%------------------------------------------------------------------------------
% File     : ITP237_3 : TPTP v8.2.0. Released v8.0.0.
% Domain   : Interactive Theorem Proving
% Problem  : Sledgehammer problem VEBT_Pred 00156_006356
% 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    : 0069_VEBT_Pred_00156_006356 [Des22]

% Status   : Theorem
% Rating   : 1.00 v8.1.0
% Syntax   : Number of formulae    : 13787 (4062 unt;2840 typ;   0 def)
%            Number of atoms       : 27862 (9098 equ)
%            Maximal formula atoms :   73 (   2 avg)
%            Number of connectives : 17886 (2308   ~; 495   |;1455   &)
%                                         (2850 <=>;10778  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   35 (   5 avg)
%            Maximal term depth    :   37 (   2 avg)
%            Number of FOOLs       : 1670 (1116 fml; 554 var)
%            Number of X terms     :  843 (   0  []; 733 ite; 110 let)
%            Number of types       :  589 ( 588 usr)
%            Number of type conns  : 2733 (1763   >; 970   *;   0   +;   0  <<)
%            Number of predicates  :  198 ( 195 usr;   4 prp; 0-4 aty)
%            Number of functors    : 2072 (2072 usr; 502 con; 0-8 aty)
%            Number of variables   : 26534 (25744   !; 790   ?;26534   :)
% 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 22:30:32.828
%------------------------------------------------------------------------------
% Could-be-implicit typings (588)
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    aTP_Lamm_aat: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__aau____,type,
    aTP_Lamm_aau: nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__aav____,type,
    aTP_Lamm_aav: nat > fun_na7928631014926962227_nat_o ).

tff(sy_c_ATP_058Lamm__aaw____,type,
    aTP_Lamm_aaw: fun_na5914473681949881652_nat_o ).

tff(sy_c_ATP_058Lamm__aax____,type,
    aTP_Lamm_aax: nat > fun_na7928631014926962227_nat_o ).

tff(sy_c_ATP_058Lamm__aay____,type,
    aTP_Lamm_aay: fun_na5914473681949881652_nat_o ).

tff(sy_c_ATP_058Lamm__aaz____,type,
    aTP_Lamm_aaz: nat > fun_nat_fun_nat_o ).

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

tff(sy_c_ATP_058Lamm__aba____,type,
    aTP_Lamm_aba: fun_nat_rat > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__abb____,type,
    aTP_Lamm_abb: set_real > fun_real_o ).

tff(sy_c_ATP_058Lamm__abc____,type,
    aTP_Lamm_abc: int > fun_in5570076352459579423_int_o ).

tff(sy_c_ATP_058Lamm__abd____,type,
    aTP_Lamm_abd: rat > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__abe____,type,
    aTP_Lamm_abe: int > fun_in5570076352459579423_int_o ).

tff(sy_c_ATP_058Lamm__abf____,type,
    aTP_Lamm_abf: rat > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__abg____,type,
    aTP_Lamm_abg: rat > fun_Pr3238374608701176162_int_o ).

tff(sy_c_ATP_058Lamm__abh____,type,
    aTP_Lamm_abh: fun_fun_nat_rat_o ).

tff(sy_c_ATP_058Lamm__abi____,type,
    aTP_Lamm_abi: fun_int_int ).

tff(sy_c_ATP_058Lamm__abj____,type,
    aTP_Lamm_abj: fun_num_fun_num_int ).

tff(sy_c_ATP_058Lamm__abk____,type,
    aTP_Lamm_abk: fun_Pr3238374608701176162_int_o ).

tff(sy_c_ATP_058Lamm__abl____,type,
    aTP_Lamm_abl: fun_na1476630412504255114at_nat ).

tff(sy_c_ATP_058Lamm__abm____,type,
    aTP_Lamm_abm: nat > fun_nat_set_nat ).

tff(sy_c_ATP_058Lamm__abn____,type,
    aTP_Lamm_abn: nat > fun_nat_set_nat ).

tff(sy_c_ATP_058Lamm__abo____,type,
    aTP_Lamm_abo: nat > fun_nat_set_nat ).

tff(sy_c_ATP_058Lamm__abp____,type,
    aTP_Lamm_abp: extended_enat > fun_nat_o ).

tff(sy_c_ATP_058Lamm__abq____,type,
    aTP_Lamm_abq: extended_enat > fun_nat_o ).

tff(sy_c_ATP_058Lamm__abr____,type,
    aTP_Lamm_abr: real > fun_re8658731967611918585t_real ).

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

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

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

tff(sy_c_ATP_058Lamm__af____,type,
    aTP_Lamm_af: fun_real_o > fun_fu9050450510664235267real_o ).

tff(sy_c_ATP_058Lamm__ag____,type,
    aTP_Lamm_ag: fun_list_nat_o > fun_fu1994797156054922755_nat_o ).

tff(sy_c_ATP_058Lamm__ah____,type,
    aTP_Lamm_ah: fun_set_nat_o > fun_fu2408719596384163971_nat_o ).

tff(sy_c_ATP_058Lamm__ai____,type,
    aTP_Lamm_ai: fun_nat_o > fun_fu4001317426178549763_nat_o ).

tff(sy_c_ATP_058Lamm__aj____,type,
    aTP_Lamm_aj: fun_int_o > fun_fu3883539188510166275_int_o ).

tff(sy_c_ATP_058Lamm__ak____,type,
    aTP_Lamm_ak: fun_complex_o > fun_fu1358370060594899587plex_o ).

tff(sy_c_ATP_058Lamm__al____,type,
    aTP_Lamm_al: fun_real_o > fun_fu9050450510664235267real_o ).

tff(sy_c_ATP_058Lamm__am____,type,
    aTP_Lamm_am: fun_list_nat_o > fun_fu1994797156054922755_nat_o ).

tff(sy_c_ATP_058Lamm__an____,type,
    aTP_Lamm_an: fun_set_nat_o > fun_fu2408719596384163971_nat_o ).

tff(sy_c_ATP_058Lamm__ao____,type,
    aTP_Lamm_ao: fun_nat_o > fun_fu4001317426178549763_nat_o ).

tff(sy_c_ATP_058Lamm__ap____,type,
    aTP_Lamm_ap: fun_int_o > fun_fu3883539188510166275_int_o ).

tff(sy_c_ATP_058Lamm__aq____,type,
    aTP_Lamm_aq: fun_complex_o > fun_fu1358370060594899587plex_o ).

tff(sy_c_ATP_058Lamm__ar____,type,
    aTP_Lamm_ar: fun_se2170385732985038782_nat_o ).

tff(sy_c_ATP_058Lamm__as____,type,
    aTP_Lamm_as: set_complex > fun_set_complex_o ).

tff(sy_c_ATP_058Lamm__at____,type,
    aTP_Lamm_at: set_int > fun_set_int_o ).

tff(sy_c_ATP_058Lamm__au____,type,
    aTP_Lamm_au: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__av____,type,
    aTP_Lamm_av: fun_nat_fun_nat_o ).

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

tff(sy_c_ATP_058Lamm__ax____,type,
    aTP_Lamm_ax: int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__ay____,type,
    aTP_Lamm_ay: int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__az____,type,
    aTP_Lamm_az: int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__ba____,type,
    aTP_Lamm_ba: set_real > fun_fu4757868175734369219real_o ).

tff(sy_c_ATP_058Lamm__bb____,type,
    aTP_Lamm_bb: set_real > fun_fu8602227297639178947real_o ).

tff(sy_c_ATP_058Lamm__bc____,type,
    aTP_Lamm_bc: set_real > fun_fu4472351043248340291real_o ).

tff(sy_c_ATP_058Lamm__bd____,type,
    aTP_Lamm_bd: set_nat > fun_fu6436183194761443139_nat_o ).

tff(sy_c_ATP_058Lamm__be____,type,
    aTP_Lamm_be: set_nat > fun_fu8304507701009772099_nat_o ).

tff(sy_c_ATP_058Lamm__bf____,type,
    aTP_Lamm_bf: set_nat > fun_fu3780232916559739075_nat_o ).

tff(sy_c_ATP_058Lamm__bg____,type,
    aTP_Lamm_bg: set_int > fun_fu7493154764834273859_int_o ).

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

tff(sy_c_ATP_058Lamm__bi____,type,
    aTP_Lamm_bi: set_int > fun_fu2620374121714387907_int_o ).

tff(sy_c_ATP_058Lamm__bj____,type,
    aTP_Lamm_bj: set_complex > fun_fu3074144567237359299plex_o ).

tff(sy_c_ATP_058Lamm__bk____,type,
    aTP_Lamm_bk: set_Pr1261947904930325089at_nat > fun_fu3275390434952296369_nat_o ).

tff(sy_c_ATP_058Lamm__bl____,type,
    aTP_Lamm_bl: set_real > fun_fu9050450510664235267real_o ).

tff(sy_c_ATP_058Lamm__bm____,type,
    aTP_Lamm_bm: set_list_nat > fun_fu1994797156054922755_nat_o ).

tff(sy_c_ATP_058Lamm__bn____,type,
    aTP_Lamm_bn: set_set_nat > fun_fu2408719596384163971_nat_o ).

tff(sy_c_ATP_058Lamm__bo____,type,
    aTP_Lamm_bo: set_nat > fun_fu4001317426178549763_nat_o ).

tff(sy_c_ATP_058Lamm__bp____,type,
    aTP_Lamm_bp: set_int > fun_fu3883539188510166275_int_o ).

tff(sy_c_ATP_058Lamm__bq____,type,
    aTP_Lamm_bq: fun_list_nat_o ).

tff(sy_c_ATP_058Lamm__br____,type,
    aTP_Lamm_br: fun_set_nat_o ).

tff(sy_c_ATP_058Lamm__bs____,type,
    aTP_Lamm_bs: fun_real_o ).

tff(sy_c_ATP_058Lamm__bt____,type,
    aTP_Lamm_bt: fun_nat_o ).

tff(sy_c_ATP_058Lamm__bu____,type,
    aTP_Lamm_bu: fun_int_o ).

tff(sy_c_ATP_058Lamm__bv____,type,
    aTP_Lamm_bv: fun_real_real ).

tff(sy_c_ATP_058Lamm__bw____,type,
    aTP_Lamm_bw: fun_rat_rat ).

tff(sy_c_ATP_058Lamm__bx____,type,
    aTP_Lamm_bx: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__by____,type,
    aTP_Lamm_by: fun_int_int ).

tff(sy_c_ATP_058Lamm__bz____,type,
    aTP_Lamm_bz: fun_complex_complex ).

tff(sy_c_ATP_058Lamm__ca____,type,
    aTP_Lamm_ca: fun_real_real ).

tff(sy_c_ATP_058Lamm__cb____,type,
    aTP_Lamm_cb: fun_rat_rat ).

tff(sy_c_ATP_058Lamm__cc____,type,
    aTP_Lamm_cc: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__cd____,type,
    aTP_Lamm_cd: fun_int_int ).

tff(sy_c_ATP_058Lamm__ce____,type,
    aTP_Lamm_ce: fun_real_fun_real_o ).

tff(sy_c_ATP_058Lamm__cf____,type,
    aTP_Lamm_cf: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__cg____,type,
    aTP_Lamm_cg: fun_Co2081214437782444048eger_o ).

tff(sy_c_ATP_058Lamm__ch____,type,
    aTP_Lamm_ch: fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__ci____,type,
    aTP_Lamm_ci: set_Pr4811707699266497531nteger > fun_Co2081214437782444048eger_o ).

tff(sy_c_ATP_058Lamm__cj____,type,
    aTP_Lamm_cj: set_Pr448751882837621926eger_o > fun_Co480733329897759773un_o_o ).

tff(sy_c_ATP_058Lamm__ck____,type,
    aTP_Lamm_ck: set_Pr8693737435421807431at_nat > fun_Pr2028404973800837934_nat_o ).

tff(sy_c_ATP_058Lamm__cl____,type,
    aTP_Lamm_cl: set_Pr1261947904930325089at_nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__cm____,type,
    aTP_Lamm_cm: set_Pr958786334691620121nt_int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__cn____,type,
    aTP_Lamm_cn: fun_nat_o > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__co____,type,
    aTP_Lamm_co: fun_nat_nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__cp____,type,
    aTP_Lamm_cp: real > fun_real_fun_real_o ).

tff(sy_c_ATP_058Lamm__cq____,type,
    aTP_Lamm_cq: rat > fun_rat_fun_rat_o ).

tff(sy_c_ATP_058Lamm__cr____,type,
    aTP_Lamm_cr: fun_nat_fun_real_o ).

tff(sy_c_ATP_058Lamm__cs____,type,
    aTP_Lamm_cs: fun_na1554462459066888140plex_o ).

tff(sy_c_ATP_058Lamm__ct____,type,
    aTP_Lamm_ct: set_complex > fun_na2418085665696872764plex_o ).

tff(sy_c_ATP_058Lamm__cu____,type,
    aTP_Lamm_cu: set_VEBT_VEBT > fun_na5873943405026509934VEBT_o ).

tff(sy_c_ATP_058Lamm__cv____,type,
    aTP_Lamm_cv: set_o > fun_nat_fun_list_o_o ).

tff(sy_c_ATP_058Lamm__cw____,type,
    aTP_Lamm_cw: set_nat > fun_na7234455742697417626_nat_o ).

tff(sy_c_ATP_058Lamm__cx____,type,
    aTP_Lamm_cx: set_int > fun_na7306406234920378942_int_o ).

tff(sy_c_ATP_058Lamm__cy____,type,
    aTP_Lamm_cy: set_complex > fun_na2418085665696872764plex_o ).

tff(sy_c_ATP_058Lamm__cz____,type,
    aTP_Lamm_cz: set_VEBT_VEBT > fun_na5873943405026509934VEBT_o ).

tff(sy_c_ATP_058Lamm__da____,type,
    aTP_Lamm_da: set_o > fun_nat_fun_list_o_o ).

tff(sy_c_ATP_058Lamm__db____,type,
    aTP_Lamm_db: set_nat > fun_na7234455742697417626_nat_o ).

tff(sy_c_ATP_058Lamm__dc____,type,
    aTP_Lamm_dc: set_int > fun_na7306406234920378942_int_o ).

tff(sy_c_ATP_058Lamm__dd____,type,
    aTP_Lamm_dd: nat > fun_co1148497455491383662plex_o ).

tff(sy_c_ATP_058Lamm__de____,type,
    aTP_Lamm_de: set_real > fun_fu7444430165986413063real_o ).

tff(sy_c_ATP_058Lamm__df____,type,
    aTP_Lamm_df: set_real > fun_fu9068519094125542729real_o ).

tff(sy_c_ATP_058Lamm__dg____,type,
    aTP_Lamm_dg: set_nat > fun_fu6176141739774794687_nat_o ).

tff(sy_c_ATP_058Lamm__dh____,type,
    aTP_Lamm_dh: set_nat > fun_fu6433034133342731997_nat_o ).

tff(sy_c_ATP_058Lamm__di____,type,
    aTP_Lamm_di: set_int > fun_fu7994259084669964039_int_o ).

tff(sy_c_ATP_058Lamm__dj____,type,
    aTP_Lamm_dj: set_int > fun_fu3930813892005118153_int_o ).

tff(sy_c_ATP_058Lamm__dk____,type,
    aTP_Lamm_dk: set_complex > fun_fu2445682883699234307plex_o ).

tff(sy_c_ATP_058Lamm__dl____,type,
    aTP_Lamm_dl: set_complex > fun_fu2142003765521078979plex_o ).

tff(sy_c_ATP_058Lamm__dm____,type,
    aTP_Lamm_dm: set_real > fun_fu3022329567491847689real_o ).

tff(sy_c_ATP_058Lamm__dn____,type,
    aTP_Lamm_dn: set_real > fun_fu5533088652784741581real_o ).

tff(sy_c_ATP_058Lamm__do____,type,
    aTP_Lamm_do: set_nat > fun_fu6621322437453427393_nat_o ).

tff(sy_c_ATP_058Lamm__dp____,type,
    aTP_Lamm_dp: set_nat > fun_fu6813129056075673441_nat_o ).

tff(sy_c_ATP_058Lamm__dq____,type,
    aTP_Lamm_dq: set_int > fun_fu4185471554550717449_int_o ).

tff(sy_c_ATP_058Lamm__dr____,type,
    aTP_Lamm_dr: set_int > fun_fu4228744305898656589_int_o ).

tff(sy_c_ATP_058Lamm__ds____,type,
    aTP_Lamm_ds: set_complex > fun_fu806425790188207109plex_o ).

tff(sy_c_ATP_058Lamm__dt____,type,
    aTP_Lamm_dt: set_complex > fun_fu7523448966510524487plex_o ).

tff(sy_c_ATP_058Lamm__du____,type,
    aTP_Lamm_du: set_real > fun_fu6510078244439976541real_o ).

tff(sy_c_ATP_058Lamm__dv____,type,
    aTP_Lamm_dv: set_real > fun_fu5587483295140630005real_o ).

tff(sy_c_ATP_058Lamm__dw____,type,
    aTP_Lamm_dw: set_nat > fun_fu7145897819039579925_nat_o ).

tff(sy_c_ATP_058Lamm__dx____,type,
    aTP_Lamm_dx: set_nat > fun_fu1806770557662746761_nat_o ).

tff(sy_c_ATP_058Lamm__dy____,type,
    aTP_Lamm_dy: set_real > fun_fu3022329567491847689real_o ).

tff(sy_c_ATP_058Lamm__dz____,type,
    aTP_Lamm_dz: set_real > fun_fu5533088652784741581real_o ).

tff(sy_c_ATP_058Lamm__ea____,type,
    aTP_Lamm_ea: set_nat > fun_fu6621322437453427393_nat_o ).

tff(sy_c_ATP_058Lamm__eb____,type,
    aTP_Lamm_eb: set_nat > fun_fu6813129056075673441_nat_o ).

tff(sy_c_ATP_058Lamm__ec____,type,
    aTP_Lamm_ec: set_int > fun_fu4185471554550717449_int_o ).

tff(sy_c_ATP_058Lamm__ed____,type,
    aTP_Lamm_ed: set_int > fun_fu4228744305898656589_int_o ).

tff(sy_c_ATP_058Lamm__ee____,type,
    aTP_Lamm_ee: set_complex > fun_fu806425790188207109plex_o ).

tff(sy_c_ATP_058Lamm__ef____,type,
    aTP_Lamm_ef: set_complex > fun_fu7523448966510524487plex_o ).

tff(sy_c_ATP_058Lamm__eg____,type,
    aTP_Lamm_eg: set_real > fun_fu6510078244439976541real_o ).

tff(sy_c_ATP_058Lamm__eh____,type,
    aTP_Lamm_eh: set_real > fun_fu5587483295140630005real_o ).

tff(sy_c_ATP_058Lamm__ei____,type,
    aTP_Lamm_ei: set_nat > fun_fu7145897819039579925_nat_o ).

tff(sy_c_ATP_058Lamm__ej____,type,
    aTP_Lamm_ej: set_nat > fun_fu1806770557662746761_nat_o ).

tff(sy_c_ATP_058Lamm__ek____,type,
    aTP_Lamm_ek: set_int > fun_fu3473604023002724957_int_o ).

tff(sy_c_ATP_058Lamm__el____,type,
    aTP_Lamm_el: set_int > fun_fu7714913403879666293_int_o ).

tff(sy_c_ATP_058Lamm__em____,type,
    aTP_Lamm_em: set_complex > fun_fu2497916604042137945plex_o ).

tff(sy_c_ATP_058Lamm__en____,type,
    aTP_Lamm_en: set_complex > fun_fu6925816717872577647plex_o ).

tff(sy_c_ATP_058Lamm__eo____,type,
    aTP_Lamm_eo: set_real > fun_fu1852998413150313061real_o ).

tff(sy_c_ATP_058Lamm__ep____,type,
    aTP_Lamm_ep: set_real > fun_fu5575653946431489541real_o ).

tff(sy_c_ATP_058Lamm__eq____,type,
    aTP_Lamm_eq: set_nat > fun_fu5477517762487726877_nat_o ).

tff(sy_c_ATP_058Lamm__er____,type,
    aTP_Lamm_er: set_nat > fun_fu6075382464776002713_nat_o ).

tff(sy_c_ATP_058Lamm__es____,type,
    aTP_Lamm_es: fun_real_fun_real_o ).

tff(sy_c_ATP_058Lamm__et____,type,
    aTP_Lamm_et: fun_rat_fun_rat_o ).

tff(sy_c_ATP_058Lamm__eu____,type,
    aTP_Lamm_eu: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__ev____,type,
    aTP_Lamm_ev: fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__ew____,type,
    aTP_Lamm_ew: fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__ex____,type,
    aTP_Lamm_ex: num > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__ey____,type,
    aTP_Lamm_ey: num > fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__ez____,type,
    aTP_Lamm_ez: num > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__fa____,type,
    aTP_Lamm_fa: fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__fb____,type,
    aTP_Lamm_fb: fun_complex_complex ).

tff(sy_c_ATP_058Lamm__fc____,type,
    aTP_Lamm_fc: fun_nat_real ).

tff(sy_c_ATP_058Lamm__fd____,type,
    aTP_Lamm_fd: real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__fe____,type,
    aTP_Lamm_fe: int > fun_fu121957152452960295t_real ).

tff(sy_c_ATP_058Lamm__ff____,type,
    aTP_Lamm_ff: complex > fun_fu7714627026455246123x_real ).

tff(sy_c_ATP_058Lamm__fg____,type,
    aTP_Lamm_fg: real > fun_fu2598743018846156927al_rat ).

tff(sy_c_ATP_058Lamm__fh____,type,
    aTP_Lamm_fh: nat > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__fi____,type,
    aTP_Lamm_fi: int > fun_fu1008360527747082367nt_rat ).

tff(sy_c_ATP_058Lamm__fj____,type,
    aTP_Lamm_fj: complex > fun_fu2990765671291504771ex_rat ).

tff(sy_c_ATP_058Lamm__fk____,type,
    aTP_Lamm_fk: real > fun_fu3214020956574444655al_nat ).

tff(sy_c_ATP_058Lamm__fl____,type,
    aTP_Lamm_fl: int > fun_fu8905809029841901679nt_nat ).

tff(sy_c_ATP_058Lamm__fm____,type,
    aTP_Lamm_fm: complex > fun_fu705315015666092147ex_nat ).

tff(sy_c_ATP_058Lamm__fn____,type,
    aTP_Lamm_fn: real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__fo____,type,
    aTP_Lamm_fo: int > fun_fu121957152452960295t_real ).

tff(sy_c_ATP_058Lamm__fp____,type,
    aTP_Lamm_fp: complex > fun_fu7714627026455246123x_real ).

tff(sy_c_ATP_058Lamm__fq____,type,
    aTP_Lamm_fq: real > fun_fu2598743018846156927al_rat ).

tff(sy_c_ATP_058Lamm__fr____,type,
    aTP_Lamm_fr: nat > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__fs____,type,
    aTP_Lamm_fs: int > fun_fu1008360527747082367nt_rat ).

tff(sy_c_ATP_058Lamm__ft____,type,
    aTP_Lamm_ft: complex > fun_fu2990765671291504771ex_rat ).

tff(sy_c_ATP_058Lamm__fu____,type,
    aTP_Lamm_fu: real > fun_fu3214020956574444655al_nat ).

tff(sy_c_ATP_058Lamm__fv____,type,
    aTP_Lamm_fv: int > fun_fu8905809029841901679nt_nat ).

tff(sy_c_ATP_058Lamm__fw____,type,
    aTP_Lamm_fw: complex > fun_fu705315015666092147ex_nat ).

tff(sy_c_ATP_058Lamm__fx____,type,
    aTP_Lamm_fx: fun_int_int > fun_int_int ).

tff(sy_c_ATP_058Lamm__fy____,type,
    aTP_Lamm_fy: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__fz____,type,
    aTP_Lamm_fz: set_int > fun_fu7328815047503086531_int_o ).

tff(sy_c_ATP_058Lamm__ga____,type,
    aTP_Lamm_ga: set_int > fun_fu8404755328348021045al_int ).

tff(sy_c_ATP_058Lamm__gb____,type,
    aTP_Lamm_gb: fun_real_fun_int_int > fun_int_fun_real_int ).

tff(sy_c_ATP_058Lamm__gc____,type,
    aTP_Lamm_gc: set_real > fun_fu6324116013759063093nt_int ).

tff(sy_c_ATP_058Lamm__gd____,type,
    aTP_Lamm_gd: set_int > fun_fu5646853623633916995_int_o ).

tff(sy_c_ATP_058Lamm__ge____,type,
    aTP_Lamm_ge: set_int > fun_fu4584101141160326545at_int ).

tff(sy_c_ATP_058Lamm__gf____,type,
    aTP_Lamm_gf: fun_nat_fun_int_int > fun_int_fun_nat_int ).

tff(sy_c_ATP_058Lamm__gg____,type,
    aTP_Lamm_gg: set_nat > fun_fu7802804735324696557nt_int ).

tff(sy_c_ATP_058Lamm__gh____,type,
    aTP_Lamm_gh: set_int > fun_fu810243669753863363_int_o ).

tff(sy_c_ATP_058Lamm__gi____,type,
    aTP_Lamm_gi: set_int > fun_fu8001260946965626035ex_int ).

tff(sy_c_ATP_058Lamm__gj____,type,
    aTP_Lamm_gj: fun_co333780937270390912nt_int > fun_in3657390099950231424ex_int ).

tff(sy_c_ATP_058Lamm__gk____,type,
    aTP_Lamm_gk: set_complex > fun_fu2339071201290874307plex_o ).

tff(sy_c_ATP_058Lamm__gl____,type,
    aTP_Lamm_gl: set_complex > fun_fu7424888833511495217nt_int ).

tff(sy_c_ATP_058Lamm__gm____,type,
    aTP_Lamm_gm: set_complex > fun_fu3043630359712370243plex_o ).

tff(sy_c_ATP_058Lamm__gn____,type,
    aTP_Lamm_gn: set_complex > fun_fu7433039812794084273omplex ).

tff(sy_c_ATP_058Lamm__go____,type,
    aTP_Lamm_go: fun_re7208826350674946562omplex > fun_co286204877037288194omplex ).

tff(sy_c_ATP_058Lamm__gp____,type,
    aTP_Lamm_gp: set_real > fun_fu1853463631764337203omplex ).

tff(sy_c_ATP_058Lamm__gq____,type,
    aTP_Lamm_gq: set_complex > fun_fu5273811730930669507plex_o ).

tff(sy_c_ATP_058Lamm__gr____,type,
    aTP_Lamm_gr: set_complex > fun_fu6938252901338612493omplex ).

tff(sy_c_ATP_058Lamm__gs____,type,
    aTP_Lamm_gs: fun_na1591161823130732966omplex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__gt____,type,
    aTP_Lamm_gt: set_nat > fun_fu3656911019502402539omplex ).

tff(sy_c_ATP_058Lamm__gu____,type,
    aTP_Lamm_gu: set_complex > fun_fu4149201653251398851plex_o ).

tff(sy_c_ATP_058Lamm__gv____,type,
    aTP_Lamm_gv: set_complex > fun_fu7724386853639975089omplex ).

tff(sy_c_ATP_058Lamm__gw____,type,
    aTP_Lamm_gw: fun_in1307110748760623106omplex > fun_co7673900812888889602omplex ).

tff(sy_c_ATP_058Lamm__gx____,type,
    aTP_Lamm_gx: set_int > fun_fu260611630652282675omplex ).

tff(sy_c_ATP_058Lamm__gy____,type,
    aTP_Lamm_gy: set_nat > fun_fu5289434066392675523_nat_o ).

tff(sy_c_ATP_058Lamm__gz____,type,
    aTP_Lamm_gz: set_nat > fun_fu8695033588447641837al_nat ).

tff(sy_c_ATP_058Lamm__ha____,type,
    aTP_Lamm_ha: fun_real_fun_nat_nat > fun_nat_fun_real_nat ).

tff(sy_c_ATP_058Lamm__hb____,type,
    aTP_Lamm_hb: set_real > fun_fu5553606195750802321at_nat ).

tff(sy_c_ATP_058Lamm__hc____,type,
    aTP_Lamm_hc: set_nat > fun_fu927436805355353155_nat_o ).

tff(sy_c_ATP_058Lamm__hd____,type,
    aTP_Lamm_hd: set_nat > fun_fu6398909621167755501nt_nat ).

tff(sy_c_ATP_058Lamm__he____,type,
    aTP_Lamm_he: fun_int_fun_nat_nat > fun_nat_fun_int_nat ).

tff(sy_c_ATP_058Lamm__hf____,type,
    aTP_Lamm_hf: set_int > fun_fu3180206027003385489at_nat ).

tff(sy_c_ATP_058Lamm__hg____,type,
    aTP_Lamm_hg: set_nat > fun_fu1580565752866428867_nat_o ).

tff(sy_c_ATP_058Lamm__hh____,type,
    aTP_Lamm_hh: set_nat > fun_fu8020931516743756395ex_nat ).

tff(sy_c_ATP_058Lamm__hi____,type,
    aTP_Lamm_hi: fun_co636942507509095880at_nat > fun_na8457291176974909128ex_nat ).

tff(sy_c_ATP_058Lamm__hj____,type,
    aTP_Lamm_hj: set_complex > fun_fu3351286688164852109at_nat ).

tff(sy_c_ATP_058Lamm__hk____,type,
    aTP_Lamm_hk: set_nat > fun_fu5253574275612187245l_real ).

tff(sy_c_ATP_058Lamm__hl____,type,
    aTP_Lamm_hl: fun_re4238438383274954274t_real > fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__hm____,type,
    aTP_Lamm_hm: set_real > fun_fu5321668254738230545t_real ).

tff(sy_c_ATP_058Lamm__hn____,type,
    aTP_Lamm_hn: fun_int_int > fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__ho____,type,
    aTP_Lamm_ho: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__hp____,type,
    aTP_Lamm_hp: fun_real_real > fun_fu4478067837645853357l_real ).

tff(sy_c_ATP_058Lamm__hq____,type,
    aTP_Lamm_hq: fun_int_real > fun_fu7030473000426307245t_real ).

tff(sy_c_ATP_058Lamm__hr____,type,
    aTP_Lamm_hr: set_complex > fun_fu1358370060594899587plex_o ).

tff(sy_c_ATP_058Lamm__hs____,type,
    aTP_Lamm_hs: fun_complex_real > fun_fu3043978019632300333x_real ).

tff(sy_c_ATP_058Lamm__ht____,type,
    aTP_Lamm_ht: fun_real_rat > fun_fu5551332503302594393al_rat ).

tff(sy_c_ATP_058Lamm__hu____,type,
    aTP_Lamm_hu: fun_nat_rat > fun_fu3162257917499486297at_rat ).

tff(sy_c_ATP_058Lamm__hv____,type,
    aTP_Lamm_hv: fun_int_rat > fun_fu3922030663826099033nt_rat ).

tff(sy_c_ATP_058Lamm__hw____,type,
    aTP_Lamm_hw: fun_complex_rat > fun_fu4396998534562958809ex_rat ).

tff(sy_c_ATP_058Lamm__hx____,type,
    aTP_Lamm_hx: fun_real_nat > fun_fu2707946803925496145al_nat ).

tff(sy_c_ATP_058Lamm__hy____,type,
    aTP_Lamm_hy: fun_int_nat > fun_fu1078644964449000785nt_nat ).

tff(sy_c_ATP_058Lamm__hz____,type,
    aTP_Lamm_hz: fun_complex_nat > fun_fu1553612835185860561ex_nat ).

tff(sy_c_ATP_058Lamm__ia____,type,
    aTP_Lamm_ia: fun_real_real > fun_real_o ).

tff(sy_c_ATP_058Lamm__ib____,type,
    aTP_Lamm_ib: fun_int_real > fun_int_o ).

tff(sy_c_ATP_058Lamm__ic____,type,
    aTP_Lamm_ic: fun_complex_real > fun_complex_o ).

tff(sy_c_ATP_058Lamm__id____,type,
    aTP_Lamm_id: fun_real_rat > fun_real_o ).

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

tff(sy_c_ATP_058Lamm__if____,type,
    aTP_Lamm_if: fun_complex_rat > fun_complex_o ).

tff(sy_c_ATP_058Lamm__ig____,type,
    aTP_Lamm_ig: fun_real_nat > fun_real_o ).

tff(sy_c_ATP_058Lamm__ih____,type,
    aTP_Lamm_ih: fun_int_nat > fun_int_o ).

tff(sy_c_ATP_058Lamm__ii____,type,
    aTP_Lamm_ii: fun_complex_nat > fun_complex_o ).

tff(sy_c_ATP_058Lamm__ij____,type,
    aTP_Lamm_ij: fun_real_int > fun_real_o ).

tff(sy_c_ATP_058Lamm__ik____,type,
    aTP_Lamm_ik: fun_re6978948838062957830nteger > fun_fu8643405727515436425nteger ).

tff(sy_c_ATP_058Lamm__il____,type,
    aTP_Lamm_il: fun_nat_Code_integer > fun_fu2499394402704830801nteger ).

tff(sy_c_ATP_058Lamm__im____,type,
    aTP_Lamm_im: fun_int_Code_integer > fun_fu3628599891403802633nteger ).

tff(sy_c_ATP_058Lamm__in____,type,
    aTP_Lamm_in: fun_real_real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__io____,type,
    aTP_Lamm_io: fun_int_real > fun_fu121957152452960295t_real ).

tff(sy_c_ATP_058Lamm__ip____,type,
    aTP_Lamm_ip: fun_real_rat > fun_fu2598743018846156927al_rat ).

tff(sy_c_ATP_058Lamm__iq____,type,
    aTP_Lamm_iq: fun_nat_rat > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__ir____,type,
    aTP_Lamm_ir: fun_int_rat > fun_fu1008360527747082367nt_rat ).

tff(sy_c_ATP_058Lamm__is____,type,
    aTP_Lamm_is: fun_real_int > fun_fu3737244727311182119al_int ).

tff(sy_c_ATP_058Lamm__it____,type,
    aTP_Lamm_it: fun_nat_int > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__iu____,type,
    aTP_Lamm_iu: fun_nat_complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__iv____,type,
    aTP_Lamm_iv: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__iw____,type,
    aTP_Lamm_iw: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__ix____,type,
    aTP_Lamm_ix: fun_nat_complex > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__iy____,type,
    aTP_Lamm_iy: fun_nat_rat > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__iz____,type,
    aTP_Lamm_iz: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__ja____,type,
    aTP_Lamm_ja: fun_Pr2602117396471556196at_nat > fun_fu7732904369943091395at_nat ).

tff(sy_c_ATP_058Lamm__jb____,type,
    aTP_Lamm_jb: fun_real_nat > fun_fu3214020956574444655al_nat ).

tff(sy_c_ATP_058Lamm__jc____,type,
    aTP_Lamm_jc: fun_set_nat_nat > fun_fu1590586450949517859at_nat ).

tff(sy_c_ATP_058Lamm__jd____,type,
    aTP_Lamm_jd: fun_int_nat > fun_fu8905809029841901679nt_nat ).

tff(sy_c_ATP_058Lamm__je____,type,
    aTP_Lamm_je: fun_nat_nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__jf____,type,
    aTP_Lamm_jf: fun_nat_nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__jg____,type,
    aTP_Lamm_jg: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__jh____,type,
    aTP_Lamm_jh: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ji____,type,
    aTP_Lamm_ji: fun_nat_real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__jj____,type,
    aTP_Lamm_jj: complex > fun_na1142814060743658952omplex ).

tff(sy_c_ATP_058Lamm__jk____,type,
    aTP_Lamm_jk: rat > fun_nat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__jl____,type,
    aTP_Lamm_jl: int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__jm____,type,
    aTP_Lamm_jm: real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__jn____,type,
    aTP_Lamm_jn: fun_nat_nat > fun_na3244735409550470327at_nat ).

tff(sy_c_ATP_058Lamm__jo____,type,
    aTP_Lamm_jo: fun_nat_real > fun_na7718645245721260691t_real ).

tff(sy_c_ATP_058Lamm__jp____,type,
    aTP_Lamm_jp: complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__jq____,type,
    aTP_Lamm_jq: rat > fun_rat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__jr____,type,
    aTP_Lamm_jr: extended_enat > fun_Ex2441982661509597004d_enat ).

tff(sy_c_ATP_058Lamm__js____,type,
    aTP_Lamm_js: int > fun_int_fun_nat_int ).

tff(sy_c_ATP_058Lamm__jt____,type,
    aTP_Lamm_jt: code_integer > fun_Co6226381647958292740nteger ).

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

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

tff(sy_c_ATP_058Lamm__jw____,type,
    aTP_Lamm_jw: fun_nat_rat > fun_nat_rat ).

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

tff(sy_c_ATP_058Lamm__jy____,type,
    aTP_Lamm_jy: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__jz____,type,
    aTP_Lamm_jz: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__ka____,type,
    aTP_Lamm_ka: fun_nat_real > fun_nat_real ).

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

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

tff(sy_c_ATP_058Lamm__kd____,type,
    aTP_Lamm_kd: fun_nat_nat > fun_nat_fun_nat_nat ).

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

tff(sy_c_ATP_058Lamm__kf____,type,
    aTP_Lamm_kf: fun_nat_rat > fun_nat_rat ).

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

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

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

tff(sy_c_ATP_058Lamm__kj____,type,
    aTP_Lamm_kj: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__kk____,type,
    aTP_Lamm_kk: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__kl____,type,
    aTP_Lamm_kl: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__km____,type,
    aTP_Lamm_km: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__kn____,type,
    aTP_Lamm_kn: fun_nat_nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ko____,type,
    aTP_Lamm_ko: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__kp____,type,
    aTP_Lamm_kp: complex > fun_na1591161823130732966omplex ).

tff(sy_c_ATP_058Lamm__kq____,type,
    aTP_Lamm_kq: rat > fun_nat_fun_rat_rat ).

tff(sy_c_ATP_058Lamm__kr____,type,
    aTP_Lamm_kr: real > fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__ks____,type,
    aTP_Lamm_ks: int > fun_nat_fun_int_int ).

tff(sy_c_ATP_058Lamm__kt____,type,
    aTP_Lamm_kt: code_integer > fun_na7078387940644931716nteger ).

tff(sy_c_ATP_058Lamm__ku____,type,
    aTP_Lamm_ku: nat > fun_nat_fun_nat_nat ).

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

tff(sy_c_ATP_058Lamm__kw____,type,
    aTP_Lamm_kw: fun_na1142814060743658952omplex ).

tff(sy_c_ATP_058Lamm__kx____,type,
    aTP_Lamm_kx: fun_nat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__ky____,type,
    aTP_Lamm_ky: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__kz____,type,
    aTP_Lamm_kz: fun_na7542918316964655096d_enat ).

tff(sy_c_ATP_058Lamm__la____,type,
    aTP_Lamm_la: fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__lb____,type,
    aTP_Lamm_lb: fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__lc____,type,
    aTP_Lamm_lc: fun_na8135612474337863287nteger ).

tff(sy_c_ATP_058Lamm__ld____,type,
    aTP_Lamm_ld: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__le____,type,
    aTP_Lamm_le: complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__lf____,type,
    aTP_Lamm_lf: rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__lg____,type,
    aTP_Lamm_lg: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__lh____,type,
    aTP_Lamm_lh: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__li____,type,
    aTP_Lamm_li: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__lj____,type,
    aTP_Lamm_lj: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__lk____,type,
    aTP_Lamm_lk: fun_nat_int ).

tff(sy_c_ATP_058Lamm__ll____,type,
    aTP_Lamm_ll: fun_int_int ).

tff(sy_c_ATP_058Lamm__lm____,type,
    aTP_Lamm_lm: real > fun_fu8384527901749275179omplex ).

tff(sy_c_ATP_058Lamm__ln____,type,
    aTP_Lamm_ln: nat > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__lo____,type,
    aTP_Lamm_lo: int > fun_fu3613808716698379563omplex ).

tff(sy_c_ATP_058Lamm__lp____,type,
    aTP_Lamm_lp: complex > fun_fu5956788528830221615omplex ).

tff(sy_c_ATP_058Lamm__lq____,type,
    aTP_Lamm_lq: real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__lr____,type,
    aTP_Lamm_lr: nat > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__ls____,type,
    aTP_Lamm_ls: int > fun_fu121957152452960295t_real ).

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

tff(sy_c_ATP_058Lamm__lu____,type,
    aTP_Lamm_lu: real > fun_fu2598743018846156927al_rat ).

tff(sy_c_ATP_058Lamm__lv____,type,
    aTP_Lamm_lv: nat > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__lw____,type,
    aTP_Lamm_lw: real > fun_fu8384527901749275179omplex ).

tff(sy_c_ATP_058Lamm__lx____,type,
    aTP_Lamm_lx: nat > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__ly____,type,
    aTP_Lamm_ly: int > fun_fu3613808716698379563omplex ).

tff(sy_c_ATP_058Lamm__lz____,type,
    aTP_Lamm_lz: complex > fun_fu5956788528830221615omplex ).

tff(sy_c_ATP_058Lamm__ma____,type,
    aTP_Lamm_ma: real > fun_fu3077872570260440615l_real ).

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

tff(sy_c_ATP_058Lamm__mc____,type,
    aTP_Lamm_mc: int > fun_fu121957152452960295t_real ).

tff(sy_c_ATP_058Lamm__md____,type,
    aTP_Lamm_md: complex > fun_fu7714627026455246123x_real ).

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

tff(sy_c_ATP_058Lamm__mf____,type,
    aTP_Lamm_mf: nat > fun_fu8328862064278946247at_rat ).

tff(sy_c_ATP_058Lamm__mg____,type,
    aTP_Lamm_mg: set_nat > fun_fu8695033588447641837al_nat ).

tff(sy_c_ATP_058Lamm__mh____,type,
    aTP_Lamm_mh: set_real > fun_fu5553606195750802321at_nat ).

tff(sy_c_ATP_058Lamm__mi____,type,
    aTP_Lamm_mi: set_nat > fun_fu6398909621167755501nt_nat ).

tff(sy_c_ATP_058Lamm__mj____,type,
    aTP_Lamm_mj: set_int > fun_fu3180206027003385489at_nat ).

tff(sy_c_ATP_058Lamm__mk____,type,
    aTP_Lamm_mk: set_nat > fun_fu8020931516743756395ex_nat ).

tff(sy_c_ATP_058Lamm__ml____,type,
    aTP_Lamm_ml: set_complex > fun_fu3351286688164852109at_nat ).

tff(sy_c_ATP_058Lamm__mm____,type,
    aTP_Lamm_mm: set_nat > fun_fu3562227594288721901al_int ).

tff(sy_c_ATP_058Lamm__mn____,type,
    aTP_Lamm_mn: fun_real_fun_nat_int > fun_nat_fun_real_int ).

tff(sy_c_ATP_058Lamm__mo____,type,
    aTP_Lamm_mo: set_real > fun_fu3518430294936654993at_int ).

tff(sy_c_ATP_058Lamm__mp____,type,
    aTP_Lamm_mp: set_nat > fun_fu2556788080963300715ex_int ).

tff(sy_c_ATP_058Lamm__mq____,type,
    aTP_Lamm_mq: fun_co7370231399432144932at_int > fun_na5967208032043182372ex_int ).

tff(sy_c_ATP_058Lamm__mr____,type,
    aTP_Lamm_mr: set_complex > fun_fu3636980702898289805at_int ).

tff(sy_c_ATP_058Lamm__ms____,type,
    aTP_Lamm_ms: set_int > fun_fu8404755328348021045al_int ).

tff(sy_c_ATP_058Lamm__mt____,type,
    aTP_Lamm_mt: set_real > fun_fu6324116013759063093nt_int ).

tff(sy_c_ATP_058Lamm__mu____,type,
    aTP_Lamm_mu: set_int > fun_fu8001260946965626035ex_int ).

tff(sy_c_ATP_058Lamm__mv____,type,
    aTP_Lamm_mv: set_complex > fun_fu7424888833511495217nt_int ).

tff(sy_c_ATP_058Lamm__mw____,type,
    aTP_Lamm_mw: set_real > fun_fu8783008166737851587real_o ).

tff(sy_c_ATP_058Lamm__mx____,type,
    aTP_Lamm_mx: set_real > fun_fu8018396694123807121at_nat ).

tff(sy_c_ATP_058Lamm__my____,type,
    aTP_Lamm_my: fun_nat_fun_real_nat > fun_real_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__mz____,type,
    aTP_Lamm_mz: set_nat > fun_fu91202020541382083_nat_o ).

tff(sy_c_ATP_058Lamm__na____,type,
    aTP_Lamm_na: set_nat > fun_fu1169599456156577517al_nat ).

tff(sy_c_ATP_058Lamm__nb____,type,
    aTP_Lamm_nb: set_int > fun_fu9148247276982919825at_nat ).

tff(sy_c_ATP_058Lamm__nc____,type,
    aTP_Lamm_nc: fun_nat_fun_int_nat > fun_int_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__nd____,type,
    aTP_Lamm_nd: set_nat > fun_fu3143578834292514029nt_nat ).

tff(sy_c_ATP_058Lamm__ne____,type,
    aTP_Lamm_ne: set_complex > fun_fu7903722123778649997at_nat ).

tff(sy_c_ATP_058Lamm__nf____,type,
    aTP_Lamm_nf: fun_na8457291176974909128ex_nat > fun_co636942507509095880at_nat ).

tff(sy_c_ATP_058Lamm__ng____,type,
    aTP_Lamm_ng: set_nat > fun_fu8082241605084145771ex_nat ).

