiom,
    ! [Mb: nat,Nb: nat,Ns: list_nat,Q2: nat] :
      ( ( cons_nat(Mb,cons_nat(Nb,Ns)) = upt(Mb,Q2) )
    <=> ( cons_nat(Nb,Ns) = upt(aa_nat_nat(suc,Mb),Q2) ) ) ).

% upt_conv_Cons_Cons
tff(fact_10177_upt__0,axiom,
    ! [I3: nat] : upt(I3,zero_zero_nat) = nil_nat ).

% upt_0
tff(fact_10178_atLeast__upt,axiom,
    ! [Nb: nat] : aa_nat_set_nat(set_ord_lessThan_nat,Nb) = set_nat2(upt(zero_zero_nat,Nb)) ).

% atLeast_upt
tff(fact_10179_atLeastLessThan__upt,axiom,
    ! [I3: nat,J: nat] : aa_nat_set_nat(set_or4665077453230672383an_nat(I3),J) = set_nat2(upt(I3,J)) ).

% atLeastLessThan_upt
tff(fact_10180_greaterThanAtMost__upt,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_set_nat(set_or6659071591806873216st_nat(Nb),Mb) = set_nat2(upt(aa_nat_nat(suc,Nb),aa_nat_nat(suc,Mb))) ).

% greaterThanAtMost_upt
tff(fact_10181_greaterThanLessThan__upt,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_set_nat(set_or5834768355832116004an_nat(Nb),Mb) = set_nat2(upt(aa_nat_nat(suc,Nb),Mb)) ).

% greaterThanLessThan_upt
tff(fact_10182_atLeastAtMost__upt,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_set_nat(set_or1269000886237332187st_nat(Nb),Mb) = set_nat2(upt(Nb,aa_nat_nat(suc,Mb))) ).

% atLeastAtMost_upt
tff(fact_10183_atMost__upto,axiom,
    ! [Nb: nat] : aa_nat_set_nat(set_ord_atMost_nat,Nb) = set_nat2(upt(zero_zero_nat,aa_nat_nat(suc,Nb))) ).

% atMost_upto
tff(fact_10184_upt__conv__Cons,axiom,
    ! [I3: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),J)
     => ( upt(I3,J) = cons_nat(I3,upt(aa_nat_nat(suc,I3),J)) ) ) ).

% upt_conv_Cons
tff(fact_10185_map__decr__upt,axiom,
    ! [Mb: nat,Nb: nat] : map_nat_nat(aTP_Lamm_uh,upt(aa_nat_nat(suc,Mb),aa_nat_nat(suc,Nb))) = upt(Mb,Nb) ).

% map_decr_upt
tff(fact_10186_upt__add__eq__append,axiom,
    ! [I3: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I3),J)
     => ( upt(I3,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K)) = append_nat(upt(I3,J),upt(J,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K))) ) ) ).

% upt_add_eq_append
tff(fact_10187_upt__eq__Cons__conv,axiom,
    ! [I3: nat,J: nat,Xb: nat,Xs: list_nat] :
      ( ( upt(I3,J) = cons_nat(Xb,Xs) )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),J)
        & ( I3 = Xb )
        & ( upt(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I3),one_one_nat),J) = Xs ) ) ) ).

% upt_eq_Cons_conv
tff(fact_10188_upt__rec,axiom,
    ! [I3: nat,J: nat] :
      upt(I3,J) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),J),cons_nat(I3,upt(aa_nat_nat(suc,I3),J)),nil_nat) ).

% upt_rec
tff(fact_10189_upt__Suc,axiom,
    ! [I3: nat,J: nat] :
      upt(I3,aa_nat_nat(suc,J)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I3),J),append_nat(upt(I3,J),cons_nat(J,nil_nat)),nil_nat) ).

% upt_Suc
tff(fact_10190_upt__Suc__append,axiom,
    ! [I3: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I3),J)
     => ( upt(I3,aa_nat_nat(suc,J)) = append_nat(upt(I3,J),cons_nat(J,nil_nat)) ) ) ).

% upt_Suc_append
tff(fact_10191_list__encode_Opelims,axiom,
    ! [Xb: list_nat,Yb: nat] :
      ( ( nat_list_encode(Xb) = Yb )
     => ( aa_list_nat_o(accp_list_nat(nat_list_encode_rel),Xb)
       => ( ( ( Xb = nil_nat )
           => ( ( Yb = zero_zero_nat )
             => ~ aa_list_nat_o(accp_list_nat(nat_list_encode_rel),nil_nat) ) )
         => ~ ! [X3: nat,Xs2: list_nat] :
                ( ( Xb = cons_nat(X3,Xs2) )
               => ( ( Yb = aa_nat_nat(suc,aa_Pro4243173770044397973at_nat(nat_prod_encode,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),nat_list_encode(Xs2)))) )
                 => ~ aa_list_nat_o(accp_list_nat(nat_list_encode_rel),cons_nat(X3,Xs2)) ) ) ) ) ) ).

% list_encode.pelims
tff(fact_10192_sum__list__upt,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( groups4561878855575611511st_nat(upt(Mb,Nb)) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_ar),aa_nat_set_nat(set_or4665077453230672383an_nat(Mb),Nb)) ) ) ).

% sum_list_upt
tff(fact_10193_card__length__sum__list__rec,axiom,
    ! [Mb: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Mb)
     => ( finite_card_list_nat(collect_list_nat(aa_nat5883842100386203267_nat_o(aTP_Lamm_ui(Mb),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_uj(Mb),N2)))),finite_card_list_nat(collect_list_nat(aa_nat5883842100386203267_nat_o(aTP_Lamm_uk(Mb),N2)))) ) ) ).

% card_length_sum_list_rec
tff(fact_10194_card__length__sum__list,axiom,
    ! [Mb: nat,N2: nat] : finite_card_list_nat(collect_list_nat(aa_nat5883842100386203267_nat_o(aTP_Lamm_ui(Mb),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),Mb)),one_one_nat)),N2) ).

% card_length_sum_list
tff(fact_10195_sorted__upt,axiom,
    ! [Mb: nat,Nb: nat] : aa_list_nat_o(sorted_wrt_nat(ord_less_eq_nat),upt(Mb,Nb)) ).

% sorted_upt
tff(fact_10196_sorted__wrt__upt,axiom,
    ! [Mb: nat,Nb: nat] : aa_list_nat_o(sorted_wrt_nat(ord_less_nat),upt(Mb,Nb)) ).

% sorted_wrt_upt
tff(fact_10197_sorted__wrt__less__idx,axiom,
    ! [Ns: list_nat,I3: nat] :
      ( aa_list_nat_o(sorted_wrt_nat(ord_less_nat),Ns)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_nat(Ns))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I3),aa_nat_nat(nth_nat(Ns),I3)) ) ) ).

% sorted_wrt_less_idx
tff(fact_10198_sorted__wrt__upto,axiom,
    ! [I3: int,J: int] : aa_list_int_o(sorted_wrt_int(ord_less_int),upto(I3,J)) ).

% sorted_wrt_upto
tff(fact_10199_sorted__upto,axiom,
    ! [Mb: int,Nb: int] : aa_list_int_o(sorted_wrt_int(ord_less_eq_int),upto(Mb,Nb)) ).

% sorted_upto
tff(fact_10200_tl__upt,axiom,
    ! [Mb: nat,Nb: nat] : tl_nat(upt(Mb,Nb)) = upt(aa_nat_nat(suc,Mb),Nb) ).

% tl_upt
tff(fact_10201_hd__upt,axiom,
    ! [I3: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),J)
     => ( hd_nat(upt(I3,J)) = I3 ) ) ).

% hd_upt
tff(fact_10202_min__Suc__Suc,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(suc,Mb)),aa_nat_nat(suc,Nb)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Mb),Nb)) ).

% min_Suc_Suc
tff(fact_10203_min__0R,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Nb),zero_zero_nat) = zero_zero_nat ).

% min_0R
tff(fact_10204_min__0L,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,zero_zero_nat),Nb) = zero_zero_nat ).

% min_0L
tff(fact_10205_min__Suc__numeral,axiom,
    ! [Nb: nat,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(suc,Nb)),aa_num_nat(numeral_numeral_nat,K)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Nb),aa_num_nat(pred_numeral,K))) ).

% min_Suc_numeral
tff(fact_10206_min__numeral__Suc,axiom,
    ! [K: num,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(suc,Nb)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_num_nat(pred_numeral,K)),Nb)) ).

% min_numeral_Suc
tff(fact_10207_concat__bit__assoc__sym,axiom,
    ! [Mb: nat,Nb: nat,K: int,L: int,R2: int] : aa_int_int(aa_int_fun_int_int(bit_concat_bit(Mb),aa_int_int(aa_int_fun_int_int(bit_concat_bit(Nb),K),L)),R2) = aa_int_int(aa_int_fun_int_int(bit_concat_bit(aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Mb),Nb)),K),aa_int_int(aa_int_fun_int_int(bit_concat_bit(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),L),R2)) ).

% concat_bit_assoc_sym
tff(fact_10208_min__diff,axiom,
    ! [Mb: nat,I3: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),I3)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),I3)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Mb),Nb)),I3) ).

% min_diff
tff(fact_10209_ATP_Olambda__1,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_ge,Uu) = aa_real_real(divide_divide_real(aa_real_real(times_times_real(aa_nat_real(aa_real_fun_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_1
tff(fact_10210_ATP_Olambda__2,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_jp,Uu) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(divide_divide_real(one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_nat(suc,Uu)) ).

% ATP.lambda_2
tff(fact_10211_ATP_Olambda__3,axiom,
    ! [Uu: real] :
      ( aa_real_o(aTP_Lamm_pv,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),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))
        & ( aa_real_real(cos_real,Uu) = zero_zero_real ) ) ) ).

% ATP.lambda_3
tff(fact_10212_ATP_Olambda__4,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_hy,Uu) = aa_real_real(times_times_real(aa_nat_real(cos_coeff,Uu)),aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),Uu)) ).

% ATP.lambda_4
tff(fact_10213_ATP_Olambda__5,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_sz,Uu) = aa_real_real(divide_divide_real(aa_real_real(cos_real,Uu)),aa_real_real(sin_real,Uu)) ).

% ATP.lambda_5
tff(fact_10214_ATP_Olambda__6,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_tb,Uu) = aa_real_real(aa_nat_fun_real_real(root,Uu),aa_nat_real(semiri5074537144036343181t_real,Uu)) ).

% ATP.lambda_6
tff(fact_10215_ATP_Olambda__7,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_uh,Uu) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),aa_nat_nat(suc,zero_zero_nat)) ).

% ATP.lambda_7
tff(fact_10216_ATP_Olambda__8,axiom,
    ! [Uu: complex] : aa_complex_complex(aTP_Lamm_kh,Uu) = aa_complex_complex(plus_plus_complex(Uu),one_one_complex) ).

% ATP.lambda_8
tff(fact_10217_ATP_Olambda__9,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_kj,Uu) = aa_real_real(plus_plus_real(Uu),one_one_real) ).

% ATP.lambda_9
tff(fact_10218_ATP_Olambda__10,axiom,
    ! [Uu: rat] : aa_rat_rat(aTP_Lamm_kl,Uu) = aa_rat_rat(plus_plus_rat(Uu),one_one_rat) ).

% ATP.lambda_10
tff(fact_10219_ATP_Olambda__11,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_kk,Uu) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),one_one_nat) ).

% ATP.lambda_11
tff(fact_10220_ATP_Olambda__12,axiom,
    ! [Uu: int] : aa_int_int(aTP_Lamm_ki,Uu) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu),one_one_int) ).

% ATP.lambda_12
tff(fact_10221_ATP_Olambda__13,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_te,Uu) = aa_real_real(divide_divide_real(one_one_real),aa_nat_real(semiri5074537144036343181t_real,Uu)) ).

% ATP.lambda_13
tff(fact_10222_ATP_Olambda__14,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_sn,Uu) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_gf,Uu)) ).

% ATP.lambda_14
tff(fact_10223_ATP_Olambda__15,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_tg,Uu) = aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uu))) ).

% ATP.lambda_15
tff(fact_10224_ATP_Olambda__16,axiom,
    ! [Uu: num] : aa_num_option_num(aTP_Lamm_su,Uu) = some_num(aa_num_num(bit1,Uu)) ).

% ATP.lambda_16
tff(fact_10225_ATP_Olambda__17,axiom,
    ! [Uu: num] : aa_num_option_num(aTP_Lamm_sq,Uu) = some_num(aa_num_num(bit0,Uu)) ).

% ATP.lambda_17
tff(fact_10226_ATP_Olambda__18,axiom,
    ! [Uu: nat] : aa_nat8803788312118491388at_nat(aTP_Lamm_cb,Uu) = product_Pair_nat_nat(aa_nat_nat(suc,Uu)) ).

% ATP.lambda_18
tff(fact_10227_ATP_Olambda__19,axiom,
    ! [Uu: real] :
      ( aa_real_o(aTP_Lamm_ud,Uu)
    <=> ? [I5: int,N4: nat] :
          ( ( Uu = aa_real_real(divide_divide_real(aa_int_real(ring_1_of_int_real,I5)),aa_nat_real(semiri5074537144036343181t_real,N4)) )
          & ( N4 != zero_zero_nat ) ) ) ).

% ATP.lambda_19
tff(fact_10228_ATP_Olambda__20,axiom,
    ! [Uu: real] :
      ( aa_real_o(aTP_Lamm_ue,Uu)
    <=> ? [I5: int,J3: int] :
          ( ( Uu = aa_real_real(divide_divide_real(aa_int_real(ring_1_of_int_real,I5)),aa_int_real(ring_1_of_int_real,J3)) )
          & ( J3 != zero_zero_int ) ) ) ).

% ATP.lambda_20
tff(fact_10229_ATP_Olambda__21,axiom,
    ! [Uu: nat] : aa_nat_option_num(aTP_Lamm_ss,Uu) = some_num(one) ).

% ATP.lambda_21
tff(fact_10230_ATP_Olambda__22,axiom,
    ! [Uu: num,Uua: nat] : aa_nat_option_num(aTP_Lamm_sx(Uu),Uua) = case_num_option_num(some_num(one),aTP_Lamm_sv(Uua),aTP_Lamm_sw(Uua),Uu) ).

% ATP.lambda_22
tff(fact_10231_ATP_Olambda__23,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_complex(aTP_Lamm_ne(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_23
tff(fact_10232_ATP_Olambda__24,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_nh,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_24
tff(fact_10233_ATP_Olambda__25,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_rat(aTP_Lamm_ng(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_25
tff(fact_10234_ATP_Olambda__26,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_nf,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_26
tff(fact_10235_ATP_Olambda__27,axiom,
    ! [Uu: fun_nat_real,Uua: nat] :
      aa_nat_real(aTP_Lamm_jq(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_27
tff(fact_10236_ATP_Olambda__28,axiom,
    ! [Uu: int,Uua: int] :
      aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_py,Uu),Uua) = $ite(Uu = zero_zero_int,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,zero_zero_int),one_one_int),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_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_28
tff(fact_10237_ATP_Olambda__29,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_complex(aTP_Lamm_na(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_29
tff(fact_10238_ATP_Olambda__30,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_nd,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_30
tff(fact_10239_ATP_Olambda__31,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_rat(aTP_Lamm_nc(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_31
tff(fact_10240_ATP_Olambda__32,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_nb,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_32
tff(fact_10241_ATP_Olambda__33,axiom,
    ! [Uu: nat,Uua: num] : aa_num_option_num(aa_nat3713591252005393295on_num(aTP_Lamm_sy,Uu),Uua) = case_nat_option_num(none_num,aTP_Lamm_sx(Uua),Uu) ).