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

tff(sy_c_ATP_058Lamm__ni____,type,
    aTP_Lamm_ni: fun_real_complex > fun_fu3532930458344041007omplex ).

tff(sy_c_ATP_058Lamm__nj____,type,
    aTP_Lamm_nj: fun_nat_complex > fun_fu4302481349826706223omplex ).

tff(sy_c_ATP_058Lamm__nk____,type,
    aTP_Lamm_nk: fun_int_complex > fun_fu3558165918884697647omplex ).

tff(sy_c_ATP_058Lamm__nl____,type,
    aTP_Lamm_nl: fun_complex_complex > fun_fu3719809307426920623omplex ).

tff(sy_c_ATP_058Lamm__nm____,type,
    aTP_Lamm_nm: fun_real_real > fun_fu4478067837645853357l_real ).

tff(sy_c_ATP_058Lamm__nn____,type,
    aTP_Lamm_nn: fun_nat_real > fun_fu4002475321494201261t_real ).

tff(sy_c_ATP_058Lamm__no____,type,
    aTP_Lamm_no: fun_int_real > fun_fu7030473000426307245t_real ).

tff(sy_c_ATP_058Lamm__np____,type,
    aTP_Lamm_np: fun_complex_real > fun_fu3043978019632300333x_real ).

tff(sy_c_ATP_058Lamm__nq____,type,
    aTP_Lamm_nq: fun_real_rat > fun_fu5551332503302594393al_rat ).

tff(sy_c_ATP_058Lamm__nr____,type,
    aTP_Lamm_nr: fun_nat_rat > fun_fu3162257917499486297at_rat ).

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

tff(sy_c_ATP_058Lamm__nt____,type,
    aTP_Lamm_nt: fun_real_complex > fun_real_o ).

tff(sy_c_ATP_058Lamm__nu____,type,
    aTP_Lamm_nu: fun_int_complex > fun_int_o ).

tff(sy_c_ATP_058Lamm__nv____,type,
    aTP_Lamm_nv: fun_complex_complex > fun_complex_o ).

tff(sy_c_ATP_058Lamm__nw____,type,
    aTP_Lamm_nw: fun_real_real > fun_real_o ).

tff(sy_c_ATP_058Lamm__nx____,type,
    aTP_Lamm_nx: fun_int_real > fun_int_o ).

tff(sy_c_ATP_058Lamm__ny____,type,
    aTP_Lamm_ny: fun_complex_real > fun_complex_o ).

tff(sy_c_ATP_058Lamm__nz____,type,
    aTP_Lamm_nz: fun_real_rat > fun_real_o ).

tff(sy_c_ATP_058Lamm__oa____,type,
    aTP_Lamm_oa: fun_int_rat > fun_int_o ).

tff(sy_c_ATP_058Lamm__ob____,type,
    aTP_Lamm_ob: fun_complex_rat > fun_complex_o ).

tff(sy_c_ATP_058Lamm__oc____,type,
    aTP_Lamm_oc: fun_real_nat > fun_real_o ).

tff(sy_c_ATP_058Lamm__od____,type,
    aTP_Lamm_od: fun_int_real > fun_int_real ).

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

tff(sy_c_ATP_058Lamm__of____,type,
    aTP_Lamm_of: fun_complex_complex > fun_complex_complex ).

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

tff(sy_c_ATP_058Lamm__oh____,type,
    aTP_Lamm_oh: fun_nat_int > fun_na7654559027617327379at_int ).

tff(sy_c_ATP_058Lamm__oi____,type,
    aTP_Lamm_oi: fun_nat_complex > fun_na1591161823130732966omplex ).

tff(sy_c_ATP_058Lamm__oj____,type,
    aTP_Lamm_oj: fun_nat_real > fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__ok____,type,
    aTP_Lamm_ok: fun_nat_rat > fun_nat_fun_rat_rat ).

tff(sy_c_ATP_058Lamm__ol____,type,
    aTP_Lamm_ol: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__om____,type,
    aTP_Lamm_om: fun_nat_int > fun_nat_fun_int_int ).

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

tff(sy_c_ATP_058Lamm__oo____,type,
    aTP_Lamm_oo: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__op____,type,
    aTP_Lamm_op: code_integer > fun_nat_Code_integer ).

tff(sy_c_ATP_058Lamm__oq____,type,
    aTP_Lamm_oq: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__or____,type,
    aTP_Lamm_or: int > fun_nat_int ).

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

tff(sy_c_ATP_058Lamm__ot____,type,
    aTP_Lamm_ot: real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__ou____,type,
    aTP_Lamm_ou: code_integer > fun_na8135612474337863287nteger ).

tff(sy_c_ATP_058Lamm__ov____,type,
    aTP_Lamm_ov: nat > fun_nat_fun_nat_nat ).

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

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

tff(sy_c_ATP_058Lamm__oy____,type,
    aTP_Lamm_oy: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__oz____,type,
    aTP_Lamm_oz: fun_nat_nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__pa____,type,
    aTP_Lamm_pa: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__pb____,type,
    aTP_Lamm_pb: fun_nat_int ).

tff(sy_c_ATP_058Lamm__pc____,type,
    aTP_Lamm_pc: fun_Pr672294278902479040t_real > fun_Pr672294278902479040t_real ).

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

tff(sy_c_ATP_058Lamm__pe____,type,
    aTP_Lamm_pe: fun_set_nat_real > fun_set_nat_real ).

tff(sy_c_ATP_058Lamm__pf____,type,
    aTP_Lamm_pf: fun_int_real > fun_int_real ).

tff(sy_c_ATP_058Lamm__pg____,type,
    aTP_Lamm_pg: fun_complex_real > fun_complex_real ).

tff(sy_c_ATP_058Lamm__ph____,type,
    aTP_Lamm_ph: fun_nat_real > fun_nat_real ).

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

tff(sy_c_ATP_058Lamm__pj____,type,
    aTP_Lamm_pj: fun_complex_complex > fun_complex_real ).

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

tff(sy_c_ATP_058Lamm__pl____,type,
    aTP_Lamm_pl: fun_Pr672294278902479040t_real > fun_fu4217534345154246979t_real ).

tff(sy_c_ATP_058Lamm__pm____,type,
    aTP_Lamm_pm: fun_real_real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__pn____,type,
    aTP_Lamm_pn: fun_set_nat_real > fun_fu5354524256550068955t_real ).

tff(sy_c_ATP_058Lamm__po____,type,
    aTP_Lamm_po: fun_int_real > fun_fu121957152452960295t_real ).

tff(sy_c_ATP_058Lamm__pp____,type,
    aTP_Lamm_pp: fun_Pr4050773816208733250omplex > fun_fu7180905015247952065t_real ).

tff(sy_c_ATP_058Lamm__pq____,type,
    aTP_Lamm_pq: fun_real_complex > fun_fu6344110373156359337l_real ).

tff(sy_c_ATP_058Lamm__pr____,type,
    aTP_Lamm_pr: fun_set_nat_complex > fun_fu1617006900395023197t_real ).

tff(sy_c_ATP_058Lamm__ps____,type,
    aTP_Lamm_ps: fun_int_complex > fun_fu3594690983434520489t_real ).

tff(sy_c_ATP_058Lamm__pt____,type,
    aTP_Lamm_pt: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__pu____,type,
    aTP_Lamm_pu: fun_nat_complex > fun_fu7631928937237476593t_real ).

tff(sy_c_ATP_058Lamm__pv____,type,
    aTP_Lamm_pv: fun_nat_real ).

tff(sy_c_ATP_058Lamm__pw____,type,
    aTP_Lamm_pw: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__px____,type,
    aTP_Lamm_px: complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__py____,type,
    aTP_Lamm_py: rat > fun_ra5513060666784058567at_rat ).

tff(sy_c_ATP_058Lamm__pz____,type,
    aTP_Lamm_pz: rat > fun_ra5513060666784058567at_rat ).

tff(sy_c_ATP_058Lamm__qa____,type,
    aTP_Lamm_qa: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__qb____,type,
    aTP_Lamm_qb: real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__qc____,type,
    aTP_Lamm_qc: nat > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__qd____,type,
    aTP_Lamm_qd: nat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__qe____,type,
    aTP_Lamm_qe: fun_nat_fun_nat_int ).

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

tff(sy_c_ATP_058Lamm__qg____,type,
    aTP_Lamm_qg: fun_nat_fun_nat_real ).

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

tff(sy_c_ATP_058Lamm__qi____,type,
    aTP_Lamm_qi: nat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__qj____,type,
    aTP_Lamm_qj: fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__qk____,type,
    aTP_Lamm_qk: nat > fun_nat_Code_integer ).

tff(sy_c_ATP_058Lamm__ql____,type,
    aTP_Lamm_ql: fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__qm____,type,
    aTP_Lamm_qm: nat > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__qn____,type,
    aTP_Lamm_qn: nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__qo____,type,
    aTP_Lamm_qo: nat > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__qp____,type,
    aTP_Lamm_qp: real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__qq____,type,
    aTP_Lamm_qq: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__qr____,type,
    aTP_Lamm_qr: set_nat > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__qs____,type,
    aTP_Lamm_qs: set_nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__qt____,type,
    aTP_Lamm_qt: set_nat > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__qu____,type,
    aTP_Lamm_qu: fun_nat_o > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__qv____,type,
    aTP_Lamm_qv: fun_nat_o > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__qw____,type,
    aTP_Lamm_qw: fun_nat_o > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__qx____,type,
    aTP_Lamm_qx: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__qy____,type,
    aTP_Lamm_qy: fun_se2170385732985038782_nat_o ).

tff(sy_c_ATP_058Lamm__qz____,type,
    aTP_Lamm_qz: fun_rat_fun_rat_o ).

tff(sy_c_ATP_058Lamm__ra____,type,
    aTP_Lamm_ra: fun_num_fun_num_o ).

tff(sy_c_ATP_058Lamm__rb____,type,
    aTP_Lamm_rb: fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__rc____,type,
    aTP_Lamm_rc: fun_real_fun_real_o ).

tff(sy_c_ATP_058Lamm__rd____,type,
    aTP_Lamm_rd: fun_real_fun_real_o ).

tff(sy_c_ATP_058Lamm__re____,type,
    aTP_Lamm_re: fun_rat_fun_rat_o ).

tff(sy_c_ATP_058Lamm__rf____,type,
    aTP_Lamm_rf: fun_num_fun_num_o ).

tff(sy_c_ATP_058Lamm__rg____,type,
    aTP_Lamm_rg: fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__rh____,type,
    aTP_Lamm_rh: fun_nat_fun_nat_nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ri____,type,
    aTP_Lamm_ri: fun_nat_fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__rj____,type,
    aTP_Lamm_rj: fun_nat_fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__rk____,type,
    aTP_Lamm_rk: fun_nat_fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__rl____,type,
    aTP_Lamm_rl: fun_nat_fun_nat_real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__rm____,type,
    aTP_Lamm_rm: fun_nat_fun_nat_real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__rn____,type,
    aTP_Lamm_rn: fun_nat_fun_nat_nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ro____,type,
    aTP_Lamm_ro: fun_nat_fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__rp____,type,
    aTP_Lamm_rp: fun_nat_fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__rq____,type,
    aTP_Lamm_rq: fun_nat_fun_nat_int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__rr____,type,
    aTP_Lamm_rr: fun_nat_fun_nat_int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__rs____,type,
    aTP_Lamm_rs: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__rt____,type,
    aTP_Lamm_rt: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ru____,type,
    aTP_Lamm_ru: fun_nat_complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__rv____,type,
    aTP_Lamm_rv: fun_nat_complex > fun_co8378349014452122532t_real ).

tff(sy_c_ATP_058Lamm__rw____,type,
    aTP_Lamm_rw: fun_nat_rat > fun_rat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__rx____,type,
    aTP_Lamm_rx: fun_nat_int > fun_int_fun_nat_int ).

tff(sy_c_ATP_058Lamm__ry____,type,
    aTP_Lamm_ry: nat > fun_nat_fun_nat_o ).

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

tff(sy_c_ATP_058Lamm__sa____,type,
    aTP_Lamm_sa: fun_nat_fun_nat_nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__sb____,type,
    aTP_Lamm_sb: fun_nat_fun_nat_real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__sc____,type,
    aTP_Lamm_sc: fun_nat_fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__sd____,type,
    aTP_Lamm_sd: fun_nat_fun_nat_nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__se____,type,
    aTP_Lamm_se: fun_nat_fun_nat_int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__sf____,type,
    aTP_Lamm_sf: fun_nat_fun_nat_int > fun_nat_int ).

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

tff(sy_c_ATP_058Lamm__sh____,type,
    aTP_Lamm_sh: fun_nat_complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__si____,type,
    aTP_Lamm_si: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__sj____,type,
    aTP_Lamm_sj: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__sk____,type,
    aTP_Lamm_sk: fun_nat_real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__sl____,type,
    aTP_Lamm_sl: fun_nat_int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__sm____,type,
    aTP_Lamm_sm: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__sn____,type,
    aTP_Lamm_sn: fun_nat_complex > fun_co3194742168270691230omplex ).

tff(sy_c_ATP_058Lamm__so____,type,
    aTP_Lamm_so: nat > fun_fu1584491925252458057omplex ).

tff(sy_c_ATP_058Lamm__sp____,type,
    aTP_Lamm_sp: fun_nat_rat > fun_ra2740371568465858076at_rat ).

tff(sy_c_ATP_058Lamm__sq____,type,
    aTP_Lamm_sq: nat > fun_fu5304001865264412401at_rat ).

tff(sy_c_ATP_058Lamm__sr____,type,
    aTP_Lamm_sr: fun_nat_int > fun_in281396392327853336at_int ).

tff(sy_c_ATP_058Lamm__ss____,type,
    aTP_Lamm_ss: nat > fun_fu4902120768161409089at_int ).

tff(sy_c_ATP_058Lamm__st____,type,
    aTP_Lamm_st: fun_nat_real > fun_re7819166435457029784t_real ).

tff(sy_c_ATP_058Lamm__su____,type,
    aTP_Lamm_su: nat > fun_fu6810213084838398785t_real ).

tff(sy_c_ATP_058Lamm__sv____,type,
    aTP_Lamm_sv: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__sw____,type,
    aTP_Lamm_sw: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__sx____,type,
    aTP_Lamm_sx: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__sy____,type,
    aTP_Lamm_sy: fun_nat_complex > fun_co6883864892131739507omplex ).

tff(sy_c_ATP_058Lamm__sz____,type,
    aTP_Lamm_sz: nat > fun_fu1584491925252458057omplex ).

tff(sy_c_ATP_058Lamm__ta____,type,
    aTP_Lamm_ta: fun_nat_rat > fun_ra5513060666784058567at_rat ).

tff(sy_c_ATP_058Lamm__tb____,type,
    aTP_Lamm_tb: nat > fun_fu5304001865264412401at_rat ).

tff(sy_c_ATP_058Lamm__tc____,type,
    aTP_Lamm_tc: fun_nat_int > fun_in6793278964266839919at_int ).

tff(sy_c_ATP_058Lamm__td____,type,
    aTP_Lamm_td: nat > fun_fu4902120768161409089at_int ).

tff(sy_c_ATP_058Lamm__te____,type,
    aTP_Lamm_te: fun_nat_real > fun_re4756235295228751087t_real ).

tff(sy_c_ATP_058Lamm__tf____,type,
    aTP_Lamm_tf: nat > fun_fu6810213084838398785t_real ).

tff(sy_c_ATP_058Lamm__tg____,type,
    aTP_Lamm_tg: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__th____,type,
    aTP_Lamm_th: fun_nat_fun_nat_nat ).

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

tff(sy_c_ATP_058Lamm__tj____,type,
    aTP_Lamm_tj: nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__tk____,type,
    aTP_Lamm_tk: fun_nat_complex > fun_na1142814060743658952omplex ).

tff(sy_c_ATP_058Lamm__tl____,type,
    aTP_Lamm_tl: real > fun_fu7631928937237476593t_real ).

tff(sy_c_ATP_058Lamm__tm____,type,
    aTP_Lamm_tm: fun_nat_complex > fun_na1554462459066888140plex_o ).

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

tff(sy_c_ATP_058Lamm__to____,type,
    aTP_Lamm_to: complex > fun_na6632006986671646835omplex ).

tff(sy_c_ATP_058Lamm__tp____,type,
    aTP_Lamm_tp: rat > fun_na1148497095348402375at_rat ).

tff(sy_c_ATP_058Lamm__tq____,type,
    aTP_Lamm_tq: int > fun_na2382631619178260847at_int ).

tff(sy_c_ATP_058Lamm__tr____,type,
    aTP_Lamm_tr: real > fun_na5225841814499866095t_real ).

tff(sy_c_ATP_058Lamm__ts____,type,
    aTP_Lamm_ts: complex > fun_na6632006986671646835omplex ).

tff(sy_c_ATP_058Lamm__tt____,type,
    aTP_Lamm_tt: rat > fun_na1148497095348402375at_rat ).

tff(sy_c_ATP_058Lamm__tu____,type,
    aTP_Lamm_tu: int > fun_na2382631619178260847at_int ).

tff(sy_c_ATP_058Lamm__tv____,type,
    aTP_Lamm_tv: real > fun_na5225841814499866095t_real ).

tff(sy_c_ATP_058Lamm__tw____,type,
    aTP_Lamm_tw: fun_nat_complex > fun_fu4528473944208963392omplex ).

tff(sy_c_ATP_058Lamm__tx____,type,
    aTP_Lamm_tx: fun_nat_complex > fun_fu6652553491520710942omplex ).

tff(sy_c_ATP_058Lamm__ty____,type,
    aTP_Lamm_ty: fun_nat_rat > fun_fu6228432884689642900at_rat ).

tff(sy_c_ATP_058Lamm__tz____,type,
    aTP_Lamm_tz: fun_nat_rat > fun_fu4710970720072501148at_rat ).

tff(sy_c_ATP_058Lamm__ua____,type,
    aTP_Lamm_ua: fun_nat_int > fun_fu1567874110255095356at_int ).

tff(sy_c_ATP_058Lamm__ub____,type,
    aTP_Lamm_ub: fun_nat_int > fun_fu5519318738670804632at_int ).

tff(sy_c_ATP_058Lamm__uc____,type,
    aTP_Lamm_uc: fun_nat_real > fun_fu163975356400620604t_real ).

tff(sy_c_ATP_058Lamm__ud____,type,
    aTP_Lamm_ud: fun_nat_real > fun_fu8064956510386623256t_real ).

tff(sy_c_ATP_058Lamm__ue____,type,
    aTP_Lamm_ue: complex > fun_na1142814060743658952omplex ).

tff(sy_c_ATP_058Lamm__uf____,type,
    aTP_Lamm_uf: rat > fun_nat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__ug____,type,
    aTP_Lamm_ug: int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__uh____,type,
    aTP_Lamm_uh: real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__ui____,type,
    aTP_Lamm_ui: real > fun_fu4063443379060642216t_real ).

tff(sy_c_ATP_058Lamm__uj____,type,
    aTP_Lamm_uj: real > fun_fu5589980185390252312t_real ).

tff(sy_c_ATP_058Lamm__uk____,type,
    aTP_Lamm_uk: real > fun_fu4806068833880117316t_real ).

tff(sy_c_ATP_058Lamm__ul____,type,
    aTP_Lamm_ul: real > fun_fu2115079063411676732t_real ).

tff(sy_c_ATP_058Lamm__um____,type,
    aTP_Lamm_um: real > fun_fu4540635375566031512t_real ).

tff(sy_c_ATP_058Lamm__un____,type,
    aTP_Lamm_un: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__uo____,type,
    aTP_Lamm_uo: fun_nat_nat > fun_fu4413992040155564420at_nat ).

tff(sy_c_ATP_058Lamm__up____,type,
    aTP_Lamm_up: fun_nat_nat > fun_fu4413992040155564420at_nat ).

tff(sy_c_ATP_058Lamm__uq____,type,
    aTP_Lamm_uq: real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__ur____,type,
    aTP_Lamm_ur: nat > fun_fu6579618355643354601at_rat ).

tff(sy_c_ATP_058Lamm__us____,type,
    aTP_Lamm_us: nat > fun_fu6579618355643354601at_rat ).

tff(sy_c_ATP_058Lamm__ut____,type,
    aTP_Lamm_ut: nat > fun_fu960360787657196261at_int ).

tff(sy_c_ATP_058Lamm__uu____,type,
    aTP_Lamm_uu: nat > fun_fu960360787657196261at_int ).

tff(sy_c_ATP_058Lamm__uv____,type,
    aTP_Lamm_uv: nat > fun_fu5935951186182374353at_nat ).

tff(sy_c_ATP_058Lamm__uw____,type,
    aTP_Lamm_uw: nat > fun_fu5935951186182374353at_nat ).

tff(sy_c_ATP_058Lamm__ux____,type,
    aTP_Lamm_ux: nat > fun_fu1620311477532340197t_real ).

tff(sy_c_ATP_058Lamm__uy____,type,
    aTP_Lamm_uy: nat > fun_fu1620311477532340197t_real ).

tff(sy_c_ATP_058Lamm__uz____,type,
    aTP_Lamm_uz: nat > fun_fu3593947265352647275omplex ).

tff(sy_c_ATP_058Lamm__va____,type,
    aTP_Lamm_va: nat > fun_fu3593947265352647275omplex ).

tff(sy_c_ATP_058Lamm__vb____,type,
    aTP_Lamm_vb: nat > fun_fu1620311477532340197t_real ).

tff(sy_c_ATP_058Lamm__vc____,type,
    aTP_Lamm_vc: nat > fun_fu6579618355643354601at_rat ).

tff(sy_c_ATP_058Lamm__vd____,type,
    aTP_Lamm_vd: nat > fun_fu5935951186182374353at_nat ).

tff(sy_c_ATP_058Lamm__ve____,type,
    aTP_Lamm_ve: nat > fun_fu960360787657196261at_int ).

tff(sy_c_ATP_058Lamm__vf____,type,
    aTP_Lamm_vf: fun_nat_real > fun_fu6899955116716589935t_real ).

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

tff(sy_c_ATP_058Lamm__vh____,type,
    aTP_Lamm_vh: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__vi____,type,
    aTP_Lamm_vi: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__vj____,type,
    aTP_Lamm_vj: nat > fun_co6147893885205795665omplex ).

tff(sy_c_ATP_058Lamm__vk____,type,
    aTP_Lamm_vk: nat > fun_in1521351555827773387at_int ).

tff(sy_c_ATP_058Lamm__vl____,type,
    aTP_Lamm_vl: nat > fun_Co8502721031311864414nteger ).

tff(sy_c_ATP_058Lamm__vm____,type,
    aTP_Lamm_vm: nat > fun_ra3995598502166916815at_rat ).

tff(sy_c_ATP_058Lamm__vn____,type,
    aTP_Lamm_vn: nat > fun_re8542384036533109323t_real ).

tff(sy_c_ATP_058Lamm__vo____,type,
    aTP_Lamm_vo: nat > fun_nat_complex ).

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

tff(sy_c_ATP_058Lamm__vq____,type,
    aTP_Lamm_vq: fun_nat_fun_nat_int ).

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

tff(sy_c_ATP_058Lamm__vs____,type,
    aTP_Lamm_vs: fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__vt____,type,
    aTP_Lamm_vt: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__vu____,type,
    aTP_Lamm_vu: nat > fun_nat_complex ).

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

tff(sy_c_ATP_058Lamm__vw____,type,
    aTP_Lamm_vw: fun_nat_fun_nat_int ).

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

tff(sy_c_ATP_058Lamm__vy____,type,
    aTP_Lamm_vy: fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__vz____,type,
    aTP_Lamm_vz: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__wa____,type,
    aTP_Lamm_wa: fun_nat_real ).

tff(sy_c_ATP_058Lamm__wb____,type,
    aTP_Lamm_wb: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__wc____,type,
    aTP_Lamm_wc: complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__wd____,type,
    aTP_Lamm_wd: rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__we____,type,
    aTP_Lamm_we: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__wf____,type,
    aTP_Lamm_wf: complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__wg____,type,
    aTP_Lamm_wg: rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__wh____,type,
    aTP_Lamm_wh: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__wi____,type,
    aTP_Lamm_wi: complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__wj____,type,
    aTP_Lamm_wj: rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__wk____,type,
    aTP_Lamm_wk: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__wl____,type,
    aTP_Lamm_wl: nat > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__wm____,type,
    aTP_Lamm_wm: nat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__wn____,type,
    aTP_Lamm_wn: fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__wo____,type,
    aTP_Lamm_wo: rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__wp____,type,
    aTP_Lamm_wp: code_integer > fun_nat_Code_integer ).

tff(sy_c_ATP_058Lamm__wq____,type,
    aTP_Lamm_wq: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__wr____,type,
    aTP_Lamm_wr: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ws____,type,
    aTP_Lamm_ws: int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__wt____,type,
    aTP_Lamm_wt: nat > fun_co3732688428756473980omplex ).

tff(sy_c_ATP_058Lamm__wu____,type,
    aTP_Lamm_wu: nat > fun_co3732688428756473980omplex ).

tff(sy_c_ATP_058Lamm__wv____,type,
    aTP_Lamm_wv: nat > fun_ra8505079968215247908at_rat ).

tff(sy_c_ATP_058Lamm__ww____,type,
    aTP_Lamm_ww: nat > fun_ra8505079968215247908at_rat ).

tff(sy_c_ATP_058Lamm__wx____,type,
    aTP_Lamm_wx: nat > fun_re6135230694098607860t_real ).

tff(sy_c_ATP_058Lamm__wy____,type,
    aTP_Lamm_wy: nat > fun_re6135230694098607860t_real ).

tff(sy_c_ATP_058Lamm__wz____,type,
    aTP_Lamm_wz: complex > fun_na1591161823130732966omplex ).

tff(sy_c_ATP_058Lamm__xa____,type,
    aTP_Lamm_xa: rat > fun_nat_fun_rat_rat ).

tff(sy_c_ATP_058Lamm__xb____,type,
    aTP_Lamm_xb: real > fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__xc____,type,
    aTP_Lamm_xc: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__xd____,type,
    aTP_Lamm_xd: real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__xe____,type,
    aTP_Lamm_xe: nat > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__xf____,type,
    aTP_Lamm_xf: nat > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__xg____,type,
    aTP_Lamm_xg: nat > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__xh____,type,
    aTP_Lamm_xh: nat > fun_int_fun_nat_int ).

tff(sy_c_ATP_058Lamm__xi____,type,
    aTP_Lamm_xi: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__xj____,type,
    aTP_Lamm_xj: fun_nat_real > fun_se2100927751953696370t_real ).

tff(sy_c_ATP_058Lamm__xk____,type,
    aTP_Lamm_xk: fun_nat_real ).

tff(sy_c_ATP_058Lamm__xl____,type,
    aTP_Lamm_xl: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__xm____,type,
    aTP_Lamm_xm: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__xn____,type,
    aTP_Lamm_xn: int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__xo____,type,
    aTP_Lamm_xo: int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__xp____,type,
    aTP_Lamm_xp: fun_real_fun_real_o ).

tff(sy_c_ATP_058Lamm__xq____,type,
    aTP_Lamm_xq: fun_real_fun_real_o ).

tff(sy_c_ATP_058Lamm__xr____,type,
    aTP_Lamm_xr: fun_real_fun_real_o ).

tff(sy_c_ATP_058Lamm__xs____,type,
    aTP_Lamm_xs: fun_real_o ).

tff(sy_c_ATP_058Lamm__xt____,type,
    aTP_Lamm_xt: fun_real_fun_int_o ).

tff(sy_c_ATP_058Lamm__xu____,type,
    aTP_Lamm_xu: rat > fun_int_o ).

tff(sy_c_ATP_058Lamm__xv____,type,
    aTP_Lamm_xv: fun_nat_nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__xw____,type,
    aTP_Lamm_xw: code_integer > fun_Co3544761417380347684eger_o ).

tff(sy_c_ATP_058Lamm__xx____,type,
    aTP_Lamm_xx: fun_nat_o ).

tff(sy_c_ATP_058Lamm__xy____,type,
    aTP_Lamm_xy: set_nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__xz____,type,
    aTP_Lamm_xz: set_nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__ya____,type,
    aTP_Lamm_ya: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__yb____,type,
    aTP_Lamm_yb: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__yc____,type,
    aTP_Lamm_yc: complex > fun_na1554462459066888140plex_o ).

tff(sy_c_ATP_058Lamm__yd____,type,
    aTP_Lamm_yd: code_integer > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__ye____,type,
    aTP_Lamm_ye: code_integer > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__yf____,type,
    aTP_Lamm_yf: nat > fun_na8563630686737515764_nat_o ).

tff(sy_c_ATP_058Lamm__yg____,type,
    aTP_Lamm_yg: code_integer > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__yh____,type,
    aTP_Lamm_yh: fun_Co3583066429914935172er_nat ).

tff(sy_c_ATP_058Lamm__yi____,type,
    aTP_Lamm_yi: fun_Co1092983284983208416er_int ).

tff(sy_c_ATP_058Lamm__yj____,type,
    aTP_Lamm_yj: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__yk____,type,
    aTP_Lamm_yk: fun_Co8521658056209057742er_num ).

tff(sy_c_ATP_058Lamm__yl____,type,
    aTP_Lamm_yl: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ym____,type,
    aTP_Lamm_ym: int > fun_int_int ).

tff(sy_c_ATP_058Lamm__yn____,type,
    aTP_Lamm_yn: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__yo____,type,
    aTP_Lamm_yo: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__yp____,type,
    aTP_Lamm_yp: fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__yq____,type,
    aTP_Lamm_yq: fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__yr____,type,
    aTP_Lamm_yr: fun_real_real > fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__ys____,type,
    aTP_Lamm_ys: fun_re7681423839302404990l_real ).

tff(sy_c_ATP_058Lamm__yt____,type,
    aTP_Lamm_yt: fun_re4238438383274954274t_real > fun_real_real ).

tff(sy_c_ATP_058Lamm__yu____,type,
    aTP_Lamm_yu: fun_real_real > fun_re7681423839302404990l_real ).

tff(sy_c_ATP_058Lamm__yv____,type,
    aTP_Lamm_yv: fun_real_real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__yw____,type,
    aTP_Lamm_yw: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__yx____,type,
    aTP_Lamm_yx: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__yy____,type,
    aTP_Lamm_yy: fun_nat_real > fun_real_real ).

tff(sy_c_ATP_058Lamm__yz____,type,
    aTP_Lamm_yz: fun_na6724076748692969250l_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__za____,type,
    aTP_Lamm_za: fun_na6724076748692969250l_real > fun_re8542384036533109323t_real ).

tff(sy_c_ATP_058Lamm__zb____,type,
    aTP_Lamm_zb: fun_na6724076748692969250l_real > fun_re8542384036533109323t_real ).

tff(sy_c_ATP_058Lamm__zc____,type,
    aTP_Lamm_zc: fun_na6724076748692969250l_real > fun_na5225841814499866095t_real ).

tff(sy_c_ATP_058Lamm__zd____,type,
    aTP_Lamm_zd: nat > fun_fu3612569900928539882l_real ).

tff(sy_c_ATP_058Lamm__ze____,type,
    aTP_Lamm_ze: fun_na6724076748692969250l_real > fun_na5225841814499866095t_real ).

tff(sy_c_ATP_058Lamm__zf____,type,
    aTP_Lamm_zf: fun_real_real ).

tff(sy_c_ATP_058Lamm__zg____,type,
    aTP_Lamm_zg: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__zh____,type,
    aTP_Lamm_zh: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__zi____,type,
    aTP_Lamm_zi: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__zj____,type,
    aTP_Lamm_zj: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__zk____,type,
    aTP_Lamm_zk: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__zl____,type,
    aTP_Lamm_zl: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__zm____,type,
    aTP_Lamm_zm: fun_nat_real ).

tff(sy_c_ATP_058Lamm__zn____,type,
    aTP_Lamm_zn: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__zo____,type,
    aTP_Lamm_zo: real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__zp____,type,
    aTP_Lamm_zp: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__zq____,type,
    aTP_Lamm_zq: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__zr____,type,
    aTP_Lamm_zr: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__zs____,type,
    aTP_Lamm_zs: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__zt____,type,
    aTP_Lamm_zt: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__zu____,type,
    aTP_Lamm_zu: nat > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__zv____,type,
    aTP_Lamm_zv: fun_nat_o > fun_nat_o ).

tff(sy_c_ATP_058Lamm__zw____,type,
    aTP_Lamm_zw: real > fun_real_fun_real_o ).

tff(sy_c_ATP_058Lamm__zx____,type,
    aTP_Lamm_zx: fun_nat_set_nat ).

tff(sy_c_ATP_058Lamm__zy____,type,
    aTP_Lamm_zy: nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__zz____,type,
    aTP_Lamm_zz: nat > fun_li6914387844523535256_nat_o ).

tff(sy_c_Archimedean__Field_Oceiling_001t__Rat__Orat,type,
    archim2889992004027027881ng_rat: fun_rat_int ).

tff(sy_c_Archimedean__Field_Oceiling_001t__Real__Oreal,type,
    archim7802044766580827645g_real: fun_real_int ).

tff(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor_001t__Rat__Orat,type,
    archim3151403230148437115or_rat: fun_rat_int ).

tff(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor_001t__Real__Oreal,type,
    archim6058952711729229775r_real: fun_real_int ).

tff(sy_c_Archimedean__Field_Ofrac_001t__Rat__Orat,type,
    archimedean_frac_rat: fun_rat_rat ).

tff(sy_c_Archimedean__Field_Ofrac_001t__Real__Oreal,type,
    archim2898591450579166408c_real: fun_real_real ).

tff(sy_c_Archimedean__Field_Oround_001t__Rat__Orat,type,
    archim7778729529865785530nd_rat: fun_rat_int ).

tff(sy_c_Archimedean__Field_Oround_001t__Real__Oreal,type,
    archim8280529875227126926d_real: fun_real_int ).

tff(sy_c_BNF__Cardinal__Order__Relation_OnatLeq,type,
    bNF_Ca8665028551170535155natLeq: set_Pr1261947904930325089at_nat ).

tff(sy_c_BNF__Cardinal__Order__Relation_OnatLess,type,
    bNF_Ca8459412986667044542atLess: set_Pr1261947904930325089at_nat ).

tff(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Code____Numeral__Ointeger_001_Eo_001_Eo,type,
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    bNF_re3379532845092657523nteger: ( fun_in4891666191172872833eger_o * fun_in4891666191172872833eger_o ) > fun_fu726873822431485118eger_o ).

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tff(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001_Eo_001_Eo,type,
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tff(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001t__Nat__Onat_001t__Nat__Onat,type,
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tff(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001t__Num__Onum_001t__Num__Onum,type,
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tff(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001t__fun_It__Nat__Onat_M_Eo_J_001t__fun_It__Nat__Onat_M_Eo_J,type,
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tff(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001t__fun_It__Nat__Onat_Mt__Nat__Onat_J_001t__fun_It__Nat__Onat_Mt__Nat__Onat_J,type,
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tff(sy_c_BNF__Wellorder__Relation_Owo__rel_001t__Nat__Onat,type,
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tff(sy_c_List_Olistrel_001t__Int__Oint_001t__Nat__Onat,type,
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tff(sy_c_List_Olistrel_001t__Int__Oint_001t__VEBT____Definitions__OVEBT,type,
    listre8491537028387690453T_VEBT: set_Pr8044002425091019955T_VEBT > set_Pr5325845658263174057T_VEBT ).

tff(sy_c_List_Olistrel_001t__VEBT____Definitions__OVEBT_001_Eo,type,
    listrel_VEBT_VEBT_o: set_Pr3175402225741728619VEBT_o > set_Pr7508168486584781291list_o ).

tff(sy_c_List_Olistrel_001t__VEBT____Definitions__OVEBT_001t__Int__Oint,type,
    listre5898179758603845167BT_int: set_Pr5066593544530342725BT_int > set_Pr4080907618048478043st_int ).

tff(sy_c_List_Olistrel_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
    listre5900670229112895443BT_nat: set_Pr7556676689462069481BT_nat > set_Pr8894456036836396799st_nat ).

tff(sy_c_List_Olistrel_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
    listre1230615542750757617T_VEBT: set_Pr6192946355708809607T_VEBT > set_Pr1916528119006554503T_VEBT ).

tff(sy_c_List_Omeasures_001t__Code____Numeral__Ointeger,type,
    measur8870801148506250077nteger: list_f8684324736568799034er_nat > set_Pr4811707699266497531nteger ).

tff(sy_c_List_Omeasures_001t__Int__Oint,type,
    measures_int: list_fun_int_nat > set_Pr958786334691620121nt_int ).

tff(sy_c_List_Omeasures_001t__Nat__Onat,type,
    measures_nat: list_fun_nat_nat > set_Pr1261947904930325089at_nat ).

tff(sy_c_List_Omeasures_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    measur2679027848233739777at_nat: list_f8256639991604547178at_nat > set_Pr8693737435421807431at_nat ).

tff(sy_c_List_Onth_001_Eo,type,
    nth_o: list_o > fun_nat_o ).

tff(sy_c_List_Onth_001t__Code____Numeral__Ointeger,type,
    nth_Code_integer: list_Code_integer > fun_nat_Code_integer ).

tff(sy_c_List_Onth_001t__Int__Oint,type,
    nth_int: list_int > fun_nat_int ).

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

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

tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J,type,
    nth_Pr8522763379788166057eger_o: ( list_P8526636022914148096eger_o * nat ) > produc6271795597528267376eger_o ).

tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
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tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__Int__Oint_M_Eo_J,type,
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tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__Int__Oint_Mt__Nat__Onat_J,type,
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tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__Int__Oint_Mt__VEBT____Definitions__OVEBT_J,type,
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tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__Nat__Onat_M_Eo_J,type,
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tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Int__Oint_J,type,
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tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__Nat__Onat_Mt__VEBT____Definitions__OVEBT_J,type,
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tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_M_Eo_J,type,
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tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Int__Oint_J,type,
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tff(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
    nth_Pr1791586995822124652BT_nat: ( list_P7037539587688870467BT_nat * nat ) > produc9072475918466114483BT_nat ).

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

tff(sy_c_List_Onth_001t__Set__Oset_It__Nat__Onat_J,type,
    nth_set_nat: ( list_set_nat * nat ) > set_nat ).

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

tff(sy_c_List_Oproduct_001t__Code____Numeral__Ointeger_001_Eo,type,
    produc3607205314601156340eger_o: ( list_Code_integer * list_o ) > list_P8526636022914148096eger_o ).

tff(sy_c_List_Oproduct_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_List_Oproduct_001t__Int__Oint_001_Eo,type,
    product_int_o: ( list_int * list_o ) > list_P5087981734274514673_int_o ).

tff(sy_c_List_Oproduct_001t__Int__Oint_001t__Int__Oint,type,
    product_int_int: ( list_int * list_int ) > list_P5707943133018811711nt_int ).

tff(sy_c_List_Oproduct_001t__Int__Oint_001t__Nat__Onat,type,
    product_int_nat: ( list_int * list_nat ) > list_P8198026277950538467nt_nat ).

tff(sy_c_List_Oproduct_001t__Int__Oint_001t__VEBT____Definitions__OVEBT,type,
    produc662631939642741121T_VEBT: ( list_int * list_VEBT_VEBT ) > list_P7524865323317820941T_VEBT ).

tff(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001_Eo,type,
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tff(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001t__Int__Oint,type,
    produc7292646706713671643BT_int: ( list_VEBT_VEBT * list_int ) > list_P4547456442757143711BT_int ).

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

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

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

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

tff(sy_c_List_Oreplicate_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    replic4235873036481779905at_nat: ( nat * product_prod_nat_nat ) > list_P6011104703257516679at_nat ).

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

tff(sy_c_List_Oreplicate_001t__Set__Oset_It__Nat__Onat_J,type,
    replicate_set_nat: ( nat * set_nat ) > list_set_nat ).

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

tff(sy_c_List_Orotate1_001_Eo,type,
    rotate1_o: list_o > list_o ).

tff(sy_c_List_Orotate1_001t__Int__Oint,type,
    rotate1_int: list_int > list_int ).

tff(sy_c_List_Orotate1_001t__Nat__Onat,type,
    rotate1_nat: list_nat > list_nat ).

tff(sy_c_List_Orotate1_001t__VEBT____Definitions__OVEBT,type,
    rotate1_VEBT_VEBT: list_VEBT_VEBT > list_VEBT_VEBT ).

tff(sy_c_List_Osorted__wrt_001t__Int__Oint,type,
    sorted_wrt_int: ( fun_int_fun_int_o * list_int ) > $o ).

tff(sy_c_List_Osorted__wrt_001t__Nat__Onat,type,
    sorted_wrt_nat: fun_nat_fun_nat_o > fun_list_nat_o ).

tff(sy_c_List_Otake_001t__Nat__Onat,type,
    take_nat: ( nat * list_nat ) > list_nat ).

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

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

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

tff(sy_c_List_Oupto__rel,type,
    upto_rel: fun_Pr9018117161326028078_int_o ).

tff(sy_c_List_Ozip_001t__Code____Numeral__Ointeger_001_Eo,type,
    zip_Code_integer_o: ( list_Code_integer * list_o ) > list_P8526636022914148096eger_o ).

tff(sy_c_List_Ozip_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
    zip_Co3543743374963494515nteger: ( list_Code_integer * list_Code_integer ) > list_P5578671422887162913nteger ).

tff(sy_c_List_Ozip_001t__Int__Oint_001_Eo,type,
    zip_int_o: ( list_int * list_o ) > list_P5087981734274514673_int_o ).

tff(sy_c_List_Ozip_001t__Int__Oint_001t__Int__Oint,type,
    zip_int_int: ( list_int * list_int ) > list_P5707943133018811711nt_int ).

tff(sy_c_List_Ozip_001t__Int__Oint_001t__Nat__Onat,type,
    zip_int_nat: ( list_int * list_nat ) > list_P8198026277950538467nt_nat ).

tff(sy_c_List_Ozip_001t__Int__Oint_001t__VEBT____Definitions__OVEBT,type,
    zip_int_VEBT_VEBT: ( list_int * list_VEBT_VEBT ) > list_P7524865323317820941T_VEBT ).

tff(sy_c_List_Ozip_001t__VEBT____Definitions__OVEBT_001_Eo,type,
    zip_VEBT_VEBT_o: ( list_VEBT_VEBT * list_o ) > list_P3126845725202233233VEBT_o ).

tff(sy_c_List_Ozip_001t__VEBT____Definitions__OVEBT_001t__Int__Oint,type,
    zip_VEBT_VEBT_int: ( list_VEBT_VEBT * list_int ) > list_P4547456442757143711BT_int ).

tff(sy_c_List_Ozip_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
    zip_VEBT_VEBT_nat: ( list_VEBT_VEBT * list_nat ) > list_P7037539587688870467BT_nat ).

tff(sy_c_List_Ozip_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
    zip_VE537291747668921783T_VEBT: ( list_VEBT_VEBT * list_VEBT_VEBT ) > list_P7413028617227757229T_VEBT ).

tff(sy_c_Nat_OSuc,type,
    suc: fun_nat_nat ).

tff(sy_c_Nat_Ocompow_001t__fun_It__Nat__Onat_Mt__Nat__Onat_J,type,
    compow_fun_nat_nat: ( nat * fun_nat_nat ) > fun_nat_nat ).

tff(sy_c_Nat_Onat_Ocase__nat_001_Eo,type,
    case_nat_o: ( $o * fun_nat_o ) > fun_nat_o ).