% ATP.lambda_33
tff(fact_10242_ATP_Olambda__34,axiom,
    ! [Uu: nat,Uua: num] : aa_num_option_num(aTP_Lamm_sv(Uu),Uua) = case_o6005452278849405969um_num(none_num,aTP_Lamm_sq,bit_take_bit_num(Uu,Uua)) ).

% ATP.lambda_34
tff(fact_10243_ATP_Olambda__35,axiom,
    ! [Uu: num,Uua: nat] : aa_nat_option_num(aTP_Lamm_sr(Uu),Uua) = case_o6005452278849405969um_num(none_num,aTP_Lamm_sq,bit_take_bit_num(Uua,Uu)) ).

% ATP.lambda_35
tff(fact_10244_ATP_Olambda__36,axiom,
    ! [Uu: fun_nat_fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_ns(Uu),Uua) = groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_nr(Uu),Uua),aa_nat_set_nat(set_ord_atMost_nat,Uua)) ).

% ATP.lambda_36
tff(fact_10245_ATP_Olambda__37,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_nq(Uu),Uua) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_np(Uu),Uua)),aa_nat_set_nat(set_ord_atMost_nat,Uua)) ).

% ATP.lambda_37
tff(fact_10246_ATP_Olambda__38,axiom,
    ! [Uu: fun_nat_fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_op(Uu),Uua) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_oo(Uu),Uua)),aa_nat_set_nat(set_ord_atMost_nat,Uua)) ).

% ATP.lambda_38
tff(fact_10247_ATP_Olambda__39,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_on(Uu),Uua) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_np(Uu),Uua)),aa_nat_set_nat(set_ord_atMost_nat,Uua)) ).

% ATP.lambda_39
tff(fact_10248_ATP_Olambda__40,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_gb,Uu),Uua) = aa_real_real(times_times_real(aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_real_real(divide_divide_real(one_one_real),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),one_one_nat))))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(minus_minus_real(Uu),one_one_real)),aa_nat_nat(suc,Uua))) ).

% ATP.lambda_40
tff(fact_10249_ATP_Olambda__41,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_je,Uu),Uua) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),one_one_nat)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,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_41
tff(fact_10250_ATP_Olambda__42,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_js,Uu),Uua) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,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_42
tff(fact_10251_ATP_Olambda__43,axiom,
    ! [Uu: real,Uua: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_pu,Uu),Uua)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(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(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
        & ( aa_real_real(sin_real,Uua) = Uu ) ) ) ).

% ATP.lambda_43
tff(fact_10252_ATP_Olambda__44,axiom,
    ! [Uu: real,Uua: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_pt,Uu),Uua)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Uua)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,Uua),aa_real_real(divide_divide_real(pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
        & ( aa_real_real(tan_real,Uua) = Uu ) ) ) ).

% ATP.lambda_44
tff(fact_10253_ATP_Olambda__45,axiom,
    ! [Uu: code_integer,Uua: code_integer] :
      aa_Code_integer_int(aa_Cod1242632294114111009er_int(aTP_Lamm_qs,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_45
tff(fact_10254_ATP_Olambda__46,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_fv,Uu),Uua) = $let(
        m3: complex,
        m3:= aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_nat_complex(semiri8010041392384452111omplex,Uu)),
        $ite(Uua = zero_zero_nat,m3,aa_complex_complex(plus_plus_complex(m3),one_one_complex)) ) ).

% ATP.lambda_46
tff(fact_10255_ATP_Olambda__47,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_fx,Uu),Uua) = $let(
        m3: real,
        m3:= aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Uu)),
        $ite(Uua = zero_zero_nat,m3,aa_real_real(plus_plus_real(m3),one_one_real)) ) ).

% ATP.lambda_47
tff(fact_10256_ATP_Olambda__48,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_rat(aa_nat_fun_nat_rat(aTP_Lamm_fz,Uu),Uua) = $let(
        m3: rat,
        m3:= aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),aa_nat_rat(semiri681578069525770553at_rat,Uu)),
        $ite(Uua = zero_zero_nat,m3,aa_rat_rat(plus_plus_rat(m3),one_one_rat)) ) ).

% ATP.lambda_48
tff(fact_10257_ATP_Olambda__49,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_fy,Uu),Uua) = $let(
        m3: nat,
        m3:= 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(semiri1316708129612266289at_nat,Uu)),
        $ite(Uua = zero_zero_nat,m3,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,m3),one_one_nat)) ) ).

% ATP.lambda_49
tff(fact_10258_ATP_Olambda__50,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_fw,Uu),Uua) = $let(
        m3: int,
        m3:= 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(semiri1314217659103216013at_int,Uu)),
        $ite(Uua = zero_zero_nat,m3,aa_int_int(aa_int_fun_int_int(plus_plus_int,m3),one_one_int)) ) ).

% ATP.lambda_50
tff(fact_10259_ATP_Olambda__51,axiom,
    ! [Uu: real,Uua: int] :
      ( aa_int_o(aTP_Lamm_pw(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_51
tff(fact_10260_ATP_Olambda__52,axiom,
    ! [Uu: rat,Uua: int] :
      ( aa_int_o(aTP_Lamm_px(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_52
tff(fact_10261_ATP_Olambda__53,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_gf,Uu),Uua) = aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_real_real(times_times_real(aa_real_real(divide_divide_real(one_one_real),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,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_53
tff(fact_10262_ATP_Olambda__54,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_so,Uu),Uua) = aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,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_54
tff(fact_10263_ATP_Olambda__55,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_ta(Uu),Uua) = aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(Uu,Uua)) ).

% ATP.lambda_55
tff(fact_10264_ATP_Olambda__56,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_ok(Uu),Uua) = aa_nat_complex(gbinomial_complex(aa_complex_complex(plus_plus_complex(Uu),aa_nat_complex(semiri8010041392384452111omplex,Uua))),Uua) ).

% ATP.lambda_56
tff(fact_10265_ATP_Olambda__57,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_om,Uu),Uua) = aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(plus_plus_real(Uu),aa_nat_real(semiri5074537144036343181t_real,Uua))),Uua) ).

% ATP.lambda_57
tff(fact_10266_ATP_Olambda__58,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_ol(Uu),Uua) = aa_nat_rat(gbinomial_rat(aa_rat_rat(plus_plus_rat(Uu),aa_nat_rat(semiri681578069525770553at_rat,Uua))),Uua) ).

% ATP.lambda_58
tff(fact_10267_ATP_Olambda__59,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_km(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_59
tff(fact_10268_ATP_Olambda__60,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ko,Uu),Uua) = aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(gbinomial_real,Uu),Uua)),aa_real_real(minus_minus_real(aa_real_real(divide_divide_real(Uu),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,Uua))) ).

% ATP.lambda_60
tff(fact_10269_ATP_Olambda__61,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_kn(Uu),Uua) = aa_rat_rat(times_times_rat(aa_nat_rat(gbinomial_rat(Uu),Uua)),aa_rat_rat(minus_minus_rat(aa_rat_rat(divide_divide_rat(Uu),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))),aa_nat_rat(semiri681578069525770553at_rat,Uua))) ).

% ATP.lambda_61
tff(fact_10270_ATP_Olambda__62,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_pg(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_62
tff(fact_10271_ATP_Olambda__63,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_pi,Uu),Uua) = aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(gbinomial_real,Uu),Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)) ).

% ATP.lambda_63
tff(fact_10272_ATP_Olambda__64,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_ph(Uu),Uua) = aa_rat_rat(times_times_rat(aa_nat_rat(gbinomial_rat(Uu),Uua)),aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Uua)) ).

% ATP.lambda_64
tff(fact_10273_ATP_Olambda__65,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_rg,Uu),Uua)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Uu),Uua)
        & ( Uu != Uua ) ) ) ).

% ATP.lambda_65
tff(fact_10274_ATP_Olambda__66,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_pq,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_66
tff(fact_10275_ATP_Olambda__67,axiom,
    ! [Uu: set_int,Uua: int] :
      ( aa_int_o(aTP_Lamm_tz(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_67
tff(fact_10276_ATP_Olambda__68,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_iw,Uu),Uua) = aa_real_real(divide_divide_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uua)),aa_nat_real(semiri2265585572941072030t_real,Uua)) ).

% ATP.lambda_68
tff(fact_10277_ATP_Olambda__69,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_uc,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_69
tff(fact_10278_ATP_Olambda__70,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_oj,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_70
tff(fact_10279_ATP_Olambda__71,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_oi,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_71
tff(fact_10280_ATP_Olambda__72,axiom,
    ! [Uu: nat,Uua: complex] :
      ( aa_complex_o(aTP_Lamm_dl(Uu),Uua)
    <=> ( aa_nat_complex(power_power_complex(Uua),Uu) = one_one_complex ) ) ).

% ATP.lambda_72
tff(fact_10281_ATP_Olambda__73,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_tl,Uu),Uua) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(plus_plus_real(one_one_real),aa_real_real(divide_divide_real(Uu),aa_nat_real(semiri5074537144036343181t_real,Uua)))),Uua) ).

% ATP.lambda_73
tff(fact_10282_ATP_Olambda__74,axiom,
    ! [Uu: real,Uua: real] : aa_real_real(aTP_Lamm_ts(Uu),Uua) = aa_real_real(powr_real(aa_real_real(plus_plus_real(one_one_real),aa_real_real(times_times_real(Uu),Uua))),aa_real_real(divide_divide_real(one_one_real),Uua)) ).

% ATP.lambda_74
tff(fact_10283_ATP_Olambda__75,axiom,
    ! [Uu: real,Uua: real] : aa_real_real(aTP_Lamm_tu(Uu),Uua) = aa_real_real(powr_real(aa_real_real(plus_plus_real(one_one_real),aa_real_real(divide_divide_real(Uu),Uua))),Uua) ).

% ATP.lambda_75
tff(fact_10284_ATP_Olambda__76,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ga,Uu),Uua) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(one_one_real),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,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_76
tff(fact_10285_ATP_Olambda__77,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_mo(Uu),Uua) = aa_real_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_77
tff(fact_10286_ATP_Olambda__78,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_mp(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_78
tff(fact_10287_ATP_Olambda__79,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_mq(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_79
tff(fact_10288_ATP_Olambda__80,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_mr(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_80
tff(fact_10289_ATP_Olambda__81,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_em(Uu),Uua) = aa_real_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_81
tff(fact_10290_ATP_Olambda__82,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_ej(Uu),Uua) = aa_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_82
tff(fact_10291_ATP_Olambda__83,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_el(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_83
tff(fact_10292_ATP_Olambda__84,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_ek(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_84
tff(fact_10293_ATP_Olambda__85,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_ea(Uu),Uua) = aa_real_real(minus_minus_real(aa_nat_real(Uu,aa_nat_nat(suc,Uua))),aa_nat_real(Uu,Uua)) ).

% ATP.lambda_85
tff(fact_10294_ATP_Olambda__86,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_dy(Uu),Uua) = aa_rat_rat(minus_minus_rat(aa_nat_rat(Uu,aa_nat_nat(suc,Uua))),aa_nat_rat(Uu,Uua)) ).

% ATP.lambda_86
tff(fact_10295_ATP_Olambda__87,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_dz(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_87
tff(fact_10296_ATP_Olambda__88,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat_complex(aTP_Lamm_gc(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_88
tff(fact_10297_ATP_Olambda__89,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_gd(Uu),Uua) = aa_real_real(times_times_real(aa_nat_real(Uu,Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),Uua)) ).

% ATP.lambda_89
tff(fact_10298_ATP_Olambda__90,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_in(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_90
tff(fact_10299_ATP_Olambda__91,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_eh(Uu),Uua) = aa_real_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_91
tff(fact_10300_ATP_Olambda__92,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_ef(Uu),Uua) = aa_rat_rat(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_92
tff(fact_10301_ATP_Olambda__93,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_eg(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_93
tff(fact_10302_ATP_Olambda__94,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat_complex(aTP_Lamm_eb(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_94
tff(fact_10303_ATP_Olambda__95,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_ee(Uu),Uua) = aa_real_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_95
tff(fact_10304_ATP_Olambda__96,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_ec(Uu),Uua) = aa_rat_rat(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_96
tff(fact_10305_ATP_Olambda__97,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_ed(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_97
tff(fact_10306_ATP_Olambda__98,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_ha(Uu),Uua) = aa_real_real(minus_minus_real(aa_nat_real(Uu,Uua)),aa_nat_real(Uu,aa_nat_nat(suc,Uua))) ).

% ATP.lambda_98
tff(fact_10307_ATP_Olambda__99,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_gy(Uu),Uua) = aa_rat_rat(minus_minus_rat(aa_nat_rat(Uu,Uua)),aa_nat_rat(Uu,aa_nat_nat(suc,Uua))) ).

% ATP.lambda_99
tff(fact_10308_ATP_Olambda__100,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_gz(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_100
tff(fact_10309_ATP_Olambda__101,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_od(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_101
tff(fact_10310_ATP_Olambda__102,axiom,
    ! [Uu: fun_nat_fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_of(Uu),Uua) = groups705719431365010083at_int(aa_nat_fun_nat_int(Uu,Uua),aa_nat_set_nat(set_ord_lessThan_nat,Uua)) ).

% ATP.lambda_102
tff(fact_10311_ATP_Olambda__103,axiom,
    ! [Uu: fun_nat_fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_oa(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_103
tff(fact_10312_ATP_Olambda__104,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_nx(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_104
tff(fact_10313_ATP_Olambda__105,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_tr(Uu),Uua) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_ta(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_105
tff(fact_10314_ATP_Olambda__106,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_tq(Uu),Uua) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_ta(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_106
tff(fact_10315_ATP_Olambda__107,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_rs,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_107
tff(fact_10316_ATP_Olambda__108,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_nv(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_108
tff(fact_10317_ATP_Olambda__109,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_nw,Uu),Uua) = aa_real_real(times_times_real(aa_real_real(inverse_inverse_real,aa_nat_real(semiri2265585572941072030t_real,Uua))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uua)) ).

% ATP.lambda_109
tff(fact_10318_ATP_Olambda__110,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_jg(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_110
tff(fact_10319_ATP_Olambda__111,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_jf,Uu),Uua) = aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uua))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uua)) ).

% ATP.lambda_111
tff(fact_10320_ATP_Olambda__112,axiom,
    ! [Uu: num,Uua: num] : aa_num_int(aTP_Lamm_sp(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_112
tff(fact_10321_ATP_Olambda__113,axiom,
    ! [Uu: nat,Uua: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_rw,Uu),Uua) = aa_real_real(times_times_real(aa_real_real(sgn_sgn_real,Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,Uua)),Uu)) ).

% ATP.lambda_113
tff(fact_10322_ATP_Olambda__114,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_jd,Uu),Uua) = aa_real_real(times_times_real(aa_nat_real(sin_coeff,Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uua)) ).

% ATP.lambda_114
tff(fact_10323_ATP_Olambda__115,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_iv,Uu),Uua) = aa_real_real(times_times_real(aa_nat_real(cos_coeff,Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uua)) ).

% ATP.lambda_115
tff(fact_10324_ATP_Olambda__116,axiom,
    ! [Uu: code_integer,Uua: code_integer] :
      aa_Code_integer_num(aa_Cod6128727250572145423er_num(aTP_Lamm_qt,Uu),Uua) = $let(
        l2: num,
        l2:= code_num_of_integer(Uu),
        $let(
          l3: num,
          l3:= aa_num_num(plus_plus_num(l2),l2),
          $ite(Uua = zero_z3403309356797280102nteger,l3,aa_num_num(plus_plus_num(l3),one)) ) ) ).