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

tff(sy_c_Nat_Onat_Opred,type,
    pred: fun_nat_nat ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Code____Numeral__Ointeger,type,
    semiri4939895301339042750nteger: fun_nat_Code_integer ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Complex__Ocomplex,type,
    semiri8010041392384452111omplex: fun_nat_complex ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Extended____Nat__Oenat,type,
    semiri4216267220026989637d_enat: fun_na8797796813480665199d_enat ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
    semiri1314217659103216013at_int: fun_nat_int ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
    semiri1316708129612266289at_nat: fun_nat_nat ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Rat__Orat,type,
    semiri681578069525770553at_rat: fun_nat_rat ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Real__Oreal,type,
    semiri5074537144036343181t_real: fun_nat_real ).

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

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

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

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

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

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

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Real__Oreal_J,type,
    size_size_list_real: list_real > nat ).

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

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

tff(sy_c_Nat_Osize__class_Osize_001t__Num__Onum,type,
    size_size_num: fun_num_nat ).

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

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

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

tff(sy_c_Nat_Osize__class_Osize_001t__VEBT____Definitions__OVEBT,type,
    size_size_VEBT_VEBT: fun_VEBT_VEBT_nat ).

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

tff(sy_c_Nat__Bijection_Oprod__decode__aux__rel,type,
    nat_pr5047031295181774490ux_rel: fun_Pr2028404973800837934_nat_o ).

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

tff(sy_c_Nat__Bijection_Oset__encode,type,
    nat_set_encode: fun_set_nat_nat ).

tff(sy_c_Nat__Bijection_Otriangle,type,
    nat_triangle: fun_nat_nat ).

tff(sy_c_NthRoot_Oroot,type,
    root: fun_na6724076748692969250l_real ).

tff(sy_c_NthRoot_Osqrt,type,
    sqrt: fun_real_real ).

tff(sy_c_Num_OBitM,type,
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tff(sy_c_Num_Oinc,type,
    inc: fun_num_num ).

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

tff(sy_c_Num_Oneg__numeral__class_Odbl_001t__Complex__Ocomplex,type,
    neg_nu7009210354673126013omplex: fun_complex_complex ).

tff(sy_c_Num_Oneg__numeral__class_Odbl_001t__Int__Oint,type,
    neg_numeral_dbl_int: fun_int_int ).

tff(sy_c_Num_Oneg__numeral__class_Odbl_001t__Rat__Orat,type,
    neg_numeral_dbl_rat: fun_rat_rat ).

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

tff(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Complex__Ocomplex,type,
    neg_nu6511756317524482435omplex: fun_complex_complex ).

tff(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Int__Oint,type,
    neg_nu3811975205180677377ec_int: fun_int_int ).

tff(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Rat__Orat,type,
    neg_nu3179335615603231917ec_rat: fun_rat_rat ).

tff(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Real__Oreal,type,
    neg_nu6075765906172075777c_real: fun_real_real ).

tff(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Complex__Ocomplex,type,
    neg_nu8557863876264182079omplex: fun_complex_complex ).

tff(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Int__Oint,type,
    neg_nu5851722552734809277nc_int: fun_int_int ).

tff(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Rat__Orat,type,
    neg_nu5219082963157363817nc_rat: fun_rat_rat ).

tff(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Real__Oreal,type,
    neg_nu8295874005876285629c_real: fun_real_real ).

tff(sy_c_Num_Oneg__numeral__class_Osub_001t__Code____Numeral__Ointeger,type,
    neg_nu5755505904847501662nteger: ( num * num ) > code_integer ).

tff(sy_c_Num_Oneg__numeral__class_Osub_001t__Int__Oint,type,
    neg_numeral_sub_int: num > fun_num_int ).

tff(sy_c_Num_Onum_OBit0,type,
    bit0: fun_num_num ).

tff(sy_c_Num_Onum_OBit1,type,
    bit1: fun_num_num ).

tff(sy_c_Num_Onum_OOne,type,
    one: num ).

tff(sy_c_Num_Onum_Osize__num,type,
    size_num: fun_num_nat ).

tff(sy_c_Num_Onum__of__nat,type,
    num_of_nat: fun_nat_num ).

tff(sy_c_Num_Onumeral__class_Onumeral_001t__Code____Numeral__Ointeger,type,
    numera6620942414471956472nteger: fun_num_Code_integer ).

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

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

tff(sy_c_Num_Onumeral__class_Onumeral_001t__Int__Oint,type,
    numeral_numeral_int: fun_num_int ).

tff(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat,type,
    numeral_numeral_nat: fun_num_nat ).

tff(sy_c_Num_Onumeral__class_Onumeral_001t__Rat__Orat,type,
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tff(sy_c_Num_Onumeral__class_Onumeral_001t__Real__Oreal,type,
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tff(sy_c_Num_Opow,type,
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tff(sy_c_Num_Opred__numeral,type,
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tff(sy_c_Num_Osqr,type,
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tff(sy_c_Option_Ooption_ONone_001t__Nat__Onat,type,
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tff(sy_c_Option_Ooption_ONone_001t__Num__Onum,type,
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tff(sy_c_Option_Ooption_OSome_001_Eo,type,
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tff(sy_c_Option_Ooption_OSome_001t__Int__Oint,type,
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tff(sy_c_Option_Ooption_OSome_001t__Nat__Onat,type,
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tff(sy_c_Option_Ooption_OSome_001t__Num__Onum,type,
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tff(sy_c_Option_Ooption_Osize__option_001t__Nat__Onat,type,
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tff(sy_c_Option_Ooption_Osize__option_001t__Num__Onum,type,
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tff(sy_c_Option_Ooption_Othe_001t__Nat__Onat,type,
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tff(sy_c_Option_Ooption_Othe_001t__Num__Onum,type,
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tff(sy_c_Order__Relation_OunderS_001t__Nat__Onat,type,
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tff(sy_c_Orderings_Obot__class_Obot_001t__Filter__Ofilter_It__Nat__Onat_J,type,
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tff(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
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tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Complex__Ocomplex_J,type,
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tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
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tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
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tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Num__Onum_J,type,
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tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Real__Oreal_J,type,
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tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
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tff(sy_c_Orderings_Oord__class_OLeast_001t__Nat__Onat,type,
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tff(sy_c_Orderings_Oord__class_OLeast_001t__Real__Oreal,type,
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tff(sy_c_Orderings_Oord__class_Oless_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Orderings_Oord__class_Oless_001t__Filter__Ofilter_It__Nat__Onat_J,type,
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tff(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
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tff(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
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tff(sy_c_Orderings_Oord__class_Oless_001t__Num__Onum,type,
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tff(sy_c_Orderings_Oord__class_Oless_001t__Rat__Orat,type,
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tff(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
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tff(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Int__Oint_J,type,
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tff(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
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tff(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Real__Oreal_J,type,
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tff(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
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tff(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Num__Onum,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Rat__Orat,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_Eo_J,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001_Eo,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Complex__Ocomplex,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Extended____Nat__Oenat,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Int__Oint,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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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,
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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,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Real__Oreal_001t__Real__Oreal_001_Eo,type,
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tff(sy_c_Product__Type_Oprod_Ofst_001t__Int__Oint_001t__Int__Oint,type,
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tff(sy_c_Product__Type_Oprod_Ofst_001t__Nat__Onat_001t__Nat__Onat,type,
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tff(sy_c_Product__Type_Oprod_Osnd_001t__Int__Oint_001t__Int__Oint,type,
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tff(sy_c_Product__Type_Oprod_Osnd_001t__Nat__Onat_001t__Nat__Onat,type,
    product_snd_nat_nat: fun_Pr2602117396471556196at_nat ).

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

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

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

tff(sy_c_Rat_Ofield__char__0__class_ORats_001t__Real__Oreal,type,
    field_5140801741446780682s_real: set_real ).

tff(sy_c_Rat_Ofield__char__0__class_Oof__rat_001t__Real__Oreal,type,
    field_7254667332652039916t_real: fun_rat_real ).

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

tff(sy_c_Rat_Opcr__rat,type,
    pcr_rat: fun_Pr7537129237977221859_rat_o ).

tff(sy_c_Rat_Opositive,type,
    positive: fun_rat_o ).

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

tff(sy_c_Rat_Oratrel,type,
    ratrel: fun_Pr9018117161326028078_int_o ).

tff(sy_c_Real_OReal,type,
    real2: fun_nat_rat > real ).

tff(sy_c_Real_Ocauchy,type,
    cauchy: fun_nat_rat > $o ).

tff(sy_c_Real_Opcr__real,type,
    pcr_real: fun_fu1771228864347509049real_o ).

tff(sy_c_Real_Opositive,type,
    positive2: fun_real_o ).

tff(sy_c_Real_Orealrel,type,
    realrel: fun_fu6714665071556184448_rat_o ).

tff(sy_c_Real_Orep__real,type,
    rep_real: real > fun_nat_rat ).

tff(sy_c_Real_Ovanishes,type,
    vanishes: fun_nat_rat > $o ).

tff(sy_c_Real__Vector__Spaces_Odist__class_Odist_001t__Complex__Ocomplex,type,
    real_V3694042436643373181omplex: complex > fun_complex_real ).

tff(sy_c_Real__Vector__Spaces_Odist__class_Odist_001t__Real__Oreal,type,
    real_V975177566351809787t_real: fun_re7681423839302404990l_real ).

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_Relation_OField_001t__Nat__Onat,type,
    field_nat: set_Pr1261947904930325089at_nat > set_nat ).

tff(sy_c_Rings_Oalgebraic__semidom__class_Ocoprime_001t__Int__Oint,type,
    algebr932160517623751201me_int: fun_int_fun_int_o ).

tff(sy_c_Rings_Oalgebraic__semidom__class_Ocoprime_001t__Nat__Onat,type,
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tff(sy_c_Rings_Odivide__class_Odivide_001t__Code____Numeral__Ointeger,type,
    divide6298287555418463151nteger: fun_Co1981900195452921617nteger ).

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: fun_re7681423839302404990l_real ).

tff(sy_c_Rings_Odvd__class_Odvd_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Rings_Odvd__class_Odvd_001t__Complex__Ocomplex,type,
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tff(sy_c_Rings_Odvd__class_Odvd_001t__Int__Oint,type,
    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: fun_Co1981900195452921617nteger ).

tff(sy_c_Rings_Omodulo__class_Omodulo_001t__Int__Oint,type,
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tff(sy_c_Rings_Omodulo__class_Omodulo_001t__Nat__Onat,type,
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tff(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Complex__Ocomplex,type,
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tff(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Int__Oint,type,
    zero_n2684676970156552555ol_int: fun_o_int ).

tff(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Nat__Onat,type,
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tff(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Rat__Orat,type,
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tff(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Real__Oreal,type,
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tff(sy_c_Series_Osuminf_001t__Complex__Ocomplex,type,
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tff(sy_c_Series_Osuminf_001t__Int__Oint,type,
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tff(sy_c_Series_Osuminf_001t__Nat__Onat,type,
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tff(sy_c_Series_Osuminf_001t__Real__Oreal,type,
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tff(sy_c_Series_Osummable_001t__Complex__Ocomplex,type,
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tff(sy_c_Series_Osummable_001t__Int__Oint,type,
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tff(sy_c_Series_Osummable_001t__Nat__Onat,type,
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tff(sy_c_Series_Osummable_001t__Real__Oreal,type,
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tff(sy_c_Series_Osums_001t__Complex__Ocomplex,type,
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tff(sy_c_Series_Osums_001t__Int__Oint,type,
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tff(sy_c_Series_Osums_001t__Nat__Onat,type,
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tff(sy_c_Series_Osums_001t__Real__Oreal,type,
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tff(sy_c_Set_OCollect_001t__Code____Numeral__Ointeger,type,
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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_I_Eo_J,type,
    collect_list_o: fun_list_o_o > set_list_o ).

tff(sy_c_Set_OCollect_001t__List__Olist_It__Complex__Ocomplex_J,type,
    collect_list_complex: fun_list_complex_o > set_list_complex ).

tff(sy_c_Set_OCollect_001t__List__Olist_It__Int__Oint_J,type,
    collect_list_int: fun_list_int_o > set_list_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__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
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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__Complex__Ocomplex_Mt__Complex__Ocomplex_J,type,
    collec8663557070575231912omplex: fun_Pr7716844367077540446plex_o > set_Pr5085853215250843933omplex ).

tff(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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tff(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
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tff(sy_c_Set_OCollect_001t__Rat__Orat,type,
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tff(sy_c_Set_OCollect_001t__Real__Oreal,type,
    collect_real: fun_real_o > set_real ).

tff(sy_c_Set_OCollect_001t__Set__Oset_It__Complex__Ocomplex_J,type,
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tff(sy_c_Set_OCollect_001t__Set__Oset_It__Int__Oint_J,type,
    collect_set_int: fun_set_int_o > set_set_int ).

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__Int__Oint_001t__Nat__Onat,type,
    image_int_nat: ( fun_int_nat * set_int ) > set_nat ).

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__Real__Oreal,type,
    image_nat_real: ( fun_nat_real * set_nat ) > set_real ).

tff(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
    image_nat_set_nat: ( fun_nat_set_nat * set_nat ) > set_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__Filter__Ofilter_It__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J_J,type,
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tff(sy_c_Set_Oimage_001t__Real__Oreal_001t__Filter__Ofilter_It__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
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tff(sy_c_Set_Oimage_001t__Real__Oreal_001t__Real__Oreal,type,
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tff(sy_c_Set_Oimage_001t__String__Ochar_001t__Nat__Onat,type,
    image_char_nat: ( fun_char_nat * set_char ) > set_nat ).

tff(sy_c_Set_Oinsert_001t__Int__Oint,type,
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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,
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tff(sy_c_Set_Ovimage_001t__Nat__Onat_001t__Nat__Onat,type,
    vimage_nat_nat: ( fun_nat_nat * set_nat ) > set_nat ).

tff(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Complex__Ocomplex,type,
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tff(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Int__Oint,type,
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tff(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Nat__Onat,type,
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tff(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Rat__Orat,type,
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tff(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Real__Oreal,type,
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tff(sy_c_Set__Interval_Ofold__atLeastAtMost__nat__rel_001t__Nat__Onat,type,
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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,
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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,
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tff(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Set__Oset_It__Int__Oint_J,type,
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tff(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Set__Oset_It__Nat__Onat_J,type,
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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,
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tff(sy_c_Set__Interval_Oord__class_OatLeast_001t__Nat__Onat,type,
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tff(sy_c_Set__Interval_Oord__class_OatLeast_001t__Real__Oreal,type,
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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__Int__Oint_J,type,
    set_or58775011639299419et_int: set_int > set_set_int ).

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 * 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 * nat ) > set_nat ).

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

tff(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Nat__Onat,type,
    set_or1210151606488870762an_nat: fun_nat_set_nat ).

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

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

tff(sy_c_Set__Interval_Oord__class_OlessThan_001t__Nat__Onat,type,
    set_ord_lessThan_nat: fun_nat_set_nat ).

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

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

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

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

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

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

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_Ounique__euclidean__semiring__with__bit__operations__class_Ochar__of_001t__Nat__Onat,type,
    unique3096191561947761185of_nat: fun_nat_char ).

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

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

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

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

tff(sy_c_Topological__Spaces_Otopological__space__class_Oconvergent_001t__Real__Oreal,type,
    topolo7531315842566124627t_real: fun_nat_real > $o ).

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

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

tff(sy_c_Topological__Spaces_Ouniformity__class_Ouniformity_001t__Complex__Ocomplex,type,
    topolo896644834953643431omplex: filter6041513312241820739omplex ).

tff(sy_c_Topological__Spaces_Ouniformity__class_Ouniformity_001t__Real__Oreal,type,
    topolo1511823702728130853y_real: filter2146258269922977983l_real ).

tff(sy_c_Transcendental_Oarccos,type,
    arccos: fun_real_real ).

tff(sy_c_Transcendental_Oarcosh_001t__Real__Oreal,type,
    arcosh_real: fun_real_real ).

tff(sy_c_Transcendental_Oarcsin,type,
    arcsin: fun_real_real ).

tff(sy_c_Transcendental_Oarctan,type,
    arctan: fun_real_real ).

tff(sy_c_Transcendental_Oarsinh_001t__Real__Oreal,type,
    arsinh_real: fun_real_real ).

tff(sy_c_Transcendental_Oartanh_001t__Real__Oreal,type,
    artanh_real: fun_real_real ).

tff(sy_c_Transcendental_Ocos_001t__Complex__Ocomplex,type,
    cos_complex: fun_complex_complex ).

tff(sy_c_Transcendental_Ocos_001t__Real__Oreal,type,
    cos_real: fun_real_real ).

tff(sy_c_Transcendental_Ocos__coeff,type,
    cos_coeff: fun_nat_real ).

tff(sy_c_Transcendental_Ocosh_001t__Real__Oreal,type,
    cosh_real: fun_real_real ).

tff(sy_c_Transcendental_Ocot_001t__Real__Oreal,type,
    cot_real: fun_real_real ).

tff(sy_c_Transcendental_Oexp_001t__Complex__Ocomplex,type,
    exp_complex: fun_complex_complex ).

tff(sy_c_Transcendental_Oexp_001t__Real__Oreal,type,
    exp_real: fun_real_real ).

tff(sy_c_Transcendental_Oln__class_Oln_001t__Real__Oreal,type,
    ln_ln_real: fun_real_real ).

tff(sy_c_Transcendental_Olog,type,
    log: fun_re7681423839302404990l_real ).

tff(sy_c_Transcendental_Opi,type,
    pi: real ).

tff(sy_c_Transcendental_Opowr_001t__Real__Oreal,type,
    powr_real: fun_re7681423839302404990l_real ).

tff(sy_c_Transcendental_Opowr__real,type,
    powr_real2: fun_re7681423839302404990l_real ).

tff(sy_c_Transcendental_Osin_001t__Complex__Ocomplex,type,
    sin_complex: fun_complex_complex ).

tff(sy_c_Transcendental_Osin_001t__Real__Oreal,type,
    sin_real: fun_real_real ).

tff(sy_c_Transcendental_Osin__coeff,type,
    sin_coeff: fun_nat_real ).

tff(sy_c_Transcendental_Osinh_001t__Real__Oreal,type,
    sinh_real: fun_real_real ).

tff(sy_c_Transcendental_Otan_001t__Complex__Ocomplex,type,
    tan_complex: fun_complex_complex ).

tff(sy_c_Transcendental_Otan_001t__Real__Oreal,type,
    tan_real: fun_real_real ).

tff(sy_c_Transcendental_Otanh_001t__Real__Oreal,type,
    tanh_real: fun_real_real ).

tff(sy_c_Transfer_Obi__total_001t__Int__Oint_001t__Code____Numeral__Ointeger,type,
    bi_tot1331153423839324337nteger: fun_in4891666191172872833eger_o > $o ).

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

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

tff(sy_c_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_OminNull__rel,type,
    vEBT_V6963167321098673237ll_rel: fun_VE2962686792283380846VEBT_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_VEBT__MinMax_OVEBT__internal_Oadd,type,
    vEBT_VEBT_add: fun_op7210469962420818650on_nat ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Ogreater,type,
    vEBT_VEBT_greater: fun_op5794009454290973322_nat_o ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Oless,type,
    vEBT_VEBT_less: fun_op5794009454290973322_nat_o ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Olesseq,type,
    vEBT_VEBT_lesseq: fun_op5794009454290973322_nat_o ).

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

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

tff(sy_c_VEBT__MinMax_OVEBT__internal_Omul,type,
    vEBT_VEBT_mul: fun_op7210469962420818650on_nat ).

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

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

tff(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    vEBT_V1502963449132264192at_nat: ( fun_Pr2477239776880934807at_nat * option4927543243414619207at_nat * option4927543243414619207at_nat ) > option4927543243414619207at_nat ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift__rel_001t__Nat__Onat,type,
    vEBT_V3895251965096974666el_nat: fun_Pr1283576600272279410_nat_o ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift__rel_001t__Num__Onum,type,
    vEBT_V452583751252753300el_num: fun_Pr4822914654802553862_num_o ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift__rel_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    vEBT_V7235779383477046023at_nat: fun_Pr4779616527126863150_nat_o ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Opower,type,
    vEBT_VEBT_power: fun_op7210469962420818650on_nat ).

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

tff(sy_c_VEBT__MinMax_Ovebt__maxt__rel,type,
    vEBT_vebt_maxt_rel: fun_VE2962686792283380846VEBT_o ).

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

tff(sy_c_VEBT__MinMax_Ovebt__mint__rel,type,
    vEBT_vebt_mint_rel: fun_VE2962686792283380846VEBT_o ).

tff(sy_c_VEBT__Pred_Ois__pred__in__set,type,
    vEBT_is_pred_in_set: set_nat > fun_nat_fun_nat_o ).

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

tff(sy_c_VEBT__Pred_Ovebt__pred__rel,type,
    vEBT_vebt_pred_rel: fun_Pr8877776288780692068_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 > fun_Pr7546709192626087934_nat_o ).

tff(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    accp_P4694664897847628410at_nat: ( fun_Pr8463018485500633518_nat_o * produc9217583074051984497at_nat ) > $o ).

tff(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Option__Ooption_It__Nat__Onat_J_Mt__Option__Ooption_It__Nat__Onat_J_J_J,type,
    accp_P5526603486790338531on_nat: ( fun_Pr1283576600272279410_nat_o * produc7004362771917097900on_nat ) > $o ).

tff(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__fun_It__Num__Onum_Mt__fun_It__Num__Onum_Mt__Num__Onum_J_J_Mt__Product____Type__Oprod_It__Option__Ooption_It__Num__Onum_J_Mt__Option__Ooption_It__Num__Onum_J_J_J,type,
    accp_P1225658569124232473on_num: ( fun_Pr4822914654802553862_num_o * produc2703417854250991842on_num ) > $o ).

tff(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_Mt__Product____Type__Oprod_It__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
    accp_P4760198183577772864at_nat: ( fun_Pr4779616527126863150_nat_o * produc1260151622200656695at_nat ) > $o ).

tff(sy_c_Wellfounded_Oaccp_001t__VEBT____Definitions__OVEBT,type,
    accp_VEBT_VEBT: fun_VE2962686792283380846VEBT_o > fun_VEBT_VEBT_o ).

tff(sy_c_Wellfounded_Ofinite__psubset_001t__Complex__Ocomplex,type,
    finite8643634255014194347omplex: set_Pr6308028481084910985omplex ).

tff(sy_c_Wellfounded_Ofinite__psubset_001t__Int__Oint,type,
    finite_psubset_int: set_Pr2522554150109002629et_int ).

tff(sy_c_Wellfounded_Ofinite__psubset_001t__Nat__Onat,type,
    finite_psubset_nat: set_Pr5488025237498180813et_nat ).

tff(sy_c_Wellfounded_Omeasure_001t__Code____Numeral__Ointeger,type,
    measure_Code_integer: fun_Code_integer_nat > set_Pr4811707699266497531nteger ).

tff(sy_c_Wellfounded_Omeasure_001t__Int__Oint,type,
    measure_int: fun_int_nat > set_Pr958786334691620121nt_int ).

tff(sy_c_Wellfounded_Omeasure_001t__Nat__Onat,type,
    measure_nat: fun_nat_nat > set_Pr1261947904930325089at_nat ).

tff(sy_c_Wellfounded_Omeasure_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    measur8038558561449204169at_nat: fun_Pr2602117396471556196at_nat > set_Pr8693737435421807431at_nat ).

tff(sy_c_Wellfounded_Opred__nat,type,
    pred_nat: set_Pr1261947904930325089at_nat ).

tff(sy_c_Wellfounded_Owf_001t__Nat__Onat,type,
    wf_nat: set_Pr1261947904930325089at_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__Nat__Onat,type,
    aa_o_nat: ( fun_o_nat * $o ) > nat ).

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

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

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

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

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

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

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

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

tff(sy_c_aa_001t__Code____Numeral__Ointeger_001t__fun_I_Eo_M_Eo_J,type,
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% Relevant facts (10935)
tff(fact_0_True,axiom,
    b ).

% True
tff(fact_1_calculation,axiom,
    vEBT_vebt_pred(vEBT_Leaf(a,b),xa) = some_nat(one_one_nat) ).

% calculation
tff(fact_2_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_3_deg1Leaf,axiom,
    ! [Ta: vEBT_VEBT] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),one_one_nat)
    <=> ? [A: $o,B: $o] : Ta = vEBT_Leaf((A),(B)) ) ).

% deg1Leaf
tff(fact_4_deg__1__Leaf,axiom,
    ! [Ta: vEBT_VEBT] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),one_one_nat)
     => ? [A2: $o,B2: $o] : Ta = vEBT_Leaf((A2),(B2)) ) ).

% deg_1_Leaf
tff(fact_5_deg__1__Leafy,axiom,
    ! [Ta: vEBT_VEBT,Na: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Na)
     => ( ( Na = one_one_nat )
       => ? [A2: $o,B2: $o] : Ta = vEBT_Leaf((A2),(B2)) ) ) ).

% deg_1_Leafy
tff(fact_6_one__natural_Orsp,axiom,
    one_one_nat = one_one_nat ).

% one_natural.rsp
tff(fact_7_one__reorient,axiom,
    ! [Xb: complex] :
      ( ( one_one_complex = Xb )
    <=> ( Xb = one_one_complex ) ) ).

% one_reorient
tff(fact_8_one__reorient,axiom,
    ! [Xb: real] :
      ( ( one_one_real = Xb )
    <=> ( Xb = one_one_real ) ) ).

% one_reorient
tff(fact_9_one__reorient,axiom,
    ! [Xb: rat] :
      ( ( one_one_rat = Xb )
    <=> ( Xb = one_one_rat ) ) ).

% one_reorient
tff(fact_10_one__reorient,axiom,
    ! [Xb: nat] :
      ( ( one_one_nat = Xb )
    <=> ( Xb = one_one_nat ) ) ).

% one_reorient
tff(fact_11_one__reorient,axiom,
    ! [Xb: int] :
      ( ( one_one_int = Xb )
    <=> ( Xb = one_one_int ) ) ).

% one_reorient
tff(fact_12_set__vebt__set__vebt_H__valid,axiom,
    ! [Ta: vEBT_VEBT,Na: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Na)
     => ( vEBT_set_vebt(Ta) = vEBT_VEBT_set_vebt(Ta) ) ) ).

% set_vebt_set_vebt'_valid
tff(fact_13_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_14_euclidean__size__1,axiom,
    aa_Code_integer_nat(euclid6377331345833325938nteger,one_one_Code_integer) = one_one_nat ).

% euclidean_size_1
tff(fact_15_euclidean__size__1,axiom,
    aa_int_nat(euclid4774559944035922753ze_int,one_one_int) = one_one_nat ).

% euclidean_size_1
tff(fact_16_euclidean__size__1,axiom,
    aa_nat_nat(euclid4777050414544973029ze_nat,one_one_nat) = one_one_nat ).

% euclidean_size_1
tff(fact_17_fact__1,axiom,
    aa_nat_complex(semiri5044797733671781792omplex,one_one_nat) = one_one_complex ).

% fact_1
tff(fact_18_fact__1,axiom,
    aa_nat_rat(semiri773545260158071498ct_rat,one_one_nat) = one_one_rat ).

% fact_1
tff(fact_19_fact__1,axiom,
    aa_nat_int(semiri1406184849735516958ct_int,one_one_nat) = one_one_int ).

% fact_1
tff(fact_20_fact__1,axiom,
    aa_nat_nat(semiri1408675320244567234ct_nat,one_one_nat) = one_one_nat ).

% fact_1
tff(fact_21_fact__1,axiom,
    aa_nat_real(semiri2265585572941072030t_real,one_one_nat) = one_one_real ).

% fact_1
tff(fact_22_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_23_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_24_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_25_member__correct,axiom,
    ! [Ta: vEBT_VEBT,Na: nat,Xb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Na)
     => ( aa_nat_o(vEBT_vebt_member(Ta),Xb)
      <=> aa_set_nat_o(member_nat(Xb),vEBT_set_vebt(Ta)) ) ) ).

% member_correct
tff(fact_26_maxt__sound,axiom,
    ! [Ta: vEBT_VEBT,Na: nat,Xb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Na)
     => ( aa_nat_o(vEBT_VEBT_max_in_set(vEBT_VEBT_set_vebt(Ta)),Xb)
       => ( vEBT_vebt_maxt(Ta) = some_nat(Xb) ) ) ) ).

% maxt_sound
tff(fact_27_maxt__corr,axiom,
    ! [Ta: vEBT_VEBT,Na: nat,Xb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Na)
     => ( ( vEBT_vebt_maxt(Ta) = some_nat(Xb) )
       => aa_nat_o(vEBT_VEBT_max_in_set(vEBT_VEBT_set_vebt(Ta)),Xb) ) ) ).

% maxt_corr
tff(fact_28_option_Oinject,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( some_nat(X2) = some_nat(Y2) )
    <=> ( X2 = Y2 ) ) ).

% option.inject
tff(fact_29_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_30_option_Oinject,axiom,
    ! [X2: num,Y2: num] :
      ( ( some_num(X2) = some_num(Y2) )
    <=> ( X2 = Y2 ) ) ).

% option.inject
tff(fact_31_Leaf__0__not,axiom,
    ! [Aa: $o,Ba: $o] : ~ aa_nat_o(vEBT_invar_vebt(vEBT_Leaf((Aa),(Ba))),zero_zero_nat) ).

% Leaf_0_not
tff(fact_32_set__vebt__finite,axiom,
    ! [Ta: vEBT_VEBT,Na: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Na)
     => aa_set_nat_o(finite_finite_nat,vEBT_VEBT_set_vebt(Ta)) ) ).

% set_vebt_finite
tff(fact_33_mint__sound,axiom,
    ! [Ta: vEBT_VEBT,Na: nat,Xb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Na)
     => ( aa_nat_o(vEBT_VEBT_min_in_set(vEBT_VEBT_set_vebt(Ta)),Xb)
       => ( vEBT_vebt_mint(Ta) = some_nat(Xb) ) ) ) ).

% mint_sound
tff(fact_34_mint__corr,axiom,
    ! [Ta: vEBT_VEBT,Na: nat,Xb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Na)
     => ( ( vEBT_vebt_mint(Ta) = some_nat(Xb) )
       => aa_nat_o(vEBT_VEBT_min_in_set(vEBT_VEBT_set_vebt(Ta)),Xb) ) ) ).

% mint_corr
tff(fact_35_valid__0__not,axiom,
    ! [Ta: vEBT_VEBT] : ~ aa_nat_o(vEBT_invar_vebt(Ta),zero_zero_nat) ).

% valid_0_not
tff(fact_36_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_37_deg__deg__n,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Na: nat] :
      ( aa_nat_o(vEBT_invar_vebt(vEBT_Node(Info,Deg,TreeList,Summary)),Na)
     => ( Deg = Na ) ) ).

% deg_deg_n
tff(fact_38_fact__0,axiom,
    aa_nat_complex(semiri5044797733671781792omplex,zero_zero_nat) = one_one_complex ).

% fact_0
tff(fact_39_fact__0,axiom,
    aa_nat_rat(semiri773545260158071498ct_rat,zero_zero_nat) = one_one_rat ).

% fact_0
tff(fact_40_fact__0,axiom,
    aa_nat_int(semiri1406184849735516958ct_int,zero_zero_nat) = one_one_int ).

% fact_0
tff(fact_41_fact__0,axiom,
    aa_nat_nat(semiri1408675320244567234ct_nat,zero_zero_nat) = one_one_nat ).

% fact_0
tff(fact_42_fact__0,axiom,
    aa_nat_real(semiri2265585572941072030t_real,zero_zero_nat) = one_one_real ).

% fact_0
tff(fact_43_mint__member,axiom,
    ! [Ta: vEBT_VEBT,Na: nat,Maxi: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Na)
     => ( ( vEBT_vebt_mint(Ta) = some_nat(Maxi) )
       => aa_nat_o(vEBT_vebt_member(Ta),Maxi) ) ) ).

% mint_member
tff(fact_44_maxt__member,axiom,
    ! [Ta: vEBT_VEBT,Na: nat,Maxi: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Na)
     => ( ( vEBT_vebt_maxt(Ta) = some_nat(Maxi) )
       => aa_nat_o(vEBT_vebt_member(Ta),Maxi) ) ) ).

% maxt_member
tff(fact_45_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_46_euclidean__size__eq__0__iff,axiom,
    ! [Ba: code_integer] :
      ( ( aa_Code_integer_nat(euclid6377331345833325938nteger,Ba) = zero_zero_nat )
    <=> ( Ba = zero_z3403309356797280102nteger ) ) ).

% euclidean_size_eq_0_iff
tff(fact_47_euclidean__size__eq__0__iff,axiom,
    ! [Ba: int] :
      ( ( aa_int_nat(euclid4774559944035922753ze_int,Ba) = zero_zero_nat )
    <=> ( Ba = zero_zero_int ) ) ).

% euclidean_size_eq_0_iff
tff(fact_48_euclidean__size__eq__0__iff,axiom,
    ! [Ba: nat] :
      ( ( aa_nat_nat(euclid4777050414544973029ze_nat,Ba) = zero_zero_nat )
    <=> ( Ba = zero_zero_nat ) ) ).

% euclidean_size_eq_0_iff
tff(fact_49_size__0,axiom,
    aa_Code_integer_nat(euclid6377331345833325938nteger,zero_z3403309356797280102nteger) = zero_zero_nat ).

% size_0
tff(fact_50_size__0,axiom,
    aa_int_nat(euclid4774559944035922753ze_int,zero_zero_int) = zero_zero_nat ).

% size_0
tff(fact_51_size__0,axiom,
    aa_nat_nat(euclid4777050414544973029ze_nat,zero_zero_nat) = zero_zero_nat ).

% size_0
tff(fact_52_zero__natural_Orsp,axiom,
    zero_zero_nat = zero_zero_nat ).

% zero_natural.rsp
tff(fact_53_zero__reorient,axiom,
    ! [Xb: literal] :
      ( ( zero_zero_literal = Xb )
    <=> ( Xb = zero_zero_literal ) ) ).

% zero_reorient
tff(fact_54_zero__reorient,axiom,
    ! [Xb: real] :
      ( ( zero_zero_real = Xb )
    <=> ( Xb = zero_zero_real ) ) ).

% zero_reorient
tff(fact_55_zero__reorient,axiom,
    ! [Xb: rat] :
      ( ( zero_zero_rat = Xb )
    <=> ( Xb = zero_zero_rat ) ) ).

% zero_reorient
tff(fact_56_zero__reorient,axiom,
    ! [Xb: nat] :
      ( ( zero_zero_nat = Xb )
    <=> ( Xb = zero_zero_nat ) ) ).

% zero_reorient
tff(fact_57_zero__reorient,axiom,
    ! [Xb: int] :
      ( ( zero_zero_int = Xb )
    <=> ( Xb = zero_zero_int ) ) ).

% zero_reorient
tff(fact_58_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_59_VEBT_Oexhaust,axiom,
    ! [Y: vEBT_VEBT] :
      ( ! [X112: option4927543243414619207at_nat,X122: nat,X132: list_VEBT_VEBT,X142: vEBT_VEBT] : Y != vEBT_Node(X112,X122,X132,X142)
     => ~ ! [X212: $o,X222: $o] : Y != vEBT_Leaf((X212),(X222)) ) ).

% VEBT.exhaust
tff(fact_60_fact__nonzero,axiom,
    ! [Na: nat] : aa_nat_rat(semiri773545260158071498ct_rat,Na) != zero_zero_rat ).

% fact_nonzero
tff(fact_61_fact__nonzero,axiom,
    ! [Na: nat] : aa_nat_int(semiri1406184849735516958ct_int,Na) != zero_zero_int ).

% fact_nonzero
tff(fact_62_fact__nonzero,axiom,
    ! [Na: nat] : aa_nat_nat(semiri1408675320244567234ct_nat,Na) != zero_zero_nat ).

% fact_nonzero
tff(fact_63_fact__nonzero,axiom,
    ! [Na: nat] : aa_nat_real(semiri2265585572941072030t_real,Na) != zero_zero_real ).

% fact_nonzero
tff(fact_64_vebt__member_Osimps_I1_J,axiom,
    ! [Aa: $o,Ba: $o,Xb: nat] :
      ( aa_nat_o(vEBT_vebt_member(vEBT_Leaf((Aa),(Ba))),Xb)
    <=> $ite(
          Xb = zero_zero_nat,
          (Aa),
          $ite(Xb = one_one_nat,(Ba),$false) ) ) ).

% vebt_member.simps(1)
tff(fact_65_maxt__corr__help,axiom,
    ! [Ta: vEBT_VEBT,Na: nat,Maxi: nat,Xb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Na)
     => ( ( vEBT_vebt_maxt(Ta) = some_nat(Maxi) )
       => ( aa_nat_o(vEBT_vebt_member(Ta),Xb)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Maxi) ) ) ) ).

% maxt_corr_help
tff(fact_66_mint__corr__help,axiom,
    ! [Ta: vEBT_VEBT,Na: nat,Mini: nat,Xb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Na)
     => ( ( vEBT_vebt_mint(Ta) = some_nat(Mini) )
       => ( aa_nat_o(vEBT_vebt_member(Ta),Xb)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mini),Xb) ) ) ) ).

% mint_corr_help
tff(fact_67_obtain__set__pred,axiom,
    ! [Z: nat,Xb: nat,A3: set_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z),Xb)
     => ( aa_nat_o(vEBT_VEBT_min_in_set(A3),Z)
       => ( aa_set_nat_o(finite_finite_nat,A3)
         => ? [X_1: nat] : aa_nat_o(aa_nat_fun_nat_o(vEBT_is_pred_in_set(A3),Xb),X_1) ) ) ) ).

% obtain_set_pred
tff(fact_68_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_69_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_70_mem__Collect__eq,axiom,
    ! [Aa: product_prod_nat_nat,P: fun_Pr4796943543331071514_nat_o] :
      ( member8440522571783428010at_nat(Aa,collec3392354462482085612at_nat(P))
    <=> aa_Pro6150637604445964947_nat_o(P,Aa) ) ).

% mem_Collect_eq
tff(fact_71_mem__Collect__eq,axiom,
    ! [Aa: real,P: fun_real_o] :
      ( member_real(Aa,collect_real(P))
    <=> aa_real_o(P,Aa) ) ).

% mem_Collect_eq
tff(fact_72_mem__Collect__eq,axiom,
    ! [Aa: list_nat,P: fun_list_nat_o] :
      ( member_list_nat(Aa,collect_list_nat(P))
    <=> aa_list_nat_o(P,Aa) ) ).

% mem_Collect_eq
tff(fact_73_mem__Collect__eq,axiom,
    ! [Aa: set_nat,P: fun_set_nat_o] :
      ( member_set_nat(Aa,collect_set_nat(P))
    <=> aa_set_nat_o(P,Aa) ) ).

% mem_Collect_eq
tff(fact_74_mem__Collect__eq,axiom,
    ! [Aa: nat,P: fun_nat_o] :
      ( aa_set_nat_o(member_nat(Aa),collect_nat(P))
    <=> aa_nat_o(P,Aa) ) ).

% mem_Collect_eq
tff(fact_75_mem__Collect__eq,axiom,
    ! [Aa: int,P: fun_int_o] :
      ( aa_set_int_o(member_int(Aa),collect_int(P))
    <=> aa_int_o(P,Aa) ) ).

% mem_Collect_eq
tff(fact_76_Collect__mem__eq,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] : collec3392354462482085612at_nat(aTP_Lamm_a(A3)) = A3 ).

% Collect_mem_eq
tff(fact_77_Collect__mem__eq,axiom,
    ! [A3: set_real] : collect_real(aTP_Lamm_aa(A3)) = A3 ).

% Collect_mem_eq
tff(fact_78_Collect__mem__eq,axiom,
    ! [A3: set_list_nat] : collect_list_nat(aTP_Lamm_ab(A3)) = A3 ).

% Collect_mem_eq
tff(fact_79_Collect__mem__eq,axiom,
    ! [A3: set_set_nat] : collect_set_nat(aTP_Lamm_ac(A3)) = A3 ).

% Collect_mem_eq
tff(fact_80_Collect__mem__eq,axiom,
    ! [A3: set_nat] : collect_nat(aTP_Lamm_ad(A3)) = A3 ).

% Collect_mem_eq
tff(fact_81_Collect__mem__eq,axiom,
    ! [A3: set_int] : collect_int(aTP_Lamm_ae(A3)) = A3 ).

% Collect_mem_eq
tff(fact_82_Collect__cong,axiom,
    ! [P: fun_real_o,Q: fun_real_o] :
      ( ! [X: real] :
          ( aa_real_o(P,X)
        <=> aa_real_o(Q,X) )
     => ( collect_real(P) = collect_real(Q) ) ) ).

% Collect_cong
tff(fact_83_Collect__cong,axiom,
    ! [P: fun_list_nat_o,Q: fun_list_nat_o] :
      ( ! [X: list_nat] :
          ( aa_list_nat_o(P,X)
        <=> aa_list_nat_o(Q,X) )
     => ( collect_list_nat(P) = collect_list_nat(Q) ) ) ).

% Collect_cong
tff(fact_84_Collect__cong,axiom,
    ! [P: fun_set_nat_o,Q: fun_set_nat_o] :
      ( ! [X: set_nat] :
          ( aa_set_nat_o(P,X)
        <=> aa_set_nat_o(Q,X) )
     => ( collect_set_nat(P) = collect_set_nat(Q) ) ) ).

% Collect_cong
tff(fact_85_Collect__cong,axiom,
    ! [P: fun_nat_o,Q: fun_nat_o] :
      ( ! [X: nat] :
          ( aa_nat_o(P,X)
        <=> aa_nat_o(Q,X) )
     => ( collect_nat(P) = collect_nat(Q) ) ) ).

% Collect_cong
tff(fact_86_Collect__cong,axiom,
    ! [P: fun_int_o,Q: fun_int_o] :
      ( ! [X: int] :
          ( aa_int_o(P,X)
        <=> aa_int_o(Q,X) )
     => ( collect_int(P) = collect_int(Q) ) ) ).

% Collect_cong
tff(fact_87_pred__none__empty,axiom,
    ! [Xs: set_nat,Aa: nat] :
      ( ~ ? [X_1: nat] : aa_nat_o(aa_nat_fun_nat_o(vEBT_is_pred_in_set(Xs),Aa),X_1)
     => ( aa_set_nat_o(finite_finite_nat,Xs)
       => ~ ? [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),Xs)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Aa) ) ) ) ).

% pred_none_empty
tff(fact_88_zero__neq__one,axiom,
    zero_zero_complex != one_one_complex ).

% zero_neq_one
tff(fact_89_zero__neq__one,axiom,
    zero_zero_real != one_one_real ).

% zero_neq_one
tff(fact_90_zero__neq__one,axiom,
    zero_zero_rat != one_one_rat ).

% zero_neq_one
tff(fact_91_zero__neq__one,axiom,
    zero_zero_nat != one_one_nat ).

% zero_neq_one
tff(fact_92_zero__neq__one,axiom,
    zero_zero_int != one_one_int ).

% zero_neq_one
tff(fact_93_deg__not__0,axiom,
    ! [Ta: vEBT_VEBT,Na: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ).

% deg_not_0
tff(fact_94_deg__SUcn__Node,axiom,
    ! [Tree: vEBT_VEBT,Na: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Tree),aa_nat_nat(suc,aa_nat_nat(suc,Na)))
     => ? [Info2: option4927543243414619207at_nat,TreeList2: list_VEBT_VEBT,S: vEBT_VEBT] : Tree = vEBT_Node(Info2,aa_nat_nat(suc,aa_nat_nat(suc,Na)),TreeList2,S) ) ).

% deg_SUcn_Node
tff(fact_95_max__in__set__def,axiom,
    ! [Xs: set_nat,Xb: nat] :
      ( aa_nat_o(vEBT_VEBT_max_in_set(Xs),Xb)
    <=> ( aa_set_nat_o(member_nat(Xb),Xs)
        & ! [X4: nat] :
            ( aa_set_nat_o(member_nat(X4),Xs)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X4),Xb) ) ) ) ).

% max_in_set_def
tff(fact_96_Suc,axiom,
    sucX = aa_nat_nat(suc,nat3) ).

% Suc
tff(fact_97_min__in__set__def,axiom,
    ! [Xs: set_nat,Xb: nat] :
      ( aa_nat_o(vEBT_VEBT_min_in_set(Xs),Xb)
    <=> ( aa_set_nat_o(member_nat(Xb),Xs)
        & ! [X4: nat] :
            ( aa_set_nat_o(member_nat(X4),Xs)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),X4) ) ) ) ).

% min_in_set_def
tff(fact_98_pred__member,axiom,
    ! [Ta: vEBT_VEBT,Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(vEBT_is_pred_in_set(vEBT_VEBT_set_vebt(Ta)),Xb),Y)
    <=> ( aa_nat_o(vEBT_vebt_member(Ta),Y)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xb)
        & ! [Z2: nat] :
            ( ( aa_nat_o(vEBT_vebt_member(Ta),Z2)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z2),Xb) )
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z2),Y) ) ) ) ).

% pred_member
tff(fact_99_le__zero__eq,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),zero_zero_nat)
    <=> ( Na = zero_zero_nat ) ) ).

% le_zero_eq
tff(fact_100_not__gr__zero,axiom,
    ! [Na: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
    <=> ( Na = zero_zero_nat ) ) ).

% not_gr_zero
tff(fact_101__092_060open_0620_A_092_060le_062_Ax_A_092_060and_062_Ax_A_061_ASuc_AsucX_092_060close_062,axiom,
    ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),xa)
    & ( xa = aa_nat_nat(suc,sucX) ) ) ).

% \<open>0 \<le> x \<and> x = Suc sucX\<close>
tff(fact_102_fact__Suc__0,axiom,
    aa_nat_complex(semiri5044797733671781792omplex,aa_nat_nat(suc,zero_zero_nat)) = one_one_complex ).

% fact_Suc_0
tff(fact_103_fact__Suc__0,axiom,
    aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(suc,zero_zero_nat)) = one_one_rat ).

% fact_Suc_0
tff(fact_104_fact__Suc__0,axiom,
    aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(suc,zero_zero_nat)) = one_one_int ).

% fact_Suc_0
tff(fact_105_fact__Suc__0,axiom,
    aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(suc,zero_zero_nat)) = one_one_nat ).

% fact_Suc_0
tff(fact_106_fact__Suc__0,axiom,
    aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(suc,zero_zero_nat)) = one_one_real ).

% fact_Suc_0
tff(fact_107_euclidean__size__greater__0__iff,axiom,
    ! [Ba: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_Code_integer_nat(euclid6377331345833325938nteger,Ba))
    <=> ( Ba != zero_z3403309356797280102nteger ) ) ).

% euclidean_size_greater_0_iff
tff(fact_108_euclidean__size__greater__0__iff,axiom,
    ! [Ba: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_int_nat(euclid4774559944035922753ze_int,Ba))
    <=> ( Ba != zero_zero_int ) ) ).

% euclidean_size_greater_0_iff
tff(fact_109_euclidean__size__greater__0__iff,axiom,
    ! [Ba: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(euclid4777050414544973029ze_nat,Ba))
    <=> ( Ba != zero_zero_nat ) ) ).

% euclidean_size_greater_0_iff
tff(fact_110_less__shift,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
    <=> aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_less,some_nat(Xb)),some_nat(Y)) ) ).

% less_shift
tff(fact_111_lesseq__shift,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y)
    <=> aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_lesseq,some_nat(Xb)),some_nat(Y)) ) ).

% lesseq_shift
tff(fact_112_linorder__neqE__linordered__idom,axiom,
    ! [Xb: real,Y: real] :
      ( ( Xb != Y )
     => ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xb) ) ) ).

% linorder_neqE_linordered_idom
tff(fact_113_linorder__neqE__linordered__idom,axiom,
    ! [Xb: rat,Y: rat] :
      ( ( Xb != Y )
     => ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Xb) ) ) ).

% linorder_neqE_linordered_idom
tff(fact_114_linorder__neqE__linordered__idom,axiom,
    ! [Xb: int,Y: int] :
      ( ( Xb != Y )
     => ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Xb) ) ) ).

% linorder_neqE_linordered_idom
tff(fact_115_fact__mono__nat,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1408675320244567234ct_nat,M)),aa_nat_nat(semiri1408675320244567234ct_nat,Na)) ) ).

% fact_mono_nat
tff(fact_116_fact__ge__self,axiom,
    ! [Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),aa_nat_nat(semiri1408675320244567234ct_nat,Na)) ).

% fact_ge_self
tff(fact_117_fact__ge__Suc__0__nat,axiom,
    ! [Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(semiri1408675320244567234ct_nat,Na)) ).

% fact_ge_Suc_0_nat
tff(fact_118_fact__mono,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri773545260158071498ct_rat,M)),aa_nat_rat(semiri773545260158071498ct_rat,Na)) ) ).

% fact_mono
tff(fact_119_fact__mono,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1406184849735516958ct_int,M)),aa_nat_int(semiri1406184849735516958ct_int,Na)) ) ).

% fact_mono
tff(fact_120_fact__mono,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1408675320244567234ct_nat,M)),aa_nat_nat(semiri1408675320244567234ct_nat,Na)) ) ).

% fact_mono
tff(fact_121_fact__mono,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri2265585572941072030t_real,M)),aa_nat_real(semiri2265585572941072030t_real,Na)) ) ).

% fact_mono
tff(fact_122_finite__has__maximal2,axiom,
    ! [A3: set_real,Aa: real] :
      ( finite_finite_real(A3)
     => ( member_real(Aa,A3)
       => ? [X: real] :
            ( member_real(X,A3)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),X)
            & ! [Xa: real] :
                ( member_real(Xa,A3)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Xa)
                 => ( X = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_123_finite__has__maximal2,axiom,
    ! [A3: set_set_nat,Aa: set_nat] :
      ( finite1152437895449049373et_nat(A3)
     => ( member_set_nat(Aa,A3)
       => ? [X: set_nat] :
            ( member_set_nat(X,A3)
            & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Aa),X)
            & ! [Xa: set_nat] :
                ( member_set_nat(Xa,A3)
               => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),Xa)
                 => ( X = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_124_finite__has__maximal2,axiom,
    ! [A3: set_set_int,Aa: set_int] :
      ( finite6197958912794628473et_int(A3)
     => ( member_set_int(Aa,A3)
       => ? [X: set_int] :
            ( member_set_int(X,A3)
            & aa_set_int_o(ord_less_eq_set_int(Aa),X)
            & ! [Xa: set_int] :
                ( member_set_int(Xa,A3)
               => ( aa_set_int_o(ord_less_eq_set_int(X),Xa)
                 => ( X = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_125_finite__has__maximal2,axiom,
    ! [A3: set_rat,Aa: rat] :
      ( finite_finite_rat(A3)
     => ( member_rat(Aa,A3)
       => ? [X: rat] :
            ( member_rat(X,A3)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),X)
            & ! [Xa: rat] :
                ( member_rat(Xa,A3)
               => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Xa)
                 => ( X = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_126_finite__has__maximal2,axiom,
    ! [A3: set_num,Aa: num] :
      ( finite_finite_num(A3)
     => ( member_num(Aa,A3)
       => ? [X: num] :
            ( member_num(X,A3)
            & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Aa),X)
            & ! [Xa: num] :
                ( member_num(Xa,A3)
               => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Xa)
                 => ( X = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_127_finite__has__maximal2,axiom,
    ! [A3: set_nat,Aa: nat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_nat_o(member_nat(Aa),A3)
       => ? [X: nat] :
            ( aa_set_nat_o(member_nat(X),A3)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),X)
            & ! [Xa: nat] :
                ( aa_set_nat_o(member_nat(Xa),A3)
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Xa)
                 => ( X = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_128_finite__has__maximal2,axiom,
    ! [A3: set_int,Aa: int] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( aa_set_int_o(member_int(Aa),A3)
       => ? [X: int] :
            ( aa_set_int_o(member_int(X),A3)
            & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),X)
            & ! [Xa: int] :
                ( aa_set_int_o(member_int(Xa),A3)
               => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Xa)
                 => ( X = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_129_finite__has__minimal2,axiom,
    ! [A3: set_real,Aa: real] :
      ( finite_finite_real(A3)
     => ( member_real(Aa,A3)
       => ? [X: real] :
            ( member_real(X,A3)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Aa)
            & ! [Xa: real] :
                ( member_real(Xa,A3)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),X)
                 => ( X = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_130_finite__has__minimal2,axiom,
    ! [A3: set_set_nat,Aa: set_nat] :
      ( finite1152437895449049373et_nat(A3)
     => ( member_set_nat(Aa,A3)
       => ? [X: set_nat] :
            ( member_set_nat(X,A3)
            & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),Aa)
            & ! [Xa: set_nat] :
                ( member_set_nat(Xa,A3)
               => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Xa),X)
                 => ( X = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_131_finite__has__minimal2,axiom,
    ! [A3: set_set_int,Aa: set_int] :
      ( finite6197958912794628473et_int(A3)
     => ( member_set_int(Aa,A3)
       => ? [X: set_int] :
            ( member_set_int(X,A3)
            & aa_set_int_o(ord_less_eq_set_int(X),Aa)
            & ! [Xa: set_int] :
                ( member_set_int(Xa,A3)
               => ( aa_set_int_o(ord_less_eq_set_int(Xa),X)
                 => ( X = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_132_finite__has__minimal2,axiom,
    ! [A3: set_rat,Aa: rat] :
      ( finite_finite_rat(A3)
     => ( member_rat(Aa,A3)
       => ? [X: rat] :
            ( member_rat(X,A3)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Aa)
            & ! [Xa: rat] :
                ( member_rat(Xa,A3)
               => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),X)
                 => ( X = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_133_finite__has__minimal2,axiom,
    ! [A3: set_num,Aa: num] :
      ( finite_finite_num(A3)
     => ( member_num(Aa,A3)
       => ? [X: num] :
            ( member_num(X,A3)
            & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Aa)
            & ! [Xa: num] :
                ( member_num(Xa,A3)
               => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),X)
                 => ( X = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_134_finite__has__minimal2,axiom,
    ! [A3: set_nat,Aa: nat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_nat_o(member_nat(Aa),A3)
       => ? [X: nat] :
            ( aa_set_nat_o(member_nat(X),A3)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Aa)
            & ! [Xa: nat] :
                ( aa_set_nat_o(member_nat(Xa),A3)
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),X)
                 => ( X = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_135_finite__has__minimal2,axiom,
    ! [A3: set_int,Aa: int] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( aa_set_int_o(member_int(Aa),A3)
       => ? [X: int] :
            ( aa_set_int_o(member_int(X),A3)
            & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Aa)
            & ! [Xa: int] :
                ( aa_set_int_o(member_int(Xa),A3)
               => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),X)
                 => ( X = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_136_VEBT__internal_OminNull_Osimps_I5_J,axiom,
    ! [Uz: product_prod_nat_nat,Va: nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT] : ~ aa_VEBT_VEBT_o(vEBT_VEBT_minNull,vEBT_Node(some_P7363390416028606310at_nat(Uz),Va,Vb,Vc)) ).

% VEBT_internal.minNull.simps(5)
tff(fact_137_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_138_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_139_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_140_is__pred__in__set__def,axiom,
    ! [Xs: set_nat,Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(vEBT_is_pred_in_set(Xs),Xb),Y)
    <=> ( aa_set_nat_o(member_nat(Y),Xs)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xb)
        & ! [X4: nat] :
            ( aa_set_nat_o(member_nat(X4),Xs)
           => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),Xb)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X4),Y) ) ) ) ) ).

% is_pred_in_set_def
tff(fact_141_fact__less__mono,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri773545260158071498ct_rat,M)),aa_nat_rat(semiri773545260158071498ct_rat,Na)) ) ) ).

% fact_less_mono
tff(fact_142_fact__less__mono,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1406184849735516958ct_int,M)),aa_nat_int(semiri1406184849735516958ct_int,Na)) ) ) ).

% fact_less_mono
tff(fact_143_fact__less__mono,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1408675320244567234ct_nat,M)),aa_nat_nat(semiri1408675320244567234ct_nat,Na)) ) ) ).

% fact_less_mono
tff(fact_144_fact__less__mono,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri2265585572941072030t_real,M)),aa_nat_real(semiri2265585572941072030t_real,Na)) ) ) ).

% fact_less_mono
tff(fact_145_zero__less__iff__neq__zero,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
    <=> ( Na != zero_zero_nat ) ) ).

% zero_less_iff_neq_zero
tff(fact_146_gr__implies__not__zero,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
     => ( Na != zero_zero_nat ) ) ).

% gr_implies_not_zero
tff(fact_147_not__less__zero,axiom,
    ! [Na: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),zero_zero_nat) ).

% not_less_zero
tff(fact_148_gr__zeroI,axiom,
    ! [Na: nat] :
      ( ( Na != zero_zero_nat )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ).

% gr_zeroI
tff(fact_149_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_150_vebt__buildup_Ocases,axiom,
    ! [Xb: nat] :
      ( ( Xb != zero_zero_nat )
     => ( ( Xb != aa_nat_nat(suc,zero_zero_nat) )
       => ~ ! [Va2: nat] : Xb != aa_nat_nat(suc,aa_nat_nat(suc,Va2)) ) ) ).

% vebt_buildup.cases
tff(fact_151_fact__less__mono__nat,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1408675320244567234ct_nat,M)),aa_nat_nat(semiri1408675320244567234ct_nat,Na)) ) ) ).

% fact_less_mono_nat
tff(fact_152_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_153_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_154_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_155_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_156_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_157_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_158_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_159_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_160_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_161_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_162_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_163_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_164_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_165_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_166_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_167_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_168_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_169_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_170_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_171_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_172_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_173_fact__gt__zero,axiom,
    ! [Na: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(semiri773545260158071498ct_rat,Na)) ).

% fact_gt_zero
tff(fact_174_fact__gt__zero,axiom,
    ! [Na: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(semiri1406184849735516958ct_int,Na)) ).

% fact_gt_zero
tff(fact_175_fact__gt__zero,axiom,
    ! [Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(semiri1408675320244567234ct_nat,Na)) ).

% fact_gt_zero
tff(fact_176_fact__gt__zero,axiom,
    ! [Na: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(semiri2265585572941072030t_real,Na)) ).

% fact_gt_zero
tff(fact_177_fact__not__neg,axiom,
    ! [Na: nat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri773545260158071498ct_rat,Na)),zero_zero_rat) ).

% fact_not_neg
tff(fact_178_fact__not__neg,axiom,
    ! [Na: nat] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1406184849735516958ct_int,Na)),zero_zero_int) ).

% fact_not_neg
tff(fact_179_fact__not__neg,axiom,
    ! [Na: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Na)),zero_zero_nat) ).

% fact_not_neg
tff(fact_180_fact__not__neg,axiom,
    ! [Na: nat] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri2265585572941072030t_real,Na)),zero_zero_real) ).

% fact_not_neg
tff(fact_181_fact__ge__zero,axiom,
    ! [Na: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(semiri773545260158071498ct_rat,Na)) ).

% fact_ge_zero
tff(fact_182_fact__ge__zero,axiom,
    ! [Na: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(semiri1406184849735516958ct_int,Na)) ).

% fact_ge_zero
tff(fact_183_fact__ge__zero,axiom,
    ! [Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(semiri1408675320244567234ct_nat,Na)) ).

% fact_ge_zero
tff(fact_184_fact__ge__zero,axiom,
    ! [Na: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(semiri2265585572941072030t_real,Na)) ).

% fact_ge_zero
tff(fact_185_fact__ge__1,axiom,
    ! [Na: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_nat_rat(semiri773545260158071498ct_rat,Na)) ).

% fact_ge_1
tff(fact_186_fact__ge__1,axiom,
    ! [Na: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_nat_int(semiri1406184849735516958ct_int,Na)) ).

% fact_ge_1
tff(fact_187_fact__ge__1,axiom,
    ! [Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),aa_nat_nat(semiri1408675320244567234ct_nat,Na)) ).

% fact_ge_1
tff(fact_188_fact__ge__1,axiom,
    ! [Na: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_nat_real(semiri2265585572941072030t_real,Na)) ).

% fact_ge_1
tff(fact_189_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_190_invar__vebt_Ointros_I1_J,axiom,
    ! [Aa: $o,Ba: $o] : aa_nat_o(vEBT_invar_vebt(vEBT_Leaf((Aa),(Ba))),aa_nat_nat(suc,zero_zero_nat)) ).

% invar_vebt.intros(1)
tff(fact_191_greater__shift,axiom,
    ! [Y: nat,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xb)
    <=> aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_greater,some_nat(Xb)),some_nat(Y)) ) ).

% greater_shift
tff(fact_192_less__one,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),one_one_nat)
    <=> ( Na = zero_zero_nat ) ) ).

% less_one
tff(fact_193_zero__less__Suc,axiom,
    ! [Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(suc,Na)) ).

% zero_less_Suc
tff(fact_194_less__Suc0,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(suc,zero_zero_nat))
    <=> ( Na = zero_zero_nat ) ) ).

% less_Suc0
tff(fact_195_buildup__gives__valid,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => aa_nat_o(vEBT_invar_vebt(vEBT_vebt_buildup(Na)),Na) ) ).

% buildup_gives_valid
tff(fact_196_bot__nat__0_Oextremum,axiom,
    ! [Aa: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Aa) ).

% bot_nat_0.extremum
tff(fact_197_le0,axiom,
    ! [Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Na) ).

% le0
tff(fact_198_Suc__le__mono,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Na)),aa_nat_nat(suc,M))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M) ) ).

% Suc_le_mono
tff(fact_199_bot__nat__0_Onot__eq__extremum,axiom,
    ! [Aa: nat] :
      ( ( Aa != zero_zero_nat )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Aa) ) ).

% bot_nat_0.not_eq_extremum
tff(fact_200_neq0__conv,axiom,
    ! [Na: nat] :
      ( ( Na != zero_zero_nat )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ).

% neq0_conv
tff(fact_201_less__nat__zero__code,axiom,
    ! [Na: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),zero_zero_nat) ).

% less_nat_zero_code
tff(fact_202_lessI,axiom,
    ! [Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(suc,Na)) ).

% lessI
tff(fact_203_old_Onat_Oinject,axiom,
    ! [Nata: nat,Nat: nat] :
      ( ( aa_nat_nat(suc,Nata) = aa_nat_nat(suc,Nat) )
    <=> ( Nata = Nat ) ) ).

% old.nat.inject
tff(fact_204_nat_Oinject,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( aa_nat_nat(suc,X2) = aa_nat_nat(suc,Y2) )
    <=> ( X2 = Y2 ) ) ).

% nat.inject
tff(fact_205_Suc__less__eq,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ).

% Suc_less_eq
tff(fact_206_Suc__mono,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,Na)) ) ).

% Suc_mono
tff(fact_207_finite__psubset__induct,axiom,
    ! [A3: set_nat,P: fun_set_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( ! [A4: set_nat] :
            ( aa_set_nat_o(finite_finite_nat,A4)
           => ( ! [B3: set_nat] :
                  ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,B3),A4)
                 => aa_set_nat_o(P,B3) )
             => aa_set_nat_o(P,A4) ) )
       => aa_set_nat_o(P,A3) ) ) ).

% finite_psubset_induct
tff(fact_208_finite__psubset__induct,axiom,
    ! [A3: set_int,P: fun_set_int_o] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( ! [A4: set_int] :
            ( aa_set_int_o(finite_finite_int,A4)
           => ( ! [B3: set_int] :
                  ( aa_set_int_o(ord_less_set_int(B3),A4)
                 => aa_set_int_o(P,B3) )
             => aa_set_int_o(P,A4) ) )
       => aa_set_int_o(P,A3) ) ) ).

% finite_psubset_induct
tff(fact_209_finite__psubset__induct,axiom,
    ! [A3: set_complex,P: fun_set_complex_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A3)
     => ( ! [A4: set_complex] :
            ( aa_set_complex_o(finite3207457112153483333omplex,A4)
           => ( ! [B3: set_complex] :
                  ( aa_set_complex_o(ord_less_set_complex(B3),A4)
                 => aa_set_complex_o(P,B3) )
             => aa_set_complex_o(P,A4) ) )
       => aa_set_complex_o(P,A3) ) ) ).

% finite_psubset_induct
tff(fact_210_rev__finite__subset,axiom,
    ! [B4: set_nat,A3: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,B4)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A3),B4)
       => aa_set_nat_o(finite_finite_nat,A3) ) ) ).

% rev_finite_subset
tff(fact_211_rev__finite__subset,axiom,
    ! [B4: set_complex,A3: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A3),B4)
       => aa_set_complex_o(finite3207457112153483333omplex,A3) ) ) ).

% rev_finite_subset
tff(fact_212_rev__finite__subset,axiom,
    ! [B4: set_int,A3: set_int] :
      ( aa_set_int_o(finite_finite_int,B4)
     => ( aa_set_int_o(ord_less_eq_set_int(A3),B4)
       => aa_set_int_o(finite_finite_int,A3) ) ) ).

% rev_finite_subset
tff(fact_213_infinite__super,axiom,
    ! [S2: set_nat,T: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S2),T)
     => ( ~ aa_set_nat_o(finite_finite_nat,S2)
       => ~ aa_set_nat_o(finite_finite_nat,T) ) ) ).

% infinite_super
tff(fact_214_infinite__super,axiom,
    ! [S2: set_complex,T: set_complex] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T)
     => ( ~ aa_set_complex_o(finite3207457112153483333omplex,S2)
       => ~ aa_set_complex_o(finite3207457112153483333omplex,T) ) ) ).

% infinite_super
tff(fact_215_infinite__super,axiom,
    ! [S2: set_int,T: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(S2),T)
     => ( ~ aa_set_int_o(finite_finite_int,S2)
       => ~ aa_set_int_o(finite_finite_int,T) ) ) ).

% infinite_super
tff(fact_216_finite__subset,axiom,
    ! [A3: set_nat,B4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A3),B4)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => aa_set_nat_o(finite_finite_nat,A3) ) ) ).

% finite_subset
tff(fact_217_finite__subset,axiom,
    ! [A3: set_complex,B4: set_complex] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(A3),B4)
     => ( aa_set_complex_o(finite3207457112153483333omplex,B4)
       => aa_set_complex_o(finite3207457112153483333omplex,A3) ) ) ).

% finite_subset
tff(fact_218_finite__subset,axiom,
    ! [A3: set_int,B4: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A3),B4)
     => ( aa_set_int_o(finite_finite_int,B4)
       => aa_set_int_o(finite_finite_int,A3) ) ) ).

% finite_subset
tff(fact_219_vebt__buildup_Osimps_I1_J,axiom,
    vEBT_vebt_buildup(zero_zero_nat) = vEBT_Leaf($false,$false) ).

% vebt_buildup.simps(1)
tff(fact_220_n__not__Suc__n,axiom,
    ! [Na: nat] : Na != aa_nat_nat(suc,Na) ).

% n_not_Suc_n
tff(fact_221_Suc__inject,axiom,
    ! [Xb: nat,Y: nat] :
      ( ( aa_nat_nat(suc,Xb) = aa_nat_nat(suc,Y) )
     => ( Xb = Y ) ) ).

% Suc_inject
tff(fact_222_linorder__neqE__nat,axiom,
    ! [Xb: nat,Y: nat] :
      ( ( Xb != Y )
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xb) ) ) ).

% linorder_neqE_nat
tff(fact_223_infinite__descent,axiom,
    ! [P: fun_nat_o,Na: nat] :
      ( ! [N: nat] :
          ( ~ aa_nat_o(P,N)
         => ? [M2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),N)
              & ~ aa_nat_o(P,M2) ) )
     => aa_nat_o(P,Na) ) ).

% infinite_descent
tff(fact_224_nat__less__induct,axiom,
    ! [P: fun_nat_o,Na: nat] :
      ( ! [N: nat] :
          ( ! [M2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),N)
             => aa_nat_o(P,M2) )
         => aa_nat_o(P,N) )
     => aa_nat_o(P,Na) ) ).

% nat_less_induct
tff(fact_225_less__irrefl__nat,axiom,
    ! [Na: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),Na) ).

% less_irrefl_nat
tff(fact_226_less__not__refl3,axiom,
    ! [S3: nat,Ta: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,S3),Ta)
     => ( S3 != Ta ) ) ).

% less_not_refl3
tff(fact_227_less__not__refl2,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M)
     => ( M != Na ) ) ).

% less_not_refl2
tff(fact_228_less__not__refl,axiom,
    ! [Na: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),Na) ).

% less_not_refl
tff(fact_229_nat__neq__iff,axiom,
    ! [M: nat,Na: nat] :
      ( ( M != Na )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M) ) ) ).

% nat_neq_iff
tff(fact_230_Nat_Oex__has__greatest__nat,axiom,
    ! [P: fun_nat_o,K: nat,Ba: nat] :
      ( aa_nat_o(P,K)
     => ( ! [Y3: nat] :
            ( aa_nat_o(P,Y3)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y3),Ba) )
       => ? [X: nat] :
            ( aa_nat_o(P,X)
            & ! [Y4: nat] :
                ( aa_nat_o(P,Y4)
               => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y4),X) ) ) ) ) ).

% Nat.ex_has_greatest_nat
tff(fact_231_nat__le__linear,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
      | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M) ) ).

% nat_le_linear
tff(fact_232_le__antisym,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
       => ( M = Na ) ) ) ).

% le_antisym
tff(fact_233_eq__imp__le,axiom,
    ! [M: nat,Na: nat] :
      ( ( M = Na )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ).

% eq_imp_le
tff(fact_234_le__trans,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),K)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),K) ) ) ).

% le_trans
tff(fact_235_le__refl,axiom,
    ! [Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),Na) ).

% le_refl
tff(fact_236_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_237_not0__implies__Suc,axiom,
    ! [Na: nat] :
      ( ( Na != zero_zero_nat )
     => ? [M3: nat] : Na = aa_nat_nat(suc,M3) ) ).

% not0_implies_Suc
tff(fact_238_Zero__not__Suc,axiom,
    ! [M: nat] : zero_zero_nat != aa_nat_nat(suc,M) ).

% Zero_not_Suc
tff(fact_239_Zero__neq__Suc,axiom,
    ! [M: nat] : zero_zero_nat != aa_nat_nat(suc,M) ).

% Zero_neq_Suc
tff(fact_240_Suc__neq__Zero,axiom,
    ! [M: nat] : aa_nat_nat(suc,M) != zero_zero_nat ).

% Suc_neq_Zero
tff(fact_241_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_242_diff__induct,axiom,
    ! [P: fun_nat_fun_nat_o,M: nat,Na: nat] :
      ( ! [X: nat] : aa_nat_o(aa_nat_fun_nat_o(P,X),zero_zero_nat)
     => ( ! [Y3: nat] : aa_nat_o(aa_nat_fun_nat_o(P,zero_zero_nat),aa_nat_nat(suc,Y3))
       => ( ! [X: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(P,X),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(P,aa_nat_nat(suc,X)),aa_nat_nat(suc,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(P,M),Na) ) ) ) ).

% diff_induct
tff(fact_243_nat__induct,axiom,
    ! [P: fun_nat_o,Na: nat] :
      ( aa_nat_o(P,zero_zero_nat)
     => ( ! [N: nat] :
            ( aa_nat_o(P,N)
           => aa_nat_o(P,aa_nat_nat(suc,N)) )
       => aa_nat_o(P,Na) ) ) ).

% nat_induct
tff(fact_244_old_Onat_Oexhaust,axiom,
    ! [Y: nat] :
      ( ( Y != zero_zero_nat )
     => ~ ! [Nat2: nat] : Y != aa_nat_nat(suc,Nat2) ) ).

% old.nat.exhaust
tff(fact_245_nat_OdiscI,axiom,
    ! [Nata: nat,X2: nat] :
      ( ( Nata = aa_nat_nat(suc,X2) )
     => ( Nata != zero_zero_nat ) ) ).

% nat.discI
tff(fact_246_old_Onat_Odistinct_I1_J,axiom,
    ! [Nat: nat] : zero_zero_nat != aa_nat_nat(suc,Nat) ).

% old.nat.distinct(1)
tff(fact_247_old_Onat_Odistinct_I2_J,axiom,
    ! [Nat: nat] : aa_nat_nat(suc,Nat) != zero_zero_nat ).

% old.nat.distinct(2)
tff(fact_248_nat_Odistinct_I1_J,axiom,
    ! [X2: nat] : zero_zero_nat != aa_nat_nat(suc,X2) ).

% nat.distinct(1)
tff(fact_249_not__less__less__Suc__eq,axiom,
    ! [Na: nat,M: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(suc,M))
      <=> ( Na = M ) ) ) ).

% not_less_less_Suc_eq
tff(fact_250_strict__inc__induct,axiom,
    ! [I: nat,J: nat,P: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => ( ! [I2: nat] :
            ( ( J = aa_nat_nat(suc,I2) )
           => aa_nat_o(P,I2) )
       => ( ! [I2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J)
             => ( aa_nat_o(P,aa_nat_nat(suc,I2))
               => aa_nat_o(P,I2) ) )
         => aa_nat_o(P,I) ) ) ) ).

% strict_inc_induct
tff(fact_251_less__Suc__induct,axiom,
    ! [I: nat,J: nat,P: fun_nat_fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => ( ! [I2: nat] : aa_nat_o(aa_nat_fun_nat_o(P,I2),aa_nat_nat(suc,I2))
       => ( ! [I2: nat,J2: nat,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,I),J) ) ) ) ).

% less_Suc_induct
tff(fact_252_less__trans__Suc,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),K)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,I)),K) ) ) ).

% less_trans_Suc
tff(fact_253_Suc__less__SucD,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,Na))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ).

% Suc_less_SucD
tff(fact_254_less__antisym,axiom,
    ! [Na: nat,M: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(suc,M))
       => ( M = Na ) ) ) ).

% less_antisym
tff(fact_255_Suc__less__eq2,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Na)),M)
    <=> ? [M4: nat] :
          ( ( M = aa_nat_nat(suc,M4) )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M4) ) ) ).

% Suc_less_eq2
tff(fact_256_All__less__Suc,axiom,
    ! [Na: nat,P: fun_nat_o] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),aa_nat_nat(suc,Na))
         => aa_nat_o(P,I3) )
    <=> ( aa_nat_o(P,Na)
        & ! [I3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),Na)
           => aa_nat_o(P,I3) ) ) ) ).

% All_less_Suc
tff(fact_257_not__less__eq,axiom,
    ! [M: nat,Na: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(suc,M)) ) ).

% not_less_eq
tff(fact_258_less__Suc__eq,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(suc,Na))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
        | ( M = Na ) ) ) ).

% less_Suc_eq
tff(fact_259_Ex__less__Suc,axiom,
    ! [Na: nat,P: fun_nat_o] :
      ( ? [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),aa_nat_nat(suc,Na))
          & aa_nat_o(P,I3) )
    <=> ( aa_nat_o(P,Na)
        | ? [I3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),Na)
            & aa_nat_o(P,I3) ) ) ) ).

% Ex_less_Suc
tff(fact_260_less__SucI,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(suc,Na)) ) ).

% less_SucI
tff(fact_261_less__SucE,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(suc,Na))
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
       => ( M = Na ) ) ) ).

% less_SucE
tff(fact_262_Suc__lessI,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
     => ( ( aa_nat_nat(suc,M) != Na )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,M)),Na) ) ) ).

% Suc_lessI
tff(fact_263_Suc__lessE,axiom,
    ! [I: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,I)),K)
     => ~ ! [J2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J2)
           => ( K != aa_nat_nat(suc,J2) ) ) ) ).

% Suc_lessE
tff(fact_264_Suc__lessD,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,M)),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ).

% Suc_lessD
tff(fact_265_Nat_OlessE,axiom,
    ! [I: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),K)
     => ( ( K != aa_nat_nat(suc,I) )
       => ~ ! [J2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J2)
             => ( K != aa_nat_nat(suc,J2) ) ) ) ) ).

% Nat.lessE
tff(fact_266_infinite__descent0,axiom,
    ! [P: fun_nat_o,Na: nat] :
      ( aa_nat_o(P,zero_zero_nat)
     => ( ! [N: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
           => ( ~ aa_nat_o(P,N)
             => ? [M2: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),N)
                  & ~ aa_nat_o(P,M2) ) ) )
       => aa_nat_o(P,Na) ) ) ).

% infinite_descent0
tff(fact_267_gr__implies__not0,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
     => ( Na != zero_zero_nat ) ) ).

% gr_implies_not0
tff(fact_268_less__zeroE,axiom,
    ! [Na: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),zero_zero_nat) ).

% less_zeroE
tff(fact_269_not__less0,axiom,
    ! [Na: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),zero_zero_nat) ).

% not_less0
tff(fact_270_not__gr0,axiom,
    ! [Na: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
    <=> ( Na = zero_zero_nat ) ) ).

% not_gr0
tff(fact_271_gr0I,axiom,
    ! [Na: nat] :
      ( ( Na != zero_zero_nat )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ).

% gr0I
tff(fact_272_bot__nat__0_Oextremum__strict,axiom,
    ! [Aa: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),zero_zero_nat) ).

% bot_nat_0.extremum_strict
tff(fact_273_transitive__stepwise__le,axiom,
    ! [M: nat,Na: nat,R: fun_nat_fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( ! [X: nat] : aa_nat_o(aa_nat_fun_nat_o(R,X),X)
       => ( ! [X: nat,Y3: nat,Z3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(R,X),Y3)
             => ( aa_nat_o(aa_nat_fun_nat_o(R,Y3),Z3)
               => aa_nat_o(aa_nat_fun_nat_o(R,X),Z3) ) )
         => ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(R,N),aa_nat_nat(suc,N))
           => aa_nat_o(aa_nat_fun_nat_o(R,M),Na) ) ) ) ) ).

% transitive_stepwise_le
tff(fact_274_nat__induct__at__least,axiom,
    ! [M: nat,Na: nat,P: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( aa_nat_o(P,M)
       => ( ! [N: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
             => ( aa_nat_o(P,N)
               => aa_nat_o(P,aa_nat_nat(suc,N)) ) )
         => aa_nat_o(P,Na) ) ) ) ).

% nat_induct_at_least
tff(fact_275_full__nat__induct,axiom,
    ! [P: fun_nat_o,Na: nat] :
      ( ! [N: nat] :
          ( ! [M2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,M2)),N)
             => aa_nat_o(P,M2) )
         => aa_nat_o(P,N) )
     => aa_nat_o(P,Na) ) ).

% full_nat_induct
tff(fact_276_not__less__eq__eq,axiom,
    ! [M: nat,Na: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Na)),M) ) ).

% not_less_eq_eq
tff(fact_277_Suc__n__not__le__n,axiom,
    ! [Na: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Na)),Na) ).

% Suc_n_not_le_n
tff(fact_278_le__Suc__eq,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,Na))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
        | ( M = aa_nat_nat(suc,Na) ) ) ) ).

% le_Suc_eq
tff(fact_279_Suc__le__D,axiom,
    ! [Na: nat,M5: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Na)),M5)
     => ? [M3: nat] : M5 = aa_nat_nat(suc,M3) ) ).

% Suc_le_D
tff(fact_280_le__SucI,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,Na)) ) ).

% le_SucI
tff(fact_281_le__SucE,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,Na))
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
       => ( M = aa_nat_nat(suc,Na) ) ) ) ).

% le_SucE
tff(fact_282_Suc__leD,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,M)),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ).

% Suc_leD
tff(fact_283_le__0__eq,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),zero_zero_nat)
    <=> ( Na = zero_zero_nat ) ) ).

% le_0_eq
tff(fact_284_bot__nat__0_Oextremum__uniqueI,axiom,
    ! [Aa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),zero_zero_nat)
     => ( Aa = zero_zero_nat ) ) ).

% bot_nat_0.extremum_uniqueI
tff(fact_285_bot__nat__0_Oextremum__unique,axiom,
    ! [Aa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),zero_zero_nat)
    <=> ( Aa = zero_zero_nat ) ) ).

% bot_nat_0.extremum_unique
tff(fact_286_less__eq__nat_Osimps_I1_J,axiom,
    ! [Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Na) ).

% less_eq_nat.simps(1)
tff(fact_287_less__mono__imp__le__mono,axiom,
    ! [F: fun_nat_nat,I: nat,J: nat] :
      ( ! [I2: nat,J2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,I2)),aa_nat_nat(F,J2)) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,I)),aa_nat_nat(F,J)) ) ) ).

% less_mono_imp_le_mono
tff(fact_288_le__neq__implies__less,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( ( M != Na )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ) ).

% le_neq_implies_less
tff(fact_289_less__or__eq__imp__le,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
        | ( M = Na ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ).

% less_or_eq_imp_le
tff(fact_290_le__eq__less__or__eq,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
        | ( M = Na ) ) ) ).

% le_eq_less_or_eq
tff(fact_291_less__imp__le__nat,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ).

% less_imp_le_nat
tff(fact_292_nat__less__le,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
        & ( M != Na ) ) ) ).

% nat_less_le
tff(fact_293_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_real,Na: nat,M: nat] :
      ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,N)),aa_nat_real(F,aa_nat_nat(suc,N)))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,Na)),aa_nat_real(F,M))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_294_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_rat,Na: nat,M: nat] :
      ( ! [N: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,N)),aa_nat_rat(F,aa_nat_nat(suc,N)))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,Na)),aa_nat_rat(F,M))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_295_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_num,Na: nat,M: 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,Na)),aa_nat_num(F,M))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_296_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_nat,Na: nat,M: nat] :
      ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,N)),aa_nat_nat(F,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,Na)),aa_nat_nat(F,M))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_297_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_int,Na: nat,M: nat] :
      ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,N)),aa_nat_int(F,aa_nat_nat(suc,N)))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,Na)),aa_nat_int(F,M))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_298_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_real,Na: nat,N2: nat] :
      ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,N)),aa_nat_real(F,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),N2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,Na)),aa_nat_real(F,N2)) ) ) ).

% lift_Suc_mono_less
tff(fact_299_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_rat,Na: nat,N2: nat] :
      ( ! [N: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,N)),aa_nat_rat(F,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),N2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,Na)),aa_nat_rat(F,N2)) ) ) ).

% lift_Suc_mono_less
tff(fact_300_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_num,Na: nat,N2: 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,Na),N2)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_nat_num(F,Na)),aa_nat_num(F,N2)) ) ) ).

% lift_Suc_mono_less
tff(fact_301_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_nat,Na: nat,N2: nat] :
      ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,N)),aa_nat_nat(F,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),N2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,Na)),aa_nat_nat(F,N2)) ) ) ).

% lift_Suc_mono_less
tff(fact_302_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_int,Na: nat,N2: nat] :
      ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,N)),aa_nat_int(F,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),N2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,Na)),aa_nat_int(F,N2)) ) ) ).

% lift_Suc_mono_less
tff(fact_303_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_set_int,Na: nat,N2: nat] :
      ( ! [N: nat] : aa_set_int_o(ord_less_eq_set_int(aa_nat_set_int(F,aa_nat_nat(suc,N))),aa_nat_set_int(F,N))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),N2)
       => aa_set_int_o(ord_less_eq_set_int(aa_nat_set_int(F,N2)),aa_nat_set_int(F,Na)) ) ) ).

% lift_Suc_antimono_le
tff(fact_304_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_rat,Na: nat,N2: nat] :
      ( ! [N: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,aa_nat_nat(suc,N))),aa_nat_rat(F,N))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),N2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,N2)),aa_nat_rat(F,Na)) ) ) ).

% lift_Suc_antimono_le
tff(fact_305_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_num,Na: nat,N2: 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,Na),N2)
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,N2)),aa_nat_num(F,Na)) ) ) ).

% lift_Suc_antimono_le
tff(fact_306_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_nat,Na: nat,N2: nat] :
      ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,aa_nat_nat(suc,N))),aa_nat_nat(F,N))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),N2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,N2)),aa_nat_nat(F,Na)) ) ) ).

% lift_Suc_antimono_le
tff(fact_307_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_int,Na: nat,N2: nat] :
      ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,aa_nat_nat(suc,N))),aa_nat_int(F,N))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),N2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,N2)),aa_nat_int(F,Na)) ) ) ).

% lift_Suc_antimono_le
tff(fact_308_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_set_int,Na: nat,N2: nat] :
      ( ! [N: nat] : aa_set_int_o(ord_less_eq_set_int(aa_nat_set_int(F,N)),aa_nat_set_int(F,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),N2)
       => aa_set_int_o(ord_less_eq_set_int(aa_nat_set_int(F,Na)),aa_nat_set_int(F,N2)) ) ) ).

% lift_Suc_mono_le
tff(fact_309_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_rat,Na: nat,N2: nat] :
      ( ! [N: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,N)),aa_nat_rat(F,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),N2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,Na)),aa_nat_rat(F,N2)) ) ) ).

% lift_Suc_mono_le
tff(fact_310_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_num,Na: nat,N2: 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,Na),N2)
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,Na)),aa_nat_num(F,N2)) ) ) ).

% lift_Suc_mono_le
tff(fact_311_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_nat,Na: nat,N2: nat] :
      ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,N)),aa_nat_nat(F,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),N2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,Na)),aa_nat_nat(F,N2)) ) ) ).

% lift_Suc_mono_le
tff(fact_312_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_int,Na: nat,N2: nat] :
      ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,N)),aa_nat_int(F,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),N2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,Na)),aa_nat_int(F,N2)) ) ) ).

% lift_Suc_mono_le
tff(fact_313_Ex__less__Suc2,axiom,
    ! [Na: nat,P: fun_nat_o] :
      ( ? [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),aa_nat_nat(suc,Na))
          & aa_nat_o(P,I3) )
    <=> ( aa_nat_o(P,zero_zero_nat)
        | ? [I3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),Na)
            & aa_nat_o(P,aa_nat_nat(suc,I3)) ) ) ) ).

% Ex_less_Suc2
tff(fact_314_gr0__conv__Suc,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
    <=> ? [M6: nat] : Na = aa_nat_nat(suc,M6) ) ).

% gr0_conv_Suc
tff(fact_315_All__less__Suc2,axiom,
    ! [Na: nat,P: fun_nat_o] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),aa_nat_nat(suc,Na))
         => aa_nat_o(P,I3) )
    <=> ( aa_nat_o(P,zero_zero_nat)
        & ! [I3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),Na)
           => aa_nat_o(P,aa_nat_nat(suc,I3)) ) ) ) ).

% All_less_Suc2
tff(fact_316_gr0__implies__Suc,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ? [M3: nat] : Na = aa_nat_nat(suc,M3) ) ).

% gr0_implies_Suc
tff(fact_317_less__Suc__eq__0__disj,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(suc,Na))
    <=> ( ( M = zero_zero_nat )
        | ? [J3: nat] :
            ( ( M = aa_nat_nat(suc,J3) )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),Na) ) ) ) ).

% less_Suc_eq_0_disj
tff(fact_318_le__imp__less__Suc,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(suc,Na)) ) ).

% le_imp_less_Suc
tff(fact_319_less__eq__Suc__le,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Na)),M) ) ).

% less_eq_Suc_le
tff(fact_320_less__Suc__eq__le,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(suc,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ).

% less_Suc_eq_le
tff(fact_321_le__less__Suc__eq,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(suc,M))
      <=> ( Na = M ) ) ) ).

% le_less_Suc_eq
tff(fact_322_Suc__le__lessD,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,M)),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ).

% Suc_le_lessD
tff(fact_323_inc__induct,axiom,
    ! [I: nat,J: nat,P: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( aa_nat_o(P,J)
       => ( ! [N: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),N)
             => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),J)
               => ( aa_nat_o(P,aa_nat_nat(suc,N))
                 => aa_nat_o(P,N) ) ) )
         => aa_nat_o(P,I) ) ) ) ).

% inc_induct
tff(fact_324_dec__induct,axiom,
    ! [I: nat,J: nat,P: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( aa_nat_o(P,I)
       => ( ! [N: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),N)
             => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),J)
               => ( aa_nat_o(P,N)
                 => aa_nat_o(P,aa_nat_nat(suc,N)) ) ) )
         => aa_nat_o(P,J) ) ) ) ).

% dec_induct
tff(fact_325_Suc__le__eq,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,M)),Na)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ).

% Suc_le_eq
tff(fact_326_Suc__leI,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,M)),Na) ) ).

% Suc_leI
tff(fact_327_ex__least__nat__le,axiom,
    ! [P: fun_nat_o,Na: nat] :
      ( aa_nat_o(P,Na)
     => ( ~ aa_nat_o(P,zero_zero_nat)
       => ? [K2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Na)
            & ! [I4: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),K2)
               => ~ aa_nat_o(P,I4) )
            & aa_nat_o(P,K2) ) ) ) ).

% ex_least_nat_le
tff(fact_328_One__nat__def,axiom,
    one_one_nat = aa_nat_nat(suc,zero_zero_nat) ).