% ATP.lambda_116
tff(fact_10325_ATP_Olambda__117,axiom,
    ! [Uu: code_integer,Uua: code_integer] :
      aa_Code_integer_nat(aa_Cod5420483313623307717er_nat(aTP_Lamm_qu,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_117
tff(fact_10326_ATP_Olambda__118,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_complex(aTP_Lamm_mx(Uu),Uua) = aa_nat_complex(gbinomial_complex(aa_nat_complex(semiri8010041392384452111omplex,Uua)),Uu) ).

% ATP.lambda_118
tff(fact_10327_ATP_Olambda__119,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_mz,Uu),Uua) = aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_nat_real(semiri5074537144036343181t_real,Uua)),Uu) ).

% ATP.lambda_119
tff(fact_10328_ATP_Olambda__120,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_rat(aTP_Lamm_my(Uu),Uua) = aa_nat_rat(gbinomial_rat(aa_nat_rat(semiri681578069525770553at_rat,Uua)),Uu) ).

% ATP.lambda_120
tff(fact_10329_ATP_Olambda__121,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_bx,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_121
tff(fact_10330_ATP_Olambda__122,axiom,
    ! [Uu: int,Uua: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_bw,Uu),Uua) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(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_122
tff(fact_10331_ATP_Olambda__123,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_bz,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_123
tff(fact_10332_ATP_Olambda__124,axiom,
    ! [Uu: int,Uua: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_by,Uu),Uua) = aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(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_124
tff(fact_10333_ATP_Olambda__125,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_jz(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_125
tff(fact_10334_ATP_Olambda__126,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ka,Uu),Uua) = aa_real_real(plus_plus_real(Uu),aa_real_real(divide_divide_real(aa_nat_real(semiri5074537144036343181t_real,Uua)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ).

% ATP.lambda_126
tff(fact_10335_ATP_Olambda__127,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_kb(Uu),Uua) = aa_rat_rat(plus_plus_rat(Uu),aa_rat_rat(divide_divide_rat(aa_nat_rat(semiri681578069525770553at_rat,Uua)),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))) ).

% ATP.lambda_127
tff(fact_10336_ATP_Olambda__128,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_pr,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_128
tff(fact_10337_ATP_Olambda__129,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_tm,Uu),Uua) = aa_real_real(times_times_real(Uu),aa_real_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_129
tff(fact_10338_ATP_Olambda__130,axiom,
    ! [Uu: int,Uua: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_ca,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_130
tff(fact_10339_ATP_Olambda__131,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_tk,Uu),Uua) = aa_real_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_131
tff(fact_10340_ATP_Olambda__132,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_th,Uu),Uua) = aa_real_real(plus_plus_real(Uu),aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uua)))) ).

% ATP.lambda_132
tff(fact_10341_ATP_Olambda__133,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_lo(Uu),Uua) = aa_complex_complex(minus_minus_complex(Uu),aa_nat_complex(semiri8010041392384452111omplex,Uua)) ).

% ATP.lambda_133
tff(fact_10342_ATP_Olambda__134,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_lq,Uu),Uua) = aa_real_real(minus_minus_real(Uu),aa_nat_real(semiri5074537144036343181t_real,Uua)) ).

% ATP.lambda_134
tff(fact_10343_ATP_Olambda__135,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_lp(Uu),Uua) = aa_rat_rat(minus_minus_rat(Uu),aa_nat_rat(semiri681578069525770553at_rat,Uua)) ).

% ATP.lambda_135
tff(fact_10344_ATP_Olambda__136,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_lr,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),aa_nat_nat(semiri1316708129612266289at_nat,Uua)) ).

% ATP.lambda_136
tff(fact_10345_ATP_Olambda__137,axiom,
    ! [Uu: int,Uua: nat] : aa_nat_int(aTP_Lamm_ls(Uu),Uua) = aa_int_int(aa_int_fun_int_int(minus_minus_int,Uu),aa_nat_int(semiri1314217659103216013at_int,Uua)) ).

% ATP.lambda_137
tff(fact_10346_ATP_Olambda__138,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_mg,Uu),Uua) = aa_real_real(plus_plus_real(Uu),aa_nat_real(semiri5074537144036343181t_real,Uua)) ).

% ATP.lambda_138
tff(fact_10347_ATP_Olambda__139,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_mh(Uu),Uua) = aa_rat_rat(plus_plus_rat(Uu),aa_nat_rat(semiri681578069525770553at_rat,Uua)) ).

% ATP.lambda_139
tff(fact_10348_ATP_Olambda__140,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_mi,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),aa_nat_nat(semiri1316708129612266289at_nat,Uua)) ).

% ATP.lambda_140
tff(fact_10349_ATP_Olambda__141,axiom,
    ! [Uu: int,Uua: nat] : aa_nat_int(aTP_Lamm_mj(Uu),Uua) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu),aa_nat_int(semiri1314217659103216013at_int,Uua)) ).

% ATP.lambda_141
tff(fact_10350_ATP_Olambda__142,axiom,
    ! [Uu: real,Uua: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_ni,Uu),Uua)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Uua),Uu) ) ).

% ATP.lambda_142
tff(fact_10351_ATP_Olambda__143,axiom,
    ! [Uu: set_nat,Uua: set_nat] :
      ( aa_set_nat_o(aTP_Lamm_nj(Uu),Uua)
    <=> aa_set_nat_o(ord_less_eq_set_nat(Uua),Uu) ) ).

% ATP.lambda_143
tff(fact_10352_ATP_Olambda__144,axiom,
    ! [Uu: rat,Uua: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(aTP_Lamm_nk,Uu),Uua)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Uua),Uu) ) ).

% ATP.lambda_144
tff(fact_10353_ATP_Olambda__145,axiom,
    ! [Uu: num,Uua: num] :
      ( aa_num_o(aa_num_fun_num_o(aTP_Lamm_nl,Uu),Uua)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Uua),Uu) ) ).

% ATP.lambda_145
tff(fact_10354_ATP_Olambda__146,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_nn,Uu),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uua),Uu) ) ).

% ATP.lambda_146
tff(fact_10355_ATP_Olambda__147,axiom,
    ! [Uu: int,Uua: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_nm,Uu),Uua)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Uua),Uu) ) ).

% ATP.lambda_147
tff(fact_10356_ATP_Olambda__148,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_rv,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Uua),Uu) ).

% ATP.lambda_148
tff(fact_10357_ATP_Olambda__149,axiom,
    ! [Uu: real,Uua: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_gs,Uu),Uua)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Uua),Uu) ) ).

% ATP.lambda_149
tff(fact_10358_ATP_Olambda__150,axiom,
    ! [Uu: set_nat,Uua: set_nat] :
      ( aa_set_nat_o(aTP_Lamm_go(Uu),Uua)
    <=> aa_set_nat_o(ord_less_set_nat(Uua),Uu) ) ).

% ATP.lambda_150
tff(fact_10359_ATP_Olambda__151,axiom,
    ! [Uu: rat,Uua: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(aTP_Lamm_gp,Uu),Uua)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Uua),Uu) ) ).

% ATP.lambda_151
tff(fact_10360_ATP_Olambda__152,axiom,
    ! [Uu: num,Uua: num] :
      ( aa_num_o(aa_num_fun_num_o(aTP_Lamm_gq,Uu),Uua)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Uua),Uu) ) ).

% ATP.lambda_152
tff(fact_10361_ATP_Olambda__153,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_ei,Uu),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uua),Uu) ) ).

% ATP.lambda_153
tff(fact_10362_ATP_Olambda__154,axiom,
    ! [Uu: int,Uua: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_gr,Uu),Uua)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Uua),Uu) ) ).

% ATP.lambda_154
tff(fact_10363_ATP_Olambda__155,axiom,
    ! [Uu: real,Uua: real] : aa_real_real(aTP_Lamm_az(Uu),Uua) = aa_real_real(times_times_real(Uua),Uu) ).

% ATP.lambda_155
tff(fact_10364_ATP_Olambda__156,axiom,
    ! [Uu: rat,Uua: rat] : aa_rat_rat(aTP_Lamm_ba(Uu),Uua) = aa_rat_rat(times_times_rat(Uua),Uu) ).

% ATP.lambda_156
tff(fact_10365_ATP_Olambda__157,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_bb,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),Uu) ).

% ATP.lambda_157
tff(fact_10366_ATP_Olambda__158,axiom,
    ! [Uu: int,Uua: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_bc,Uu),Uua) = aa_int_int(aa_int_fun_int_int(times_times_int,Uua),Uu) ).

% ATP.lambda_158
tff(fact_10367_ATP_Olambda__159,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_rt,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uu) ).

% ATP.lambda_159
tff(fact_10368_ATP_Olambda__160,axiom,
    ! [Uu: nat,Uua: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_rx,Uu),Uua) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uu) ).

% ATP.lambda_160
tff(fact_10369_ATP_Olambda__161,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(plus_plus_nat,Uua),Uu) ).

% ATP.lambda_161
tff(fact_10370_ATP_Olambda__162,axiom,
    ! [Uu: int,Uua: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_ru,Uu),Uua) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uua),Uu) ).

% ATP.lambda_162
tff(fact_10371_ATP_Olambda__163,axiom,
    ! [Uu: real,Uua: real] : aa_real_real(aTP_Lamm_rz(Uu),Uua) = aa_real_real(powr_real(Uua),Uu) ).

% ATP.lambda_163
tff(fact_10372_ATP_Olambda__164,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_164
tff(fact_10373_ATP_Olambda__165,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_165
tff(fact_10374_ATP_Olambda__166,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_166
tff(fact_10375_ATP_Olambda__167,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_167
tff(fact_10376_ATP_Olambda__168,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_rj,Uu),Uua) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uua),Uu) ).

% ATP.lambda_168
tff(fact_10377_ATP_Olambda__169,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_nu,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uua),Uu) ).

% ATP.lambda_169
tff(fact_10378_ATP_Olambda__170,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_tf,Uu),Uua) = aa_real_real(aa_nat_fun_real_real(root,Uua),Uu) ).

% ATP.lambda_170
tff(fact_10379_ATP_Olambda__171,axiom,
    ! [Uu: set_complex,Uua: complex] :
      ( aa_complex_o(aTP_Lamm_a(Uu),Uua)
    <=> member_complex(Uua,Uu) ) ).

% ATP.lambda_171
tff(fact_10380_ATP_Olambda__172,axiom,
    ! [Uu: set_real,Uua: real] :
      ( aa_real_o(aTP_Lamm_aa(Uu),Uua)
    <=> member_real(Uua,Uu) ) ).

% ATP.lambda_172
tff(fact_10381_ATP_Olambda__173,axiom,
    ! [Uu: set_list_nat,Uua: list_nat] :
      ( aa_list_nat_o(aTP_Lamm_ab(Uu),Uua)
    <=> member_list_nat(Uua,Uu) ) ).

% ATP.lambda_173
tff(fact_10382_ATP_Olambda__174,axiom,
    ! [Uu: set_set_nat,Uua: set_nat] :
      ( aa_set_nat_o(aTP_Lamm_ac(Uu),Uua)
    <=> member_set_nat(Uua,Uu) ) ).

% ATP.lambda_174
tff(fact_10383_ATP_Olambda__175,axiom,
    ! [Uu: set_nat,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_ad(Uu),Uua)
    <=> aa_set_nat_o(member_nat(Uua),Uu) ) ).

% ATP.lambda_175
tff(fact_10384_ATP_Olambda__176,axiom,
    ! [Uu: set_int,Uua: int] :
      ( aa_int_o(aTP_Lamm_ae(Uu),Uua)
    <=> member_int(Uua,Uu) ) ).

% ATP.lambda_176
tff(fact_10385_ATP_Olambda__177,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_hx(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_177
tff(fact_10386_ATP_Olambda__178,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_hw(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_178
tff(fact_10387_ATP_Olambda__179,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat_complex(aTP_Lamm_jj(Uu),Uua) = aa_nat_complex(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_179
tff(fact_10388_ATP_Olambda__180,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_dn(Uu),Uua) = aa_nat_real(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_180
tff(fact_10389_ATP_Olambda__181,axiom,
    ! [Uu: fun_nat_o,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_tv(Uu),Uua)
    <=> aa_nat_o(Uu,aa_nat_nat(suc,Uua)) ) ).

% ATP.lambda_181
tff(fact_10390_ATP_Olambda__182,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_dw(Uu),Uua) = aa_nat_rat(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_182
tff(fact_10391_ATP_Olambda__183,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_dm(Uu),Uua) = aa_nat_nat(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_183
tff(fact_10392_ATP_Olambda__184,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_dx(Uu),Uua) = aa_nat_int(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_184
tff(fact_10393_ATP_Olambda__185,axiom,
    ! [Uu: nat,Uua: num] : aa_num_option_num(aTP_Lamm_sw(Uu),Uua) = some_num(case_option_num_num(one,bit1,bit_take_bit_num(Uu,Uua))) ).

% ATP.lambda_185
tff(fact_10394_ATP_Olambda__186,axiom,
    ! [Uu: num,Uua: nat] : aa_nat_option_num(aTP_Lamm_st(Uu),Uua) = some_num(case_option_num_num(one,bit1,bit_take_bit_num(Uua,Uu))) ).

% ATP.lambda_186
tff(fact_10395_ATP_Olambda__187,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(aTP_Lamm_rr,Uu),Uua) = produc2626176000494625587at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_rq(Uu),Uua)) ).

% ATP.lambda_187
tff(fact_10396_ATP_Olambda__188,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(aTP_Lamm_rp,Uu),Uua) = produc2626176000494625587at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_ro(Uu),Uua)) ).

% ATP.lambda_188
tff(fact_10397_ATP_Olambda__189,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(aTP_Lamm_rn,Uu),Uua) = produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aTP_Lamm_rm(Uu),Uua)) ).

% ATP.lambda_189
tff(fact_10398_ATP_Olambda__190,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(aTP_Lamm_rl,Uu),Uua) = produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aTP_Lamm_rk(Uu),Uua)) ).

% ATP.lambda_190
tff(fact_10399_ATP_Olambda__191,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(aTP_Lamm_ri,Uu),Uua) = produc2626176000494625587at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_rh(Uu),Uua)) ).

% ATP.lambda_191
tff(fact_10400_ATP_Olambda__192,axiom,
    ! [Uu: fun_nat_real,Uua: real] : aa_real_real(aTP_Lamm_sg(Uu),Uua) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_sf(Uu),Uua)) ).

% ATP.lambda_192
tff(fact_10401_ATP_Olambda__193,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_complex(aTP_Lamm_ps(Uu),Uua) = aa_real_complex(cis,aa_real_real(divide_divide_real(aa_real_real(times_times_real(aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)),aa_nat_real(semiri5074537144036343181t_real,Uua))),aa_nat_real(semiri5074537144036343181t_real,Uu))) ).

% ATP.lambda_193
tff(fact_10402_ATP_Olambda__194,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_194
tff(fact_10403_ATP_Olambda__195,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat3829984526574172007omplex(aTP_Lamm_md(Uu),Uua) = times_times_complex(aa_complex_complex(minus_minus_complex(Uu),aa_nat_complex(semiri8010041392384452111omplex,Uua))) ).

% ATP.lambda_195
tff(fact_10404_ATP_Olambda__196,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_fun_real_real(aTP_Lamm_mf(Uu),Uua) = times_times_real(aa_real_real(minus_minus_real(Uu),aa_nat_real(semiri5074537144036343181t_real,Uua))) ).

% ATP.lambda_196
tff(fact_10405_ATP_Olambda__197,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_fun_rat_rat(aTP_Lamm_me(Uu),Uua) = times_times_rat(aa_rat_rat(minus_minus_rat(Uu),aa_nat_rat(semiri681578069525770553at_rat,Uua))) ).