% One_nat_def
tff(fact_329_ex__least__nat__less,axiom,
    ! [P: fun_nat_o,Na: nat] :
      ( aa_nat_o(P,Na)
     => ( ~ aa_nat_o(P,zero_zero_nat)
       => ? [K2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K2),Na)
            & ! [I4: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I4),K2)
               => ~ aa_nat_o(P,I4) )
            & aa_nat_o(P,aa_nat_nat(suc,K2)) ) ) ) ).

% ex_least_nat_less
tff(fact_330_nat__induct__non__zero,axiom,
    ! [Na: nat,P: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_o(P,one_one_nat)
       => ( ! [N: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
             => ( aa_nat_o(P,N)
               => aa_nat_o(P,aa_nat_nat(suc,N)) ) )
         => aa_nat_o(P,Na) ) ) ) ).

% nat_induct_non_zero
tff(fact_331_buildup__nothing__in__leaf,axiom,
    ! [Na: nat,Xb: nat] : ~ aa_nat_o(vEBT_V5719532721284313246member(vEBT_vebt_buildup(Na)),Xb) ).

% buildup_nothing_in_leaf
tff(fact_332_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_333_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_334_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_335_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_336_buildup__gives__empty,axiom,
    ! [Na: nat] : vEBT_VEBT_set_vebt(vEBT_vebt_buildup(Na)) = bot_bot_set_nat ).

% buildup_gives_empty
tff(fact_337_dual__order_Orefl,axiom,
    ! [Aa: set_int] : aa_set_int_o(ord_less_eq_set_int(Aa),Aa) ).

% dual_order.refl
tff(fact_338_dual__order_Orefl,axiom,
    ! [Aa: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),Aa) ).

% dual_order.refl
tff(fact_339_dual__order_Orefl,axiom,
    ! [Aa: num] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Aa),Aa) ).

% dual_order.refl
tff(fact_340_dual__order_Orefl,axiom,
    ! [Aa: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),Aa) ).

% dual_order.refl
tff(fact_341_dual__order_Orefl,axiom,
    ! [Aa: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),Aa) ).

% dual_order.refl
tff(fact_342_order__refl,axiom,
    ! [Xb: set_int] : aa_set_int_o(ord_less_eq_set_int(Xb),Xb) ).

% order_refl
tff(fact_343_order__refl,axiom,
    ! [Xb: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Xb) ).

% order_refl
tff(fact_344_order__refl,axiom,
    ! [Xb: num] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Xb) ).

% order_refl
tff(fact_345_order__refl,axiom,
    ! [Xb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Xb) ).

% order_refl
tff(fact_346_order__refl,axiom,
    ! [Xb: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Xb) ).

% order_refl
tff(fact_347_buildup__nothing__in__min__max,axiom,
    ! [Na: nat,Xb: nat] : ~ aa_nat_o(vEBT_VEBT_membermima(vEBT_vebt_buildup(Na)),Xb) ).

% buildup_nothing_in_min_max
tff(fact_348_finite__nat__set__iff__bounded__le,axiom,
    ! [N3: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,N3)
    <=> ? [M6: nat] :
        ! [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),N3)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X4),M6) ) ) ).

% finite_nat_set_iff_bounded_le
tff(fact_349_infinite__nat__iff__unbounded__le,axiom,
    ! [S2: set_nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,S2)
    <=> ! [M6: nat] :
        ? [N4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M6),N4)
          & aa_set_nat_o(member_nat(N4),S2) ) ) ).

% infinite_nat_iff_unbounded_le
tff(fact_350_finite__nat__set__iff__bounded,axiom,
    ! [N3: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,N3)
    <=> ? [M6: nat] :
        ! [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),N3)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),M6) ) ) ).

% finite_nat_set_iff_bounded
tff(fact_351_infinite__nat__iff__unbounded,axiom,
    ! [S2: set_nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,S2)
    <=> ! [M6: nat] :
        ? [N4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M6),N4)
          & aa_set_nat_o(member_nat(N4),S2) ) ) ).

% infinite_nat_iff_unbounded
tff(fact_352_bounded__nat__set__is__finite,axiom,
    ! [N3: set_nat,Na: nat] :
      ( ! [X: nat] :
          ( aa_set_nat_o(member_nat(X),N3)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Na) )
     => aa_set_nat_o(finite_finite_nat,N3) ) ).

% bounded_nat_set_is_finite
tff(fact_353_unbounded__k__infinite,axiom,
    ! [K: nat,S2: set_nat] :
      ( ! [M3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),M3)
         => ? [N5: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M3),N5)
              & aa_set_nat_o(member_nat(N5),S2) ) )
     => ~ aa_set_nat_o(finite_finite_nat,S2) ) ).

% unbounded_k_infinite
tff(fact_354_member__valid__both__member__options,axiom,
    ! [Tree: vEBT_VEBT,Na: nat,Xb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Tree),Na)
     => ( 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_355_bot_Oextremum,axiom,
    ! [Aa: filter_nat] : ord_le2510731241096832064er_nat(bot_bot_filter_nat,Aa) ).

% bot.extremum
tff(fact_356_bot_Oextremum,axiom,
    ! [Aa: set_real] : ord_less_eq_set_real(bot_bot_set_real,Aa) ).

% bot.extremum
tff(fact_357_bot_Oextremum,axiom,
    ! [Aa: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,bot_bot_set_nat),Aa) ).

% bot.extremum
tff(fact_358_bot_Oextremum,axiom,
    ! [Aa: set_int] : aa_set_int_o(ord_less_eq_set_int(bot_bot_set_int),Aa) ).

% bot.extremum
tff(fact_359_bot_Oextremum,axiom,
    ! [Aa: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,bot_bot_nat),Aa) ).

% bot.extremum
tff(fact_360_bot_Oextremum__unique,axiom,
    ! [Aa: filter_nat] :
      ( ord_le2510731241096832064er_nat(Aa,bot_bot_filter_nat)
    <=> ( Aa = bot_bot_filter_nat ) ) ).

% bot.extremum_unique
tff(fact_361_bot_Oextremum__unique,axiom,
    ! [Aa: set_real] :
      ( ord_less_eq_set_real(Aa,bot_bot_set_real)
    <=> ( Aa = bot_bot_set_real ) ) ).

% bot.extremum_unique
tff(fact_362_bot_Oextremum__unique,axiom,
    ! [Aa: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Aa),bot_bot_set_nat)
    <=> ( Aa = bot_bot_set_nat ) ) ).

% bot.extremum_unique
tff(fact_363_bot_Oextremum__unique,axiom,
    ! [Aa: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(Aa),bot_bot_set_int)
    <=> ( Aa = bot_bot_set_int ) ) ).

% bot.extremum_unique
tff(fact_364_bot_Oextremum__unique,axiom,
    ! [Aa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),bot_bot_nat)
    <=> ( Aa = bot_bot_nat ) ) ).

% bot.extremum_unique
tff(fact_365_bot_Oextremum__uniqueI,axiom,
    ! [Aa: filter_nat] :
      ( ord_le2510731241096832064er_nat(Aa,bot_bot_filter_nat)
     => ( Aa = bot_bot_filter_nat ) ) ).

% bot.extremum_uniqueI
tff(fact_366_bot_Oextremum__uniqueI,axiom,
    ! [Aa: set_real] :
      ( ord_less_eq_set_real(Aa,bot_bot_set_real)
     => ( Aa = bot_bot_set_real ) ) ).

% bot.extremum_uniqueI
tff(fact_367_bot_Oextremum__uniqueI,axiom,
    ! [Aa: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Aa),bot_bot_set_nat)
     => ( Aa = bot_bot_set_nat ) ) ).

% bot.extremum_uniqueI
tff(fact_368_bot_Oextremum__uniqueI,axiom,
    ! [Aa: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(Aa),bot_bot_set_int)
     => ( Aa = bot_bot_set_int ) ) ).

% bot.extremum_uniqueI
tff(fact_369_bot_Oextremum__uniqueI,axiom,
    ! [Aa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),bot_bot_nat)
     => ( Aa = bot_bot_nat ) ) ).

% bot.extremum_uniqueI
tff(fact_370_bot_Oextremum__strict,axiom,
    ! [Aa: filter_nat] : ~ ord_less_filter_nat(Aa,bot_bot_filter_nat) ).

% bot.extremum_strict
tff(fact_371_bot_Oextremum__strict,axiom,
    ! [Aa: set_real] : ~ ord_less_set_real(Aa,bot_bot_set_real) ).

% bot.extremum_strict
tff(fact_372_bot_Oextremum__strict,axiom,
    ! [Aa: set_nat] : ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,Aa),bot_bot_set_nat) ).

% bot.extremum_strict
tff(fact_373_bot_Oextremum__strict,axiom,
    ! [Aa: set_int] : ~ aa_set_int_o(ord_less_set_int(Aa),bot_bot_set_int) ).

% bot.extremum_strict
tff(fact_374_bot_Oextremum__strict,axiom,
    ! [Aa: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),bot_bot_nat) ).

% bot.extremum_strict
tff(fact_375_bot_Onot__eq__extremum,axiom,
    ! [Aa: filter_nat] :
      ( ( Aa != bot_bot_filter_nat )
    <=> ord_less_filter_nat(bot_bot_filter_nat,Aa) ) ).

% bot.not_eq_extremum
tff(fact_376_bot_Onot__eq__extremum,axiom,
    ! [Aa: set_real] :
      ( ( Aa != bot_bot_set_real )
    <=> ord_less_set_real(bot_bot_set_real,Aa) ) ).

% bot.not_eq_extremum
tff(fact_377_bot_Onot__eq__extremum,axiom,
    ! [Aa: set_nat] :
      ( ( Aa != bot_bot_set_nat )
    <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,bot_bot_set_nat),Aa) ) ).

% bot.not_eq_extremum
tff(fact_378_bot_Onot__eq__extremum,axiom,
    ! [Aa: set_int] :
      ( ( Aa != bot_bot_set_int )
    <=> aa_set_int_o(ord_less_set_int(bot_bot_set_int),Aa) ) ).

% bot.not_eq_extremum
tff(fact_379_bot_Onot__eq__extremum,axiom,
    ! [Aa: nat] :
      ( ( Aa != bot_bot_nat )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,bot_bot_nat),Aa) ) ).

% bot.not_eq_extremum
tff(fact_380_finite__transitivity__chain,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,R: fun_Pr2028404973800837934_nat_o] :
      ( finite6177210948735845034at_nat(A3)
     => ( ! [X: product_prod_nat_nat] : ~ aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(R,X),X)
       => ( ! [X: product_prod_nat_nat,Y3: product_prod_nat_nat,Z3: product_prod_nat_nat] :
              ( aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(R,X),Y3)
             => ( aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(R,Y3),Z3)
               => aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(R,X),Z3) ) )
         => ( ! [X: product_prod_nat_nat] :
                ( member8440522571783428010at_nat(X,A3)
               => ? [Y4: product_prod_nat_nat] :
                    ( member8440522571783428010at_nat(Y4,A3)
                    & aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(R,X),Y4) ) )
           => ( A3 = bot_bo2099793752762293965at_nat ) ) ) ) ) ).

% finite_transitivity_chain
tff(fact_381_finite__transitivity__chain,axiom,
    ! [A3: set_set_nat,R: fun_se2170385732985038782_nat_o] :
      ( finite1152437895449049373et_nat(A3)
     => ( ! [X: set_nat] : ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(R,X),X)
       => ( ! [X: set_nat,Y3: set_nat,Z3: set_nat] :
              ( aa_set_nat_o(aa_set7861427066675100071_nat_o(R,X),Y3)
             => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(R,Y3),Z3)
               => aa_set_nat_o(aa_set7861427066675100071_nat_o(R,X),Z3) ) )
         => ( ! [X: set_nat] :
                ( member_set_nat(X,A3)
               => ? [Y4: set_nat] :
                    ( member_set_nat(Y4,A3)
                    & aa_set_nat_o(aa_set7861427066675100071_nat_o(R,X),Y4) ) )
           => ( A3 = bot_bot_set_set_nat ) ) ) ) ) ).

% finite_transitivity_chain
tff(fact_382_finite__transitivity__chain,axiom,
    ! [A3: set_complex,R: fun_co1148497455491383662plex_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A3)
     => ( ! [X: complex] : ~ aa_complex_o(aa_com5808801942499186519plex_o(R,X),X)
       => ( ! [X: complex,Y3: complex,Z3: complex] :
              ( aa_complex_o(aa_com5808801942499186519plex_o(R,X),Y3)
             => ( aa_complex_o(aa_com5808801942499186519plex_o(R,Y3),Z3)
               => aa_complex_o(aa_com5808801942499186519plex_o(R,X),Z3) ) )
         => ( ! [X: complex] :
                ( aa_set_complex_o(member_complex(X),A3)
               => ? [Y4: complex] :
                    ( aa_set_complex_o(member_complex(Y4),A3)
                    & aa_complex_o(aa_com5808801942499186519plex_o(R,X),Y4) ) )
           => ( A3 = bot_bot_set_complex ) ) ) ) ) ).

% finite_transitivity_chain
tff(fact_383_finite__transitivity__chain,axiom,
    ! [A3: set_real,R: fun_real_fun_real_o] :
      ( finite_finite_real(A3)
     => ( ! [X: real] : ~ aa_real_o(aa_real_fun_real_o(R,X),X)
       => ( ! [X: real,Y3: real,Z3: real] :
              ( aa_real_o(aa_real_fun_real_o(R,X),Y3)
             => ( aa_real_o(aa_real_fun_real_o(R,Y3),Z3)
               => aa_real_o(aa_real_fun_real_o(R,X),Z3) ) )
         => ( ! [X: real] :
                ( member_real(X,A3)
               => ? [Y4: real] :
                    ( member_real(Y4,A3)
                    & aa_real_o(aa_real_fun_real_o(R,X),Y4) ) )
           => ( A3 = bot_bot_set_real ) ) ) ) ) ).

% finite_transitivity_chain
tff(fact_384_finite__transitivity__chain,axiom,
    ! [A3: set_nat,R: fun_nat_fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( ! [X: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(R,X),X)
       => ( ! [X: nat,Y3: nat,Z3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(R,X),Y3)
             => ( aa_nat_o(aa_nat_fun_nat_o(R,Y3),Z3)
               => aa_nat_o(aa_nat_fun_nat_o(R,X),Z3) ) )
         => ( ! [X: nat] :
                ( aa_set_nat_o(member_nat(X),A3)
               => ? [Y4: nat] :
                    ( aa_set_nat_o(member_nat(Y4),A3)
                    & aa_nat_o(aa_nat_fun_nat_o(R,X),Y4) ) )
           => ( A3 = bot_bot_set_nat ) ) ) ) ) ).

% finite_transitivity_chain
tff(fact_385_finite__transitivity__chain,axiom,
    ! [A3: set_int,R: fun_int_fun_int_o] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( ! [X: int] : ~ aa_int_o(aa_int_fun_int_o(R,X),X)
       => ( ! [X: int,Y3: int,Z3: int] :
              ( aa_int_o(aa_int_fun_int_o(R,X),Y3)
             => ( aa_int_o(aa_int_fun_int_o(R,Y3),Z3)
               => aa_int_o(aa_int_fun_int_o(R,X),Z3) ) )
         => ( ! [X: int] :
                ( aa_set_int_o(member_int(X),A3)
               => ? [Y4: int] :
                    ( aa_set_int_o(member_int(Y4),A3)
                    & aa_int_o(aa_int_fun_int_o(R,X),Y4) ) )
           => ( A3 = bot_bot_set_int ) ) ) ) ) ).

% finite_transitivity_chain
tff(fact_386_finite_OemptyI,axiom,
    aa_set_complex_o(finite3207457112153483333omplex,bot_bot_set_complex) ).

% finite.emptyI
tff(fact_387_finite_OemptyI,axiom,
    finite_finite_real(bot_bot_set_real) ).

% finite.emptyI
tff(fact_388_finite_OemptyI,axiom,
    aa_set_nat_o(finite_finite_nat,bot_bot_set_nat) ).

% finite.emptyI
tff(fact_389_finite_OemptyI,axiom,
    aa_set_int_o(finite_finite_int,bot_bot_set_int) ).

% finite.emptyI
tff(fact_390_infinite__imp__nonempty,axiom,
    ! [S2: set_complex] :
      ( ~ aa_set_complex_o(finite3207457112153483333omplex,S2)
     => ( S2 != bot_bot_set_complex ) ) ).

% infinite_imp_nonempty
tff(fact_391_infinite__imp__nonempty,axiom,
    ! [S2: set_real] :
      ( ~ finite_finite_real(S2)
     => ( S2 != bot_bot_set_real ) ) ).

% infinite_imp_nonempty
tff(fact_392_infinite__imp__nonempty,axiom,
    ! [S2: set_nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,S2)
     => ( S2 != bot_bot_set_nat ) ) ).

% infinite_imp_nonempty
tff(fact_393_infinite__imp__nonempty,axiom,
    ! [S2: set_int] :
      ( ~ aa_set_int_o(finite_finite_int,S2)
     => ( S2 != bot_bot_set_int ) ) ).

% infinite_imp_nonempty
tff(fact_394_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_395_finite__has__minimal,axiom,
    ! [A3: set_real] :
      ( finite_finite_real(A3)
     => ( ( A3 != bot_bot_set_real )
       => ? [X: real] :
            ( member_real(X,A3)
            & ! [Xa: real] :
                ( member_real(Xa,A3)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),X)
                 => ( X = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_396_finite__has__minimal,axiom,
    ! [A3: set_set_int] :
      ( finite6197958912794628473et_int(A3)
     => ( ( A3 != bot_bot_set_set_int )
       => ? [X: set_int] :
            ( member_set_int(X,A3)
            & ! [Xa: set_int] :
                ( member_set_int(Xa,A3)
               => ( aa_set_int_o(ord_less_eq_set_int(Xa),X)
                 => ( X = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_397_finite__has__minimal,axiom,
    ! [A3: set_rat] :
      ( finite_finite_rat(A3)
     => ( ( A3 != bot_bot_set_rat )
       => ? [X: rat] :
            ( member_rat(X,A3)
            & ! [Xa: rat] :
                ( member_rat(Xa,A3)
               => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),X)
                 => ( X = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_398_finite__has__minimal,axiom,
    ! [A3: set_num] :
      ( finite_finite_num(A3)
     => ( ( A3 != bot_bot_set_num )
       => ? [X: num] :
            ( member_num(X,A3)
            & ! [Xa: num] :
                ( member_num(Xa,A3)
               => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),X)
                 => ( X = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_399_finite__has__minimal,axiom,
    ! [A3: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( ( A3 != bot_bot_set_nat )
       => ? [X: nat] :
            ( aa_set_nat_o(member_nat(X),A3)
            & ! [Xa: nat] :
                ( aa_set_nat_o(member_nat(Xa),A3)
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),X)
                 => ( X = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_400_finite__has__minimal,axiom,
    ! [A3: set_int] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( ( A3 != bot_bot_set_int )
       => ? [X: int] :
            ( aa_set_int_o(member_int(X),A3)
            & ! [Xa: int] :
                ( aa_set_int_o(member_int(Xa),A3)
               => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),X)
                 => ( X = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_401_finite__has__maximal,axiom,
    ! [A3: set_real] :
      ( finite_finite_real(A3)
     => ( ( A3 != bot_bot_set_real )
       => ? [X: real] :
            ( member_real(X,A3)
            & ! [Xa: real] :
                ( member_real(Xa,A3)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Xa)
                 => ( X = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_402_finite__has__maximal,axiom,
    ! [A3: set_set_int] :
      ( finite6197958912794628473et_int(A3)
     => ( ( A3 != bot_bot_set_set_int )
       => ? [X: set_int] :
            ( member_set_int(X,A3)
            & ! [Xa: set_int] :
                ( member_set_int(Xa,A3)
               => ( aa_set_int_o(ord_less_eq_set_int(X),Xa)
                 => ( X = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_403_finite__has__maximal,axiom,
    ! [A3: set_rat] :
      ( finite_finite_rat(A3)
     => ( ( A3 != bot_bot_set_rat )
       => ? [X: rat] :
            ( member_rat(X,A3)
            & ! [Xa: rat] :
                ( member_rat(Xa,A3)
               => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Xa)
                 => ( X = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_404_finite__has__maximal,axiom,
    ! [A3: set_num] :
      ( finite_finite_num(A3)
     => ( ( A3 != bot_bot_set_num )
       => ? [X: num] :
            ( member_num(X,A3)
            & ! [Xa: num] :
                ( member_num(Xa,A3)
               => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Xa)
                 => ( X = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_405_finite__has__maximal,axiom,
    ! [A3: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( ( A3 != bot_bot_set_nat )
       => ? [X: nat] :
            ( aa_set_nat_o(member_nat(X),A3)
            & ! [Xa: nat] :
                ( aa_set_nat_o(member_nat(Xa),A3)
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Xa)
                 => ( X = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_406_finite__has__maximal,axiom,
    ! [A3: set_int] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( ( A3 != bot_bot_set_int )
       => ? [X: int] :
            ( aa_set_int_o(member_int(X),A3)
            & ! [Xa: int] :
                ( aa_set_int_o(member_int(Xa),A3)
               => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Xa)
                 => ( X = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_407_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_408_nle__le,axiom,
    ! [Aa: rat,Ba: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),Ba)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ba),Aa)
        & ( Ba != Aa ) ) ) ).

% nle_le
tff(fact_409_nle__le,axiom,
    ! [Aa: num,Ba: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Aa),Ba)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ba),Aa)
        & ( Ba != Aa ) ) ) ).

% nle_le
tff(fact_410_nle__le,axiom,
    ! [Aa: nat,Ba: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),Ba)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ba),Aa)
        & ( Ba != Aa ) ) ) ).

% nle_le
tff(fact_411_nle__le,axiom,
    ! [Aa: int,Ba: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),Ba)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ba),Aa)
        & ( Ba != Aa ) ) ) ).

% nle_le
tff(fact_412_le__cases3,axiom,
    ! [Xb: rat,Y: rat,Z: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y)
       => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Z) )
     => ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),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),Y) )
         => ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z),Y)
             => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Xb) )
           => ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),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),Y) ) ) ) ) ) ) ).

% le_cases3
tff(fact_413_le__cases3,axiom,
    ! [Xb: num,Y: num,Z: num] :
      ( ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Y)
       => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),Z) )
     => ( ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),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),Y) )
         => ( ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Z),Y)
             => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),Xb) )
           => ( ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),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),Y) ) ) ) ) ) ) ).

% le_cases3
tff(fact_414_le__cases3,axiom,
    ! [Xb: nat,Y: nat,Z: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y)
       => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Z) )
     => ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),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),Y) )
         => ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z),Y)
             => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Xb) )
           => ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),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),Y) ) ) ) ) ) ) ).

% le_cases3
tff(fact_415_le__cases3,axiom,
    ! [Xb: int,Y: int,Z: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y)
       => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Z) )
     => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),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),Y) )
         => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),Y)
             => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Xb) )
           => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Z)
               => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,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),Y) ) ) ) ) ) ) ).

% le_cases3
tff(fact_416_order__class_Oorder__eq__iff,axiom,
    ! [Xb: set_int,Y: set_int] :
      ( ( Xb = Y )
    <=> ( aa_set_int_o(ord_less_eq_set_int(Xb),Y)
        & aa_set_int_o(ord_less_eq_set_int(Y),Xb) ) ) ).

% order_class.order_eq_iff
tff(fact_417_order__class_Oorder__eq__iff,axiom,
    ! [Xb: rat,Y: rat] :
      ( ( Xb = Y )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Xb) ) ) ).

% order_class.order_eq_iff
tff(fact_418_order__class_Oorder__eq__iff,axiom,
    ! [Xb: num,Y: num] :
      ( ( Xb = Y )
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Y)
        & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),Xb) ) ) ).

% order_class.order_eq_iff
tff(fact_419_order__class_Oorder__eq__iff,axiom,
    ! [Xb: nat,Y: nat] :
      ( ( Xb = Y )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Xb) ) ) ).

% order_class.order_eq_iff
tff(fact_420_order__class_Oorder__eq__iff,axiom,
    ! [Xb: int,Y: int] :
      ( ( Xb = Y )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Xb) ) ) ).

% order_class.order_eq_iff
tff(fact_421_ord__eq__le__trans,axiom,
    ! [Aa: set_int,Ba: set_int,C: set_int] :
      ( ( Aa = Ba )
     => ( aa_set_int_o(ord_less_eq_set_int(Ba),C)
       => aa_set_int_o(ord_less_eq_set_int(Aa),C) ) ) ).

% ord_eq_le_trans
tff(fact_422_ord__eq__le__trans,axiom,
    ! [Aa: rat,Ba: rat,C: rat] :
      ( ( Aa = Ba )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ba),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),C) ) ) ).

% ord_eq_le_trans
tff(fact_423_ord__eq__le__trans,axiom,
    ! [Aa: num,Ba: num,C: num] :
      ( ( Aa = Ba )
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ba),C)
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Aa),C) ) ) ).

% ord_eq_le_trans
tff(fact_424_ord__eq__le__trans,axiom,
    ! [Aa: nat,Ba: nat,C: nat] :
      ( ( Aa = Ba )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ba),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),C) ) ) ).

% ord_eq_le_trans
tff(fact_425_ord__eq__le__trans,axiom,
    ! [Aa: int,Ba: int,C: int] :
      ( ( Aa = Ba )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ba),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),C) ) ) ).

% ord_eq_le_trans
tff(fact_426_ord__le__eq__trans,axiom,
    ! [Aa: set_int,Ba: set_int,C: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(Aa),Ba)
     => ( ( Ba = C )
       => aa_set_int_o(ord_less_eq_set_int(Aa),C) ) ) ).

% ord_le_eq_trans
tff(fact_427_ord__le__eq__trans,axiom,
    ! [Aa: rat,Ba: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),Ba)
     => ( ( Ba = C )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),C) ) ) ).

% ord_le_eq_trans
tff(fact_428_ord__le__eq__trans,axiom,
    ! [Aa: num,Ba: num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Aa),Ba)
     => ( ( Ba = C )
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Aa),C) ) ) ).

% ord_le_eq_trans
tff(fact_429_ord__le__eq__trans,axiom,
    ! [Aa: nat,Ba: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),Ba)
     => ( ( Ba = C )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),C) ) ) ).

% ord_le_eq_trans
tff(fact_430_ord__le__eq__trans,axiom,
    ! [Aa: int,Ba: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),Ba)
     => ( ( Ba = C )
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),C) ) ) ).

% ord_le_eq_trans
tff(fact_431_order__antisym,axiom,
    ! [Xb: set_int,Y: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(Xb),Y)
     => ( aa_set_int_o(ord_less_eq_set_int(Y),Xb)
       => ( Xb = Y ) ) ) ).

% order_antisym
tff(fact_432_order__antisym,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Xb)
       => ( Xb = Y ) ) ) ).

% order_antisym
tff(fact_433_order__antisym,axiom,
    ! [Xb: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Y)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),Xb)
       => ( Xb = Y ) ) ) ).

% order_antisym
tff(fact_434_order__antisym,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Xb)
       => ( Xb = Y ) ) ) ).

% order_antisym
tff(fact_435_order__antisym,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Xb)
       => ( Xb = Y ) ) ) ).

% order_antisym
tff(fact_436_order_Otrans,axiom,
    ! [Aa: set_int,Ba: set_int,C: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(Aa),Ba)
     => ( aa_set_int_o(ord_less_eq_set_int(Ba),C)
       => aa_set_int_o(ord_less_eq_set_int(Aa),C) ) ) ).

% order.trans
tff(fact_437_order_Otrans,axiom,
    ! [Aa: rat,Ba: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),Ba)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ba),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),C) ) ) ).

% order.trans
tff(fact_438_order_Otrans,axiom,
    ! [Aa: num,Ba: num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Aa),Ba)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ba),C)
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Aa),C) ) ) ).

% order.trans
tff(fact_439_order_Otrans,axiom,
    ! [Aa: nat,Ba: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),Ba)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ba),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),C) ) ) ).

% order.trans
tff(fact_440_order_Otrans,axiom,
    ! [Aa: int,Ba: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),Ba)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ba),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),C) ) ) ).

% order.trans
tff(fact_441_order__trans,axiom,
    ! [Xb: set_int,Y: set_int,Z: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(Xb),Y)
     => ( aa_set_int_o(ord_less_eq_set_int(Y),Z)
       => aa_set_int_o(ord_less_eq_set_int(Xb),Z) ) ) ).

% order_trans
tff(fact_442_order__trans,axiom,
    ! [Xb: rat,Y: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Z)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Z) ) ) ).

% order_trans
tff(fact_443_order__trans,axiom,
    ! [Xb: num,Y: num,Z: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Y)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),Z)
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Z) ) ) ).

% order_trans
tff(fact_444_order__trans,axiom,
    ! [Xb: nat,Y: nat,Z: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Z)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Z) ) ) ).

% order_trans
tff(fact_445_order__trans,axiom,
    ! [Xb: int,Y: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Z)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Z) ) ) ).

% order_trans
tff(fact_446_linorder__wlog,axiom,
    ! [P: fun_rat_fun_rat_o,Aa: rat,Ba: rat] :
      ( ! [A2: rat,B2: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2)
         => aa_rat_o(aa_rat_fun_rat_o(P,A2),B2) )
     => ( ! [A2: rat,B2: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(P,B2),A2)
           => aa_rat_o(aa_rat_fun_rat_o(P,A2),B2) )
       => aa_rat_o(aa_rat_fun_rat_o(P,Aa),Ba) ) ) ).

% linorder_wlog
tff(fact_447_linorder__wlog,axiom,
    ! [P: fun_num_fun_num_o,Aa: num,Ba: num] :
      ( ! [A2: num,B2: num] :
          ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A2),B2)
         => aa_num_o(aa_num_fun_num_o(P,A2),B2) )
     => ( ! [A2: num,B2: num] :
            ( aa_num_o(aa_num_fun_num_o(P,B2),A2)
           => aa_num_o(aa_num_fun_num_o(P,A2),B2) )
       => aa_num_o(aa_num_fun_num_o(P,Aa),Ba) ) ) ).

% linorder_wlog
tff(fact_448_linorder__wlog,axiom,
    ! [P: fun_nat_fun_nat_o,Aa: nat,Ba: nat] :
      ( ! [A2: nat,B2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B2)
         => aa_nat_o(aa_nat_fun_nat_o(P,A2),B2) )
     => ( ! [A2: nat,B2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(P,B2),A2)
           => aa_nat_o(aa_nat_fun_nat_o(P,A2),B2) )
       => aa_nat_o(aa_nat_fun_nat_o(P,Aa),Ba) ) ) ).

% linorder_wlog
tff(fact_449_linorder__wlog,axiom,
    ! [P: fun_int_fun_int_o,Aa: int,Ba: int] :
      ( ! [A2: int,B2: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B2)
         => aa_int_o(aa_int_fun_int_o(P,A2),B2) )
     => ( ! [A2: int,B2: int] :
            ( aa_int_o(aa_int_fun_int_o(P,B2),A2)
           => aa_int_o(aa_int_fun_int_o(P,A2),B2) )
       => aa_int_o(aa_int_fun_int_o(P,Aa),Ba) ) ) ).

% linorder_wlog
tff(fact_450_dual__order_Oeq__iff,axiom,
    ! [Aa: set_int,Ba: set_int] :
      ( ( Aa = Ba )
    <=> ( aa_set_int_o(ord_less_eq_set_int(Ba),Aa)
        & aa_set_int_o(ord_less_eq_set_int(Aa),Ba) ) ) ).

% dual_order.eq_iff
tff(fact_451_dual__order_Oeq__iff,axiom,
    ! [Aa: rat,Ba: rat] :
      ( ( Aa = Ba )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ba),Aa)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),Ba) ) ) ).

% dual_order.eq_iff
tff(fact_452_dual__order_Oeq__iff,axiom,
    ! [Aa: num,Ba: num] :
      ( ( Aa = Ba )
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ba),Aa)
        & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Aa),Ba) ) ) ).

% dual_order.eq_iff
tff(fact_453_dual__order_Oeq__iff,axiom,
    ! [Aa: nat,Ba: nat] :
      ( ( Aa = Ba )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ba),Aa)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),Ba) ) ) ).

% dual_order.eq_iff
tff(fact_454_dual__order_Oeq__iff,axiom,
    ! [Aa: int,Ba: int] :
      ( ( Aa = Ba )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ba),Aa)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),Ba) ) ) ).

% dual_order.eq_iff
tff(fact_455_dual__order_Oantisym,axiom,
    ! [Ba: set_int,Aa: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(Ba),Aa)
     => ( aa_set_int_o(ord_less_eq_set_int(Aa),Ba)
       => ( Aa = Ba ) ) ) ).

% dual_order.antisym
tff(fact_456_dual__order_Oantisym,axiom,
    ! [Ba: rat,Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ba),Aa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),Ba)
       => ( Aa = Ba ) ) ) ).

% dual_order.antisym
tff(fact_457_dual__order_Oantisym,axiom,
    ! [Ba: num,Aa: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ba),Aa)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Aa),Ba)
       => ( Aa = Ba ) ) ) ).

% dual_order.antisym
tff(fact_458_dual__order_Oantisym,axiom,
    ! [Ba: nat,Aa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ba),Aa)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),Ba)
       => ( Aa = Ba ) ) ) ).

% dual_order.antisym
tff(fact_459_dual__order_Oantisym,axiom,
    ! [Ba: int,Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ba),Aa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),Ba)
       => ( Aa = Ba ) ) ) ).

% dual_order.antisym
tff(fact_460_dual__order_Otrans,axiom,
    ! [Ba: set_int,Aa: set_int,C: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(Ba),Aa)
     => ( aa_set_int_o(ord_less_eq_set_int(C),Ba)
       => aa_set_int_o(ord_less_eq_set_int(C),Aa) ) ) ).

% dual_order.trans
tff(fact_461_dual__order_Otrans,axiom,
    ! [Ba: rat,Aa: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ba),Aa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),Ba)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),Aa) ) ) ).

% dual_order.trans
tff(fact_462_dual__order_Otrans,axiom,
    ! [Ba: num,Aa: num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ba),Aa)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),Ba)
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),Aa) ) ) ).

% dual_order.trans
tff(fact_463_dual__order_Otrans,axiom,
    ! [Ba: nat,Aa: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ba),Aa)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),Ba)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),Aa) ) ) ).

% dual_order.trans
tff(fact_464_dual__order_Otrans,axiom,
    ! [Ba: int,Aa: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ba),Aa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),Ba)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),Aa) ) ) ).

% dual_order.trans
tff(fact_465_antisym,axiom,
    ! [Aa: set_int,Ba: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(Aa),Ba)
     => ( aa_set_int_o(ord_less_eq_set_int(Ba),Aa)
       => ( Aa = Ba ) ) ) ).

% antisym
tff(fact_466_antisym,axiom,
    ! [Aa: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),Ba)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ba),Aa)
       => ( Aa = Ba ) ) ) ).

% antisym
tff(fact_467_antisym,axiom,
    ! [Aa: num,Ba: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Aa),Ba)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ba),Aa)
       => ( Aa = Ba ) ) ) ).

% antisym
tff(fact_468_antisym,axiom,
    ! [Aa: nat,Ba: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),Ba)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ba),Aa)
       => ( Aa = Ba ) ) ) ).

% antisym
tff(fact_469_antisym,axiom,
    ! [Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),Ba)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ba),Aa)
       => ( Aa = Ba ) ) ) ).

% antisym
tff(fact_470_Orderings_Oorder__eq__iff,axiom,
    ! [Aa: set_int,Ba: set_int] :
      ( ( Aa = Ba )
    <=> ( aa_set_int_o(ord_less_eq_set_int(Aa),Ba)
        & aa_set_int_o(ord_less_eq_set_int(Ba),Aa) ) ) ).

% Orderings.order_eq_iff
tff(fact_471_Orderings_Oorder__eq__iff,axiom,
    ! [Aa: rat,Ba: rat] :
      ( ( Aa = Ba )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),Ba)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ba),Aa) ) ) ).

% Orderings.order_eq_iff
tff(fact_472_Orderings_Oorder__eq__iff,axiom,
    ! [Aa: num,Ba: num] :
      ( ( Aa = Ba )
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Aa),Ba)
        & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ba),Aa) ) ) ).

% Orderings.order_eq_iff
tff(fact_473_Orderings_Oorder__eq__iff,axiom,
    ! [Aa: nat,Ba: nat] :
      ( ( Aa = Ba )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),Ba)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ba),Aa) ) ) ).

% Orderings.order_eq_iff
tff(fact_474_Orderings_Oorder__eq__iff,axiom,
    ! [Aa: int,Ba: int] :
      ( ( Aa = Ba )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),Ba)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ba),Aa) ) ) ).

% Orderings.order_eq_iff
tff(fact_475_order__subst1,axiom,
    ! [Aa: rat,F: fun_rat_rat,Ba: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),aa_rat_rat(F,Ba))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ba),C)
       => ( ! [X: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,X)),aa_rat_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),aa_rat_rat(F,C)) ) ) ) ).

% order_subst1
tff(fact_476_order__subst1,axiom,
    ! [Aa: rat,F: fun_num_rat,Ba: num,C: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),aa_num_rat(F,Ba))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ba),C)
       => ( ! [X: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,X)),aa_num_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),aa_num_rat(F,C)) ) ) ) ).

% order_subst1
tff(fact_477_order__subst1,axiom,
    ! [Aa: rat,F: fun_nat_rat,Ba: nat,C: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),aa_nat_rat(F,Ba))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ba),C)
       => ( ! [X: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X)),aa_nat_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),aa_nat_rat(F,C)) ) ) ) ).

% order_subst1
tff(fact_478_order__subst1,axiom,
    ! [Aa: rat,F: fun_int_rat,Ba: int,C: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),aa_int_rat(F,Ba))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ba),C)
       => ( ! [X: int,Y3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,X)),aa_int_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),aa_int_rat(F,C)) ) ) ) ).

% order_subst1
tff(fact_479_order__subst1,axiom,
    ! [Aa: num,F: fun_rat_num,Ba: rat,C: rat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Aa),aa_rat_num(F,Ba))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ba),C)
       => ( ! [X: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,X)),aa_rat_num(F,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Aa),aa_rat_num(F,C)) ) ) ) ).

% order_subst1
tff(fact_480_order__subst1,axiom,
    ! [Aa: num,F: fun_num_num,Ba: num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Aa),aa_num_num(F,Ba))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ba),C)
       => ( ! [X: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,X)),aa_num_num(F,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Aa),aa_num_num(F,C)) ) ) ) ).

% order_subst1
tff(fact_481_order__subst1,axiom,
    ! [Aa: num,F: fun_nat_num,Ba: nat,C: nat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Aa),aa_nat_num(F,Ba))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ba),C)
       => ( ! [X: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,X)),aa_nat_num(F,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Aa),aa_nat_num(F,C)) ) ) ) ).

% order_subst1
tff(fact_482_order__subst1,axiom,
    ! [Aa: num,F: fun_int_num,Ba: int,C: int] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Aa),aa_int_num(F,Ba))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ba),C)
       => ( ! [X: int,Y3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_int_num(F,X)),aa_int_num(F,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Aa),aa_int_num(F,C)) ) ) ) ).

% order_subst1
tff(fact_483_order__subst1,axiom,
    ! [Aa: nat,F: fun_rat_nat,Ba: rat,C: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),aa_rat_nat(F,Ba))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ba),C)
       => ( ! [X: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,X)),aa_rat_nat(F,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),aa_rat_nat(F,C)) ) ) ) ).

% order_subst1
tff(fact_484_order__subst1,axiom,
    ! [Aa: nat,F: fun_num_nat,Ba: num,C: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),aa_num_nat(F,Ba))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ba),C)
       => ( ! [X: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,X)),aa_num_nat(F,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),aa_num_nat(F,C)) ) ) ) ).

% order_subst1
tff(fact_485_order__subst2,axiom,
    ! [Aa: rat,Ba: rat,F: fun_rat_rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),Ba)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,Ba)),C)
       => ( ! [X: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,X)),aa_rat_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,Aa)),C) ) ) ) ).

% order_subst2
tff(fact_486_order__subst2,axiom,
    ! [Aa: rat,Ba: rat,F: fun_rat_num,C: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),Ba)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,Ba)),C)
       => ( ! [X: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,X)),aa_rat_num(F,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,Aa)),C) ) ) ) ).

% order_subst2
tff(fact_487_order__subst2,axiom,
    ! [Aa: rat,Ba: rat,F: fun_rat_nat,C: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),Ba)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,Ba)),C)
       => ( ! [X: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,X)),aa_rat_nat(F,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,Aa)),C) ) ) ) ).

% order_subst2
tff(fact_488_order__subst2,axiom,
    ! [Aa: rat,Ba: rat,F: fun_rat_int,C: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),Ba)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,Ba)),C)
       => ( ! [X: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,X)),aa_rat_int(F,Y3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,Aa)),C) ) ) ) ).

% order_subst2
tff(fact_489_order__subst2,axiom,
    ! [Aa: num,Ba: num,F: fun_num_rat,C: rat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Aa),Ba)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,Ba)),C)
       => ( ! [X: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,X)),aa_num_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,Aa)),C) ) ) ) ).

% order_subst2
tff(fact_490_order__subst2,axiom,
    ! [Aa: num,Ba: num,F: fun_num_num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Aa),Ba)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,Ba)),C)
       => ( ! [X: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,X)),aa_num_num(F,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,Aa)),C) ) ) ) ).

% order_subst2
tff(fact_491_order__subst2,axiom,
    ! [Aa: num,Ba: num,F: fun_num_nat,C: nat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Aa),Ba)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,Ba)),C)
       => ( ! [X: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,X)),aa_num_nat(F,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,Aa)),C) ) ) ) ).

% order_subst2
tff(fact_492_order__subst2,axiom,
    ! [Aa: num,Ba: num,F: fun_num_int,C: int] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Aa),Ba)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(F,Ba)),C)
       => ( ! [X: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(F,X)),aa_num_int(F,Y3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(F,Aa)),C) ) ) ) ).

% order_subst2
tff(fact_493_order__subst2,axiom,
    ! [Aa: nat,Ba: nat,F: fun_nat_rat,C: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),Ba)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,Ba)),C)
       => ( ! [X: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X)),aa_nat_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,Aa)),C) ) ) ) ).

% order_subst2
tff(fact_494_order__subst2,axiom,
    ! [Aa: nat,Ba: nat,F: fun_nat_num,C: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),Ba)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,Ba)),C)
       => ( ! [X: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,X)),aa_nat_num(F,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,Aa)),C) ) ) ) ).

% order_subst2
tff(fact_495_order__eq__refl,axiom,
    ! [Xb: set_int,Y: set_int] :
      ( ( Xb = Y )
     => aa_set_int_o(ord_less_eq_set_int(Xb),Y) ) ).

% order_eq_refl
tff(fact_496_order__eq__refl,axiom,
    ! [Xb: rat,Y: rat] :
      ( ( Xb = Y )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y) ) ).

% order_eq_refl
tff(fact_497_order__eq__refl,axiom,
    ! [Xb: num,Y: num] :
      ( ( Xb = Y )
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Y) ) ).

% order_eq_refl
tff(fact_498_order__eq__refl,axiom,
    ! [Xb: nat,Y: nat] :
      ( ( Xb = Y )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y) ) ).

% order_eq_refl
tff(fact_499_order__eq__refl,axiom,
    ! [Xb: int,Y: int] :
      ( ( Xb = Y )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y) ) ).

% order_eq_refl
tff(fact_500_linorder__linear,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y)
      | aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Xb) ) ).

% linorder_linear
tff(fact_501_linorder__linear,axiom,
    ! [Xb: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Y)
      | aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),Xb) ) ).

% linorder_linear
tff(fact_502_linorder__linear,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y)
      | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Xb) ) ).

% linorder_linear
tff(fact_503_linorder__linear,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y)
      | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Xb) ) ).

% linorder_linear
tff(fact_504_ord__eq__le__subst,axiom,
    ! [Aa: rat,F: fun_rat_rat,Ba: rat,C: rat] :
      ( ( Aa = aa_rat_rat(F,Ba) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ba),C)
       => ( ! [X: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,X)),aa_rat_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),aa_rat_rat(F,C)) ) ) ) ).

% ord_eq_le_subst
tff(fact_505_ord__eq__le__subst,axiom,
    ! [Aa: num,F: fun_rat_num,Ba: rat,C: rat] :
      ( ( Aa = aa_rat_num(F,Ba) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ba),C)
       => ( ! [X: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,X)),aa_rat_num(F,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Aa),aa_rat_num(F,C)) ) ) ) ).