% ATP.lambda_197
tff(fact_10406_ATP_Olambda__198,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat3829984526574172007omplex(aTP_Lamm_kc(Uu),Uua) = times_times_complex(aa_complex_complex(plus_plus_complex(Uu),aa_nat_complex(semiri8010041392384452111omplex,Uua))) ).

% ATP.lambda_198
tff(fact_10407_ATP_Olambda__199,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_fun_real_real(aTP_Lamm_ke(Uu),Uua) = times_times_real(aa_real_real(plus_plus_real(Uu),aa_nat_real(semiri5074537144036343181t_real,Uua))) ).

% ATP.lambda_199
tff(fact_10408_ATP_Olambda__200,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_fun_rat_rat(aTP_Lamm_kf(Uu),Uua) = times_times_rat(aa_rat_rat(plus_plus_rat(Uu),aa_nat_rat(semiri681578069525770553at_rat,Uua))) ).

% ATP.lambda_200
tff(fact_10409_ATP_Olambda__201,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_fun_nat_nat(aTP_Lamm_kg(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_201
tff(fact_10410_ATP_Olambda__202,axiom,
    ! [Uu: int,Uua: nat] : aa_nat_fun_int_int(aTP_Lamm_kd(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_202
tff(fact_10411_ATP_Olambda__203,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_set_nat(aTP_Lamm_uf(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_203
tff(fact_10412_ATP_Olambda__204,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_tj,Uu),Uua) = aa_real_real(inverse_inverse_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uua)) ).

% ATP.lambda_204
tff(fact_10413_ATP_Olambda__205,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_rc,Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Uu),Uua)) ).

% ATP.lambda_205
tff(fact_10414_ATP_Olambda__206,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_rd,Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Uua),Uu)) ).

% ATP.lambda_206
tff(fact_10415_ATP_Olambda__207,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat_real(aTP_Lamm_is(Uu),Uua) = aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(Uu,Uua)) ).

% ATP.lambda_207
tff(fact_10416_ATP_Olambda__208,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_ow(Uu),Uua) = aa_real_real(real_V7735802525324610683m_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_208
tff(fact_10417_ATP_Olambda__209,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_td(Uu),Uua) = aa_real_real(inverse_inverse_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_209
tff(fact_10418_ATP_Olambda__210,axiom,
    ! [Uu: fun_complex_nat,Uua: complex] : aa_complex_complex(aTP_Lamm_fl(Uu),Uua) = aa_nat_complex(semiri8010041392384452111omplex,aa_complex_nat(Uu,Uua)) ).

% ATP.lambda_210
tff(fact_10419_ATP_Olambda__211,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_real(aTP_Lamm_fp(Uu),Uua) = aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(Uu,Uua)) ).

% ATP.lambda_211
tff(fact_10420_ATP_Olambda__212,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_rat(aTP_Lamm_fn(Uu),Uua) = aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(Uu,Uua)) ).

% ATP.lambda_212
tff(fact_10421_ATP_Olambda__213,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_fo(Uu),Uua) = aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(Uu,Uua)) ).

% ATP.lambda_213
tff(fact_10422_ATP_Olambda__214,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_int(aTP_Lamm_fm(Uu),Uua) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(Uu,Uua)) ).

% ATP.lambda_214
tff(fact_10423_ATP_Olambda__215,axiom,
    ! [Uu: fun_int_nat,Uua: int] : aa_int_int(aTP_Lamm_fk(Uu),Uua) = aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(Uu,Uua)) ).

% ATP.lambda_215
tff(fact_10424_ATP_Olambda__216,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat3829984526574172007omplex(aTP_Lamm_lb(Uu),Uua) = times_times_complex(aa_nat_complex(Uu,Uua)) ).

% ATP.lambda_216
tff(fact_10425_ATP_Olambda__217,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_fun_real_real(aTP_Lamm_lc(Uu),Uua) = times_times_real(aa_nat_real(Uu,Uua)) ).

% ATP.lambda_217
tff(fact_10426_ATP_Olambda__218,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_fun_rat_rat(aTP_Lamm_ld(Uu),Uua) = times_times_rat(aa_nat_rat(Uu,Uua)) ).

% ATP.lambda_218
tff(fact_10427_ATP_Olambda__219,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_fun_nat_nat(aTP_Lamm_le(Uu),Uua) = aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu,Uua)) ).

% ATP.lambda_219
tff(fact_10428_ATP_Olambda__220,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_fun_int_int(aTP_Lamm_lf(Uu),Uua) = aa_int_fun_int_int(times_times_int,aa_nat_int(Uu,Uua)) ).

% ATP.lambda_220
tff(fact_10429_ATP_Olambda__221,axiom,
    ! [Uu: fun_complex_int,Uua: complex] : aa_complex_complex(aTP_Lamm_cl(Uu),Uua) = aa_int_complex(ring_17405671764205052669omplex,aa_complex_int(Uu,Uua)) ).

% ATP.lambda_221
tff(fact_10430_ATP_Olambda__222,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_real(aTP_Lamm_cm(Uu),Uua) = aa_int_real(ring_1_of_int_real,aa_nat_int(Uu,Uua)) ).

% ATP.lambda_222
tff(fact_10431_ATP_Olambda__223,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_rat(aTP_Lamm_ks(Uu),Uua) = aa_int_rat(ring_1_of_int_rat,aa_nat_int(Uu,Uua)) ).

% ATP.lambda_223
tff(fact_10432_ATP_Olambda__224,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_kt(Uu),Uua) = aa_int_int(ring_1_of_int_int,aa_nat_int(Uu,Uua)) ).

% ATP.lambda_224
tff(fact_10433_ATP_Olambda__225,axiom,
    ! [Uu: fun_int_int,Uua: int] : aa_int_real(aTP_Lamm_cn(Uu),Uua) = aa_int_real(ring_1_of_int_real,aa_int_int(Uu,Uua)) ).

% ATP.lambda_225
tff(fact_10434_ATP_Olambda__226,axiom,
    ! [Uu: fun_int_int,Uua: int] : aa_int_rat(aTP_Lamm_co(Uu),Uua) = aa_int_rat(ring_1_of_int_rat,aa_int_int(Uu,Uua)) ).

% ATP.lambda_226
tff(fact_10435_ATP_Olambda__227,axiom,
    ! [Uu: fun_int_int,Uua: int] : aa_int_int(aTP_Lamm_cp(Uu),Uua) = aa_int_int(ring_1_of_int_int,aa_int_int(Uu,Uua)) ).

% ATP.lambda_227
tff(fact_10436_ATP_Olambda__228,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat3829984526574172007omplex(aTP_Lamm_ms(Uu),Uua) = plus_plus_complex(aa_nat_complex(Uu,Uua)) ).

% ATP.lambda_228
tff(fact_10437_ATP_Olambda__229,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_fun_real_real(aTP_Lamm_mw(Uu),Uua) = plus_plus_real(aa_nat_real(Uu,Uua)) ).

% ATP.lambda_229
tff(fact_10438_ATP_Olambda__230,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_fun_rat_rat(aTP_Lamm_mt(Uu),Uua) = plus_plus_rat(aa_nat_rat(Uu,Uua)) ).

% ATP.lambda_230
tff(fact_10439_ATP_Olambda__231,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_fun_nat_nat(aTP_Lamm_mv(Uu),Uua) = aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(Uu,Uua)) ).

% ATP.lambda_231
tff(fact_10440_ATP_Olambda__232,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_fun_int_int(aTP_Lamm_mu(Uu),Uua) = aa_int_fun_int_int(plus_plus_int,aa_nat_int(Uu,Uua)) ).

% ATP.lambda_232
tff(fact_10441_ATP_Olambda__233,axiom,
    ! [Uu: fun_real_real,Uua: real] : aa_real_real(aTP_Lamm_ub(Uu),Uua) = aa_real_real(artanh_real,aa_real_real(Uu,Uua)) ).

% ATP.lambda_233
tff(fact_10442_ATP_Olambda__234,axiom,
    ! [Uu: fun_real_real,Uua: real] : aa_real_real(aTP_Lamm_ua(Uu),Uua) = aa_real_real(arcosh_real,aa_real_real(Uu,Uua)) ).

% ATP.lambda_234
tff(fact_10443_ATP_Olambda__235,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_235
tff(fact_10444_ATP_Olambda__236,axiom,
    ! [Uu: fun_int_int,Uua: int] : aa_int_int(aTP_Lamm_cs(Uu),Uua) = aa_int_int(abs_abs_int,aa_int_int(Uu,Uua)) ).

% ATP.lambda_236
tff(fact_10445_ATP_Olambda__237,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_tp(Uu),Uua) = aa_real_real(cos_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_237
tff(fact_10446_ATP_Olambda__238,axiom,
    ! [Uu: fun_re4238438383274954274t_real,Uua: real] : aa_real_real(aTP_Lamm_sd(Uu),Uua) = suminf_real(aa_real_fun_nat_real(Uu,Uua)) ).

% ATP.lambda_238
tff(fact_10447_ATP_Olambda__239,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] :
      aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_jr(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_239
tff(fact_10448_ATP_Olambda__240,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] :
      aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_hv(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_240
tff(fact_10449_ATP_Olambda__241,axiom,
    ! [Uu: num,Uua: code_integer,Uub: code_integer] :
      aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_bg(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_241
tff(fact_10450_ATP_Olambda__242,axiom,
    ! [Uu: num,Uua: nat,Uub: nat] :
      aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_be(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_242
tff(fact_10451_ATP_Olambda__243,axiom,
    ! [Uu: num,Uua: int,Uub: int] :
      aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_bf(Uu),Uua),Uub) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Uu)),Uub),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(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))),aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(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_243
tff(fact_10452_ATP_Olambda__244,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: nat] :
      aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_hz(Uu),Uua),Uub) = $ite(Uub = Uu,aa_nat_complex(Uua,Uub),zero_zero_complex) ).

% ATP.lambda_244
tff(fact_10453_ATP_Olambda__245,axiom,
    ! [Uu: nat,Uua: fun_nat_real,Uub: nat] :
      aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_ia(Uu),Uua),Uub) = $ite(Uub = Uu,aa_nat_real(Uua,Uub),zero_zero_real) ).

% ATP.lambda_245
tff(fact_10454_ATP_Olambda__246,axiom,
    ! [Uu: nat,Uua: fun_nat_nat,Uub: nat] :
      aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_ib(Uu),Uua),Uub) = $ite(Uub = Uu,aa_nat_nat(Uua,Uub),zero_zero_nat) ).

% ATP.lambda_246
tff(fact_10455_ATP_Olambda__247,axiom,
    ! [Uu: nat,Uua: fun_nat_int,Uub: nat] :
      aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_ic(Uu),Uua),Uub) = $ite(Uub = Uu,aa_nat_int(Uua,Uub),zero_zero_int) ).

% ATP.lambda_247
tff(fact_10456_ATP_Olambda__248,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] :
      aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_qz(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_248
tff(fact_10457_ATP_Olambda__249,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] :
      aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_rf(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_249
tff(fact_10458_ATP_Olambda__250,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] :
      aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_qy(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_250
tff(fact_10459_ATP_Olambda__251,axiom,
    ! [Uu: fun_nat_fun_nat_real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_oo(Uu),Uua),Uub) = aa_nat_real(aa_nat_fun_nat_real(Uu,Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uub)) ).

% ATP.lambda_251
tff(fact_10460_ATP_Olambda__252,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_np(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(Uu,Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uub)) ).

% ATP.lambda_252
tff(fact_10461_ATP_Olambda__253,axiom,
    ! [Uu: fun_nat_fun_nat_int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_nr(Uu),Uua),Uub) = aa_nat_int(aa_nat_fun_nat_int(Uu,Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uub)) ).

% ATP.lambda_253
tff(fact_10462_ATP_Olambda__254,axiom,
    ! [Uu: fun_re4238438383274954274t_real,Uua: nat,Uub: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_sc(Uu),Uua),Uub) = aa_nat_real(aa_real_fun_nat_real(Uu,Uub),Uua) ).

% ATP.lambda_254
tff(fact_10463_ATP_Olambda__255,axiom,
    ! [Uu: fun_nat_fun_nat_real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_ob(Uu),Uua),Uub) = aa_nat_real(aa_nat_fun_nat_real(Uu,Uub),Uua) ).

% ATP.lambda_255
tff(fact_10464_ATP_Olambda__256,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ny(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(Uu,Uub),Uua) ).

% ATP.lambda_256
tff(fact_10465_ATP_Olambda__257,axiom,
    ! [Uu: fun_nat_fun_nat_int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_og(Uu),Uua),Uub) = aa_nat_int(aa_nat_fun_nat_int(Uu,Uub),Uua) ).

% ATP.lambda_257
tff(fact_10466_ATP_Olambda__258,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_oy(Uu),Uua),Uub) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_fun3023522663955195517t_real(aTP_Lamm_ox(Uu),Uua),Uub)),aa_nat_set_nat(set_ord_atMost_nat,Uub)) ).

% ATP.lambda_258
tff(fact_10467_ATP_Olambda__259,axiom,
    ! [Uu: fun_nat_complex,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_ov(Uu),Uua),Uub) = aa_set_nat_complex(groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_fun5379041630539201665omplex(aTP_Lamm_ou(Uu),Uua),Uub)),aa_nat_set_nat(set_ord_atMost_nat,Uub)) ).

% ATP.lambda_259
tff(fact_10468_ATP_Olambda__260,axiom,
    ! [Uu: nat,Uua: complex,Uub: nat] :
      aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_jm(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_260
tff(fact_10469_ATP_Olambda__261,axiom,
    ! [Uu: nat,Uua: real,Uub: nat] :
      aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_jn(Uu),Uua),Uub) = aa_real_real(times_times_real($ite(Uub = Uu,one_one_real,zero_zero_real)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uub)) ).

% ATP.lambda_261
tff(fact_10470_ATP_Olambda__262,axiom,
    ! [Uu: nat,Uua: int,Uub: nat] :
      aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_jo(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_262
tff(fact_10471_ATP_Olambda__263,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] :
      aa_Cod3168588076859124107eger_o(aa_Cod1096751537565085541eger_o(aTP_Lamm_qv(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_263
tff(fact_10472_ATP_Olambda__264,axiom,
    ! [Uu: fun_nat_fun_nat_int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_oh(Uu),Uua),Uub) = groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_og(Uu),Uub),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Uub)),Uua)) ).

% ATP.lambda_264
tff(fact_10473_ATP_Olambda__265,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_oe(Uu),Uua),Uub) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ny(Uu),Uub)),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Uub)),Uua)) ).

% ATP.lambda_265
tff(fact_10474_ATP_Olambda__266,axiom,
    ! [Uu: fun_nat_fun_nat_real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_oc(Uu),Uua),Uub) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_ob(Uu),Uub)),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Uub)),Uua)) ).

% ATP.lambda_266
tff(fact_10475_ATP_Olambda__267,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_nz(Uu),Uua),Uub) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ny(Uu),Uub)),aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Uub)),Uua)) ).

% ATP.lambda_267
tff(fact_10476_ATP_Olambda__268,axiom,
    ! [Uu: rat,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_qm(Uu),Uua),Uub)
    <=> aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(aa_int2464330468366073224_int_o(aTP_Lamm_ql(Uua),Uub)),quotient_of(Uu)) ) ).

% ATP.lambda_268
tff(fact_10477_ATP_Olambda__269,axiom,
    ! [Uu: rat,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_qk(Uu),Uua),Uub)
    <=> aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(aa_int2464330468366073224_int_o(aTP_Lamm_qj(Uua),Uub)),quotient_of(Uu)) ) ).