% ord_eq_le_subst
tff(fact_506_ord__eq__le__subst,axiom,
    ! [Aa: nat,F: fun_rat_nat,Ba: rat,C: rat] :
      ( ( Aa = aa_rat_nat(F,Ba) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ba),C)
       => ( ! [X: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,X)),aa_rat_nat(F,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),aa_rat_nat(F,C)) ) ) ) ).

% ord_eq_le_subst
tff(fact_507_ord__eq__le__subst,axiom,
    ! [Aa: int,F: fun_rat_int,Ba: rat,C: rat] :
      ( ( Aa = aa_rat_int(F,Ba) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ba),C)
       => ( ! [X: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,X)),aa_rat_int(F,Y3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),aa_rat_int(F,C)) ) ) ) ).

% ord_eq_le_subst
tff(fact_508_ord__eq__le__subst,axiom,
    ! [Aa: rat,F: fun_num_rat,Ba: num,C: num] :
      ( ( Aa = aa_num_rat(F,Ba) )
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ba),C)
       => ( ! [X: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,X)),aa_num_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),aa_num_rat(F,C)) ) ) ) ).

% ord_eq_le_subst
tff(fact_509_ord__eq__le__subst,axiom,
    ! [Aa: num,F: fun_num_num,Ba: num,C: num] :
      ( ( Aa = aa_num_num(F,Ba) )
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ba),C)
       => ( ! [X: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,X)),aa_num_num(F,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Aa),aa_num_num(F,C)) ) ) ) ).

% ord_eq_le_subst
tff(fact_510_ord__eq__le__subst,axiom,
    ! [Aa: nat,F: fun_num_nat,Ba: num,C: num] :
      ( ( Aa = aa_num_nat(F,Ba) )
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ba),C)
       => ( ! [X: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,X)),aa_num_nat(F,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),aa_num_nat(F,C)) ) ) ) ).

% ord_eq_le_subst
tff(fact_511_ord__eq__le__subst,axiom,
    ! [Aa: int,F: fun_num_int,Ba: num,C: num] :
      ( ( Aa = aa_num_int(F,Ba) )
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ba),C)
       => ( ! [X: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(F,X)),aa_num_int(F,Y3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),aa_num_int(F,C)) ) ) ) ).

% ord_eq_le_subst
tff(fact_512_ord__eq__le__subst,axiom,
    ! [Aa: rat,F: fun_nat_rat,Ba: nat,C: nat] :
      ( ( Aa = aa_nat_rat(F,Ba) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ba),C)
       => ( ! [X: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X)),aa_nat_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),aa_nat_rat(F,C)) ) ) ) ).

% ord_eq_le_subst
tff(fact_513_ord__eq__le__subst,axiom,
    ! [Aa: num,F: fun_nat_num,Ba: nat,C: nat] :
      ( ( Aa = aa_nat_num(F,Ba) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ba),C)
       => ( ! [X: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,X)),aa_nat_num(F,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Aa),aa_nat_num(F,C)) ) ) ) ).

% ord_eq_le_subst
tff(fact_514_ord__le__eq__subst,axiom,
    ! [Aa: rat,Ba: rat,F: fun_rat_rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),Ba)
     => ( ( aa_rat_rat(F,Ba) = C )
       => ( ! [X: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,X)),aa_rat_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,Aa)),C) ) ) ) ).

% ord_le_eq_subst
tff(fact_515_ord__le__eq__subst,axiom,
    ! [Aa: rat,Ba: rat,F: fun_rat_num,C: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),Ba)
     => ( ( aa_rat_num(F,Ba) = C )
       => ( ! [X: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,X)),aa_rat_num(F,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,Aa)),C) ) ) ) ).

% ord_le_eq_subst
tff(fact_516_ord__le__eq__subst,axiom,
    ! [Aa: rat,Ba: rat,F: fun_rat_nat,C: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),Ba)
     => ( ( aa_rat_nat(F,Ba) = C )
       => ( ! [X: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,X)),aa_rat_nat(F,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,Aa)),C) ) ) ) ).

% ord_le_eq_subst
tff(fact_517_ord__le__eq__subst,axiom,
    ! [Aa: rat,Ba: rat,F: fun_rat_int,C: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),Ba)
     => ( ( aa_rat_int(F,Ba) = C )
       => ( ! [X: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,X)),aa_rat_int(F,Y3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,Aa)),C) ) ) ) ).

% ord_le_eq_subst
tff(fact_518_ord__le__eq__subst,axiom,
    ! [Aa: num,Ba: num,F: fun_num_rat,C: rat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Aa),Ba)
     => ( ( aa_num_rat(F,Ba) = C )
       => ( ! [X: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,X)),aa_num_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,Aa)),C) ) ) ) ).

% ord_le_eq_subst
tff(fact_519_ord__le__eq__subst,axiom,
    ! [Aa: num,Ba: num,F: fun_num_num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Aa),Ba)
     => ( ( aa_num_num(F,Ba) = C )
       => ( ! [X: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,X)),aa_num_num(F,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,Aa)),C) ) ) ) ).

% ord_le_eq_subst
tff(fact_520_ord__le__eq__subst,axiom,
    ! [Aa: num,Ba: num,F: fun_num_nat,C: nat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Aa),Ba)
     => ( ( aa_num_nat(F,Ba) = C )
       => ( ! [X: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,X)),aa_num_nat(F,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,Aa)),C) ) ) ) ).

% ord_le_eq_subst
tff(fact_521_ord__le__eq__subst,axiom,
    ! [Aa: num,Ba: num,F: fun_num_int,C: int] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Aa),Ba)
     => ( ( aa_num_int(F,Ba) = C )
       => ( ! [X: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(F,X)),aa_num_int(F,Y3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(F,Aa)),C) ) ) ) ).

% ord_le_eq_subst
tff(fact_522_ord__le__eq__subst,axiom,
    ! [Aa: nat,Ba: nat,F: fun_nat_rat,C: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),Ba)
     => ( ( aa_nat_rat(F,Ba) = C )
       => ( ! [X: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X)),aa_nat_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,Aa)),C) ) ) ) ).

% ord_le_eq_subst
tff(fact_523_ord__le__eq__subst,axiom,
    ! [Aa: nat,Ba: nat,F: fun_nat_num,C: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),Ba)
     => ( ( aa_nat_num(F,Ba) = C )
       => ( ! [X: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,X)),aa_nat_num(F,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,Aa)),C) ) ) ) ).

% ord_le_eq_subst
tff(fact_524_linorder__le__cases,axiom,
    ! [Xb: rat,Y: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Xb) ) ).

% linorder_le_cases
tff(fact_525_linorder__le__cases,axiom,
    ! [Xb: num,Y: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Y)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),Xb) ) ).

% linorder_le_cases
tff(fact_526_linorder__le__cases,axiom,
    ! [Xb: nat,Y: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Xb) ) ).

% linorder_le_cases
tff(fact_527_linorder__le__cases,axiom,
    ! [Xb: int,Y: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Xb) ) ).

% linorder_le_cases
tff(fact_528_order__antisym__conv,axiom,
    ! [Y: set_int,Xb: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(Y),Xb)
     => ( aa_set_int_o(ord_less_eq_set_int(Xb),Y)
      <=> ( Xb = Y ) ) ) ).

% order_antisym_conv
tff(fact_529_order__antisym__conv,axiom,
    ! [Y: rat,Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Xb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y)
      <=> ( Xb = Y ) ) ) ).

% order_antisym_conv
tff(fact_530_order__antisym__conv,axiom,
    ! [Y: num,Xb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),Xb)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Y)
      <=> ( Xb = Y ) ) ) ).

% order_antisym_conv
tff(fact_531_order__antisym__conv,axiom,
    ! [Y: nat,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Xb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y)
      <=> ( Xb = Y ) ) ) ).

% order_antisym_conv
tff(fact_532_order__antisym__conv,axiom,
    ! [Y: int,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Xb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y)
      <=> ( Xb = Y ) ) ) ).

% order_antisym_conv
tff(fact_533_lt__ex,axiom,
    ! [Xb: real] :
    ? [Y3: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,Y3),Xb) ).

% lt_ex
tff(fact_534_lt__ex,axiom,
    ! [Xb: rat] :
    ? [Y3: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y3),Xb) ).

% lt_ex
tff(fact_535_lt__ex,axiom,
    ! [Xb: int] :
    ? [Y3: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,Y3),Xb) ).

% lt_ex
tff(fact_536_gt__ex,axiom,
    ! [Xb: real] :
    ? [X_1: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),X_1) ).

% gt_ex
tff(fact_537_gt__ex,axiom,
    ! [Xb: rat] :
    ? [X_1: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),X_1) ).

% gt_ex
tff(fact_538_gt__ex,axiom,
    ! [Xb: nat] :
    ? [X_1: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),X_1) ).

% gt_ex
tff(fact_539_gt__ex,axiom,
    ! [Xb: int] :
    ? [X_1: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),X_1) ).

% gt_ex
tff(fact_540_dense,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
     => ? [Z3: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Z3)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),Y) ) ) ).

% dense
tff(fact_541_dense,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
     => ? [Z3: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Z3)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z3),Y) ) ) ).

% dense
tff(fact_542_less__imp__neq,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
     => ( Xb != Y ) ) ).

% less_imp_neq
tff(fact_543_less__imp__neq,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
     => ( Xb != Y ) ) ).

% less_imp_neq
tff(fact_544_less__imp__neq,axiom,
    ! [Xb: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y)
     => ( Xb != Y ) ) ).

% less_imp_neq
tff(fact_545_less__imp__neq,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
     => ( Xb != Y ) ) ).

% less_imp_neq
tff(fact_546_less__imp__neq,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
     => ( Xb != Y ) ) ).

% less_imp_neq
tff(fact_547_order_Oasym,axiom,
    ! [Aa: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),Ba)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Ba),Aa) ) ).

% order.asym
tff(fact_548_order_Oasym,axiom,
    ! [Aa: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),Ba)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ba),Aa) ) ).

% order.asym
tff(fact_549_order_Oasym,axiom,
    ! [Aa: num,Ba: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Aa),Ba)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Ba),Aa) ) ).

% order.asym
tff(fact_550_order_Oasym,axiom,
    ! [Aa: nat,Ba: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),Ba)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ba),Aa) ) ).

% order.asym
tff(fact_551_order_Oasym,axiom,
    ! [Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),Ba)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Ba),Aa) ) ).

% order.asym
tff(fact_552_ord__eq__less__trans,axiom,
    ! [Aa: real,Ba: real,C: real] :
      ( ( Aa = Ba )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Ba),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),C) ) ) ).

% ord_eq_less_trans
tff(fact_553_ord__eq__less__trans,axiom,
    ! [Aa: rat,Ba: rat,C: rat] :
      ( ( Aa = Ba )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ba),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),C) ) ) ).

% ord_eq_less_trans
tff(fact_554_ord__eq__less__trans,axiom,
    ! [Aa: num,Ba: num,C: num] :
      ( ( Aa = Ba )
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,Ba),C)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,Aa),C) ) ) ).

% ord_eq_less_trans
tff(fact_555_ord__eq__less__trans,axiom,
    ! [Aa: nat,Ba: nat,C: nat] :
      ( ( Aa = Ba )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ba),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),C) ) ) ).

% ord_eq_less_trans
tff(fact_556_ord__eq__less__trans,axiom,
    ! [Aa: int,Ba: int,C: int] :
      ( ( Aa = Ba )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Ba),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),C) ) ) ).

% ord_eq_less_trans
tff(fact_557_ord__less__eq__trans,axiom,
    ! [Aa: real,Ba: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),Ba)
     => ( ( Ba = C )
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),C) ) ) ).

% ord_less_eq_trans
tff(fact_558_ord__less__eq__trans,axiom,
    ! [Aa: rat,Ba: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),Ba)
     => ( ( Ba = C )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),C) ) ) ).

% ord_less_eq_trans
tff(fact_559_ord__less__eq__trans,axiom,
    ! [Aa: num,Ba: num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Aa),Ba)
     => ( ( Ba = C )
       => aa_num_o(aa_num_fun_num_o(ord_less_num,Aa),C) ) ) ).

% ord_less_eq_trans
tff(fact_560_ord__less__eq__trans,axiom,
    ! [Aa: nat,Ba: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),Ba)
     => ( ( Ba = C )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),C) ) ) ).

% ord_less_eq_trans
tff(fact_561_ord__less__eq__trans,axiom,
    ! [Aa: int,Ba: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),Ba)
     => ( ( Ba = C )
       => aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),C) ) ) ).

% ord_less_eq_trans
tff(fact_562_less__induct,axiom,
    ! [P: fun_nat_o,Aa: nat] :
      ( ! [X: nat] :
          ( ! [Y4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y4),X)
             => aa_nat_o(P,Y4) )
         => aa_nat_o(P,X) )
     => aa_nat_o(P,Aa) ) ).

% less_induct
tff(fact_563_antisym__conv3,axiom,
    ! [Y: real,Xb: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xb)
     => ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
      <=> ( Xb = Y ) ) ) ).

% antisym_conv3
tff(fact_564_antisym__conv3,axiom,
    ! [Y: rat,Xb: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Xb)
     => ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
      <=> ( Xb = Y ) ) ) ).

% antisym_conv3
tff(fact_565_antisym__conv3,axiom,
    ! [Y: num,Xb: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Y),Xb)
     => ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y)
      <=> ( Xb = Y ) ) ) ).

% antisym_conv3
tff(fact_566_antisym__conv3,axiom,
    ! [Y: nat,Xb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xb)
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
      <=> ( Xb = Y ) ) ) ).

% antisym_conv3
tff(fact_567_antisym__conv3,axiom,
    ! [Y: int,Xb: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Xb)
     => ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
      <=> ( Xb = Y ) ) ) ).

% antisym_conv3
tff(fact_568_linorder__cases,axiom,
    ! [Xb: real,Y: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
     => ( ( Xb != Y )
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xb) ) ) ).

% linorder_cases
tff(fact_569_linorder__cases,axiom,
    ! [Xb: rat,Y: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
     => ( ( Xb != Y )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Xb) ) ) ).

% linorder_cases
tff(fact_570_linorder__cases,axiom,
    ! [Xb: num,Y: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y)
     => ( ( Xb != Y )
       => aa_num_o(aa_num_fun_num_o(ord_less_num,Y),Xb) ) ) ).

% linorder_cases
tff(fact_571_linorder__cases,axiom,
    ! [Xb: nat,Y: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
     => ( ( Xb != Y )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xb) ) ) ).

% linorder_cases
tff(fact_572_linorder__cases,axiom,
    ! [Xb: int,Y: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
     => ( ( Xb != Y )
       => aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Xb) ) ) ).

% linorder_cases
tff(fact_573_dual__order_Oasym,axiom,
    ! [Ba: real,Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Ba),Aa)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),Ba) ) ).

% dual_order.asym
tff(fact_574_dual__order_Oasym,axiom,
    ! [Ba: rat,Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ba),Aa)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),Ba) ) ).

% dual_order.asym
tff(fact_575_dual__order_Oasym,axiom,
    ! [Ba: num,Aa: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Ba),Aa)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Aa),Ba) ) ).

% dual_order.asym
tff(fact_576_dual__order_Oasym,axiom,
    ! [Ba: nat,Aa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ba),Aa)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),Ba) ) ).

% dual_order.asym
tff(fact_577_dual__order_Oasym,axiom,
    ! [Ba: int,Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Ba),Aa)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),Ba) ) ).

% dual_order.asym
tff(fact_578_dual__order_Oirrefl,axiom,
    ! [Aa: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),Aa) ).

% dual_order.irrefl
tff(fact_579_dual__order_Oirrefl,axiom,
    ! [Aa: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),Aa) ).

% dual_order.irrefl
tff(fact_580_dual__order_Oirrefl,axiom,
    ! [Aa: num] : ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Aa),Aa) ).

% dual_order.irrefl
tff(fact_581_dual__order_Oirrefl,axiom,
    ! [Aa: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),Aa) ).

% dual_order.irrefl
tff(fact_582_dual__order_Oirrefl,axiom,
    ! [Aa: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),Aa) ).

% dual_order.irrefl
tff(fact_583_exists__least__iff,axiom,
    ! [P: fun_nat_o] :
      ( ? [X_12: nat] : aa_nat_o(P,X_12)
    <=> ? [N4: nat] :
          ( aa_nat_o(P,N4)
          & ! [M6: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M6),N4)
             => ~ aa_nat_o(P,M6) ) ) ) ).

% exists_least_iff
tff(fact_584_linorder__less__wlog,axiom,
    ! [P: fun_real_fun_real_o,Aa: real,Ba: real] :
      ( ! [A2: real,B2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2)
         => aa_real_o(aa_real_fun_real_o(P,A2),B2) )
     => ( ! [A2: real] : aa_real_o(aa_real_fun_real_o(P,A2),A2)
       => ( ! [A2: real,B2: real] :
              ( aa_real_o(aa_real_fun_real_o(P,B2),A2)
             => aa_real_o(aa_real_fun_real_o(P,A2),B2) )
         => aa_real_o(aa_real_fun_real_o(P,Aa),Ba) ) ) ) ).

% linorder_less_wlog
tff(fact_585_linorder__less__wlog,axiom,
    ! [P: fun_rat_fun_rat_o,Aa: rat,Ba: rat] :
      ( ! [A2: rat,B2: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2)
         => aa_rat_o(aa_rat_fun_rat_o(P,A2),B2) )
     => ( ! [A2: rat] : aa_rat_o(aa_rat_fun_rat_o(P,A2),A2)
       => ( ! [A2: rat,B2: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(P,B2),A2)
             => aa_rat_o(aa_rat_fun_rat_o(P,A2),B2) )
         => aa_rat_o(aa_rat_fun_rat_o(P,Aa),Ba) ) ) ) ).

% linorder_less_wlog
tff(fact_586_linorder__less__wlog,axiom,
    ! [P: fun_num_fun_num_o,Aa: num,Ba: num] :
      ( ! [A2: num,B2: num] :
          ( aa_num_o(aa_num_fun_num_o(ord_less_num,A2),B2)
         => aa_num_o(aa_num_fun_num_o(P,A2),B2) )
     => ( ! [A2: num] : aa_num_o(aa_num_fun_num_o(P,A2),A2)
       => ( ! [A2: num,B2: num] :
              ( aa_num_o(aa_num_fun_num_o(P,B2),A2)
             => aa_num_o(aa_num_fun_num_o(P,A2),B2) )
         => aa_num_o(aa_num_fun_num_o(P,Aa),Ba) ) ) ) ).

% linorder_less_wlog
tff(fact_587_linorder__less__wlog,axiom,
    ! [P: fun_nat_fun_nat_o,Aa: nat,Ba: nat] :
      ( ! [A2: nat,B2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B2)
         => aa_nat_o(aa_nat_fun_nat_o(P,A2),B2) )
     => ( ! [A2: nat] : aa_nat_o(aa_nat_fun_nat_o(P,A2),A2)
       => ( ! [A2: nat,B2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(P,B2),A2)
             => aa_nat_o(aa_nat_fun_nat_o(P,A2),B2) )
         => aa_nat_o(aa_nat_fun_nat_o(P,Aa),Ba) ) ) ) ).

% linorder_less_wlog
tff(fact_588_linorder__less__wlog,axiom,
    ! [P: fun_int_fun_int_o,Aa: int,Ba: int] :
      ( ! [A2: int,B2: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B2)
         => aa_int_o(aa_int_fun_int_o(P,A2),B2) )
     => ( ! [A2: int] : aa_int_o(aa_int_fun_int_o(P,A2),A2)
       => ( ! [A2: int,B2: int] :
              ( aa_int_o(aa_int_fun_int_o(P,B2),A2)
             => aa_int_o(aa_int_fun_int_o(P,A2),B2) )
         => aa_int_o(aa_int_fun_int_o(P,Aa),Ba) ) ) ) ).

% linorder_less_wlog
tff(fact_589_order_Ostrict__trans,axiom,
    ! [Aa: real,Ba: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),Ba)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Ba),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),C) ) ) ).

% order.strict_trans
tff(fact_590_order_Ostrict__trans,axiom,
    ! [Aa: rat,Ba: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),Ba)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ba),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),C) ) ) ).

% order.strict_trans
tff(fact_591_order_Ostrict__trans,axiom,
    ! [Aa: num,Ba: num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Aa),Ba)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,Ba),C)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,Aa),C) ) ) ).

% order.strict_trans
tff(fact_592_order_Ostrict__trans,axiom,
    ! [Aa: nat,Ba: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),Ba)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ba),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),C) ) ) ).

% order.strict_trans
tff(fact_593_order_Ostrict__trans,axiom,
    ! [Aa: int,Ba: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),Ba)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Ba),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),C) ) ) ).

% order.strict_trans
tff(fact_594_not__less__iff__gr__or__eq,axiom,
    ! [Xb: real,Y: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xb)
        | ( Xb = Y ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_595_not__less__iff__gr__or__eq,axiom,
    ! [Xb: rat,Y: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Xb)
        | ( Xb = Y ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_596_not__less__iff__gr__or__eq,axiom,
    ! [Xb: num,Y: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,Y),Xb)
        | ( Xb = Y ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_597_not__less__iff__gr__or__eq,axiom,
    ! [Xb: nat,Y: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xb)
        | ( Xb = Y ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_598_not__less__iff__gr__or__eq,axiom,
    ! [Xb: int,Y: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Xb)
        | ( Xb = Y ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_599_dual__order_Ostrict__trans,axiom,
    ! [Ba: real,Aa: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Ba),Aa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),Ba)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,C),Aa) ) ) ).

% dual_order.strict_trans
tff(fact_600_dual__order_Ostrict__trans,axiom,
    ! [Ba: rat,Aa: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ba),Aa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),Ba)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),Aa) ) ) ).

% dual_order.strict_trans
tff(fact_601_dual__order_Ostrict__trans,axiom,
    ! [Ba: num,Aa: num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Ba),Aa)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,C),Ba)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,C),Aa) ) ) ).

% dual_order.strict_trans
tff(fact_602_dual__order_Ostrict__trans,axiom,
    ! [Ba: nat,Aa: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ba),Aa)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),Ba)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),Aa) ) ) ).

% dual_order.strict_trans
tff(fact_603_dual__order_Ostrict__trans,axiom,
    ! [Ba: int,Aa: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Ba),Aa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),Ba)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,C),Aa) ) ) ).

% dual_order.strict_trans
tff(fact_604_order_Ostrict__implies__not__eq,axiom,
    ! [Aa: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),Ba)
     => ( Aa != Ba ) ) ).

% order.strict_implies_not_eq
tff(fact_605_order_Ostrict__implies__not__eq,axiom,
    ! [Aa: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),Ba)
     => ( Aa != Ba ) ) ).

% order.strict_implies_not_eq
tff(fact_606_order_Ostrict__implies__not__eq,axiom,
    ! [Aa: num,Ba: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Aa),Ba)
     => ( Aa != Ba ) ) ).

% order.strict_implies_not_eq
tff(fact_607_order_Ostrict__implies__not__eq,axiom,
    ! [Aa: nat,Ba: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),Ba)
     => ( Aa != Ba ) ) ).

% order.strict_implies_not_eq
tff(fact_608_order_Ostrict__implies__not__eq,axiom,
    ! [Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),Ba)
     => ( Aa != Ba ) ) ).

% order.strict_implies_not_eq
tff(fact_609_dual__order_Ostrict__implies__not__eq,axiom,
    ! [Ba: real,Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Ba),Aa)
     => ( Aa != Ba ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_610_dual__order_Ostrict__implies__not__eq,axiom,
    ! [Ba: rat,Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ba),Aa)
     => ( Aa != Ba ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_611_dual__order_Ostrict__implies__not__eq,axiom,
    ! [Ba: num,Aa: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Ba),Aa)
     => ( Aa != Ba ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_612_dual__order_Ostrict__implies__not__eq,axiom,
    ! [Ba: nat,Aa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ba),Aa)
     => ( Aa != Ba ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_613_dual__order_Ostrict__implies__not__eq,axiom,
    ! [Ba: int,Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Ba),Aa)
     => ( Aa != Ba ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_614_linorder__neqE,axiom,
    ! [Xb: real,Y: real] :
      ( ( Xb != Y )
     => ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xb) ) ) ).

% linorder_neqE
tff(fact_615_linorder__neqE,axiom,
    ! [Xb: rat,Y: rat] :
      ( ( Xb != Y )
     => ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Xb) ) ) ).

% linorder_neqE
tff(fact_616_linorder__neqE,axiom,
    ! [Xb: num,Y: num] :
      ( ( Xb != Y )
     => ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,Y),Xb) ) ) ).

% linorder_neqE
tff(fact_617_linorder__neqE,axiom,
    ! [Xb: nat,Y: nat] :
      ( ( Xb != Y )
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xb) ) ) ).

% linorder_neqE
tff(fact_618_linorder__neqE,axiom,
    ! [Xb: int,Y: int] :
      ( ( Xb != Y )
     => ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Xb) ) ) ).

% linorder_neqE
tff(fact_619_order__less__asym,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xb) ) ).

% order_less_asym
tff(fact_620_order__less__asym,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Xb) ) ).

% order_less_asym
tff(fact_621_order__less__asym,axiom,
    ! [Xb: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Y),Xb) ) ).

% order_less_asym
tff(fact_622_order__less__asym,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xb) ) ).

% order_less_asym
tff(fact_623_order__less__asym,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Xb) ) ).

% order_less_asym
tff(fact_624_linorder__neq__iff,axiom,
    ! [Xb: real,Y: real] :
      ( ( Xb != Y )
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
        | aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xb) ) ) ).

% linorder_neq_iff
tff(fact_625_linorder__neq__iff,axiom,
    ! [Xb: rat,Y: rat] :
      ( ( Xb != Y )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
        | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Xb) ) ) ).

% linorder_neq_iff
tff(fact_626_linorder__neq__iff,axiom,
    ! [Xb: num,Y: num] :
      ( ( Xb != Y )
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y)
        | aa_num_o(aa_num_fun_num_o(ord_less_num,Y),Xb) ) ) ).

% linorder_neq_iff
tff(fact_627_linorder__neq__iff,axiom,
    ! [Xb: nat,Y: nat] :
      ( ( Xb != Y )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xb) ) ) ).

% linorder_neq_iff
tff(fact_628_linorder__neq__iff,axiom,
    ! [Xb: int,Y: int] :
      ( ( Xb != Y )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
        | aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Xb) ) ) ).

% linorder_neq_iff
tff(fact_629_order__less__asym_H,axiom,
    ! [Aa: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),Ba)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Ba),Aa) ) ).

% order_less_asym'
tff(fact_630_order__less__asym_H,axiom,
    ! [Aa: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),Ba)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ba),Aa) ) ).

% order_less_asym'
tff(fact_631_order__less__asym_H,axiom,
    ! [Aa: num,Ba: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Aa),Ba)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Ba),Aa) ) ).

% order_less_asym'
tff(fact_632_order__less__asym_H,axiom,
    ! [Aa: nat,Ba: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),Ba)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ba),Aa) ) ).

% order_less_asym'
tff(fact_633_order__less__asym_H,axiom,
    ! [Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),Ba)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Ba),Aa) ) ).

% order_less_asym'
tff(fact_634_order__less__trans,axiom,
    ! [Xb: real,Y: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Z)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Z) ) ) ).

% order_less_trans
tff(fact_635_order__less__trans,axiom,
    ! [Xb: rat,Y: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Z)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Z) ) ) ).

% order_less_trans
tff(fact_636_order__less__trans,axiom,
    ! [Xb: num,Y: num,Z: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,Y),Z)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Z) ) ) ).

% order_less_trans
tff(fact_637_order__less__trans,axiom,
    ! [Xb: nat,Y: nat,Z: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Z)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Z) ) ) ).

% order_less_trans
tff(fact_638_order__less__trans,axiom,
    ! [Xb: int,Y: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Z)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Z) ) ) ).

% order_less_trans
tff(fact_639_ord__eq__less__subst,axiom,
    ! [Aa: real,F: fun_real_real,Ba: real,C: real] :
      ( ( Aa = aa_real_real(F,Ba) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Ba),C)
       => ( ! [X: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,X)),aa_real_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),aa_real_real(F,C)) ) ) ) ).

% ord_eq_less_subst
tff(fact_640_ord__eq__less__subst,axiom,
    ! [Aa: rat,F: fun_real_rat,Ba: real,C: real] :
      ( ( Aa = aa_real_rat(F,Ba) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Ba),C)
       => ( ! [X: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,X)),aa_real_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),aa_real_rat(F,C)) ) ) ) ).

% ord_eq_less_subst
tff(fact_641_ord__eq__less__subst,axiom,
    ! [Aa: num,F: fun_real_num,Ba: real,C: real] :
      ( ( Aa = aa_real_num(F,Ba) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Ba),C)
       => ( ! [X: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_real_num(F,X)),aa_real_num(F,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,Aa),aa_real_num(F,C)) ) ) ) ).

% ord_eq_less_subst
tff(fact_642_ord__eq__less__subst,axiom,
    ! [Aa: nat,F: fun_real_nat,Ba: real,C: real] :
      ( ( Aa = aa_real_nat(F,Ba) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Ba),C)
       => ( ! [X: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_real_nat(F,X)),aa_real_nat(F,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),aa_real_nat(F,C)) ) ) ) ).

% ord_eq_less_subst
tff(fact_643_ord__eq__less__subst,axiom,
    ! [Aa: int,F: fun_real_int,Ba: real,C: real] :
      ( ( Aa = aa_real_int(F,Ba) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Ba),C)
       => ( ! [X: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(F,X)),aa_real_int(F,Y3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),aa_real_int(F,C)) ) ) ) ).

% ord_eq_less_subst
tff(fact_644_ord__eq__less__subst,axiom,
    ! [Aa: real,F: fun_rat_real,Ba: rat,C: rat] :
      ( ( Aa = aa_rat_real(F,Ba) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ba),C)
       => ( ! [X: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,X)),aa_rat_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),aa_rat_real(F,C)) ) ) ) ).

% ord_eq_less_subst
tff(fact_645_ord__eq__less__subst,axiom,
    ! [Aa: rat,F: fun_rat_rat,Ba: rat,C: rat] :
      ( ( Aa = aa_rat_rat(F,Ba) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ba),C)
       => ( ! [X: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,X)),aa_rat_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),aa_rat_rat(F,C)) ) ) ) ).

% ord_eq_less_subst
tff(fact_646_ord__eq__less__subst,axiom,
    ! [Aa: num,F: fun_rat_num,Ba: rat,C: rat] :
      ( ( Aa = aa_rat_num(F,Ba) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ba),C)
       => ( ! [X: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F,X)),aa_rat_num(F,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,Aa),aa_rat_num(F,C)) ) ) ) ).

% ord_eq_less_subst
tff(fact_647_ord__eq__less__subst,axiom,
    ! [Aa: nat,F: fun_rat_nat,Ba: rat,C: rat] :
      ( ( Aa = aa_rat_nat(F,Ba) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ba),C)
       => ( ! [X: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,X)),aa_rat_nat(F,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),aa_rat_nat(F,C)) ) ) ) ).

% ord_eq_less_subst
tff(fact_648_ord__eq__less__subst,axiom,
    ! [Aa: int,F: fun_rat_int,Ba: rat,C: rat] :
      ( ( Aa = aa_rat_int(F,Ba) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ba),C)
       => ( ! [X: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(F,X)),aa_rat_int(F,Y3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),aa_rat_int(F,C)) ) ) ) ).

% ord_eq_less_subst
tff(fact_649_ord__less__eq__subst,axiom,
    ! [Aa: real,Ba: real,F: fun_real_real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),Ba)
     => ( ( aa_real_real(F,Ba) = C )
       => ( ! [X: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,X)),aa_real_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,Aa)),C) ) ) ) ).

% ord_less_eq_subst
tff(fact_650_ord__less__eq__subst,axiom,
    ! [Aa: real,Ba: real,F: fun_real_rat,C: rat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),Ba)
     => ( ( aa_real_rat(F,Ba) = C )
       => ( ! [X: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,X)),aa_real_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,Aa)),C) ) ) ) ).

% ord_less_eq_subst
tff(fact_651_ord__less__eq__subst,axiom,
    ! [Aa: real,Ba: real,F: fun_real_num,C: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),Ba)
     => ( ( aa_real_num(F,Ba) = C )
       => ( ! [X: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_real_num(F,X)),aa_real_num(F,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_real_num(F,Aa)),C) ) ) ) ).

% ord_less_eq_subst
tff(fact_652_ord__less__eq__subst,axiom,
    ! [Aa: real,Ba: real,F: fun_real_nat,C: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),Ba)
     => ( ( aa_real_nat(F,Ba) = C )
       => ( ! [X: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_real_nat(F,X)),aa_real_nat(F,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_real_nat(F,Aa)),C) ) ) ) ).

% ord_less_eq_subst
tff(fact_653_ord__less__eq__subst,axiom,
    ! [Aa: real,Ba: real,F: fun_real_int,C: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),Ba)
     => ( ( aa_real_int(F,Ba) = C )
       => ( ! [X: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(F,X)),aa_real_int(F,Y3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(F,Aa)),C) ) ) ) ).

% ord_less_eq_subst
tff(fact_654_ord__less__eq__subst,axiom,
    ! [Aa: rat,Ba: rat,F: fun_rat_real,C: real] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),Ba)
     => ( ( aa_rat_real(F,Ba) = C )
       => ( ! [X: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,X)),aa_rat_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,Aa)),C) ) ) ) ).

% ord_less_eq_subst
tff(fact_655_ord__less__eq__subst,axiom,
    ! [Aa: rat,Ba: rat,F: fun_rat_rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),Ba)
     => ( ( aa_rat_rat(F,Ba) = C )
       => ( ! [X: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,X)),aa_rat_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,Aa)),C) ) ) ) ).

% ord_less_eq_subst
tff(fact_656_ord__less__eq__subst,axiom,
    ! [Aa: rat,Ba: rat,F: fun_rat_num,C: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),Ba)
     => ( ( aa_rat_num(F,Ba) = C )
       => ( ! [X: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F,X)),aa_rat_num(F,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F,Aa)),C) ) ) ) ).

% ord_less_eq_subst
tff(fact_657_ord__less__eq__subst,axiom,
    ! [Aa: rat,Ba: rat,F: fun_rat_nat,C: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),Ba)
     => ( ( aa_rat_nat(F,Ba) = C )
       => ( ! [X: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,X)),aa_rat_nat(F,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,Aa)),C) ) ) ) ).

% ord_less_eq_subst
tff(fact_658_ord__less__eq__subst,axiom,
    ! [Aa: rat,Ba: rat,F: fun_rat_int,C: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),Ba)
     => ( ( aa_rat_int(F,Ba) = C )
       => ( ! [X: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(F,X)),aa_rat_int(F,Y3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(F,Aa)),C) ) ) ) ).

% ord_less_eq_subst
tff(fact_659_order__less__irrefl,axiom,
    ! [Xb: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Xb) ).

% order_less_irrefl
tff(fact_660_order__less__irrefl,axiom,
    ! [Xb: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Xb) ).

% order_less_irrefl
tff(fact_661_order__less__irrefl,axiom,
    ! [Xb: num] : ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Xb) ).

% order_less_irrefl
tff(fact_662_order__less__irrefl,axiom,
    ! [Xb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Xb) ).

% order_less_irrefl
tff(fact_663_order__less__irrefl,axiom,
    ! [Xb: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Xb) ).

% order_less_irrefl
tff(fact_664_order__less__subst1,axiom,
    ! [Aa: real,F: fun_real_real,Ba: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),aa_real_real(F,Ba))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Ba),C)
       => ( ! [X: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,X)),aa_real_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),aa_real_real(F,C)) ) ) ) ).

% order_less_subst1
tff(fact_665_order__less__subst1,axiom,
    ! [Aa: real,F: fun_rat_real,Ba: rat,C: rat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),aa_rat_real(F,Ba))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ba),C)
       => ( ! [X: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,X)),aa_rat_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),aa_rat_real(F,C)) ) ) ) ).

% order_less_subst1
tff(fact_666_order__less__subst1,axiom,
    ! [Aa: real,F: fun_num_real,Ba: num,C: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),aa_num_real(F,Ba))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,Ba),C)
       => ( ! [X: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(F,X)),aa_num_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),aa_num_real(F,C)) ) ) ) ).

% order_less_subst1
tff(fact_667_order__less__subst1,axiom,
    ! [Aa: real,F: fun_nat_real,Ba: nat,C: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),aa_nat_real(F,Ba))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ba),C)
       => ( ! [X: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,X)),aa_nat_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),aa_nat_real(F,C)) ) ) ) ).

% order_less_subst1
tff(fact_668_order__less__subst1,axiom,
    ! [Aa: real,F: fun_int_real,Ba: int,C: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),aa_int_real(F,Ba))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Ba),C)
       => ( ! [X: int,Y3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(F,X)),aa_int_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),aa_int_real(F,C)) ) ) ) ).

% order_less_subst1
tff(fact_669_order__less__subst1,axiom,
    ! [Aa: rat,F: fun_real_rat,Ba: real,C: real] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),aa_real_rat(F,Ba))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Ba),C)
       => ( ! [X: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,X)),aa_real_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),aa_real_rat(F,C)) ) ) ) ).

% order_less_subst1
tff(fact_670_order__less__subst1,axiom,
    ! [Aa: rat,F: fun_rat_rat,Ba: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),aa_rat_rat(F,Ba))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ba),C)
       => ( ! [X: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,X)),aa_rat_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),aa_rat_rat(F,C)) ) ) ) ).

% order_less_subst1
tff(fact_671_order__less__subst1,axiom,
    ! [Aa: rat,F: fun_num_rat,Ba: num,C: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),aa_num_rat(F,Ba))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,Ba),C)
       => ( ! [X: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(F,X)),aa_num_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),aa_num_rat(F,C)) ) ) ) ).

% order_less_subst1
tff(fact_672_order__less__subst1,axiom,
    ! [Aa: rat,F: fun_nat_rat,Ba: nat,C: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),aa_nat_rat(F,Ba))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ba),C)
       => ( ! [X: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,X)),aa_nat_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),aa_nat_rat(F,C)) ) ) ) ).

% order_less_subst1
tff(fact_673_order__less__subst1,axiom,
    ! [Aa: rat,F: fun_int_rat,Ba: int,C: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),aa_int_rat(F,Ba))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Ba),C)
       => ( ! [X: int,Y3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(F,X)),aa_int_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),aa_int_rat(F,C)) ) ) ) ).

% order_less_subst1
tff(fact_674_order__less__subst2,axiom,
    ! [Aa: real,Ba: real,F: fun_real_real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),Ba)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,Ba)),C)
       => ( ! [X: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,X)),aa_real_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,Aa)),C) ) ) ) ).

% order_less_subst2
tff(fact_675_order__less__subst2,axiom,
    ! [Aa: real,Ba: real,F: fun_real_rat,C: rat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),Ba)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,Ba)),C)
       => ( ! [X: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,X)),aa_real_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,Aa)),C) ) ) ) ).

% order_less_subst2
tff(fact_676_order__less__subst2,axiom,
    ! [Aa: real,Ba: real,F: fun_real_num,C: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),Ba)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_real_num(F,Ba)),C)
       => ( ! [X: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_real_num(F,X)),aa_real_num(F,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_real_num(F,Aa)),C) ) ) ) ).

% order_less_subst2
tff(fact_677_order__less__subst2,axiom,
    ! [Aa: real,Ba: real,F: fun_real_nat,C: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),Ba)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_real_nat(F,Ba)),C)
       => ( ! [X: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_real_nat(F,X)),aa_real_nat(F,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_real_nat(F,Aa)),C) ) ) ) ).

% order_less_subst2
tff(fact_678_order__less__subst2,axiom,
    ! [Aa: real,Ba: real,F: fun_real_int,C: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),Ba)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(F,Ba)),C)
       => ( ! [X: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(F,X)),aa_real_int(F,Y3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(F,Aa)),C) ) ) ) ).

% order_less_subst2
tff(fact_679_order__less__subst2,axiom,
    ! [Aa: rat,Ba: rat,F: fun_rat_real,C: real] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),Ba)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,Ba)),C)
       => ( ! [X: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,X)),aa_rat_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,Aa)),C) ) ) ) ).

% order_less_subst2
tff(fact_680_order__less__subst2,axiom,
    ! [Aa: rat,Ba: rat,F: fun_rat_rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),Ba)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,Ba)),C)
       => ( ! [X: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,X)),aa_rat_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,Aa)),C) ) ) ) ).

% order_less_subst2
tff(fact_681_order__less__subst2,axiom,
    ! [Aa: rat,Ba: rat,F: fun_rat_num,C: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),Ba)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F,Ba)),C)
       => ( ! [X: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F,X)),aa_rat_num(F,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F,Aa)),C) ) ) ) ).

% order_less_subst2
tff(fact_682_order__less__subst2,axiom,
    ! [Aa: rat,Ba: rat,F: fun_rat_nat,C: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),Ba)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,Ba)),C)
       => ( ! [X: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,X)),aa_rat_nat(F,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,Aa)),C) ) ) ) ).

% order_less_subst2
tff(fact_683_order__less__subst2,axiom,
    ! [Aa: rat,Ba: rat,F: fun_rat_int,C: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),Ba)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(F,Ba)),C)
       => ( ! [X: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(F,X)),aa_rat_int(F,Y3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(F,Aa)),C) ) ) ) ).

% order_less_subst2
tff(fact_684_order__less__not__sym,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xb) ) ).

% order_less_not_sym
tff(fact_685_order__less__not__sym,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Xb) ) ).

% order_less_not_sym
tff(fact_686_order__less__not__sym,axiom,
    ! [Xb: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Y),Xb) ) ).

% order_less_not_sym
tff(fact_687_order__less__not__sym,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xb) ) ).

% order_less_not_sym
tff(fact_688_order__less__not__sym,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Xb) ) ).

% order_less_not_sym
tff(fact_689_order__less__imp__triv,axiom,
    ! [Xb: real,Y: real,P: $o] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xb)
       => (P) ) ) ).

% order_less_imp_triv
tff(fact_690_order__less__imp__triv,axiom,
    ! [Xb: rat,Y: rat,P: $o] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Xb)
       => (P) ) ) ).

% order_less_imp_triv
tff(fact_691_order__less__imp__triv,axiom,
    ! [Xb: num,Y: num,P: $o] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,Y),Xb)
       => (P) ) ) ).

% order_less_imp_triv
tff(fact_692_order__less__imp__triv,axiom,
    ! [Xb: nat,Y: nat,P: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xb)
       => (P) ) ) ).

% order_less_imp_triv
tff(fact_693_order__less__imp__triv,axiom,
    ! [Xb: int,Y: int,P: $o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Xb)
       => (P) ) ) ).

% order_less_imp_triv
tff(fact_694_linorder__less__linear,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
      | ( Xb = Y )
      | aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xb) ) ).

% linorder_less_linear
tff(fact_695_linorder__less__linear,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
      | ( Xb = Y )
      | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Xb) ) ).

% linorder_less_linear
tff(fact_696_linorder__less__linear,axiom,
    ! [Xb: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y)
      | ( Xb = Y )
      | aa_num_o(aa_num_fun_num_o(ord_less_num,Y),Xb) ) ).

% linorder_less_linear
tff(fact_697_linorder__less__linear,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
      | ( Xb = Y )
      | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xb) ) ).

% linorder_less_linear
tff(fact_698_linorder__less__linear,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
      | ( Xb = Y )
      | aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Xb) ) ).

% linorder_less_linear
tff(fact_699_order__less__imp__not__eq,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
     => ( Xb != Y ) ) ).

% order_less_imp_not_eq
tff(fact_700_order__less__imp__not__eq,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
     => ( Xb != Y ) ) ).

% order_less_imp_not_eq
tff(fact_701_order__less__imp__not__eq,axiom,
    ! [Xb: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y)
     => ( Xb != Y ) ) ).

% order_less_imp_not_eq
tff(fact_702_order__less__imp__not__eq,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
     => ( Xb != Y ) ) ).

% order_less_imp_not_eq
tff(fact_703_order__less__imp__not__eq,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
     => ( Xb != Y ) ) ).

% order_less_imp_not_eq
tff(fact_704_order__less__imp__not__eq2,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
     => ( Y != Xb ) ) ).

% order_less_imp_not_eq2
tff(fact_705_order__less__imp__not__eq2,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
     => ( Y != Xb ) ) ).

% order_less_imp_not_eq2
tff(fact_706_order__less__imp__not__eq2,axiom,
    ! [Xb: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y)
     => ( Y != Xb ) ) ).

% order_less_imp_not_eq2
tff(fact_707_order__less__imp__not__eq2,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
     => ( Y != Xb ) ) ).