% ATP.lambda_269
tff(fact_10478_ATP_Olambda__270,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_qi(Uu),Uua),Uub) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_qh(Uua),Uub)),quotient_of(Uu)) ).

% ATP.lambda_270
tff(fact_10479_ATP_Olambda__271,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_qg(Uu),Uua),Uub) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_qf(Uua),Uub)),quotient_of(Uu)) ).

% ATP.lambda_271
tff(fact_10480_ATP_Olambda__272,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_qe(Uu),Uua),Uub) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_qd(Uua),Uub)),quotient_of(Uu)) ).

% ATP.lambda_272
tff(fact_10481_ATP_Olambda__273,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_qc(Uu),Uua),Uub) = aa_Pro6970504127699319548nt_int(produc4245557441103728435nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_qb(Uua),Uub)),quotient_of(Uu)) ).

% ATP.lambda_273
tff(fact_10482_ATP_Olambda__274,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_sh(Uu),Uua),Uub) = aa_real_real(times_times_real(aa_real_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(aa_real_fun_nat_real(power_power_real,Uua),Uub)) ).

% ATP.lambda_274
tff(fact_10483_ATP_Olambda__275,axiom,
    ! [Uu: real,Uua: fun_na5723442673449079972x_real,Uub: nat] : aa_nat_real(aa_fun5746040925223427547t_real(aTP_Lamm_ix(Uu),Uua),Uub) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_complex_real(aa_nat480766339200631653x_real(Uua,Uub),zero_zero_complex)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uub)) ).

% ATP.lambda_275
tff(fact_10484_ATP_Olambda__276,axiom,
    ! [Uu: real,Uua: fun_na6724076748692969250l_real,Uub: nat] : aa_nat_real(aa_fun1113820866546176217t_real(aTP_Lamm_iy(Uu),Uua),Uub) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Uua,Uub),zero_zero_real)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uub)) ).

% ATP.lambda_276
tff(fact_10485_ATP_Olambda__277,axiom,
    ! [Uu: real,Uua: fun_nat_fun_rat_real,Uub: nat] : aa_nat_real(aa_fun507281639146497029t_real(aTP_Lamm_iz(Uu),Uua),Uub) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_rat_real(aa_nat_fun_rat_real(Uua,Uub),zero_zero_rat)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uub)) ).

% ATP.lambda_277
tff(fact_10486_ATP_Olambda__278,axiom,
    ! [Uu: real,Uua: fun_nat_fun_nat_real,Uub: nat] : aa_nat_real(aa_fun7039663905532832253t_real(aTP_Lamm_ja(Uu),Uua),Uub) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_nat_real(aa_nat_fun_nat_real(Uua,Uub),zero_zero_nat)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uub)) ).

% ATP.lambda_278
tff(fact_10487_ATP_Olambda__279,axiom,
    ! [Uu: real,Uua: fun_nat_fun_int_real,Uub: nat] : aa_nat_real(aa_fun241848180832411225t_real(aTP_Lamm_jb(Uu),Uua),Uub) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_int_real(aa_nat_fun_int_real(Uua,Uub),zero_zero_int)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uub)) ).

% ATP.lambda_279
tff(fact_10488_ATP_Olambda__280,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_se(Uu),Uua),Uub) = aa_real_real(times_times_real(aa_real_real(times_times_real(aa_nat_real(Uu,Uub)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uub)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uub)) ).

% ATP.lambda_280
tff(fact_10489_ATP_Olambda__281,axiom,
    ! [Uu: real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_jc(Uu),Uua),Uub) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_nat_real(Uua,Uub)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uub)) ).

% ATP.lambda_281
tff(fact_10490_ATP_Olambda__282,axiom,
    ! [Uu: nat,Uua: nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_nat5883842100386203267_nat_o(aTP_Lamm_uj(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_282
tff(fact_10491_ATP_Olambda__283,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_ju(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_283
tff(fact_10492_ATP_Olambda__284,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_jw(Uu),Uua),Uub) = aa_real_real(times_times_real(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Uub)),aa_nat_real(Uu,Uub))),aa_nat_real(aa_real_fun_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_284
tff(fact_10493_ATP_Olambda__285,axiom,
    ! [Uu: nat,Uua: nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_nat5883842100386203267_nat_o(aTP_Lamm_uk(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_285
tff(fact_10494_ATP_Olambda__286,axiom,
    ! [Uu: nat,Uua: nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_nat5883842100386203267_nat_o(aTP_Lamm_ui(Uu),Uua),Uub)
    <=> ( ( size_size_list_nat(Uub) = Uu )
        & ( groups4561878855575611511st_nat(Uub) = Uua ) ) ) ).

% ATP.lambda_286
tff(fact_10495_ATP_Olambda__287,axiom,
    ! [Uu: set_nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_qx(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_287
tff(fact_10496_ATP_Olambda__288,axiom,
    ! [Uu: nat,Uua: nat,Uub: set_nat] :
      ( aa_set_nat_o(aa_nat_fun_set_nat_o(aTP_Lamm_re(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_288
tff(fact_10497_ATP_Olambda__289,axiom,
    ! [Uu: set_nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_qw(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_289
tff(fact_10498_ATP_Olambda__290,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_jt(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_290
tff(fact_10499_ATP_Olambda__291,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_jv(Uu),Uua),Uub) = aa_real_real(times_times_real(aa_nat_real(diffs_real(Uu),Uub)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uub)) ).

% ATP.lambda_291
tff(fact_10500_ATP_Olambda__292,axiom,
    ! [Uu: complex,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_jy(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_292
tff(fact_10501_ATP_Olambda__293,axiom,
    ! [Uu: real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_jx(Uu),Uua),Uub) = aa_real_real(times_times_real(aa_nat_real(diffs_real(Uua),Uub)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uub)) ).

% ATP.lambda_293
tff(fact_10502_ATP_Olambda__294,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_oq(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_294
tff(fact_10503_ATP_Olambda__295,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_cj(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_295
tff(fact_10504_ATP_Olambda__296,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_qn(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_296
tff(fact_10505_ATP_Olambda__297,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_ck(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_297
tff(fact_10506_ATP_Olambda__298,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_qq(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_298
tff(fact_10507_ATP_Olambda__299,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_qp(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_299
tff(fact_10508_ATP_Olambda__300,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_qo(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_300
tff(fact_10509_ATP_Olambda__301,axiom,
    ! [Uu: set_complex,Uua: fun_complex_o,Uub: complex] :
      ( aa_complex_o(aa_fun2514317545224092796plex_o(aTP_Lamm_at(Uu),Uua),Uub)
    <=> ( member_complex(Uub,Uu)
        & aa_complex_o(Uua,Uub) ) ) ).

% ATP.lambda_301
tff(fact_10510_ATP_Olambda__302,axiom,
    ! [Uu: set_real,Uua: fun_real_o,Uub: real] :
      ( aa_real_o(aa_fun5868358341814070780real_o(aTP_Lamm_au(Uu),Uua),Uub)
    <=> ( member_real(Uub,Uu)
        & aa_real_o(Uua,Uub) ) ) ).

% ATP.lambda_302
tff(fact_10511_ATP_Olambda__303,axiom,
    ! [Uu: set_list_nat,Uua: fun_list_nat_o,Uub: list_nat] :
      ( aa_list_nat_o(aa_fun6155608546934989052_nat_o(aTP_Lamm_av(Uu),Uua),Uub)
    <=> ( member_list_nat(Uub,Uu)
        & aa_list_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_303
tff(fact_10512_ATP_Olambda__304,axiom,
    ! [Uu: set_set_nat,Uua: fun_set_nat_o,Uub: set_nat] :
      ( aa_set_nat_o(aa_fun751667557118977660_nat_o(aTP_Lamm_aw(Uu),Uua),Uub)
    <=> ( member_set_nat(Uub,Uu)
        & aa_set_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_304
tff(fact_10513_ATP_Olambda__305,axiom,
    ! [Uu: set_nat,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aTP_Lamm_ay(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu)
        & aa_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_305
tff(fact_10514_ATP_Olambda__306,axiom,
    ! [Uu: set_int,Uua: fun_int_o,Uub: int] :
      ( aa_int_o(aa_fun6243517787477317628_int_o(aTP_Lamm_ax(Uu),Uua),Uub)
    <=> ( member_int(Uub,Uu)
        & aa_int_o(Uua,Uub) ) ) ).

% ATP.lambda_306
tff(fact_10515_ATP_Olambda__307,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_no(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_307
tff(fact_10516_ATP_Olambda__308,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_nt(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)),Uu) ) ).

% ATP.lambda_308
tff(fact_10517_ATP_Olambda__309,axiom,
    ! [Uu: set_Pr448751882837621926eger_o,Uua: code_integer,Uub: $o] :
      ( aa_o_o(aa_Cod6379977994291289310un_o_o(aTP_Lamm_ce(Uu),Uua),(Uub))
    <=> member1379723562493234055eger_o(produc6677183202524767010eger_o(Uua,(Uub)),Uu) ) ).

% ATP.lambda_309
tff(fact_10518_ATP_Olambda__310,axiom,
    ! [Uu: set_Pr8218934625190621173um_num,Uua: num,Uub: num] :
      ( aa_num_o(aa_num_fun_num_o(aTP_Lamm_cf(Uu),Uua),Uub)
    <=> member7279096912039735102um_num(product_Pair_num_num(Uua,Uub),Uu) ) ).

% ATP.lambda_310
tff(fact_10519_ATP_Olambda__311,axiom,
    ! [Uu: set_Pr6200539531224447659at_num,Uua: nat,Uub: num] :
      ( aa_num_o(aa_nat_fun_num_o(aTP_Lamm_cg(Uu),Uua),Uub)
    <=> member9148766508732265716at_num(product_Pair_nat_num(Uua,Uub),Uu) ) ).

% ATP.lambda_311
tff(fact_10520_ATP_Olambda__312,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_ch(Uu),Uua),Uub)
    <=> member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uua),Uub),Uu) ) ).

% ATP.lambda_312
tff(fact_10521_ATP_Olambda__313,axiom,
    ! [Uu: set_Pr958786334691620121nt_int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_ci(Uu),Uua),Uub)
    <=> member5262025264175285858nt_int(aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Uua),Uub),Uu) ) ).

% ATP.lambda_313
tff(fact_10522_ATP_Olambda__314,axiom,
    ! [Uu: nat,Uua: complex,Uub: complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(aTP_Lamm_dk(Uu),Uua),Uub)
    <=> ( aa_nat_complex(power_power_complex(Uub),Uu) = Uua ) ) ).

% ATP.lambda_314
tff(fact_10523_ATP_Olambda__315,axiom,
    ! [Uu: complex,Uua: nat,Uub: complex] :
      ( aa_complex_o(aa_nat_fun_complex_o(aTP_Lamm_qr(Uu),Uua),Uub)
    <=> ( aa_nat_complex(power_power_complex(Uub),Uua) = Uu ) ) ).

% ATP.lambda_315
tff(fact_10524_ATP_Olambda__316,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_io(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_316
tff(fact_10525_ATP_Olambda__317,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ip(Uu),Uua),Uub) = aa_real_real(times_times_real(aa_nat_real(Uu,aa_nat_nat(suc,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uub)) ).

% ATP.lambda_317
tff(fact_10526_ATP_Olambda__318,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_int,Uub: nat] : aa_nat_real(aa_fun4551936527957760560t_real(aTP_Lamm_to(Uu),Uua),Uub) = aa_real_real(minus_minus_real(aa_nat_real(Uu,Uub)),aa_real_real(times_times_real(aa_int_real(ring_1_of_int_real,aa_nat_int(Uua,Uub))),aa_real_real(times_times_real(aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))) ).

% ATP.lambda_318
tff(fact_10527_ATP_Olambda__319,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_sf(Uu),Uua),Uub) = aa_real_real(times_times_real(aa_nat_real(Uu,Uub)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),aa_nat_nat(suc,Uub))) ).

% ATP.lambda_319
tff(fact_10528_ATP_Olambda__320,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_il(Uu),Uua),Uub) = aa_complex_complex(times_times_complex(aa_nat_complex(Uu,Uub)),aa_nat_complex(power_power_complex(Uua),Uub)) ).

% ATP.lambda_320
tff(fact_10529_ATP_Olambda__321,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ij(Uu),Uua),Uub) = aa_real_real(times_times_real(aa_nat_real(Uu,Uub)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uub)) ).

% ATP.lambda_321
tff(fact_10530_ATP_Olambda__322,axiom,
    ! [Uu: fun_nat_rat,Uua: rat,Uub: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aTP_Lamm_os(Uu),Uua),Uub) = aa_rat_rat(times_times_rat(aa_nat_rat(Uu,Uub)),aa_nat_rat(power_power_rat(Uua),Uub)) ).

% ATP.lambda_322
tff(fact_10531_ATP_Olambda__323,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_pn(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_323
tff(fact_10532_ATP_Olambda__324,axiom,
    ! [Uu: fun_nat_int,Uua: int,Uub: nat] : aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_ot(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_324
tff(fact_10533_ATP_Olambda__325,axiom,
    ! [Uu: fun_nat_o,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_pz(Uu),Uua),Uub)
    <=> ( aa_nat_o(Uu,Uub)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uub),Uua) ) ) ).

% ATP.lambda_325
tff(fact_10534_ATP_Olambda__326,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_ku(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_326
tff(fact_10535_ATP_Olambda__327,axiom,
    ! [Uu: fun_nat_int,Uua: fun_nat_int,Uub: nat] : aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_kv(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(Uu,Uub)),aa_nat_int(Uua,Uub)) ).

% ATP.lambda_327
tff(fact_10536_ATP_Olambda__328,axiom,
    ! [Uu: fun_int_int,Uua: fun_int_int,Uub: int] : aa_int_int(aa_fun6280345759651049192nt_int(aTP_Lamm_kw(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(Uu,Uub)),aa_int_int(Uua,Uub)) ).

% ATP.lambda_328
tff(fact_10537_ATP_Olambda__329,axiom,
    ! [Uu: fun_co8153928105989009288nteger,Uua: fun_co8153928105989009288nteger,Uub: complex] : aa_com4217951221296830281nteger(aa_fun5267417455630836558nteger(aTP_Lamm_cu(Uu),Uua),Uub) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_com4217951221296830281nteger(Uua,Uub)),aa_com4217951221296830281nteger(Uu,Uub)) ).

% ATP.lambda_329
tff(fact_10538_ATP_Olambda__330,axiom,
    ! [Uu: fun_complex_real,Uua: fun_complex_real,Uub: complex] : aa_complex_real(aa_fun8933909628229591020x_real(aTP_Lamm_cy(Uu),Uua),Uub) = aa_real_real(times_times_real(aa_complex_real(Uua,Uub)),aa_complex_real(Uu,Uub)) ).

% ATP.lambda_330
tff(fact_10539_ATP_Olambda__331,axiom,
    ! [Uu: fun_complex_rat,Uua: fun_complex_rat,Uub: complex] : aa_complex_rat(aa_fun7622890401989837636ex_rat(aTP_Lamm_db(Uu),Uua),Uub) = aa_rat_rat(times_times_rat(aa_complex_rat(Uua,Uub)),aa_complex_rat(Uu,Uub)) ).

% ATP.lambda_331
tff(fact_10540_ATP_Olambda__332,axiom,
    ! [Uu: fun_re6978948838062957830nteger,Uua: fun_re6978948838062957830nteger,Uub: real] : aa_real_Code_integer(aa_fun7923231501384339402nteger(aTP_Lamm_cv(Uu),Uua),Uub) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_real_Code_integer(Uua,Uub)),aa_real_Code_integer(Uu,Uub)) ).

% ATP.lambda_332
tff(fact_10541_ATP_Olambda__333,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_cz(Uu),Uua),Uub) = aa_real_real(times_times_real(aa_real_real(Uua,Uub)),aa_real_real(Uu,Uub)) ).

% ATP.lambda_333
tff(fact_10542_ATP_Olambda__334,axiom,
    ! [Uu: fun_real_rat,Uua: fun_real_rat,Uub: real] : aa_real_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_dc(Uu),Uua),Uub) = aa_rat_rat(times_times_rat(aa_real_rat(Uua,Uub)),aa_real_rat(Uu,Uub)) ).

% ATP.lambda_334
tff(fact_10543_ATP_Olambda__335,axiom,
    ! [Uu: fun_nat_Code_integer,Uua: fun_nat_Code_integer,Uub: nat] : aa_nat_Code_integer(aa_fun589076786913273746nteger(aTP_Lamm_cw(Uu),Uua),Uub) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(Uua,Uub)),aa_nat_Code_integer(Uu,Uub)) ).

% ATP.lambda_335
tff(fact_10544_ATP_Olambda__336,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_nat_rat,Uub: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_dd(Uu),Uua),Uub) = aa_rat_rat(times_times_rat(aa_nat_rat(Uua,Uub)),aa_nat_rat(Uu,Uub)) ).

% ATP.lambda_336
tff(fact_10545_ATP_Olambda__337,axiom,
    ! [Uu: fun_int_Code_integer,Uua: fun_int_Code_integer,Uub: int] : aa_int_Code_integer(aa_fun5304498221708291146nteger(aTP_Lamm_cx(Uu),Uua),Uub) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_int_Code_integer(Uua,Uub)),aa_int_Code_integer(Uu,Uub)) ).

% ATP.lambda_337
tff(fact_10546_ATP_Olambda__338,axiom,
    ! [Uu: fun_int_real,Uua: fun_int_real,Uub: int] : aa_int_real(aa_fun3252798400008884712t_real(aTP_Lamm_da(Uu),Uua),Uub) = aa_real_real(times_times_real(aa_int_real(Uua,Uub)),aa_int_real(Uu,Uub)) ).

% ATP.lambda_338
tff(fact_10547_ATP_Olambda__339,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_tc(Uu),Uua),Uub) = aa_real_real(minus_minus_real(aa_nat_real(Uu,Uub)),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_339
tff(fact_10548_ATP_Olambda__340,axiom,
    ! [Uu: fun_complex_nat,Uua: fun_complex_nat,Uub: complex] : aa_complex_nat(aa_fun2465681967280144180ex_nat(aTP_Lamm_df(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_340
tff(fact_10549_ATP_Olambda__341,axiom,
    ! [Uu: fun_real_nat,Uua: fun_real_nat,Uub: real] : aa_real_nat(aa_fun211602392931233840al_nat(aTP_Lamm_dg(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_341
tff(fact_10550_ATP_Olambda__342,axiom,
    ! [Uu: fun_set_nat_nat,Uua: fun_set_nat_nat,Uub: set_nat] : aa_set_nat_nat(aa_fun3386010016858254564at_nat(aTP_Lamm_dh(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_342
tff(fact_10551_ATP_Olambda__343,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_dj(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_343
tff(fact_10552_ATP_Olambda__344,axiom,
    ! [Uu: fun_int_nat,Uua: fun_int_nat,Uub: int] : aa_int_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_di(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_344
tff(fact_10553_ATP_Olambda__345,axiom,
    ! [Uu: fun_complex_complex,Uua: fun_complex_complex,Uub: complex] : aa_complex_complex(aa_fun3187328805393345264omplex(aTP_Lamm_fe(Uu),Uua),Uub) = aa_complex_complex(plus_plus_complex(aa_complex_complex(Uu,Uub)),aa_complex_complex(Uua,Uub)) ).

% ATP.lambda_345
tff(fact_10554_ATP_Olambda__346,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_fg(Uu),Uua),Uub) = aa_real_real(plus_plus_real(aa_nat_real(Uu,Uub)),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_346
tff(fact_10555_ATP_Olambda__347,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_ff(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_347
tff(fact_10556_ATP_Olambda__348,axiom,
    ! [Uu: fun_nat_int,Uua: fun_nat_int,Uub: nat] : aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_ii(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_348
tff(fact_10557_ATP_Olambda__349,axiom,
    ! [Uu: fun_int_int,Uua: fun_int_int,Uub: int] : aa_int_int(aa_fun6280345759651049192nt_int(aTP_Lamm_fd(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_349
tff(fact_10558_ATP_Olambda__350,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_sb(Uu),Uua),Uub) = aa_real_real(powr_real(aa_real_real(Uu,Uub)),aa_real_real(Uua,Uub)) ).

% ATP.lambda_350
tff(fact_10559_ATP_Olambda__351,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_qa(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(Uu,Uub)),Uua) ) ).

% ATP.lambda_351
tff(fact_10560_ATP_Olambda__352,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_cr(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(Uu,Uub)),Uua) ).

% ATP.lambda_352
tff(fact_10561_ATP_Olambda__353,axiom,
    ! [Uu: fun_nat_int,Uua: int,Uub: nat] : aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_la(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_nat_int(Uu,Uub)),Uua) ).

% ATP.lambda_353
tff(fact_10562_ATP_Olambda__354,axiom,
    ! [Uu: fun_int_int,Uua: int,Uub: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_cq(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(Uu,Uub)),Uua) ).

% ATP.lambda_354
tff(fact_10563_ATP_Olambda__355,axiom,
    ! [Uu: fun_complex_complex,Uua: complex,Uub: complex] : aa_complex_complex(aa_com3342990543769683525omplex(aTP_Lamm_fh(Uu),Uua),Uub) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(Uu,Uub)),Uua) ).

% ATP.lambda_355
tff(fact_10564_ATP_Olambda__356,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_ie(Uu),Uua),Uub) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(Uu,Uub)),Uua) ).

% ATP.lambda_356
tff(fact_10565_ATP_Olambda__357,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_fi(Uu),Uua),Uub) = aa_real_real(divide_divide_real(aa_nat_real(Uu,Uub)),Uua) ).

% ATP.lambda_357
tff(fact_10566_ATP_Olambda__358,axiom,
    ! [Uu: fun_complex_complex,Uua: complex,Uub: complex] : aa_complex_complex(aa_com3342990543769683525omplex(aTP_Lamm_es(Uu),Uua),Uub) = aa_complex_complex(times_times_complex(aa_complex_complex(Uu,Uub)),Uua) ).

% ATP.lambda_358
tff(fact_10567_ATP_Olambda__359,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_eu(Uu),Uua),Uub) = aa_real_real(times_times_real(aa_nat_real(Uu,Uub)),Uua) ).

% ATP.lambda_359
tff(fact_10568_ATP_Olambda__360,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_et(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu,Uub)),Uua) ).

% ATP.lambda_360
tff(fact_10569_ATP_Olambda__361,axiom,
    ! [Uu: fun_int_int,Uua: int,Uub: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_er(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(Uu,Uub)),Uua) ).

% ATP.lambda_361
tff(fact_10570_ATP_Olambda__362,axiom,
    ! [Uu: complex,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_jh(Uu),Uua),Uub) = aa_complex_complex(times_times_complex(aa_nat_complex(Uua,Uub)),Uu) ).

% ATP.lambda_362
tff(fact_10571_ATP_Olambda__363,axiom,
    ! [Uu: real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_ji(Uu),Uua),Uub) = aa_real_real(times_times_real(aa_nat_real(Uua,Uub)),Uu) ).

% ATP.lambda_363
tff(fact_10572_ATP_Olambda__364,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_gx(Uu),Uua),Uub) = aa_real_real(minus_minus_real(aa_nat_real(Uu,Uub)),Uua) ).

% ATP.lambda_364
tff(fact_10573_ATP_Olambda__365,axiom,
    ! [Uu: fun_nat_rat,Uua: rat,Uub: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aTP_Lamm_gw(Uu),Uua),Uub) = aa_rat_rat(minus_minus_rat(aa_nat_rat(Uu,Uub)),Uua) ).

% ATP.lambda_365
tff(fact_10574_ATP_Olambda__366,axiom,
    ! [Uu: fun_nat_int,Uua: int,Uub: nat] : aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_gv(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(Uu,Uub)),Uua) ).

% ATP.lambda_366
tff(fact_10575_ATP_Olambda__367,axiom,
    ! [Uu: fun_real_real,Uua: nat,Uub: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_ry(Uu),Uua),Uub) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(Uu,Uub)),Uua) ).

% ATP.lambda_367
tff(fact_10576_ATP_Olambda__368,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_kx(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(Uu,Uub)),Uua) ).

% ATP.lambda_368
tff(fact_10577_ATP_Olambda__369,axiom,
    ! [Uu: fun_nat_int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_ky(Uu),Uua),Uub) = aa_nat_int(power_power_int(aa_nat_int(Uu,Uub)),Uua) ).

% ATP.lambda_369
tff(fact_10578_ATP_Olambda__370,axiom,
    ! [Uu: fun_int_int,Uua: nat,Uub: int] : aa_int_int(aa_nat_fun_int_int(aTP_Lamm_kz(Uu),Uua),Uub) = aa_nat_int(power_power_int(aa_int_int(Uu,Uub)),Uua) ).

% ATP.lambda_370
tff(fact_10579_ATP_Olambda__371,axiom,
    ! [Uu: nat,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_tt(Uu),Uua),Uub) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(Uua,Uub)),Uu) ).

% ATP.lambda_371
tff(fact_10580_ATP_Olambda__372,axiom,
    ! [Uu: fun_real_real,Uua: real,Uub: real] : aa_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_sa(Uu),Uua),Uub) = aa_real_real(powr_real(aa_real_real(Uu,Uub)),Uua) ).

% ATP.lambda_372
tff(fact_10581_ATP_Olambda__373,axiom,
    ! [Uu: real,Uua: fun_nat_complex,Uub: nat] : aa_nat_real(aa_fun7054779078796324018t_real(aTP_Lamm_iu(Uu),Uua),Uub) = aa_real_real(times_times_real(aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(Uua,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uub)) ).

% ATP.lambda_373
tff(fact_10582_ATP_Olambda__374,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_fq(Uu),Uua),Uub) = aa_complex_complex(plus_plus_complex(Uu),aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,Uub)),Uua)) ).

% ATP.lambda_374
tff(fact_10583_ATP_Olambda__375,axiom,
    ! [Uu: real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_fu(Uu),Uua),Uub) = aa_real_real(plus_plus_real(Uu),aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,Uub)),Uua)) ).

% ATP.lambda_375
tff(fact_10584_ATP_Olambda__376,axiom,
    ! [Uu: rat,Uua: rat,Uub: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aTP_Lamm_fs(Uu),Uua),Uub) = aa_rat_rat(plus_plus_rat(Uu),aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,Uub)),Uua)) ).

% ATP.lambda_376
tff(fact_10585_ATP_Olambda__377,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ft(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_377
tff(fact_10586_ATP_Olambda__378,axiom,
    ! [Uu: int,Uua: int,Uub: nat] : aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_fr(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_378
tff(fact_10587_ATP_Olambda__379,axiom,
    ! [Uu: complex,Uua: nat,Uub: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_hr(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_379
tff(fact_10588_ATP_Olambda__380,axiom,
    ! [Uu: real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_hu(Uu),Uua),Uub) = aa_nat_real(aa_real_fun_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_380
tff(fact_10589_ATP_Olambda__381,axiom,
    ! [Uu: rat,Uua: nat,Uub: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aTP_Lamm_hs(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_381
tff(fact_10590_ATP_Olambda__382,axiom,
    ! [Uu: int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_ht(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_382
tff(fact_10591_ATP_Olambda__383,axiom,
    ! [Uu: real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ti(Uu),Uua),Uub) = aa_real_real(divide_divide_real(Uua),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uub)) ).

% ATP.lambda_383
tff(fact_10592_ATP_Olambda__384,axiom,
    ! [Uu: complex,Uua: nat,Uub: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_dq(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_384
tff(fact_10593_ATP_Olambda__385,axiom,
    ! [Uu: real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_dt(Uu),Uua),Uub) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)) ).

% ATP.lambda_385
tff(fact_10594_ATP_Olambda__386,axiom,
    ! [Uu: rat,Uua: nat,Uub: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aTP_Lamm_dr(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_386
tff(fact_10595_ATP_Olambda__387,axiom,
    ! [Uu: int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_ds(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_387
tff(fact_10596_ATP_Olambda__388,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_fj(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_388
tff(fact_10597_ATP_Olambda__389,axiom,
    ! [Uu: complex,Uua: fun_complex_complex,Uub: complex] : aa_complex_complex(aa_fun3187328805393345264omplex(aTP_Lamm_eo(Uu),Uua),Uub) = aa_complex_complex(times_times_complex(Uu),aa_complex_complex(Uua,Uub)) ).

% ATP.lambda_389
tff(fact_10598_ATP_Olambda__390,axiom,
    ! [Uu: complex,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_id(Uu),Uua),Uub) = aa_complex_complex(times_times_complex(Uu),aa_nat_complex(Uua,Uub)) ).

% ATP.lambda_390
tff(fact_10599_ATP_Olambda__391,axiom,
    ! [Uu: real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_eq(Uu),Uua),Uub) = aa_real_real(times_times_real(Uu),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_391
tff(fact_10600_ATP_Olambda__392,axiom,
    ! [Uu: nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_ep(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uu),aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_392
tff(fact_10601_ATP_Olambda__393,axiom,
    ! [Uu: int,Uua: fun_int_int,Uub: int] : aa_int_int(aa_fun6280345759651049192nt_int(aTP_Lamm_en(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(times_times_int,Uu),aa_int_int(Uua,Uub)) ).

% ATP.lambda_393
tff(fact_10602_ATP_Olambda__394,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ih(Uu),Uua),Uub) = aa_real_real(times_times_real(Uua),aa_nat_real(Uu,Uub)) ).

% ATP.lambda_394
tff(fact_10603_ATP_Olambda__395,axiom,
    ! [Uu: complex,Uua: fun_nat_nat,Uub: nat] : aa_nat_complex(aa_fun1958354139826840918omplex(aTP_Lamm_lh(Uu),Uua),Uub) = aa_nat_complex(power_power_complex(Uu),aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_395
tff(fact_10604_ATP_Olambda__396,axiom,
    ! [Uu: real,Uua: fun_nat_nat,Uub: nat] : aa_nat_real(aa_fun1538629612289296340t_real(aTP_Lamm_lg(Uu),Uua),Uub) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_396
tff(fact_10605_ATP_Olambda__397,axiom,
    ! [Uu: nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_li(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Uu),aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_397
tff(fact_10606_ATP_Olambda__398,axiom,
    ! [Uu: int,Uua: fun_nat_nat,Uub: nat] : aa_nat_int(aa_fun1613598493704395220at_int(aTP_Lamm_lj(Uu),Uua),Uub) = aa_nat_int(power_power_int(Uu),aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_398
tff(fact_10607_ATP_Olambda__399,axiom,
    ! [Uu: int,Uua: fun_int_nat,Uub: int] : aa_int_int(aa_fun2328901131235339916nt_int(aTP_Lamm_lk(Uu),Uua),Uub) = aa_nat_int(power_power_int(Uu),aa_int_nat(Uua,Uub)) ).