% order_less_imp_not_eq2
tff(fact_708_order__less__imp__not__eq2,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
     => ( Y != Xb ) ) ).

% order_less_imp_not_eq2
tff(fact_709_order__less__imp__not__less,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xb) ) ).

% order_less_imp_not_less
tff(fact_710_order__less__imp__not__less,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Xb) ) ).

% order_less_imp_not_less
tff(fact_711_order__less__imp__not__less,axiom,
    ! [Xb: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Y),Xb) ) ).

% order_less_imp_not_less
tff(fact_712_order__less__imp__not__less,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xb) ) ).

% order_less_imp_not_less
tff(fact_713_order__less__imp__not__less,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Xb) ) ).

% order_less_imp_not_less
tff(fact_714_bounded__Max__nat,axiom,
    ! [P: fun_nat_o,Xb: nat,M7: nat] :
      ( aa_nat_o(P,Xb)
     => ( ! [X: nat] :
            ( aa_nat_o(P,X)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),M7) )
       => ~ ! [M3: nat] :
              ( aa_nat_o(P,M3)
             => ~ ! [X3: nat] :
                    ( aa_nat_o(P,X3)
                   => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),M3) ) ) ) ) ).

% bounded_Max_nat
tff(fact_715_VEBT__internal_Onaive__member_Osimps_I1_J,axiom,
    ! [Aa: $o,Ba: $o,Xb: nat] :
      ( aa_nat_o(vEBT_V5719532721284313246member(vEBT_Leaf((Aa),(Ba))),Xb)
    <=> $ite(
          Xb = zero_zero_nat,
          (Aa),
          $ite(Xb = one_one_nat,(Ba),$false) ) ) ).

% VEBT_internal.naive_member.simps(1)
tff(fact_716_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_717_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_718_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_719_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_720_leD,axiom,
    ! [Y: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Xb)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y) ) ).

% leD
tff(fact_721_leD,axiom,
    ! [Y: set_int,Xb: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(Y),Xb)
     => ~ aa_set_int_o(ord_less_set_int(Xb),Y) ) ).

% leD
tff(fact_722_leD,axiom,
    ! [Y: rat,Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Xb)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y) ) ).

% leD
tff(fact_723_leD,axiom,
    ! [Y: num,Xb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),Xb)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y) ) ).

% leD
tff(fact_724_leD,axiom,
    ! [Y: nat,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Xb)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y) ) ).

% leD
tff(fact_725_leD,axiom,
    ! [Y: int,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Xb)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y) ) ).

% leD
tff(fact_726_leI,axiom,
    ! [Xb: real,Y: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Xb) ) ).

% leI
tff(fact_727_leI,axiom,
    ! [Xb: rat,Y: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Xb) ) ).

% leI
tff(fact_728_leI,axiom,
    ! [Xb: num,Y: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),Xb) ) ).

% leI
tff(fact_729_leI,axiom,
    ! [Xb: nat,Y: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Xb) ) ).

% leI
tff(fact_730_leI,axiom,
    ! [Xb: int,Y: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Xb) ) ).

% leI
tff(fact_731_nless__le,axiom,
    ! [Aa: real,Ba: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),Ba)
    <=> ( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),Ba)
        | ( Aa = Ba ) ) ) ).

% nless_le
tff(fact_732_nless__le,axiom,
    ! [Aa: set_int,Ba: set_int] :
      ( ~ aa_set_int_o(ord_less_set_int(Aa),Ba)
    <=> ( ~ aa_set_int_o(ord_less_eq_set_int(Aa),Ba)
        | ( Aa = Ba ) ) ) ).

% nless_le
tff(fact_733_nless__le,axiom,
    ! [Aa: rat,Ba: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),Ba)
    <=> ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),Ba)
        | ( Aa = Ba ) ) ) ).

% nless_le
tff(fact_734_nless__le,axiom,
    ! [Aa: num,Ba: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Aa),Ba)
    <=> ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Aa),Ba)
        | ( Aa = Ba ) ) ) ).

% nless_le
tff(fact_735_nless__le,axiom,
    ! [Aa: nat,Ba: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),Ba)
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),Ba)
        | ( Aa = Ba ) ) ) ).

% nless_le
tff(fact_736_nless__le,axiom,
    ! [Aa: int,Ba: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),Ba)
    <=> ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),Ba)
        | ( Aa = Ba ) ) ) ).

% nless_le
tff(fact_737_antisym__conv1,axiom,
    ! [Xb: real,Y: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
      <=> ( Xb = Y ) ) ) ).

% antisym_conv1
tff(fact_738_antisym__conv1,axiom,
    ! [Xb: set_int,Y: set_int] :
      ( ~ aa_set_int_o(ord_less_set_int(Xb),Y)
     => ( aa_set_int_o(ord_less_eq_set_int(Xb),Y)
      <=> ( Xb = Y ) ) ) ).

% antisym_conv1
tff(fact_739_antisym__conv1,axiom,
    ! [Xb: rat,Y: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y)
      <=> ( Xb = Y ) ) ) ).

% antisym_conv1
tff(fact_740_antisym__conv1,axiom,
    ! [Xb: num,Y: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Y)
      <=> ( Xb = Y ) ) ) ).

% antisym_conv1
tff(fact_741_antisym__conv1,axiom,
    ! [Xb: nat,Y: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y)
      <=> ( Xb = Y ) ) ) ).

% antisym_conv1
tff(fact_742_antisym__conv1,axiom,
    ! [Xb: int,Y: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y)
      <=> ( Xb = Y ) ) ) ).

% antisym_conv1
tff(fact_743_antisym__conv2,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
     => ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
      <=> ( Xb = Y ) ) ) ).

% antisym_conv2
tff(fact_744_antisym__conv2,axiom,
    ! [Xb: set_int,Y: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(Xb),Y)
     => ( ~ aa_set_int_o(ord_less_set_int(Xb),Y)
      <=> ( Xb = Y ) ) ) ).

% antisym_conv2
tff(fact_745_antisym__conv2,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y)
     => ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
      <=> ( Xb = Y ) ) ) ).

% antisym_conv2
tff(fact_746_antisym__conv2,axiom,
    ! [Xb: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Y)
     => ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y)
      <=> ( Xb = Y ) ) ) ).

% antisym_conv2
tff(fact_747_antisym__conv2,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y)
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
      <=> ( Xb = Y ) ) ) ).

% antisym_conv2
tff(fact_748_antisym__conv2,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y)
     => ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
      <=> ( Xb = Y ) ) ) ).

% antisym_conv2
tff(fact_749_dense__ge,axiom,
    ! [Z: real,Y: real] :
      ( ! [X: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z),X)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),X) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Z) ) ).

% dense_ge
tff(fact_750_dense__ge,axiom,
    ! [Z: rat,Y: rat] :
      ( ! [X: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),X)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),X) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Z) ) ).

% dense_ge
tff(fact_751_dense__le,axiom,
    ! [Y: real,Z: real] :
      ( ! [X: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Z) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Z) ) ).

% dense_le
tff(fact_752_dense__le,axiom,
    ! [Y: rat,Z: rat] :
      ( ! [X: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Z) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Z) ) ).

% dense_le
tff(fact_753_less__le__not__le,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
        & ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Xb) ) ) ).

% less_le_not_le
tff(fact_754_less__le__not__le,axiom,
    ! [Xb: set_int,Y: set_int] :
      ( aa_set_int_o(ord_less_set_int(Xb),Y)
    <=> ( aa_set_int_o(ord_less_eq_set_int(Xb),Y)
        & ~ aa_set_int_o(ord_less_eq_set_int(Y),Xb) ) ) ).

% less_le_not_le
tff(fact_755_less__le__not__le,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y)
        & ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Xb) ) ) ).

% less_le_not_le
tff(fact_756_less__le__not__le,axiom,
    ! [Xb: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Y)
        & ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),Xb) ) ) ).

% less_le_not_le
tff(fact_757_less__le__not__le,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y)
        & ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Xb) ) ) ).

% less_le_not_le
tff(fact_758_less__le__not__le,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y)
        & ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Xb) ) ) ).

% less_le_not_le
tff(fact_759_not__le__imp__less,axiom,
    ! [Y: real,Xb: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y) ) ).

% not_le_imp_less
tff(fact_760_not__le__imp__less,axiom,
    ! [Y: rat,Xb: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Xb)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y) ) ).

% not_le_imp_less
tff(fact_761_not__le__imp__less,axiom,
    ! [Y: num,Xb: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),Xb)
     => aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y) ) ).

% not_le_imp_less
tff(fact_762_not__le__imp__less,axiom,
    ! [Y: nat,Xb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Xb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y) ) ).

% not_le_imp_less
tff(fact_763_not__le__imp__less,axiom,
    ! [Y: int,Xb: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Xb)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y) ) ).

% not_le_imp_less
tff(fact_764_order_Oorder__iff__strict,axiom,
    ! [Aa: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),Ba)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),Ba)
        | ( Aa = Ba ) ) ) ).

% order.order_iff_strict
tff(fact_765_order_Oorder__iff__strict,axiom,
    ! [Aa: set_int,Ba: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(Aa),Ba)
    <=> ( aa_set_int_o(ord_less_set_int(Aa),Ba)
        | ( Aa = Ba ) ) ) ).

% order.order_iff_strict
tff(fact_766_order_Oorder__iff__strict,axiom,
    ! [Aa: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),Ba)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),Ba)
        | ( Aa = Ba ) ) ) ).

% order.order_iff_strict
tff(fact_767_order_Oorder__iff__strict,axiom,
    ! [Aa: num,Ba: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Aa),Ba)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,Aa),Ba)
        | ( Aa = Ba ) ) ) ).

% order.order_iff_strict
tff(fact_768_order_Oorder__iff__strict,axiom,
    ! [Aa: nat,Ba: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),Ba)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),Ba)
        | ( Aa = Ba ) ) ) ).

% order.order_iff_strict
tff(fact_769_order_Oorder__iff__strict,axiom,
    ! [Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),Ba)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),Ba)
        | ( Aa = Ba ) ) ) ).

% order.order_iff_strict
tff(fact_770_order_Ostrict__iff__order,axiom,
    ! [Aa: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),Ba)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),Ba)
        & ( Aa != Ba ) ) ) ).

% order.strict_iff_order
tff(fact_771_order_Ostrict__iff__order,axiom,
    ! [Aa: set_int,Ba: set_int] :
      ( aa_set_int_o(ord_less_set_int(Aa),Ba)
    <=> ( aa_set_int_o(ord_less_eq_set_int(Aa),Ba)
        & ( Aa != Ba ) ) ) ).

% order.strict_iff_order
tff(fact_772_order_Ostrict__iff__order,axiom,
    ! [Aa: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),Ba)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),Ba)
        & ( Aa != Ba ) ) ) ).

% order.strict_iff_order
tff(fact_773_order_Ostrict__iff__order,axiom,
    ! [Aa: num,Ba: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Aa),Ba)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Aa),Ba)
        & ( Aa != Ba ) ) ) ).

% order.strict_iff_order
tff(fact_774_order_Ostrict__iff__order,axiom,
    ! [Aa: nat,Ba: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),Ba)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),Ba)
        & ( Aa != Ba ) ) ) ).

% order.strict_iff_order
tff(fact_775_order_Ostrict__iff__order,axiom,
    ! [Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),Ba)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),Ba)
        & ( Aa != Ba ) ) ) ).

% order.strict_iff_order
tff(fact_776_order_Ostrict__trans1,axiom,
    ! [Aa: real,Ba: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),Ba)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Ba),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),C) ) ) ).

% order.strict_trans1
tff(fact_777_order_Ostrict__trans1,axiom,
    ! [Aa: set_int,Ba: set_int,C: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(Aa),Ba)
     => ( aa_set_int_o(ord_less_set_int(Ba),C)
       => aa_set_int_o(ord_less_set_int(Aa),C) ) ) ).

% order.strict_trans1
tff(fact_778_order_Ostrict__trans1,axiom,
    ! [Aa: rat,Ba: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),Ba)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ba),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),C) ) ) ).

% order.strict_trans1
tff(fact_779_order_Ostrict__trans1,axiom,
    ! [Aa: num,Ba: num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Aa),Ba)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,Ba),C)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,Aa),C) ) ) ).

% order.strict_trans1
tff(fact_780_order_Ostrict__trans1,axiom,
    ! [Aa: nat,Ba: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),Ba)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ba),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),C) ) ) ).

% order.strict_trans1
tff(fact_781_order_Ostrict__trans1,axiom,
    ! [Aa: int,Ba: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),Ba)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Ba),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),C) ) ) ).

% order.strict_trans1
tff(fact_782_order_Ostrict__trans2,axiom,
    ! [Aa: real,Ba: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),Ba)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ba),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),C) ) ) ).

% order.strict_trans2
tff(fact_783_order_Ostrict__trans2,axiom,
    ! [Aa: set_int,Ba: set_int,C: set_int] :
      ( aa_set_int_o(ord_less_set_int(Aa),Ba)
     => ( aa_set_int_o(ord_less_eq_set_int(Ba),C)
       => aa_set_int_o(ord_less_set_int(Aa),C) ) ) ).

% order.strict_trans2
tff(fact_784_order_Ostrict__trans2,axiom,
    ! [Aa: rat,Ba: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),Ba)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ba),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),C) ) ) ).

% order.strict_trans2
tff(fact_785_order_Ostrict__trans2,axiom,
    ! [Aa: num,Ba: num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Aa),Ba)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ba),C)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,Aa),C) ) ) ).

% order.strict_trans2
tff(fact_786_order_Ostrict__trans2,axiom,
    ! [Aa: nat,Ba: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),Ba)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ba),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),C) ) ) ).

% order.strict_trans2
tff(fact_787_order_Ostrict__trans2,axiom,
    ! [Aa: int,Ba: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),Ba)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ba),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),C) ) ) ).

% order.strict_trans2
tff(fact_788_order_Ostrict__iff__not,axiom,
    ! [Aa: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),Ba)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),Ba)
        & ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ba),Aa) ) ) ).

% order.strict_iff_not
tff(fact_789_order_Ostrict__iff__not,axiom,
    ! [Aa: set_int,Ba: set_int] :
      ( aa_set_int_o(ord_less_set_int(Aa),Ba)
    <=> ( aa_set_int_o(ord_less_eq_set_int(Aa),Ba)
        & ~ aa_set_int_o(ord_less_eq_set_int(Ba),Aa) ) ) ).

% order.strict_iff_not
tff(fact_790_order_Ostrict__iff__not,axiom,
    ! [Aa: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),Ba)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),Ba)
        & ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ba),Aa) ) ) ).

% order.strict_iff_not
tff(fact_791_order_Ostrict__iff__not,axiom,
    ! [Aa: num,Ba: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Aa),Ba)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Aa),Ba)
        & ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ba),Aa) ) ) ).

% order.strict_iff_not
tff(fact_792_order_Ostrict__iff__not,axiom,
    ! [Aa: nat,Ba: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),Ba)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),Ba)
        & ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ba),Aa) ) ) ).

% order.strict_iff_not
tff(fact_793_order_Ostrict__iff__not,axiom,
    ! [Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),Ba)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),Ba)
        & ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ba),Aa) ) ) ).

% order.strict_iff_not
tff(fact_794_dense__ge__bounded,axiom,
    ! [Z: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z),Xb)
     => ( ! [W: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z),W)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,W),Xb)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),W) ) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Z) ) ) ).

% dense_ge_bounded
tff(fact_795_dense__ge__bounded,axiom,
    ! [Z: rat,Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),Xb)
     => ( ! [W: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),W)
           => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,W),Xb)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),W) ) )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Z) ) ) ).

% dense_ge_bounded
tff(fact_796_dense__le__bounded,axiom,
    ! [Xb: real,Y: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
     => ( ! [W: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),W)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,W),Y)
             => 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,Y),Z) ) ) ).

% dense_le_bounded
tff(fact_797_dense__le__bounded,axiom,
    ! [Xb: rat,Y: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
     => ( ! [W: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),W)
           => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,W),Y)
             => 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,Y),Z) ) ) ).

% dense_le_bounded
tff(fact_798_dual__order_Oorder__iff__strict,axiom,
    ! [Ba: real,Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ba),Aa)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Ba),Aa)
        | ( Aa = Ba ) ) ) ).

% dual_order.order_iff_strict
tff(fact_799_dual__order_Oorder__iff__strict,axiom,
    ! [Ba: set_int,Aa: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(Ba),Aa)
    <=> ( aa_set_int_o(ord_less_set_int(Ba),Aa)
        | ( Aa = Ba ) ) ) ).

% dual_order.order_iff_strict
tff(fact_800_dual__order_Oorder__iff__strict,axiom,
    ! [Ba: rat,Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ba),Aa)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ba),Aa)
        | ( Aa = Ba ) ) ) ).

% dual_order.order_iff_strict
tff(fact_801_dual__order_Oorder__iff__strict,axiom,
    ! [Ba: num,Aa: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ba),Aa)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,Ba),Aa)
        | ( Aa = Ba ) ) ) ).

% dual_order.order_iff_strict
tff(fact_802_dual__order_Oorder__iff__strict,axiom,
    ! [Ba: nat,Aa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ba),Aa)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ba),Aa)
        | ( Aa = Ba ) ) ) ).

% dual_order.order_iff_strict
tff(fact_803_dual__order_Oorder__iff__strict,axiom,
    ! [Ba: int,Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ba),Aa)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Ba),Aa)
        | ( Aa = Ba ) ) ) ).

% dual_order.order_iff_strict
tff(fact_804_dual__order_Ostrict__iff__order,axiom,
    ! [Ba: real,Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Ba),Aa)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ba),Aa)
        & ( Aa != Ba ) ) ) ).

% dual_order.strict_iff_order
tff(fact_805_dual__order_Ostrict__iff__order,axiom,
    ! [Ba: set_int,Aa: set_int] :
      ( aa_set_int_o(ord_less_set_int(Ba),Aa)
    <=> ( aa_set_int_o(ord_less_eq_set_int(Ba),Aa)
        & ( Aa != Ba ) ) ) ).

% dual_order.strict_iff_order
tff(fact_806_dual__order_Ostrict__iff__order,axiom,
    ! [Ba: rat,Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ba),Aa)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ba),Aa)
        & ( Aa != Ba ) ) ) ).

% dual_order.strict_iff_order
tff(fact_807_dual__order_Ostrict__iff__order,axiom,
    ! [Ba: num,Aa: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Ba),Aa)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ba),Aa)
        & ( Aa != Ba ) ) ) ).

% dual_order.strict_iff_order
tff(fact_808_dual__order_Ostrict__iff__order,axiom,
    ! [Ba: nat,Aa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ba),Aa)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ba),Aa)
        & ( Aa != Ba ) ) ) ).

% dual_order.strict_iff_order
tff(fact_809_dual__order_Ostrict__iff__order,axiom,
    ! [Ba: int,Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Ba),Aa)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ba),Aa)
        & ( Aa != Ba ) ) ) ).

% dual_order.strict_iff_order
tff(fact_810_dual__order_Ostrict__trans1,axiom,
    ! [Ba: real,Aa: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ba),Aa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),Ba)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,C),Aa) ) ) ).

% dual_order.strict_trans1
tff(fact_811_dual__order_Ostrict__trans1,axiom,
    ! [Ba: set_int,Aa: set_int,C: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(Ba),Aa)
     => ( aa_set_int_o(ord_less_set_int(C),Ba)
       => aa_set_int_o(ord_less_set_int(C),Aa) ) ) ).

% dual_order.strict_trans1
tff(fact_812_dual__order_Ostrict__trans1,axiom,
    ! [Ba: rat,Aa: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ba),Aa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),Ba)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),Aa) ) ) ).

% dual_order.strict_trans1
tff(fact_813_dual__order_Ostrict__trans1,axiom,
    ! [Ba: num,Aa: num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ba),Aa)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,C),Ba)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,C),Aa) ) ) ).

% dual_order.strict_trans1
tff(fact_814_dual__order_Ostrict__trans1,axiom,
    ! [Ba: nat,Aa: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ba),Aa)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),Ba)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),Aa) ) ) ).

% dual_order.strict_trans1
tff(fact_815_dual__order_Ostrict__trans1,axiom,
    ! [Ba: int,Aa: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ba),Aa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),Ba)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,C),Aa) ) ) ).

% dual_order.strict_trans1
tff(fact_816_dual__order_Ostrict__trans2,axiom,
    ! [Ba: real,Aa: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Ba),Aa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),Ba)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,C),Aa) ) ) ).

% dual_order.strict_trans2
tff(fact_817_dual__order_Ostrict__trans2,axiom,
    ! [Ba: set_int,Aa: set_int,C: set_int] :
      ( aa_set_int_o(ord_less_set_int(Ba),Aa)
     => ( aa_set_int_o(ord_less_eq_set_int(C),Ba)
       => aa_set_int_o(ord_less_set_int(C),Aa) ) ) ).

% dual_order.strict_trans2
tff(fact_818_dual__order_Ostrict__trans2,axiom,
    ! [Ba: rat,Aa: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ba),Aa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),Ba)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),Aa) ) ) ).

% dual_order.strict_trans2
tff(fact_819_dual__order_Ostrict__trans2,axiom,
    ! [Ba: num,Aa: num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Ba),Aa)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),Ba)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,C),Aa) ) ) ).

% dual_order.strict_trans2
tff(fact_820_dual__order_Ostrict__trans2,axiom,
    ! [Ba: nat,Aa: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ba),Aa)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),Ba)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),Aa) ) ) ).

% dual_order.strict_trans2
tff(fact_821_dual__order_Ostrict__trans2,axiom,
    ! [Ba: int,Aa: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Ba),Aa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),Ba)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,C),Aa) ) ) ).

% dual_order.strict_trans2
tff(fact_822_dual__order_Ostrict__iff__not,axiom,
    ! [Ba: real,Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Ba),Aa)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ba),Aa)
        & ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),Ba) ) ) ).

% dual_order.strict_iff_not
tff(fact_823_dual__order_Ostrict__iff__not,axiom,
    ! [Ba: set_int,Aa: set_int] :
      ( aa_set_int_o(ord_less_set_int(Ba),Aa)
    <=> ( aa_set_int_o(ord_less_eq_set_int(Ba),Aa)
        & ~ aa_set_int_o(ord_less_eq_set_int(Aa),Ba) ) ) ).

% dual_order.strict_iff_not
tff(fact_824_dual__order_Ostrict__iff__not,axiom,
    ! [Ba: rat,Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ba),Aa)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ba),Aa)
        & ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),Ba) ) ) ).

% dual_order.strict_iff_not
tff(fact_825_dual__order_Ostrict__iff__not,axiom,
    ! [Ba: num,Aa: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Ba),Aa)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ba),Aa)
        & ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Aa),Ba) ) ) ).

% dual_order.strict_iff_not
tff(fact_826_dual__order_Ostrict__iff__not,axiom,
    ! [Ba: nat,Aa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ba),Aa)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ba),Aa)
        & ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),Ba) ) ) ).

% dual_order.strict_iff_not
tff(fact_827_dual__order_Ostrict__iff__not,axiom,
    ! [Ba: int,Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Ba),Aa)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ba),Aa)
        & ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),Ba) ) ) ).

% dual_order.strict_iff_not
tff(fact_828_order_Ostrict__implies__order,axiom,
    ! [Aa: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),Ba)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),Ba) ) ).

% order.strict_implies_order
tff(fact_829_order_Ostrict__implies__order,axiom,
    ! [Aa: set_int,Ba: set_int] :
      ( aa_set_int_o(ord_less_set_int(Aa),Ba)
     => aa_set_int_o(ord_less_eq_set_int(Aa),Ba) ) ).

% order.strict_implies_order
tff(fact_830_order_Ostrict__implies__order,axiom,
    ! [Aa: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),Ba)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),Ba) ) ).

% order.strict_implies_order
tff(fact_831_order_Ostrict__implies__order,axiom,
    ! [Aa: num,Ba: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Aa),Ba)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Aa),Ba) ) ).

% order.strict_implies_order
tff(fact_832_order_Ostrict__implies__order,axiom,
    ! [Aa: nat,Ba: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),Ba)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),Ba) ) ).

% order.strict_implies_order
tff(fact_833_order_Ostrict__implies__order,axiom,
    ! [Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),Ba)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),Ba) ) ).

% order.strict_implies_order
tff(fact_834_dual__order_Ostrict__implies__order,axiom,
    ! [Ba: real,Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Ba),Aa)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ba),Aa) ) ).

% dual_order.strict_implies_order
tff(fact_835_dual__order_Ostrict__implies__order,axiom,
    ! [Ba: set_int,Aa: set_int] :
      ( aa_set_int_o(ord_less_set_int(Ba),Aa)
     => aa_set_int_o(ord_less_eq_set_int(Ba),Aa) ) ).

% dual_order.strict_implies_order
tff(fact_836_dual__order_Ostrict__implies__order,axiom,
    ! [Ba: rat,Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ba),Aa)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ba),Aa) ) ).

% dual_order.strict_implies_order
tff(fact_837_dual__order_Ostrict__implies__order,axiom,
    ! [Ba: num,Aa: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Ba),Aa)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ba),Aa) ) ).

% dual_order.strict_implies_order
tff(fact_838_dual__order_Ostrict__implies__order,axiom,
    ! [Ba: nat,Aa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ba),Aa)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ba),Aa) ) ).

% dual_order.strict_implies_order
tff(fact_839_dual__order_Ostrict__implies__order,axiom,
    ! [Ba: int,Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Ba),Aa)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ba),Aa) ) ).

% dual_order.strict_implies_order
tff(fact_840_order__le__less,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
        | ( Xb = Y ) ) ) ).

% order_le_less
tff(fact_841_order__le__less,axiom,
    ! [Xb: set_int,Y: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(Xb),Y)
    <=> ( aa_set_int_o(ord_less_set_int(Xb),Y)
        | ( Xb = Y ) ) ) ).

% order_le_less
tff(fact_842_order__le__less,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
        | ( Xb = Y ) ) ) ).

% order_le_less
tff(fact_843_order__le__less,axiom,
    ! [Xb: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Y)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y)
        | ( Xb = Y ) ) ) ).

% order_le_less
tff(fact_844_order__le__less,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
        | ( Xb = Y ) ) ) ).

% order_le_less
tff(fact_845_order__le__less,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
        | ( Xb = Y ) ) ) ).

% order_le_less
tff(fact_846_order__less__le,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
        & ( Xb != Y ) ) ) ).

% order_less_le
tff(fact_847_order__less__le,axiom,
    ! [Xb: set_int,Y: set_int] :
      ( aa_set_int_o(ord_less_set_int(Xb),Y)
    <=> ( aa_set_int_o(ord_less_eq_set_int(Xb),Y)
        & ( Xb != Y ) ) ) ).

% order_less_le
tff(fact_848_order__less__le,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y)
        & ( Xb != Y ) ) ) ).

% order_less_le
tff(fact_849_order__less__le,axiom,
    ! [Xb: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Y)
        & ( Xb != Y ) ) ) ).

% order_less_le
tff(fact_850_order__less__le,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y)
        & ( Xb != Y ) ) ) ).

% order_less_le
tff(fact_851_order__less__le,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y)
        & ( Xb != Y ) ) ) ).

% order_less_le
tff(fact_852_linorder__not__le,axiom,
    ! [Xb: real,Y: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xb) ) ).

% linorder_not_le
tff(fact_853_linorder__not__le,axiom,
    ! [Xb: rat,Y: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Xb) ) ).

% linorder_not_le
tff(fact_854_linorder__not__le,axiom,
    ! [Xb: num,Y: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Y)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Y),Xb) ) ).

% linorder_not_le
tff(fact_855_linorder__not__le,axiom,
    ! [Xb: nat,Y: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xb) ) ).

% linorder_not_le
tff(fact_856_linorder__not__le,axiom,
    ! [Xb: int,Y: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Xb) ) ).

% linorder_not_le
tff(fact_857_linorder__not__less,axiom,
    ! [Xb: real,Y: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Xb) ) ).

% linorder_not_less
tff(fact_858_linorder__not__less,axiom,
    ! [Xb: rat,Y: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Xb) ) ).

% linorder_not_less
tff(fact_859_linorder__not__less,axiom,
    ! [Xb: num,Y: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),Xb) ) ).

% linorder_not_less
tff(fact_860_linorder__not__less,axiom,
    ! [Xb: nat,Y: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Xb) ) ).

% linorder_not_less
tff(fact_861_linorder__not__less,axiom,
    ! [Xb: int,Y: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Xb) ) ).

% linorder_not_less
tff(fact_862_order__less__imp__le,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y) ) ).

% order_less_imp_le
tff(fact_863_order__less__imp__le,axiom,
    ! [Xb: set_int,Y: set_int] :
      ( aa_set_int_o(ord_less_set_int(Xb),Y)
     => aa_set_int_o(ord_less_eq_set_int(Xb),Y) ) ).

% order_less_imp_le
tff(fact_864_order__less__imp__le,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y) ) ).

% order_less_imp_le
tff(fact_865_order__less__imp__le,axiom,
    ! [Xb: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Y) ) ).

% order_less_imp_le
tff(fact_866_order__less__imp__le,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y) ) ).

% order_less_imp_le
tff(fact_867_order__less__imp__le,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y) ) ).

% order_less_imp_le
tff(fact_868_order__le__neq__trans,axiom,
    ! [Aa: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),Ba)
     => ( ( Aa != Ba )
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),Ba) ) ) ).

% order_le_neq_trans
tff(fact_869_order__le__neq__trans,axiom,
    ! [Aa: set_int,Ba: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(Aa),Ba)
     => ( ( Aa != Ba )
       => aa_set_int_o(ord_less_set_int(Aa),Ba) ) ) ).

% order_le_neq_trans
tff(fact_870_order__le__neq__trans,axiom,
    ! [Aa: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),Ba)
     => ( ( Aa != Ba )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),Ba) ) ) ).

% order_le_neq_trans
tff(fact_871_order__le__neq__trans,axiom,
    ! [Aa: num,Ba: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Aa),Ba)
     => ( ( Aa != Ba )
       => aa_num_o(aa_num_fun_num_o(ord_less_num,Aa),Ba) ) ) ).

% order_le_neq_trans
tff(fact_872_order__le__neq__trans,axiom,
    ! [Aa: nat,Ba: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),Ba)
     => ( ( Aa != Ba )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),Ba) ) ) ).

% order_le_neq_trans
tff(fact_873_order__le__neq__trans,axiom,
    ! [Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),Ba)
     => ( ( Aa != Ba )
       => aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),Ba) ) ) ).

% order_le_neq_trans
tff(fact_874_order__neq__le__trans,axiom,
    ! [Aa: real,Ba: real] :
      ( ( Aa != Ba )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),Ba)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),Ba) ) ) ).

% order_neq_le_trans
tff(fact_875_order__neq__le__trans,axiom,
    ! [Aa: set_int,Ba: set_int] :
      ( ( Aa != Ba )
     => ( aa_set_int_o(ord_less_eq_set_int(Aa),Ba)
       => aa_set_int_o(ord_less_set_int(Aa),Ba) ) ) ).

% order_neq_le_trans
tff(fact_876_order__neq__le__trans,axiom,
    ! [Aa: rat,Ba: rat] :
      ( ( Aa != Ba )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),Ba)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),Ba) ) ) ).

% order_neq_le_trans
tff(fact_877_order__neq__le__trans,axiom,
    ! [Aa: num,Ba: num] :
      ( ( Aa != Ba )
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Aa),Ba)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,Aa),Ba) ) ) ).

% order_neq_le_trans
tff(fact_878_order__neq__le__trans,axiom,
    ! [Aa: nat,Ba: nat] :
      ( ( Aa != Ba )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),Ba)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),Ba) ) ) ).

% order_neq_le_trans
tff(fact_879_order__neq__le__trans,axiom,
    ! [Aa: int,Ba: int] :
      ( ( Aa != Ba )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),Ba)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),Ba) ) ) ).

% order_neq_le_trans
tff(fact_880_order__le__less__trans,axiom,
    ! [Xb: real,Y: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Z)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Z) ) ) ).

% order_le_less_trans
tff(fact_881_order__le__less__trans,axiom,
    ! [Xb: set_int,Y: set_int,Z: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(Xb),Y)
     => ( aa_set_int_o(ord_less_set_int(Y),Z)
       => aa_set_int_o(ord_less_set_int(Xb),Z) ) ) ).

% order_le_less_trans
tff(fact_882_order__le__less__trans,axiom,
    ! [Xb: rat,Y: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Z)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Z) ) ) ).

% order_le_less_trans
tff(fact_883_order__le__less__trans,axiom,
    ! [Xb: num,Y: num,Z: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Y)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,Y),Z)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Z) ) ) ).

% order_le_less_trans
tff(fact_884_order__le__less__trans,axiom,
    ! [Xb: nat,Y: nat,Z: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Z)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Z) ) ) ).

% order_le_less_trans
tff(fact_885_order__le__less__trans,axiom,
    ! [Xb: int,Y: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Z)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Z) ) ) ).

% order_le_less_trans
tff(fact_886_order__less__le__trans,axiom,
    ! [Xb: real,Y: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Z)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Z) ) ) ).

% order_less_le_trans
tff(fact_887_order__less__le__trans,axiom,
    ! [Xb: set_int,Y: set_int,Z: set_int] :
      ( aa_set_int_o(ord_less_set_int(Xb),Y)
     => ( aa_set_int_o(ord_less_eq_set_int(Y),Z)
       => aa_set_int_o(ord_less_set_int(Xb),Z) ) ) ).

% order_less_le_trans
tff(fact_888_order__less__le__trans,axiom,
    ! [Xb: rat,Y: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Z)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Z) ) ) ).

% order_less_le_trans
tff(fact_889_order__less__le__trans,axiom,
    ! [Xb: num,Y: num,Z: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),Z)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Z) ) ) ).

% order_less_le_trans
tff(fact_890_order__less__le__trans,axiom,
    ! [Xb: nat,Y: nat,Z: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Z)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Z) ) ) ).

% order_less_le_trans
tff(fact_891_order__less__le__trans,axiom,
    ! [Xb: int,Y: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Z)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Z) ) ) ).

% order_less_le_trans
tff(fact_892_order__le__less__subst1,axiom,
    ! [Aa: real,F: fun_real_real,Ba: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),aa_real_real(F,Ba))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Ba),C)
       => ( ! [X: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,X)),aa_real_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),aa_real_real(F,C)) ) ) ) ).

% order_le_less_subst1
tff(fact_893_order__le__less__subst1,axiom,
    ! [Aa: real,F: fun_rat_real,Ba: rat,C: rat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),aa_rat_real(F,Ba))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ba),C)
       => ( ! [X: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,X)),aa_rat_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),aa_rat_real(F,C)) ) ) ) ).

% order_le_less_subst1
tff(fact_894_order__le__less__subst1,axiom,
    ! [Aa: real,F: fun_num_real,Ba: num,C: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),aa_num_real(F,Ba))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,Ba),C)
       => ( ! [X: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(F,X)),aa_num_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),aa_num_real(F,C)) ) ) ) ).

% order_le_less_subst1
tff(fact_895_order__le__less__subst1,axiom,
    ! [Aa: real,F: fun_nat_real,Ba: nat,C: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),aa_nat_real(F,Ba))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ba),C)
       => ( ! [X: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,X)),aa_nat_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),aa_nat_real(F,C)) ) ) ) ).

% order_le_less_subst1
tff(fact_896_order__le__less__subst1,axiom,
    ! [Aa: real,F: fun_int_real,Ba: int,C: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),aa_int_real(F,Ba))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Ba),C)
       => ( ! [X: int,Y3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(F,X)),aa_int_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),aa_int_real(F,C)) ) ) ) ).

% order_le_less_subst1
tff(fact_897_order__le__less__subst1,axiom,
    ! [Aa: rat,F: fun_real_rat,Ba: real,C: real] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),aa_real_rat(F,Ba))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Ba),C)
       => ( ! [X: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,X)),aa_real_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),aa_real_rat(F,C)) ) ) ) ).

% order_le_less_subst1
tff(fact_898_order__le__less__subst1,axiom,
    ! [Aa: rat,F: fun_rat_rat,Ba: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),aa_rat_rat(F,Ba))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ba),C)
       => ( ! [X: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,X)),aa_rat_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),aa_rat_rat(F,C)) ) ) ) ).

% order_le_less_subst1
tff(fact_899_order__le__less__subst1,axiom,
    ! [Aa: rat,F: fun_num_rat,Ba: num,C: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),aa_num_rat(F,Ba))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,Ba),C)
       => ( ! [X: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(F,X)),aa_num_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),aa_num_rat(F,C)) ) ) ) ).

% order_le_less_subst1
tff(fact_900_order__le__less__subst1,axiom,
    ! [Aa: rat,F: fun_nat_rat,Ba: nat,C: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),aa_nat_rat(F,Ba))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ba),C)
       => ( ! [X: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,X)),aa_nat_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),aa_nat_rat(F,C)) ) ) ) ).

% order_le_less_subst1
tff(fact_901_order__le__less__subst1,axiom,
    ! [Aa: rat,F: fun_int_rat,Ba: int,C: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),aa_int_rat(F,Ba))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Ba),C)
       => ( ! [X: int,Y3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(F,X)),aa_int_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),aa_int_rat(F,C)) ) ) ) ).

% order_le_less_subst1
tff(fact_902_order__le__less__subst2,axiom,
    ! [Aa: rat,Ba: rat,F: fun_rat_real,C: real] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),Ba)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,Ba)),C)
       => ( ! [X: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_rat_real(F,X)),aa_rat_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,Aa)),C) ) ) ) ).

% order_le_less_subst2
tff(fact_903_order__le__less__subst2,axiom,
    ! [Aa: rat,Ba: rat,F: fun_rat_rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),Ba)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,Ba)),C)
       => ( ! [X: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,X)),aa_rat_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,Aa)),C) ) ) ) ).

% order_le_less_subst2
tff(fact_904_order__le__less__subst2,axiom,
    ! [Aa: rat,Ba: rat,F: fun_rat_num,C: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),Ba)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F,Ba)),C)
       => ( ! [X: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,X)),aa_rat_num(F,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F,Aa)),C) ) ) ) ).

% order_le_less_subst2
tff(fact_905_order__le__less__subst2,axiom,
    ! [Aa: rat,Ba: rat,F: fun_rat_nat,C: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),Ba)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,Ba)),C)
       => ( ! [X: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,X)),aa_rat_nat(F,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,Aa)),C) ) ) ) ).

% order_le_less_subst2
tff(fact_906_order__le__less__subst2,axiom,
    ! [Aa: rat,Ba: rat,F: fun_rat_int,C: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),Ba)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(F,Ba)),C)
       => ( ! [X: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,X)),aa_rat_int(F,Y3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(F,Aa)),C) ) ) ) ).

% order_le_less_subst2
tff(fact_907_order__le__less__subst2,axiom,
    ! [Aa: num,Ba: num,F: fun_num_real,C: real] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Aa),Ba)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(F,Ba)),C)
       => ( ! [X: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(F,X)),aa_num_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(F,Aa)),C) ) ) ) ).

% order_le_less_subst2
tff(fact_908_order__le__less__subst2,axiom,
    ! [Aa: num,Ba: num,F: fun_num_rat,C: rat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Aa),Ba)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(F,Ba)),C)
       => ( ! [X: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,X)),aa_num_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(F,Aa)),C) ) ) ) ).

% order_le_less_subst2
tff(fact_909_order__le__less__subst2,axiom,
    ! [Aa: num,Ba: num,F: fun_num_num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Aa),Ba)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(F,Ba)),C)
       => ( ! [X: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,X)),aa_num_num(F,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(F,Aa)),C) ) ) ) ).

% order_le_less_subst2
tff(fact_910_order__le__less__subst2,axiom,
    ! [Aa: num,Ba: num,F: fun_num_nat,C: nat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Aa),Ba)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(F,Ba)),C)
       => ( ! [X: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,X)),aa_num_nat(F,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(F,Aa)),C) ) ) ) ).

% order_le_less_subst2
tff(fact_911_order__le__less__subst2,axiom,
    ! [Aa: num,Ba: num,F: fun_num_int,C: int] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Aa),Ba)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(F,Ba)),C)
       => ( ! [X: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(F,X)),aa_num_int(F,Y3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(F,Aa)),C) ) ) ) ).

% order_le_less_subst2
tff(fact_912_order__less__le__subst1,axiom,
    ! [Aa: real,F: fun_rat_real,Ba: rat,C: rat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),aa_rat_real(F,Ba))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ba),C)
       => ( ! [X: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_rat_real(F,X)),aa_rat_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),aa_rat_real(F,C)) ) ) ) ).

% order_less_le_subst1
tff(fact_913_order__less__le__subst1,axiom,
    ! [Aa: rat,F: fun_rat_rat,Ba: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),aa_rat_rat(F,Ba))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ba),C)
       => ( ! [X: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,X)),aa_rat_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),aa_rat_rat(F,C)) ) ) ) ).

% order_less_le_subst1
tff(fact_914_order__less__le__subst1,axiom,
    ! [Aa: num,F: fun_rat_num,Ba: rat,C: rat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Aa),aa_rat_num(F,Ba))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ba),C)
       => ( ! [X: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,X)),aa_rat_num(F,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,Aa),aa_rat_num(F,C)) ) ) ) ).

% order_less_le_subst1
tff(fact_915_order__less__le__subst1,axiom,
    ! [Aa: nat,F: fun_rat_nat,Ba: rat,C: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),aa_rat_nat(F,Ba))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ba),C)
       => ( ! [X: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,X)),aa_rat_nat(F,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),aa_rat_nat(F,C)) ) ) ) ).

% order_less_le_subst1
tff(fact_916_order__less__le__subst1,axiom,
    ! [Aa: int,F: fun_rat_int,Ba: rat,C: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),aa_rat_int(F,Ba))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ba),C)
       => ( ! [X: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,X)),aa_rat_int(F,Y3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),aa_rat_int(F,C)) ) ) ) ).

% order_less_le_subst1
tff(fact_917_order__less__le__subst1,axiom,
    ! [Aa: real,F: fun_num_real,Ba: num,C: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),aa_num_real(F,Ba))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ba),C)
       => ( ! [X: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(F,X)),aa_num_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),aa_num_real(F,C)) ) ) ) ).

% order_less_le_subst1
tff(fact_918_order__less__le__subst1,axiom,
    ! [Aa: rat,F: fun_num_rat,Ba: num,C: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),aa_num_rat(F,Ba))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ba),C)
       => ( ! [X: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,X)),aa_num_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),aa_num_rat(F,C)) ) ) ) ).

% order_less_le_subst1
tff(fact_919_order__less__le__subst1,axiom,
    ! [Aa: num,F: fun_num_num,Ba: num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Aa),aa_num_num(F,Ba))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ba),C)
       => ( ! [X: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,X)),aa_num_num(F,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,Aa),aa_num_num(F,C)) ) ) ) ).

% order_less_le_subst1
tff(fact_920_order__less__le__subst1,axiom,
    ! [Aa: nat,F: fun_num_nat,Ba: num,C: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),aa_num_nat(F,Ba))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ba),C)
       => ( ! [X: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,X)),aa_num_nat(F,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),aa_num_nat(F,C)) ) ) ) ).

% order_less_le_subst1
tff(fact_921_order__less__le__subst1,axiom,
    ! [Aa: int,F: fun_num_int,Ba: num,C: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),aa_num_int(F,Ba))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ba),C)
       => ( ! [X: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(F,X)),aa_num_int(F,Y3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),aa_num_int(F,C)) ) ) ) ).

% order_less_le_subst1
tff(fact_922_order__less__le__subst2,axiom,
    ! [Aa: real,Ba: real,F: fun_real_real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),Ba)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,Ba)),C)
       => ( ! [X: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,X)),aa_real_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,Aa)),C) ) ) ) ).

% order_less_le_subst2
tff(fact_923_order__less__le__subst2,axiom,
    ! [Aa: rat,Ba: rat,F: fun_rat_real,C: real] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),Ba)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_rat_real(F,Ba)),C)
       => ( ! [X: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,X)),aa_rat_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,Aa)),C) ) ) ) ).

% order_less_le_subst2
tff(fact_924_order__less__le__subst2,axiom,
    ! [Aa: num,Ba: num,F: fun_num_real,C: real] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Aa),Ba)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(F,Ba)),C)
       => ( ! [X: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(F,X)),aa_num_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(F,Aa)),C) ) ) ) ).

% order_less_le_subst2
tff(fact_925_order__less__le__subst2,axiom,
    ! [Aa: nat,Ba: nat,F: fun_nat_real,C: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),Ba)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,Ba)),C)
       => ( ! [X: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,X)),aa_nat_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,Aa)),C) ) ) ) ).