% ATP.lambda_399
tff(fact_10608_ATP_Olambda__400,axiom,
    ! [Uu: real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_mk(Uu),Uua),Uub) = aa_real_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_400
tff(fact_10609_ATP_Olambda__401,axiom,
    ! [Uu: rat,Uua: nat,Uub: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aTP_Lamm_ml(Uu),Uua),Uub) = aa_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_401
tff(fact_10610_ATP_Olambda__402,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_mm(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_402
tff(fact_10611_ATP_Olambda__403,axiom,
    ! [Uu: int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_mn(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_403
tff(fact_10612_ATP_Olambda__404,axiom,
    ! [Uu: fun_nat_real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_gu(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_404
tff(fact_10613_ATP_Olambda__405,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_gt(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_405
tff(fact_10614_ATP_Olambda__406,axiom,
    ! [Uu: fun_nat_int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_lm(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_406
tff(fact_10615_ATP_Olambda__407,axiom,
    ! [Uu: fun_real_o,Uua: real,Uub: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_tx(Uu),Uua),Uub)
    <=> aa_real_o(Uu,aa_real_real(plus_plus_real(Uub),Uua)) ) ).

% ATP.lambda_407
tff(fact_10616_ATP_Olambda__408,axiom,
    ! [Uu: fun_nat_complex,Uua: nat,Uub: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_it(Uu),Uua),Uub) = aa_nat_complex(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ).

% ATP.lambda_408
tff(fact_10617_ATP_Olambda__409,axiom,
    ! [Uu: fun_nat_real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_dp(Uu),Uua),Uub) = aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ).

% ATP.lambda_409
tff(fact_10618_ATP_Olambda__410,axiom,
    ! [Uu: fun_nat_o,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_tw(Uu),Uua),Uub)
    <=> aa_nat_o(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ) ).

% ATP.lambda_410
tff(fact_10619_ATP_Olambda__411,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_do(Uu),Uua),Uub) = aa_nat_nat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ).

% ATP.lambda_411
tff(fact_10620_ATP_Olambda__412,axiom,
    ! [Uu: fun_nat_int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_ll(Uu),Uua),Uub) = aa_nat_int(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ).

% ATP.lambda_412
tff(fact_10621_ATP_Olambda__413,axiom,
    ! [Uu: fun_Pr2477239776880934807at_nat,Uua: nat,Uub: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(aTP_Lamm_br(Uu),Uua),Uub) = aa_Pro8023792847356852296at_nat(Uu,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uua),Uub)) ).

% ATP.lambda_413
tff(fact_10622_ATP_Olambda__414,axiom,
    ! [Uu: fun_Pr2028404973800837934_nat_o,Uua: nat,Uub: nat] : aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(aTP_Lamm_bs(Uu),Uua),Uub) = aa_Pro989595756709598047_nat_o(Uu,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uua),Uub)) ).

% ATP.lambda_414
tff(fact_10623_ATP_Olambda__415,axiom,
    ! [Uu: fun_Pr2478283820680522243nt_int,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_bt(Uu),Uua),Uub) = aa_Pro6970504127699319548nt_int(Uu,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Uua),Uub)) ).

% ATP.lambda_415
tff(fact_10624_ATP_Olambda__416,axiom,
    ! [Uu: fun_Pr3238374608701176162_int_o,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_bu(Uu),Uua),Uub)
    <=> aa_Pro4592068669816069595_int_o(Uu,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Uua),Uub)) ) ).

% ATP.lambda_416
tff(fact_10625_ATP_Olambda__417,axiom,
    ! [Uu: fun_Pr982841511892225912nt_int,Uua: int,Uub: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_bv(Uu),Uua),Uub) = aa_Pro6799258434465214121nt_int(Uu,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,Uua),Uub)) ).

% ATP.lambda_417
tff(fact_10626_ATP_Olambda__418,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_jk(Uu),Uua),Uub) = aa_nat_complex(Uua,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uu)) ).

% ATP.lambda_418
tff(fact_10627_ATP_Olambda__419,axiom,
    ! [Uu: nat,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_jl(Uu),Uua),Uub) = aa_nat_real(Uua,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uu)) ).

% ATP.lambda_419
tff(fact_10628_ATP_Olambda__420,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_real(aa_com20334259897932645t_real(aTP_Lamm_im(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_420
tff(fact_10629_ATP_Olambda__421,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ik(Uu),Uua),Uub) = aa_real_real(real_V7735802525324610683m_real,aa_real_real(times_times_real(aa_nat_real(Uu,Uub)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uub))) ).

% ATP.lambda_421
tff(fact_10630_ATP_Olambda__422,axiom,
    ! [Uu: fun_complex_complex,Uua: fun_complex_complex,Uub: complex] : aa_complex_real(aa_fun1953249243402654830x_real(aTP_Lamm_lx(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_422
tff(fact_10631_ATP_Olambda__423,axiom,
    ! [Uu: fun_complex_real,Uua: fun_complex_real,Uub: complex] : aa_complex_real(aa_fun8933909628229591020x_real(aTP_Lamm_lt(Uu),Uua),Uub) = aa_real_real(real_V7735802525324610683m_real,aa_real_real(minus_minus_real(aa_complex_real(Uu,Uub)),aa_complex_real(Uua,Uub))) ).

% ATP.lambda_423
tff(fact_10632_ATP_Olambda__424,axiom,
    ! [Uu: fun_real_complex,Uua: fun_real_complex,Uub: real] : aa_real_real(aa_fun2786211375485122666l_real(aTP_Lamm_ly(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_424
tff(fact_10633_ATP_Olambda__425,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_lu(Uu),Uua),Uub) = aa_real_real(real_V7735802525324610683m_real,aa_real_real(minus_minus_real(aa_real_real(Uu,Uub)),aa_real_real(Uua,Uub))) ).

% ATP.lambda_425
tff(fact_10634_ATP_Olambda__426,axiom,
    ! [Uu: fun_set_nat_complex,Uua: fun_set_nat_complex,Uub: set_nat] : aa_set_nat_real(aa_fun5767801665285160478t_real(aTP_Lamm_lz(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_426
tff(fact_10635_ATP_Olambda__427,axiom,
    ! [Uu: fun_set_nat_real,Uua: fun_set_nat_real,Uub: set_nat] : aa_set_nat_real(aa_fun3569254276315353500t_real(aTP_Lamm_lv(Uu),Uua),Uub) = aa_real_real(real_V7735802525324610683m_real,aa_real_real(minus_minus_real(aa_set_nat_real(Uu,Uub)),aa_set_nat_real(Uua,Uub))) ).

% ATP.lambda_427
tff(fact_10636_ATP_Olambda__428,axiom,
    ! [Uu: fun_nat_complex,Uua: fun_nat_complex,Uub: nat] : aa_nat_real(aa_fun7054779078796324018t_real(aTP_Lamm_mc(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_428
tff(fact_10637_ATP_Olambda__429,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_mb(Uu),Uua),Uub) = aa_real_real(real_V7735802525324610683m_real,aa_real_real(minus_minus_real(aa_nat_real(Uu,Uub)),aa_nat_real(Uua,Uub))) ).

% ATP.lambda_429
tff(fact_10638_ATP_Olambda__430,axiom,
    ! [Uu: fun_int_complex,Uua: fun_int_complex,Uub: int] : aa_int_real(aa_fun1985759105522019690t_real(aTP_Lamm_ma(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_430
tff(fact_10639_ATP_Olambda__431,axiom,
    ! [Uu: fun_int_real,Uua: fun_int_real,Uub: int] : aa_int_real(aa_fun3252798400008884712t_real(aTP_Lamm_lw(Uu),Uua),Uub) = aa_real_real(real_V7735802525324610683m_real,aa_real_real(minus_minus_real(aa_int_real(Uu,Uub)),aa_int_real(Uua,Uub))) ).

% ATP.lambda_431
tff(fact_10640_ATP_Olambda__432,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_tn(Uu),Uua),Uub) = aa_real_real(cos_real,aa_real_real(minus_minus_real(aa_nat_real(Uu,Uub)),Uua)) ).

% ATP.lambda_432
tff(fact_10641_ATP_Olambda__433,axiom,
    ! [Uu: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_gn(Uu),Uua),Uub),Uuc) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_gm(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_433
tff(fact_10642_ATP_Olambda__434,axiom,
    ! [Uu: rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_gl(Uu),Uua),Uub),Uuc) = groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_gk(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_434
tff(fact_10643_ATP_Olambda__435,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_gj(Uu),Uua),Uub),Uuc) = aa_set_nat_complex(groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_gi(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_435
tff(fact_10644_ATP_Olambda__436,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: set_nat,Uuc: nat] : aa_nat_real(aa_set1166323925324055613t_real(aa_fun1400677770452585267t_real(aTP_Lamm_fc(Uu),Uua),Uub),Uuc) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_fun3023522663955195517t_real(aTP_Lamm_fb(Uu),Uua),Uuc)),Uub) ).

% ATP.lambda_436
tff(fact_10645_ATP_Olambda__437,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: set_nat,Uuc: nat] : aa_nat_nat(aa_set6537093552120046177at_nat(aa_fun4561837934398617467at_nat(aTP_Lamm_fa(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_fun6650658898577385413at_nat(aTP_Lamm_ez(Uu),Uua),Uuc)),Uub) ).

% ATP.lambda_437
tff(fact_10646_ATP_Olambda__438,axiom,
    ! [Uu: fun_complex_complex,Uua: fun_complex_complex,Uub: set_complex,Uuc: complex] : aa_complex_complex(aa_set7743095458595710203omplex(aa_fun1583801116089702097omplex(aTP_Lamm_ey(Uu),Uua),Uub),Uuc) = groups7754918857620584856omplex(aa_com3342990543769683525omplex(aa_fun6995112137416684571omplex(aTP_Lamm_ex(Uu),Uua),Uuc),Uub) ).

% ATP.lambda_438
tff(fact_10647_ATP_Olambda__439,axiom,
    ! [Uu: fun_int_int,Uua: fun_int_int,Uub: set_int,Uuc: int] : aa_int_int(aa_set7310040872927613301nt_int(aa_fun8076508010538293575nt_int(aTP_Lamm_ew(Uu),Uua),Uub),Uuc) = groups4538972089207619220nt_int(aa_int_fun_int_int(aa_fun5911324870756489105nt_int(aTP_Lamm_ev(Uu),Uua),Uuc),Uub) ).

% ATP.lambda_439
tff(fact_10648_ATP_Olambda__440,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_pp(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_po(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_440
tff(fact_10649_ATP_Olambda__441,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_pe(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_pd(Uu),Uua),Uuc),aa_nat_set_nat(set_ord_atMost_nat,Uuc))),aa_nat_int(power_power_int(Uub),Uuc)) ).

% ATP.lambda_441
tff(fact_10650_ATP_Olambda__442,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_pc(Uu),Uua),Uub),Uuc) = aa_rat_rat(times_times_rat(groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_fun2663889555218712533at_rat(aTP_Lamm_pb(Uu),Uua),Uuc),aa_nat_set_nat(set_ord_atMost_nat,Uuc))),aa_nat_rat(power_power_rat(Uub),Uuc)) ).

% ATP.lambda_442
tff(fact_10651_ATP_Olambda__443,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_pf(Uu),Uua),Uub),Uuc) = aa_real_real(times_times_real(aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_fun3023522663955195517t_real(aTP_Lamm_ox(Uu),Uua),Uuc)),aa_nat_set_nat(set_ord_atMost_nat,Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uub),Uuc)) ).

% ATP.lambda_443
tff(fact_10652_ATP_Olambda__444,axiom,
    ! [Uu: fun_nat_complex,Uua: fun_nat_complex,Uub: complex,Uuc: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_fun3721018923249493343omplex(aTP_Lamm_pa(Uu),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(aa_set_nat_complex(groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_fun5379041630539201665omplex(aTP_Lamm_ou(Uu),Uua),Uuc)),aa_nat_set_nat(set_ord_atMost_nat,Uuc))),aa_nat_complex(power_power_complex(Uub),Uuc)) ).

% ATP.lambda_444
tff(fact_10653_ATP_Olambda__445,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_sm(Uu),Uua),Uub),Uuc) = aa_real_real(times_times_real(aa_real_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(aa_real_fun_nat_real(power_power_real,Uub),Uuc)) ).

% ATP.lambda_445
tff(fact_10654_ATP_Olambda__446,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_sk(Uu),Uua),Uub),Uuc) = aa_real_real(times_times_real(aa_real_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(aa_real_fun_nat_real(power_power_real,Uub),Uuc)) ).

% ATP.lambda_446
tff(fact_10655_ATP_Olambda__447,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: real,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_si(Uu),Uua),Uub),Uuc) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Uu,Uuc),Uua)),aa_nat_real(semiri2265585572941072030t_real,Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(minus_minus_real(Uub),Uua)),Uuc)) ).

% ATP.lambda_447
tff(fact_10656_ATP_Olambda__448,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: real,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_sj(Uu),Uua),Uub),Uuc) = aa_real_real(times_times_real(aa_real_real(divide_divide_real(aa_real_real(aa_nat_fun_real_real(Uu,Uuc),Uub)),aa_nat_real(semiri2265585572941072030t_real,Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(minus_minus_real(Uua),Uub)),Uuc)) ).

% ATP.lambda_448
tff(fact_10657_ATP_Olambda__449,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_hb(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_449
tff(fact_10658_ATP_Olambda__450,axiom,
    ! [Uu: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_hh(Uu),Uua),Uub),Uuc) = aa_real_real(minus_minus_real(aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(plus_plus_real(Uu),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uub)) ).

% ATP.lambda_450
tff(fact_10659_ATP_Olambda__451,axiom,
    ! [Uu: rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_hd(Uu),Uua),Uub),Uuc) = aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(plus_plus_rat(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_451
tff(fact_10660_ATP_Olambda__452,axiom,
    ! [Uu: int,Uua: int,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_hf(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_452
tff(fact_10661_ATP_Olambda__453,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_rh(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_453
tff(fact_10662_ATP_Olambda__454,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_gi(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_454
tff(fact_10663_ATP_Olambda__455,axiom,
    ! [Uu: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_gm(Uu),Uua),Uub),Uuc) = aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(plus_plus_real(Uua),Uu)),Uuc)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Uuc))) ).

% ATP.lambda_455
tff(fact_10664_ATP_Olambda__456,axiom,
    ! [Uu: rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_gk(Uu),Uua),Uub),Uuc) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_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_456
tff(fact_10665_ATP_Olambda__457,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_pj(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_457
tff(fact_10666_ATP_Olambda__458,axiom,
    ! [Uu: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_pm(Uu),Uua),Uub),Uuc) = aa_real_real(times_times_real(aa_real_real(times_times_real(aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_458
tff(fact_10667_ATP_Olambda__459,axiom,
    ! [Uu: rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_pl(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_459
tff(fact_10668_ATP_Olambda__460,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_oz(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_460
tff(fact_10669_ATP_Olambda__461,axiom,
    ! [Uu: int,Uua: int,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_pk(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_461
tff(fact_10670_ATP_Olambda__462,axiom,
    ! [Uu: complex,Uua: nat,Uub: complex,Uuc: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_hj(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_462
tff(fact_10671_ATP_Olambda__463,axiom,
    ! [Uu: real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_hm(Uu),Uua),Uub),Uuc) = aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uuc)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uuc)) ).

% ATP.lambda_463
tff(fact_10672_ATP_Olambda__464,axiom,
    ! [Uu: rat,Uua: nat,Uub: rat,Uuc: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aa_nat298952458717073160at_rat(aTP_Lamm_hk(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_464
tff(fact_10673_ATP_Olambda__465,axiom,
    ! [Uu: int,Uua: nat,Uub: int,Uuc: nat] : aa_nat_int(aa_int_fun_nat_int(aa_nat6704147566042551216at_int(aTP_Lamm_hl(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_465
tff(fact_10674_ATP_Olambda__466,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_hc(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_466
tff(fact_10675_ATP_Olambda__467,axiom,
    ! [Uu: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_hi(Uu),Uua),Uub),Uuc) = aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uuc)),aa_real_real(minus_minus_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(plus_plus_real(Uu),Uua)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc)))) ).

% ATP.lambda_467
tff(fact_10676_ATP_Olambda__468,axiom,
    ! [Uu: rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_he(Uu),Uua),Uub),Uuc) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(Uu),Uuc)),aa_rat_rat(minus_minus_rat(aa_nat_rat(power_power_rat(aa_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_468
tff(fact_10677_ATP_Olambda__469,axiom,
    ! [Uu: int,Uua: int,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_hg(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_469
tff(fact_10678_ATP_Olambda__470,axiom,
    ! [Uu: complex,Uua: nat,Uub: complex,Uuc: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_hn(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_470
tff(fact_10679_ATP_Olambda__471,axiom,
    ! [Uu: real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_hq(Uu),Uua),Uub),Uuc) = aa_real_real(times_times_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uuc)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uuc))) ).

% ATP.lambda_471
tff(fact_10680_ATP_Olambda__472,axiom,
    ! [Uu: rat,Uua: nat,Uub: rat,Uuc: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aa_nat298952458717073160at_rat(aTP_Lamm_ho(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_472
tff(fact_10681_ATP_Olambda__473,axiom,
    ! [Uu: int,Uua: nat,Uub: int,Uuc: nat] : aa_nat_int(aa_int_fun_nat_int(aa_nat6704147566042551216at_int(aTP_Lamm_hp(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_473
tff(fact_10682_ATP_Olambda__474,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_or(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_474
tff(fact_10683_ATP_Olambda__475,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] :
      ( aa_int_o(aa_int_fun_int_o(aa_int2464330468366073224_int_o(aTP_Lamm_ql(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_475
tff(fact_10684_ATP_Olambda__476,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aa_nat8314798964848609180_nat_o(aTP_Lamm_rm(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_476
tff(fact_10685_ATP_Olambda__477,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] :
      ( aa_int_o(aa_int_fun_int_o(aa_int2464330468366073224_int_o(aTP_Lamm_qj(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_477
tff(fact_10686_ATP_Olambda__478,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aa_nat8314798964848609180_nat_o(aTP_Lamm_rk(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_478
tff(fact_10687_ATP_Olambda__479,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_ro(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_479
tff(fact_10688_ATP_Olambda__480,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_rq(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_480
tff(fact_10689_ATP_Olambda__481,axiom,
    ! [Uu: fun_nat_complex,Uua: nat,Uub: complex,Uuc: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_iq(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_481
tff(fact_10690_ATP_Olambda__482,axiom,
    ! [Uu: fun_nat_real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_ir(Uu),Uua),Uub),Uuc) = aa_real_real(times_times_real(aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uuc),Uua))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uub),Uuc)) ).

% ATP.lambda_482
tff(fact_10691_ATP_Olambda__483,axiom,
    ! [Uu: fun_nat_complex,Uua: fun_nat_complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_fun5379041630539201665omplex(aTP_Lamm_ou(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_483
tff(fact_10692_ATP_Olambda__484,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_ox(Uu),Uua),Uub),Uuc) = aa_real_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_484
tff(fact_10693_ATP_Olambda__485,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_pb(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_485
tff(fact_10694_ATP_Olambda__486,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_po(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_486
tff(fact_10695_ATP_Olambda__487,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_pd(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_487
tff(fact_10696_ATP_Olambda__488,axiom,
    ! [Uu: fun_complex_complex,Uua: fun_complex_complex,Uub: complex,Uuc: complex] : aa_complex_complex(aa_com3342990543769683525omplex(aa_fun6995112137416684571omplex(aTP_Lamm_ex(Uu),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(aa_complex_complex(Uu,Uub)),aa_complex_complex(Uua,Uuc)) ).

% ATP.lambda_488
tff(fact_10697_ATP_Olambda__489,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_fb(Uu),Uua),Uub),Uuc) = aa_real_real(times_times_real(aa_nat_real(Uu,Uub)),aa_nat_real(Uua,Uuc)) ).

% ATP.lambda_489
tff(fact_10698_ATP_Olambda__490,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_ez(Uu),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu,Uub)),aa_nat_nat(Uua,Uuc)) ).

% ATP.lambda_490
tff(fact_10699_ATP_Olambda__491,axiom,
    ! [Uu: fun_int_int,Uua: fun_int_int,Uub: int,Uuc: int] : aa_int_int(aa_int_fun_int_int(aa_fun5911324870756489105nt_int(aTP_Lamm_ev(Uu),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(Uu,Uub)),aa_int_int(Uua,Uuc)) ).

% ATP.lambda_491
tff(fact_10700_ATP_Olambda__492,axiom,
    ! [Uu: $o,Uua: fun_int_fun_int_o,Uub: int,Uuc: int] :
      ( aa_int_o(aa_int_fun_int_o(aa_fun6133342665515662012_int_o(aTP_Lamm_cc((Uu)),Uua),Uub),Uuc)
    <=> ( (Uu)
        & aa_int_o(aa_int_fun_int_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_492
tff(fact_10701_ATP_Olambda__493,axiom,
    ! [Uu: fun_Pr2478283820680522243nt_int,Uua: fun_in1696471567231710611nt_int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_fun4152872853588681340nt_int(aTP_Lamm_bp(Uu),Uua),Uub),Uuc) = aa_Pro6970504127699319548nt_int(Uu,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(Uua,Uub),Uuc)) ).

% ATP.lambda_493
tff(fact_10702_ATP_Olambda__494,axiom,
    ! [Uu: fun_Pr3238374608701176162_int_o,Uua: fun_in1696471567231710611nt_int,Uub: int,Uuc: int] :
      ( aa_int_o(aa_int_fun_int_o(aa_fun4545072666309863515_int_o(aTP_Lamm_bl(Uu),Uua),Uub),Uuc)
    <=> aa_Pro4592068669816069595_int_o(Uu,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(Uua,Uub),Uuc)) ) ).

% ATP.lambda_494
tff(fact_10703_ATP_Olambda__495,axiom,
    ! [Uu: fun_Pr982841511892225912nt_int,Uua: fun_in1696471567231710611nt_int,Uub: int,Uuc: int] : aa_int_int(aa_int_fun_int_int(aa_fun7461304428735181397nt_int(aTP_Lamm_bm(Uu),Uua),Uub),Uuc) = aa_Pro6799258434465214121nt_int(Uu,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(Uua,Uub),Uuc)) ).

% ATP.lambda_495
tff(fact_10704_ATP_Olambda__496,axiom,
    ! [Uu: fun_o_8988054265482332260nt_int,Uua: fun_int_fun_int_o,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_fun4023131440832735197nt_int(aTP_Lamm_bn(Uu),Uua),Uub),Uuc) = aa_o_P3134438092330336605nt_int(Uu,aa_int_o(aa_int_fun_int_o(Uua,Uub),Uuc)) ).

% ATP.lambda_496
tff(fact_10705_ATP_Olambda__497,axiom,
    ! [Uu: fun_o_o,Uua: fun_int_fun_int_o,Uub: int,Uuc: int] :
      ( aa_int_o(aa_int_fun_int_o(aa_fun6133342665515662012_int_o(aTP_Lamm_bh(Uu),Uua),Uub),Uuc)
    <=> aa_o_o(Uu,aa_int_o(aa_int_fun_int_o(Uua,Uub),Uuc)) ) ).

% ATP.lambda_497
tff(fact_10706_ATP_Olambda__498,axiom,
    ! [Uu: fun_o_int,Uua: fun_int_fun_int_o,Uub: int,Uuc: int] : aa_int_int(aa_int_fun_int_int(aa_fun7323606162024495668nt_int(aTP_Lamm_bi(Uu),Uua),Uub),Uuc) = aa_o_int(Uu,aa_int_o(aa_int_fun_int_o(Uua,Uub),Uuc)) ).

% ATP.lambda_498
tff(fact_10707_ATP_Olambda__499,axiom,
    ! [Uu: fun_in4211310926729683942nt_int,Uua: fun_int_fun_int_int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_fun1034830265760468567nt_int(aTP_Lamm_bo(Uu),Uua),Uub),Uuc) = aa_int7462565939943104719nt_int(Uu,aa_int_int(aa_int_fun_int_int(Uua,Uub),Uuc)) ).

% ATP.lambda_499
tff(fact_10708_ATP_Olambda__500,axiom,
    ! [Uu: fun_int_o,Uua: fun_int_fun_int_int,Uub: int,Uuc: int] :
      ( aa_int_o(aa_int_fun_int_o(aa_fun82482788810740534_int_o(aTP_Lamm_bj(Uu),Uua),Uub),Uuc)
    <=> aa_int_o(Uu,aa_int_int(aa_int_fun_int_int(Uua,Uub),Uuc)) ) ).

% ATP.lambda_500
tff(fact_10709_ATP_Olambda__501,axiom,
    ! [Uu: fun_int_int,Uua: fun_int_fun_int_int,Uub: int,Uuc: int] : aa_int_int(aa_int_fun_int_int(aa_fun6119123525526402106nt_int(aTP_Lamm_bk(Uu),Uua),Uub),Uuc) = aa_int_int(Uu,aa_int_int(aa_int_fun_int_int(Uua,Uub),Uuc)) ).

% ATP.lambda_501
tff(fact_10710_ATP_Olambda__502,axiom,
    ! [Uu: fun_fu3275390434952296369_nat_o,Uua: fun_na5914473681949881652_nat_o,Uub: nat,Uuc: nat] : aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(aa_fun824319262034445606_nat_o(aTP_Lamm_bq(Uu),Uua),Uub),Uuc) = aa_fun359341089595923442_nat_o(Uu,aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(Uua,Uub),Uuc)) ).

% ATP.lambda_502
tff(fact_10711_ATP_Olambda__503,axiom,
    ! [Uu: fun_nat_real,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_nat4085805313368488148t_real(aTP_Lamm_dv(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_503
tff(fact_10712_ATP_Olambda__504,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_du(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_504
tff(fact_10713_ATP_Olambda__505,axiom,
    ! [Uu: fun_nat_int,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_nat1293065384339088724at_int(aTP_Lamm_ln(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_505
tff(fact_10714_ATP_Olambda__506,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_qh(Uu),Uua),Uub),Uuc) = aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(minus_minus_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_506
tff(fact_10715_ATP_Olambda__507,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_qf(Uu),Uua),Uub),Uuc) = aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(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_507
tff(fact_10716_ATP_Olambda__508,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_qd(Uu),Uua),Uub),Uuc) = aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(product_Pair_int_int,aa_int_int(aa_int_fun_int_int(times_times_int,Uu),Uub)),aa_int_int(aa_int_fun_int_int(times_times_int,Uua),Uuc))) ).

% ATP.lambda_508
tff(fact_10717_ATP_Olambda__509,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_qb(Uu),Uua),Uub),Uuc) = aa_Pro6970504127699319548nt_int(normalize,aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(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_509
tff(fact_10718_ATP_Olambda__510,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_sl(Uu),Uua),Uub),Uuc),Uud) = aa_real_real(minus_minus_real(aa_real_real(aa_nat_fun_real_real(Uua,Uuc),Uud)),aa_real_real(plus_plus_real(aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_sk(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(times_times_real(Uub),aa_real_real(divide_divide_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,Uud),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uuc))),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uuc)))))) ).

% ATP.lambda_510
tff(fact_10719_ATP_Olambda__511,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_ty(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))
        & ! [I5: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uud))
           => ( ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(Uua,I5)),X_13)
            <=> aa_nat_o(vEBT_V8194947554948674370ptions(Uub),I5) ) )
        & $ite(
            Uue = Uuf,
            ! [X4: vEBT_VEBT] :
              ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(Uua))
             => ~ ? [X9: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X4),X9) ),
            ( 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_511
tff(fact_10720_ATP_Olambda__512,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_if(Uu),Uua) = Uu ).

% ATP.lambda_512
tff(fact_10721_ATP_Olambda__513,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ig,Uu),Uua) = Uu ).

% ATP.lambda_513
tff(fact_10722_ATP_Olambda__514,axiom,
    ! [Uu: int,Uua: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_cd,Uu),Uua)
    <=> $true ) ).

% ATP.lambda_514
tff(fact_10723_ATP_Olambda__515,axiom,
    ! [Uu: complex] : aa_complex_complex(aTP_Lamm_ao,Uu) = Uu ).

% ATP.lambda_515
tff(fact_10724_ATP_Olambda__516,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_ap,Uu) = Uu ).

% ATP.lambda_516
tff(fact_10725_ATP_Olambda__517,axiom,
    ! [Uu: rat] : aa_rat_rat(aTP_Lamm_aq,Uu) = Uu ).

% ATP.lambda_517
tff(fact_10726_ATP_Olambda__518,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_ar,Uu) = Uu ).

% ATP.lambda_518
tff(fact_10727_ATP_Olambda__519,axiom,
    ! [Uu: int] : aa_int_int(aTP_Lamm_as,Uu) = Uu ).

% ATP.lambda_519
tff(fact_10728_ATP_Olambda__520,axiom,
    ! [Uu: complex] : aa_complex_complex(aTP_Lamm_aj,Uu) = zero_zero_complex ).

% ATP.lambda_520
tff(fact_10729_ATP_Olambda__521,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_ak,Uu) = zero_zero_real ).

% ATP.lambda_521
tff(fact_10730_ATP_Olambda__522,axiom,
    ! [Uu: rat] : aa_rat_rat(aTP_Lamm_al,Uu) = zero_zero_rat ).

% ATP.lambda_522
tff(fact_10731_ATP_Olambda__523,axiom,
    ! [Uu: nat] : aa_nat_complex(aTP_Lamm_gg,Uu) = zero_zero_complex ).

% ATP.lambda_523
tff(fact_10732_ATP_Olambda__524,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_de,Uu) = zero_zero_real ).

% ATP.lambda_524
tff(fact_10733_ATP_Olambda__525,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_am,Uu) = zero_zero_nat ).

% ATP.lambda_525
tff(fact_10734_ATP_Olambda__526,axiom,
    ! [Uu: nat] : aa_nat_int(aTP_Lamm_gh,Uu) = zero_zero_int ).

% ATP.lambda_526
tff(fact_10735_ATP_Olambda__527,axiom,
    ! [Uu: int] : aa_int_int(aTP_Lamm_an,Uu) = zero_zero_int ).

% ATP.lambda_527
tff(fact_10736_ATP_Olambda__528,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_kp,Uu) = one_one_nat ).

% ATP.lambda_528
tff(fact_10737_ATP_Olambda__529,axiom,
    ! [Uu: nat] : aa_nat_int(aTP_Lamm_kq,Uu) = one_one_int ).

% ATP.lambda_529
tff(fact_10738_ATP_Olambda__530,axiom,
    ! [Uu: int] : aa_int_int(aTP_Lamm_kr,Uu) = one_one_int ).

% ATP.lambda_530
tff(fact_10739_ATP_Olambda__531,axiom,
    ! [Uu: nat] :
      ( aa_nat_o(aTP_Lamm_ra,Uu)
    <=> $false ) ).

% ATP.lambda_531
tff(fact_10740_ATP_Olambda__532,axiom,
    ! [Uu: nat] :
      ( aa_nat_o(aTP_Lamm_rb,Uu)
    <=> $true ) ).

% ATP.lambda_532

% Conjectures (1)
tff(conj_0,conjecture,
    aa_nat_o(vEBT_V8194947554948674370ptions(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(mi),ma)),deg,treeList,summary)),ya) ).

%------------------------------------------------------------------------------