% order_less_le_subst2
tff(fact_926_order__less__le__subst2,axiom,
    ! [Aa: int,Ba: int,F: fun_int_real,C: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),Ba)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(F,Ba)),C)
       => ( ! [X: int,Y3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(F,X)),aa_int_real(F,Y3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(F,Aa)),C) ) ) ) ).

% order_less_le_subst2
tff(fact_927_order__less__le__subst2,axiom,
    ! [Aa: real,Ba: real,F: fun_real_rat,C: rat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),Ba)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_real_rat(F,Ba)),C)
       => ( ! [X: real,Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,X)),aa_real_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,Aa)),C) ) ) ) ).

% order_less_le_subst2
tff(fact_928_order__less__le__subst2,axiom,
    ! [Aa: rat,Ba: rat,F: fun_rat_rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),Ba)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,Ba)),C)
       => ( ! [X: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,X)),aa_rat_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,Aa)),C) ) ) ) ).

% order_less_le_subst2
tff(fact_929_order__less__le__subst2,axiom,
    ! [Aa: num,Ba: num,F: fun_num_rat,C: rat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Aa),Ba)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,Ba)),C)
       => ( ! [X: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(F,X)),aa_num_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(F,Aa)),C) ) ) ) ).

% order_less_le_subst2
tff(fact_930_order__less__le__subst2,axiom,
    ! [Aa: nat,Ba: nat,F: fun_nat_rat,C: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),Ba)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,Ba)),C)
       => ( ! [X: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,X)),aa_nat_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,Aa)),C) ) ) ) ).

% order_less_le_subst2
tff(fact_931_order__less__le__subst2,axiom,
    ! [Aa: int,Ba: int,F: fun_int_rat,C: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),Ba)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,Ba)),C)
       => ( ! [X: int,Y3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(F,X)),aa_int_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(F,Aa)),C) ) ) ) ).

% order_less_le_subst2
tff(fact_932_linorder__le__less__linear,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
      | aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xb) ) ).

% linorder_le_less_linear
tff(fact_933_linorder__le__less__linear,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y)
      | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Xb) ) ).

% linorder_le_less_linear
tff(fact_934_linorder__le__less__linear,axiom,
    ! [Xb: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Y)
      | aa_num_o(aa_num_fun_num_o(ord_less_num,Y),Xb) ) ).

% linorder_le_less_linear
tff(fact_935_linorder__le__less__linear,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y)
      | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xb) ) ).

% linorder_le_less_linear
tff(fact_936_linorder__le__less__linear,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y)
      | aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Xb) ) ).

% linorder_le_less_linear
tff(fact_937_order__le__imp__less__or__eq,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
        | ( Xb = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_938_order__le__imp__less__or__eq,axiom,
    ! [Xb: set_int,Y: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(Xb),Y)
     => ( aa_set_int_o(ord_less_set_int(Xb),Y)
        | ( Xb = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_939_order__le__imp__less__or__eq,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
        | ( Xb = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_940_order__le__imp__less__or__eq,axiom,
    ! [Xb: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Y)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y)
        | ( Xb = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_941_order__le__imp__less__or__eq,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
        | ( Xb = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_942_order__le__imp__less__or__eq,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
        | ( Xb = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_943_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_944_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_945_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_946_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_947_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_948_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_949_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_950_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_951_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_952_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_953_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_954_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_955_psubsetI,axiom,
    ! [A3: set_int,B4: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A3),B4)
     => ( ( A3 != B4 )
       => aa_set_int_o(ord_less_set_int(A3),B4) ) ) ).

% psubsetI
tff(fact_956_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_957_subset__empty,axiom,
    ! [A3: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A3),bot_bot_set_nat)
    <=> ( A3 = bot_bot_set_nat ) ) ).

% subset_empty
tff(fact_958_subset__empty,axiom,
    ! [A3: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A3),bot_bot_set_int)
    <=> ( A3 = bot_bot_set_int ) ) ).

% subset_empty
tff(fact_959_empty__subsetI,axiom,
    ! [A3: set_real] : ord_less_eq_set_real(bot_bot_set_real,A3) ).

% empty_subsetI
tff(fact_960_empty__subsetI,axiom,
    ! [A3: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,bot_bot_set_nat),A3) ).

% empty_subsetI
tff(fact_961_empty__subsetI,axiom,
    ! [A3: set_int] : aa_set_int_o(ord_less_eq_set_int(bot_bot_set_int),A3) ).

% empty_subsetI
tff(fact_962_maxt__corr__help__empty,axiom,
    ! [Ta: vEBT_VEBT,Na: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Na)
     => ( ( vEBT_vebt_maxt(Ta) = none_nat )
       => ( vEBT_VEBT_set_vebt(Ta) = bot_bot_set_nat ) ) ) ).

% maxt_corr_help_empty
tff(fact_963_mint__corr__help__empty,axiom,
    ! [Ta: vEBT_VEBT,Na: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Na)
     => ( ( vEBT_vebt_mint(Ta) = none_nat )
       => ( vEBT_VEBT_set_vebt(Ta) = bot_bot_set_nat ) ) ) ).

% mint_corr_help_empty
tff(fact_964_ex__min__if__finite,axiom,
    ! [S2: set_real] :
      ( finite_finite_real(S2)
     => ( ( S2 != bot_bot_set_real )
       => ? [X: real] :
            ( member_real(X,S2)
            & ~ ? [Xa: real] :
                  ( member_real(Xa,S2)
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),X) ) ) ) ) ).

% ex_min_if_finite
tff(fact_965_ex__min__if__finite,axiom,
    ! [S2: set_rat] :
      ( finite_finite_rat(S2)
     => ( ( S2 != bot_bot_set_rat )
       => ? [X: rat] :
            ( member_rat(X,S2)
            & ~ ? [Xa: rat] :
                  ( member_rat(Xa,S2)
                  & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),X) ) ) ) ) ).

% ex_min_if_finite
tff(fact_966_ex__min__if__finite,axiom,
    ! [S2: set_num] :
      ( finite_finite_num(S2)
     => ( ( S2 != bot_bot_set_num )
       => ? [X: num] :
            ( member_num(X,S2)
            & ~ ? [Xa: num] :
                  ( member_num(Xa,S2)
                  & aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),X) ) ) ) ) ).

% ex_min_if_finite
tff(fact_967_ex__min__if__finite,axiom,
    ! [S2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( ( S2 != bot_bot_set_nat )
       => ? [X: nat] :
            ( aa_set_nat_o(member_nat(X),S2)
            & ~ ? [Xa: nat] :
                  ( aa_set_nat_o(member_nat(Xa),S2)
                  & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),X) ) ) ) ) ).

% ex_min_if_finite
tff(fact_968_ex__min__if__finite,axiom,
    ! [S2: set_int] :
      ( aa_set_int_o(finite_finite_int,S2)
     => ( ( S2 != bot_bot_set_int )
       => ? [X: int] :
            ( aa_set_int_o(member_int(X),S2)
            & ~ ? [Xa: int] :
                  ( aa_set_int_o(member_int(Xa),S2)
                  & aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),X) ) ) ) ) ).

% ex_min_if_finite
tff(fact_969_infinite__growing,axiom,
    ! [X5: set_real] :
      ( ( X5 != bot_bot_set_real )
     => ( ! [X: real] :
            ( member_real(X,X5)
           => ? [Xa: real] :
                ( member_real(Xa,X5)
                & aa_real_o(aa_real_fun_real_o(ord_less_real,X),Xa) ) )
       => ~ finite_finite_real(X5) ) ) ).

% infinite_growing
tff(fact_970_infinite__growing,axiom,
    ! [X5: set_rat] :
      ( ( X5 != bot_bot_set_rat )
     => ( ! [X: rat] :
            ( member_rat(X,X5)
           => ? [Xa: rat] :
                ( member_rat(Xa,X5)
                & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Xa) ) )
       => ~ finite_finite_rat(X5) ) ) ).

% infinite_growing
tff(fact_971_infinite__growing,axiom,
    ! [X5: set_num] :
      ( ( X5 != bot_bot_set_num )
     => ( ! [X: num] :
            ( member_num(X,X5)
           => ? [Xa: num] :
                ( member_num(Xa,X5)
                & aa_num_o(aa_num_fun_num_o(ord_less_num,X),Xa) ) )
       => ~ finite_finite_num(X5) ) ) ).

% infinite_growing
tff(fact_972_infinite__growing,axiom,
    ! [X5: set_nat] :
      ( ( X5 != bot_bot_set_nat )
     => ( ! [X: nat] :
            ( aa_set_nat_o(member_nat(X),X5)
           => ? [Xa: nat] :
                ( aa_set_nat_o(member_nat(Xa),X5)
                & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Xa) ) )
       => ~ aa_set_nat_o(finite_finite_nat,X5) ) ) ).

% infinite_growing
tff(fact_973_infinite__growing,axiom,
    ! [X5: set_int] :
      ( ( X5 != bot_bot_set_int )
     => ( ! [X: int] :
            ( aa_set_int_o(member_int(X),X5)
           => ? [Xa: int] :
                ( aa_set_int_o(member_int(Xa),X5)
                & aa_int_o(aa_int_fun_int_o(ord_less_int,X),Xa) ) )
       => ~ aa_set_int_o(finite_finite_int,X5) ) ) ).

% infinite_growing
tff(fact_974_subsetI,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ! [X: product_prod_nat_nat] :
          ( member8440522571783428010at_nat(X,A3)
         => member8440522571783428010at_nat(X,B4) )
     => ord_le3146513528884898305at_nat(A3,B4) ) ).

% subsetI
tff(fact_975_subsetI,axiom,
    ! [A3: set_real,B4: set_real] :
      ( ! [X: real] :
          ( member_real(X,A3)
         => member_real(X,B4) )
     => ord_less_eq_set_real(A3,B4) ) ).

% subsetI
tff(fact_976_subsetI,axiom,
    ! [A3: set_set_nat,B4: set_set_nat] :
      ( ! [X: set_nat] :
          ( member_set_nat(X,A3)
         => member_set_nat(X,B4) )
     => ord_le6893508408891458716et_nat(A3,B4) ) ).

% subsetI
tff(fact_977_subsetI,axiom,
    ! [A3: set_nat,B4: set_nat] :
      ( ! [X: nat] :
          ( aa_set_nat_o(member_nat(X),A3)
         => aa_set_nat_o(member_nat(X),B4) )
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A3),B4) ) ).

% subsetI
tff(fact_978_subsetI,axiom,
    ! [A3: set_int,B4: set_int] :
      ( ! [X: int] :
          ( aa_set_int_o(member_int(X),A3)
         => aa_set_int_o(member_int(X),B4) )
     => aa_set_int_o(ord_less_eq_set_int(A3),B4) ) ).

% subsetI
tff(fact_979_subset__antisym,axiom,
    ! [A3: set_int,B4: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A3),B4)
     => ( aa_set_int_o(ord_less_eq_set_int(B4),A3)
       => ( A3 = B4 ) ) ) ).

% subset_antisym
tff(fact_980_empty__iff,axiom,
    ! [C: product_prod_nat_nat] : ~ member8440522571783428010at_nat(C,bot_bo2099793752762293965at_nat) ).

% empty_iff
tff(fact_981_empty__iff,axiom,
    ! [C: set_nat] : ~ member_set_nat(C,bot_bot_set_set_nat) ).

% empty_iff
tff(fact_982_empty__iff,axiom,
    ! [C: real] : ~ member_real(C,bot_bot_set_real) ).

% empty_iff
tff(fact_983_empty__iff,axiom,
    ! [C: nat] : ~ aa_set_nat_o(member_nat(C),bot_bot_set_nat) ).

% empty_iff
tff(fact_984_empty__iff,axiom,
    ! [C: int] : ~ aa_set_int_o(member_int(C),bot_bot_set_int) ).

% empty_iff
tff(fact_985_all__not__in__conv,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] :
      ( ! [X4: product_prod_nat_nat] : ~ member8440522571783428010at_nat(X4,A3)
    <=> ( A3 = bot_bo2099793752762293965at_nat ) ) ).

% all_not_in_conv
tff(fact_986_all__not__in__conv,axiom,
    ! [A3: set_set_nat] :
      ( ! [X4: set_nat] : ~ member_set_nat(X4,A3)
    <=> ( A3 = bot_bot_set_set_nat ) ) ).

% all_not_in_conv
tff(fact_987_all__not__in__conv,axiom,
    ! [A3: set_real] :
      ( ! [X4: real] : ~ member_real(X4,A3)
    <=> ( A3 = bot_bot_set_real ) ) ).

% all_not_in_conv
tff(fact_988_all__not__in__conv,axiom,
    ! [A3: set_nat] :
      ( ! [X4: nat] : ~ aa_set_nat_o(member_nat(X4),A3)
    <=> ( A3 = bot_bot_set_nat ) ) ).

% all_not_in_conv
tff(fact_989_all__not__in__conv,axiom,
    ! [A3: set_int] :
      ( ! [X4: int] : ~ aa_set_int_o(member_int(X4),A3)
    <=> ( A3 = bot_bot_set_int ) ) ).

% all_not_in_conv
tff(fact_990_minminNull,axiom,
    ! [Ta: vEBT_VEBT] :
      ( ( vEBT_vebt_mint(Ta) = none_nat )
     => aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Ta) ) ).

% minminNull
tff(fact_991_minNullmin,axiom,
    ! [Ta: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Ta)
     => ( vEBT_vebt_mint(Ta) = none_nat ) ) ).

% minNullmin
tff(fact_992_empty__Collect__eq,axiom,
    ! [P: fun_list_nat_o] :
      ( ( bot_bot_set_list_nat = collect_list_nat(P) )
    <=> ! [X4: list_nat] : ~ aa_list_nat_o(P,X4) ) ).

% empty_Collect_eq
tff(fact_993_empty__Collect__eq,axiom,
    ! [P: fun_set_nat_o] :
      ( ( bot_bot_set_set_nat = collect_set_nat(P) )
    <=> ! [X4: set_nat] : ~ aa_set_nat_o(P,X4) ) ).

% empty_Collect_eq
tff(fact_994_empty__Collect__eq,axiom,
    ! [P: fun_real_o] :
      ( ( bot_bot_set_real = collect_real(P) )
    <=> ! [X4: real] : ~ aa_real_o(P,X4) ) ).

% empty_Collect_eq
tff(fact_995_empty__Collect__eq,axiom,
    ! [P: fun_nat_o] :
      ( ( bot_bot_set_nat = collect_nat(P) )
    <=> ! [X4: nat] : ~ aa_nat_o(P,X4) ) ).

% empty_Collect_eq
tff(fact_996_empty__Collect__eq,axiom,
    ! [P: fun_int_o] :
      ( ( bot_bot_set_int = collect_int(P) )
    <=> ! [X4: int] : ~ aa_int_o(P,X4) ) ).

% empty_Collect_eq
tff(fact_997_Collect__empty__eq,axiom,
    ! [P: fun_list_nat_o] :
      ( ( collect_list_nat(P) = bot_bot_set_list_nat )
    <=> ! [X4: list_nat] : ~ aa_list_nat_o(P,X4) ) ).

% Collect_empty_eq
tff(fact_998_Collect__empty__eq,axiom,
    ! [P: fun_set_nat_o] :
      ( ( collect_set_nat(P) = bot_bot_set_set_nat )
    <=> ! [X4: set_nat] : ~ aa_set_nat_o(P,X4) ) ).

% Collect_empty_eq
tff(fact_999_Collect__empty__eq,axiom,
    ! [P: fun_real_o] :
      ( ( collect_real(P) = bot_bot_set_real )
    <=> ! [X4: real] : ~ aa_real_o(P,X4) ) ).

% Collect_empty_eq
tff(fact_1000_Collect__empty__eq,axiom,
    ! [P: fun_nat_o] :
      ( ( collect_nat(P) = bot_bot_set_nat )
    <=> ! [X4: nat] : ~ aa_nat_o(P,X4) ) ).

% Collect_empty_eq
tff(fact_1001_Collect__empty__eq,axiom,
    ! [P: fun_int_o] :
      ( ( collect_int(P) = bot_bot_set_int )
    <=> ! [X4: int] : ~ aa_int_o(P,X4) ) ).

% Collect_empty_eq
tff(fact_1002_not__Some__eq,axiom,
    ! [Xb: option_nat] :
      ( ! [Y5: nat] : Xb != some_nat(Y5)
    <=> ( Xb = none_nat ) ) ).

% not_Some_eq
tff(fact_1003_not__Some__eq,axiom,
    ! [Xb: option4927543243414619207at_nat] :
      ( ! [Y5: product_prod_nat_nat] : Xb != some_P7363390416028606310at_nat(Y5)
    <=> ( Xb = none_P5556105721700978146at_nat ) ) ).

% not_Some_eq
tff(fact_1004_not__Some__eq,axiom,
    ! [Xb: option_num] :
      ( ! [Y5: num] : Xb != some_num(Y5)
    <=> ( Xb = none_num ) ) ).

% not_Some_eq
tff(fact_1005_not__None__eq,axiom,
    ! [Xb: option_nat] :
      ( ( Xb != none_nat )
    <=> ? [Y5: nat] : Xb = some_nat(Y5) ) ).

% not_None_eq
tff(fact_1006_not__None__eq,axiom,
    ! [Xb: option4927543243414619207at_nat] :
      ( ( Xb != none_P5556105721700978146at_nat )
    <=> ? [Y5: product_prod_nat_nat] : Xb = some_P7363390416028606310at_nat(Y5) ) ).

% not_None_eq
tff(fact_1007_not__None__eq,axiom,
    ! [Xb: option_num] :
      ( ( Xb != none_num )
    <=> ? [Y5: num] : Xb = some_num(Y5) ) ).

% not_None_eq
tff(fact_1008_bot__set__def,axiom,
    bot_bot_set_list_nat = collect_list_nat(bot_bo8828579575001852285_nat_o) ).

% bot_set_def
tff(fact_1009_bot__set__def,axiom,
    bot_bot_set_set_nat = collect_set_nat(bot_bo3967814874310145751_nat_o) ).

% bot_set_def
tff(fact_1010_bot__set__def,axiom,
    bot_bot_set_real = collect_real(bot_bot_fun_real_o) ).

% bot_set_def
tff(fact_1011_bot__set__def,axiom,
    bot_bot_set_nat = collect_nat(bot_bot_fun_nat_o) ).

% bot_set_def
tff(fact_1012_bot__set__def,axiom,
    bot_bot_set_int = collect_int(bot_bot_fun_int_o) ).

% bot_set_def
tff(fact_1013_bot__nat__def,axiom,
    bot_bot_nat = zero_zero_nat ).

% bot_nat_def
tff(fact_1014_combine__options__cases,axiom,
    ! [Xb: option_nat,P: fun_op5794009454290973322_nat_o,Y: option_nat] :
      ( ( ( Xb = none_nat )
       => aa_option_nat_o(aa_opt4319133605541840243_nat_o(P,Xb),Y) )
     => ( ( ( Y = none_nat )
         => aa_option_nat_o(aa_opt4319133605541840243_nat_o(P,Xb),Y) )
       => ( ! [A2: nat,B2: nat] :
              ( ( Xb = some_nat(A2) )
             => ( ( Y = some_nat(B2) )
               => aa_option_nat_o(aa_opt4319133605541840243_nat_o(P,Xb),Y) ) )
         => aa_option_nat_o(aa_opt4319133605541840243_nat_o(P,Xb),Y) ) ) ) ).

% combine_options_cases
tff(fact_1015_combine__options__cases,axiom,
    ! [Xb: option_nat,P: fun_op3832135881594366065_nat_o,Y: option4927543243414619207at_nat] :
      ( ( ( Xb = none_nat )
       => aa_opt7112393969400857165_nat_o(aa_opt120907569613471666_nat_o(P,Xb),Y) )
     => ( ( ( Y = none_P5556105721700978146at_nat )
         => aa_opt7112393969400857165_nat_o(aa_opt120907569613471666_nat_o(P,Xb),Y) )
       => ( ! [A2: nat,B2: product_prod_nat_nat] :
              ( ( Xb = some_nat(A2) )
             => ( ( Y = some_P7363390416028606310at_nat(B2) )
               => aa_opt7112393969400857165_nat_o(aa_opt120907569613471666_nat_o(P,Xb),Y) ) )
         => aa_opt7112393969400857165_nat_o(aa_opt120907569613471666_nat_o(P,Xb),Y) ) ) ) ).

% combine_options_cases
tff(fact_1016_combine__options__cases,axiom,
    ! [Xb: option_nat,P: fun_op6120599277751319488_num_o,Y: option_num] :
      ( ( ( Xb = none_nat )
       => aa_option_num_o(aa_opt5077314168511351209_num_o(P,Xb),Y) )
     => ( ( ( Y = none_num )
         => aa_option_num_o(aa_opt5077314168511351209_num_o(P,Xb),Y) )
       => ( ! [A2: nat,B2: num] :
              ( ( Xb = some_nat(A2) )
             => ( ( Y = some_num(B2) )
               => aa_option_num_o(aa_opt5077314168511351209_num_o(P,Xb),Y) ) )
         => aa_option_num_o(aa_opt5077314168511351209_num_o(P,Xb),Y) ) ) ) ).

% combine_options_cases
tff(fact_1017_combine__options__cases,axiom,
    ! [Xb: option4927543243414619207at_nat,P: fun_op7326680082016885773_nat_o,Y: option_nat] :
      ( ( ( Xb = none_P5556105721700978146at_nat )
       => aa_option_nat_o(aa_opt7110402928079307526_nat_o(P,Xb),Y) )
     => ( ( ( Y = none_nat )
         => aa_option_nat_o(aa_opt7110402928079307526_nat_o(P,Xb),Y) )
       => ( ! [A2: product_prod_nat_nat,B2: nat] :
              ( ( Xb = some_P7363390416028606310at_nat(A2) )
             => ( ( Y = some_nat(B2) )
               => aa_option_nat_o(aa_opt7110402928079307526_nat_o(P,Xb),Y) ) )
         => aa_option_nat_o(aa_opt7110402928079307526_nat_o(P,Xb),Y) ) ) ) ).

% combine_options_cases
tff(fact_1018_combine__options__cases,axiom,
    ! [Xb: option4927543243414619207at_nat,P: fun_op1221328669798608686_nat_o,Y: option4927543243414619207at_nat] :
      ( ( ( Xb = none_P5556105721700978146at_nat )
       => aa_opt7112393969400857165_nat_o(aa_opt9039718665754746463_nat_o(P,Xb),Y) )
     => ( ( ( Y = none_P5556105721700978146at_nat )
         => aa_opt7112393969400857165_nat_o(aa_opt9039718665754746463_nat_o(P,Xb),Y) )
       => ( ! [A2: product_prod_nat_nat,B2: product_prod_nat_nat] :
              ( ( Xb = some_P7363390416028606310at_nat(A2) )
             => ( ( Y = some_P7363390416028606310at_nat(B2) )
               => aa_opt7112393969400857165_nat_o(aa_opt9039718665754746463_nat_o(P,Xb),Y) ) )
         => aa_opt7112393969400857165_nat_o(aa_opt9039718665754746463_nat_o(P,Xb),Y) ) ) ) ).

% combine_options_cases
tff(fact_1019_combine__options__cases,axiom,
    ! [Xb: option4927543243414619207at_nat,P: fun_op7653269905477231939_num_o,Y: option_num] :
      ( ( ( Xb = none_P5556105721700978146at_nat )
       => aa_option_num_o(aa_opt7868583491048818492_num_o(P,Xb),Y) )
     => ( ( ( Y = none_num )
         => aa_option_num_o(aa_opt7868583491048818492_num_o(P,Xb),Y) )
       => ( ! [A2: product_prod_nat_nat,B2: num] :
              ( ( Xb = some_P7363390416028606310at_nat(A2) )
             => ( ( Y = some_num(B2) )
               => aa_option_num_o(aa_opt7868583491048818492_num_o(P,Xb),Y) ) )
         => aa_option_num_o(aa_opt7868583491048818492_num_o(P,Xb),Y) ) ) ) ).

% combine_options_cases
tff(fact_1020_combine__options__cases,axiom,
    ! [Xb: option_num,P: fun_op167334234089630144_nat_o,Y: option_nat] :
      ( ( ( Xb = none_num )
       => aa_option_nat_o(aa_opt7915830422195272873_nat_o(P,Xb),Y) )
     => ( ( ( Y = none_nat )
         => aa_option_nat_o(aa_opt7915830422195272873_nat_o(P,Xb),Y) )
       => ( ! [A2: num,B2: nat] :
              ( ( Xb = some_num(A2) )
             => ( ( Y = some_nat(B2) )
               => aa_option_nat_o(aa_opt7915830422195272873_nat_o(P,Xb),Y) ) )
         => aa_option_nat_o(aa_opt7915830422195272873_nat_o(P,Xb),Y) ) ) ) ).

% combine_options_cases
tff(fact_1021_combine__options__cases,axiom,
    ! [Xb: option_num,P: fun_op5373868528577402299_nat_o,Y: option4927543243414619207at_nat] :
      ( ( ( Xb = none_num )
       => aa_opt7112393969400857165_nat_o(aa_opt1662640216596507900_nat_o(P,Xb),Y) )
     => ( ( ( Y = none_P5556105721700978146at_nat )
         => aa_opt7112393969400857165_nat_o(aa_opt1662640216596507900_nat_o(P,Xb),Y) )
       => ( ! [A2: num,B2: product_prod_nat_nat] :
              ( ( Xb = some_num(A2) )
             => ( ( Y = some_P7363390416028606310at_nat(B2) )
               => aa_opt7112393969400857165_nat_o(aa_opt1662640216596507900_nat_o(P,Xb),Y) ) )
         => aa_opt7112393969400857165_nat_o(aa_opt1662640216596507900_nat_o(P,Xb),Y) ) ) ) ).

% combine_options_cases
tff(fact_1022_combine__options__cases,axiom,
    ! [Xb: option_num,P: fun_op493924057549976310_num_o,Y: option_num] :
      ( ( ( Xb = none_num )
       => aa_option_num_o(aa_opt8674010985164783839_num_o(P,Xb),Y) )
     => ( ( ( Y = none_num )
         => aa_option_num_o(aa_opt8674010985164783839_num_o(P,Xb),Y) )
       => ( ! [A2: num,B2: num] :
              ( ( Xb = some_num(A2) )
             => ( ( Y = some_num(B2) )
               => aa_option_num_o(aa_opt8674010985164783839_num_o(P,Xb),Y) ) )
         => aa_option_num_o(aa_opt8674010985164783839_num_o(P,Xb),Y) ) ) ) ).

% combine_options_cases
tff(fact_1023_split__option__all,axiom,
    ! [P: fun_option_nat_o] :
      ( ! [X_12: option_nat] : aa_option_nat_o(P,X_12)
    <=> ( aa_option_nat_o(P,none_nat)
        & ! [X4: nat] : aa_option_nat_o(P,some_nat(X4)) ) ) ).

% split_option_all
tff(fact_1024_split__option__all,axiom,
    ! [P: fun_op8028897214545316308_nat_o] :
      ( ! [X_12: option4927543243414619207at_nat] : aa_opt7112393969400857165_nat_o(P,X_12)
    <=> ( aa_opt7112393969400857165_nat_o(P,none_P5556105721700978146at_nat)
        & ! [X4: product_prod_nat_nat] : aa_opt7112393969400857165_nat_o(P,some_P7363390416028606310at_nat(X4)) ) ) ).

% split_option_all
tff(fact_1025_split__option__all,axiom,
    ! [P: fun_option_num_o] :
      ( ! [X_12: option_num] : aa_option_num_o(P,X_12)
    <=> ( aa_option_num_o(P,none_num)
        & ! [X4: num] : aa_option_num_o(P,some_num(X4)) ) ) ).

% split_option_all
tff(fact_1026_split__option__ex,axiom,
    ! [P: fun_option_nat_o] :
      ( ? [X_12: option_nat] : aa_option_nat_o(P,X_12)
    <=> ( aa_option_nat_o(P,none_nat)
        | ? [X4: nat] : aa_option_nat_o(P,some_nat(X4)) ) ) ).

% split_option_ex
tff(fact_1027_split__option__ex,axiom,
    ! [P: fun_op8028897214545316308_nat_o] :
      ( ? [X_12: option4927543243414619207at_nat] : aa_opt7112393969400857165_nat_o(P,X_12)
    <=> ( aa_opt7112393969400857165_nat_o(P,none_P5556105721700978146at_nat)
        | ? [X4: product_prod_nat_nat] : aa_opt7112393969400857165_nat_o(P,some_P7363390416028606310at_nat(X4)) ) ) ).

% split_option_ex
tff(fact_1028_split__option__ex,axiom,
    ! [P: fun_option_num_o] :
      ( ? [X_12: option_num] : aa_option_num_o(P,X_12)
    <=> ( aa_option_num_o(P,none_num)
        | ? [X4: num] : aa_option_num_o(P,some_num(X4)) ) ) ).

% split_option_ex
tff(fact_1029_option_Oexhaust,axiom,
    ! [Y: option_nat] :
      ( ( Y != none_nat )
     => ~ ! [X23: nat] : Y != some_nat(X23) ) ).

% option.exhaust
tff(fact_1030_option_Oexhaust,axiom,
    ! [Y: option4927543243414619207at_nat] :
      ( ( Y != none_P5556105721700978146at_nat )
     => ~ ! [X23: product_prod_nat_nat] : Y != some_P7363390416028606310at_nat(X23) ) ).

% option.exhaust
tff(fact_1031_option_Oexhaust,axiom,
    ! [Y: option_num] :
      ( ( Y != none_num )
     => ~ ! [X23: num] : Y != some_num(X23) ) ).

% option.exhaust
tff(fact_1032_option_OdiscI,axiom,
    ! [Option: option_nat,X2: nat] :
      ( ( Option = some_nat(X2) )
     => ( Option != none_nat ) ) ).

% option.discI
tff(fact_1033_option_OdiscI,axiom,
    ! [Option: option4927543243414619207at_nat,X2: product_prod_nat_nat] :
      ( ( Option = some_P7363390416028606310at_nat(X2) )
     => ( Option != none_P5556105721700978146at_nat ) ) ).

% option.discI
tff(fact_1034_option_OdiscI,axiom,
    ! [Option: option_num,X2: num] :
      ( ( Option = some_num(X2) )
     => ( Option != none_num ) ) ).

% option.discI
tff(fact_1035_option_Odistinct_I1_J,axiom,
    ! [X2: nat] : none_nat != some_nat(X2) ).

% option.distinct(1)
tff(fact_1036_option_Odistinct_I1_J,axiom,
    ! [X2: product_prod_nat_nat] : none_P5556105721700978146at_nat != some_P7363390416028606310at_nat(X2) ).

% option.distinct(1)
tff(fact_1037_option_Odistinct_I1_J,axiom,
    ! [X2: num] : none_num != some_num(X2) ).

% option.distinct(1)
tff(fact_1038_ex__in__conv,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] :
      ( ? [X4: product_prod_nat_nat] : member8440522571783428010at_nat(X4,A3)
    <=> ( A3 != bot_bo2099793752762293965at_nat ) ) ).

% ex_in_conv
tff(fact_1039_ex__in__conv,axiom,
    ! [A3: set_set_nat] :
      ( ? [X4: set_nat] : member_set_nat(X4,A3)
    <=> ( A3 != bot_bot_set_set_nat ) ) ).

% ex_in_conv
tff(fact_1040_ex__in__conv,axiom,
    ! [A3: set_real] :
      ( ? [X4: real] : member_real(X4,A3)
    <=> ( A3 != bot_bot_set_real ) ) ).

% ex_in_conv
tff(fact_1041_ex__in__conv,axiom,
    ! [A3: set_nat] :
      ( ? [X4: nat] : aa_set_nat_o(member_nat(X4),A3)
    <=> ( A3 != bot_bot_set_nat ) ) ).

% ex_in_conv
tff(fact_1042_ex__in__conv,axiom,
    ! [A3: set_int] :
      ( ? [X4: int] : aa_set_int_o(member_int(X4),A3)
    <=> ( A3 != bot_bot_set_int ) ) ).

% ex_in_conv
tff(fact_1043_equals0I,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] :
      ( ! [Y3: product_prod_nat_nat] : ~ member8440522571783428010at_nat(Y3,A3)
     => ( A3 = bot_bo2099793752762293965at_nat ) ) ).

% equals0I
tff(fact_1044_equals0I,axiom,
    ! [A3: set_set_nat] :
      ( ! [Y3: set_nat] : ~ member_set_nat(Y3,A3)
     => ( A3 = bot_bot_set_set_nat ) ) ).

% equals0I
tff(fact_1045_equals0I,axiom,
    ! [A3: set_real] :
      ( ! [Y3: real] : ~ member_real(Y3,A3)
     => ( A3 = bot_bot_set_real ) ) ).

% equals0I
tff(fact_1046_equals0I,axiom,
    ! [A3: set_nat] :
      ( ! [Y3: nat] : ~ aa_set_nat_o(member_nat(Y3),A3)
     => ( A3 = bot_bot_set_nat ) ) ).

% equals0I
tff(fact_1047_equals0I,axiom,
    ! [A3: set_int] :
      ( ! [Y3: int] : ~ aa_set_int_o(member_int(Y3),A3)
     => ( A3 = bot_bot_set_int ) ) ).

% equals0I
tff(fact_1048_equals0D,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,Aa: product_prod_nat_nat] :
      ( ( A3 = bot_bo2099793752762293965at_nat )
     => ~ member8440522571783428010at_nat(Aa,A3) ) ).

% equals0D
tff(fact_1049_equals0D,axiom,
    ! [A3: set_set_nat,Aa: set_nat] :
      ( ( A3 = bot_bot_set_set_nat )
     => ~ member_set_nat(Aa,A3) ) ).

% equals0D
tff(fact_1050_equals0D,axiom,
    ! [A3: set_real,Aa: real] :
      ( ( A3 = bot_bot_set_real )
     => ~ member_real(Aa,A3) ) ).

% equals0D
tff(fact_1051_equals0D,axiom,
    ! [A3: set_nat,Aa: nat] :
      ( ( A3 = bot_bot_set_nat )
     => ~ aa_set_nat_o(member_nat(Aa),A3) ) ).

% equals0D
tff(fact_1052_equals0D,axiom,
    ! [A3: set_int,Aa: int] :
      ( ( A3 = bot_bot_set_int )
     => ~ aa_set_int_o(member_int(Aa),A3) ) ).

% equals0D
tff(fact_1053_emptyE,axiom,
    ! [Aa: product_prod_nat_nat] : ~ member8440522571783428010at_nat(Aa,bot_bo2099793752762293965at_nat) ).

% emptyE
tff(fact_1054_emptyE,axiom,
    ! [Aa: set_nat] : ~ member_set_nat(Aa,bot_bot_set_set_nat) ).

% emptyE
tff(fact_1055_emptyE,axiom,
    ! [Aa: real] : ~ member_real(Aa,bot_bot_set_real) ).

% emptyE
tff(fact_1056_emptyE,axiom,
    ! [Aa: nat] : ~ aa_set_nat_o(member_nat(Aa),bot_bot_set_nat) ).

% emptyE
tff(fact_1057_emptyE,axiom,
    ! [Aa: int] : ~ aa_set_int_o(member_int(Aa),bot_bot_set_int) ).

% emptyE
tff(fact_1058_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_1059_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_1060_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_1061_Collect__mono__iff,axiom,
    ! [P: fun_nat_o,Q: fun_nat_o] :
      ( aa_set_nat_o(aa_set7861427066675100071_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_1062_Collect__mono__iff,axiom,
    ! [P: fun_int_o,Q: fun_int_o] :
      ( aa_set_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_1063_set__eq__subset,axiom,
    ! [A3: set_int,B4: set_int] :
      ( ( A3 = B4 )
    <=> ( aa_set_int_o(ord_less_eq_set_int(A3),B4)
        & aa_set_int_o(ord_less_eq_set_int(B4),A3) ) ) ).

% set_eq_subset
tff(fact_1064_subset__trans,axiom,
    ! [A3: set_int,B4: set_int,C2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A3),B4)
     => ( aa_set_int_o(ord_less_eq_set_int(B4),C2)
       => aa_set_int_o(ord_less_eq_set_int(A3),C2) ) ) ).

% subset_trans
tff(fact_1065_Collect__mono,axiom,
    ! [P: fun_real_o,Q: fun_real_o] :
      ( ! [X: real] :
          ( aa_real_o(P,X)
         => aa_real_o(Q,X) )
     => ord_less_eq_set_real(collect_real(P),collect_real(Q)) ) ).

% Collect_mono
tff(fact_1066_Collect__mono,axiom,
    ! [P: fun_list_nat_o,Q: fun_list_nat_o] :
      ( ! [X: list_nat] :
          ( aa_list_nat_o(P,X)
         => aa_list_nat_o(Q,X) )
     => ord_le6045566169113846134st_nat(collect_list_nat(P),collect_list_nat(Q)) ) ).

% Collect_mono
tff(fact_1067_Collect__mono,axiom,
    ! [P: fun_set_nat_o,Q: fun_set_nat_o] :
      ( ! [X: set_nat] :
          ( aa_set_nat_o(P,X)
         => aa_set_nat_o(Q,X) )
     => ord_le6893508408891458716et_nat(collect_set_nat(P),collect_set_nat(Q)) ) ).

% Collect_mono
tff(fact_1068_Collect__mono,axiom,
    ! [P: fun_nat_o,Q: fun_nat_o] :
      ( ! [X: nat] :
          ( aa_nat_o(P,X)
         => aa_nat_o(Q,X) )
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,collect_nat(P)),collect_nat(Q)) ) ).

% Collect_mono
tff(fact_1069_Collect__mono,axiom,
    ! [P: fun_int_o,Q: fun_int_o] :
      ( ! [X: int] :
          ( aa_int_o(P,X)
         => aa_int_o(Q,X) )
     => aa_set_int_o(ord_less_eq_set_int(collect_int(P)),collect_int(Q)) ) ).

% Collect_mono
tff(fact_1070_subset__refl,axiom,
    ! [A3: set_int] : aa_set_int_o(ord_less_eq_set_int(A3),A3) ).

% subset_refl
tff(fact_1071_subset__iff,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ord_le3146513528884898305at_nat(A3,B4)
    <=> ! [T2: product_prod_nat_nat] :
          ( member8440522571783428010at_nat(T2,A3)
         => member8440522571783428010at_nat(T2,B4) ) ) ).

% subset_iff
tff(fact_1072_subset__iff,axiom,
    ! [A3: set_real,B4: set_real] :
      ( ord_less_eq_set_real(A3,B4)
    <=> ! [T2: real] :
          ( member_real(T2,A3)
         => member_real(T2,B4) ) ) ).

% subset_iff
tff(fact_1073_subset__iff,axiom,
    ! [A3: set_set_nat,B4: set_set_nat] :
      ( ord_le6893508408891458716et_nat(A3,B4)
    <=> ! [T2: set_nat] :
          ( member_set_nat(T2,A3)
         => member_set_nat(T2,B4) ) ) ).

% subset_iff
tff(fact_1074_subset__iff,axiom,
    ! [A3: set_nat,B4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A3),B4)
    <=> ! [T2: nat] :
          ( aa_set_nat_o(member_nat(T2),A3)
         => aa_set_nat_o(member_nat(T2),B4) ) ) ).

% subset_iff
tff(fact_1075_subset__iff,axiom,
    ! [A3: set_int,B4: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A3),B4)
    <=> ! [T2: int] :
          ( aa_set_int_o(member_int(T2),A3)
         => aa_set_int_o(member_int(T2),B4) ) ) ).

% subset_iff
tff(fact_1076_equalityD2,axiom,
    ! [A3: set_int,B4: set_int] :
      ( ( A3 = B4 )
     => aa_set_int_o(ord_less_eq_set_int(B4),A3) ) ).

% equalityD2
tff(fact_1077_equalityD1,axiom,
    ! [A3: set_int,B4: set_int] :
      ( ( A3 = B4 )
     => aa_set_int_o(ord_less_eq_set_int(A3),B4) ) ).

% equalityD1
tff(fact_1078_subset__eq,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ord_le3146513528884898305at_nat(A3,B4)
    <=> ! [X4: product_prod_nat_nat] :
          ( member8440522571783428010at_nat(X4,A3)
         => member8440522571783428010at_nat(X4,B4) ) ) ).

% subset_eq
tff(fact_1079_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_1080_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_1081_subset__eq,axiom,
    ! [A3: set_nat,B4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_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_1082_subset__eq,axiom,
    ! [A3: set_int,B4: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A3),B4)
    <=> ! [X4: int] :
          ( aa_set_int_o(member_int(X4),A3)
         => aa_set_int_o(member_int(X4),B4) ) ) ).

% subset_eq
tff(fact_1083_equalityE,axiom,
    ! [A3: set_int,B4: set_int] :
      ( ( A3 = B4 )
     => ~ ( aa_set_int_o(ord_less_eq_set_int(A3),B4)
         => ~ aa_set_int_o(ord_less_eq_set_int(B4),A3) ) ) ).

% equalityE
tff(fact_1084_subsetD,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat,C: product_prod_nat_nat] :
      ( ord_le3146513528884898305at_nat(A3,B4)
     => ( member8440522571783428010at_nat(C,A3)
       => member8440522571783428010at_nat(C,B4) ) ) ).

% subsetD
tff(fact_1085_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_1086_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_1087_subsetD,axiom,
    ! [A3: set_nat,B4: set_nat,C: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_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_1088_subsetD,axiom,
    ! [A3: set_int,B4: set_int,C: int] :
      ( aa_set_int_o(ord_less_eq_set_int(A3),B4)
     => ( aa_set_int_o(member_int(C),A3)
       => aa_set_int_o(member_int(C),B4) ) ) ).

% subsetD
tff(fact_1089_in__mono,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat,Xb: product_prod_nat_nat] :
      ( ord_le3146513528884898305at_nat(A3,B4)
     => ( member8440522571783428010at_nat(Xb,A3)
       => member8440522571783428010at_nat(Xb,B4) ) ) ).

% in_mono
tff(fact_1090_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_1091_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_1092_in__mono,axiom,
    ! [A3: set_nat,B4: set_nat,Xb: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_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_1093_in__mono,axiom,
    ! [A3: set_int,B4: set_int,Xb: int] :
      ( aa_set_int_o(ord_less_eq_set_int(A3),B4)
     => ( aa_set_int_o(member_int(Xb),A3)
       => aa_set_int_o(member_int(Xb),B4) ) ) ).

% in_mono
tff(fact_1094_vebt__pred_Osimps_I1_J,axiom,
    ! [Uu: $o,Uv: $o] : vEBT_vebt_pred(vEBT_Leaf((Uu),(Uv)),zero_zero_nat) = none_nat ).

% vebt_pred.simps(1)
tff(fact_1095_psubsetD,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat,C: product_prod_nat_nat] :
      ( ord_le7866589430770878221at_nat(A3,B4)
     => ( member8440522571783428010at_nat(C,A3)
       => member8440522571783428010at_nat(C,B4) ) ) ).

% psubsetD
tff(fact_1096_psubsetD,axiom,
    ! [A3: set_real,B4: set_real,C: real] :
      ( ord_less_set_real(A3,B4)
     => ( member_real(C,A3)
       => member_real(C,B4) ) ) ).

% psubsetD
tff(fact_1097_psubsetD,axiom,
    ! [A3: set_set_nat,B4: set_set_nat,C: set_nat] :
      ( ord_less_set_set_nat(A3,B4)
     => ( member_set_nat(C,A3)
       => member_set_nat(C,B4) ) ) ).

% psubsetD
tff(fact_1098_psubsetD,axiom,
    ! [A3: set_nat,B4: set_nat,C: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A3),B4)
     => ( aa_set_nat_o(member_nat(C),A3)
       => aa_set_nat_o(member_nat(C),B4) ) ) ).

% psubsetD
tff(fact_1099_psubsetD,axiom,
    ! [A3: set_int,B4: set_int,C: int] :
      ( aa_set_int_o(ord_less_set_int(A3),B4)
     => ( aa_set_int_o(member_int(C),A3)
       => aa_set_int_o(member_int(C),B4) ) ) ).

% psubsetD
tff(fact_1100_vebt__pred_Osimps_I5_J,axiom,
    ! [V: product_prod_nat_nat,Vd: list_VEBT_VEBT,Ve: vEBT_VEBT,Vf: nat] : vEBT_vebt_pred(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Vd,Ve),Vf) = none_nat ).

% vebt_pred.simps(5)
tff(fact_1101_vebt__pred_Osimps_I6_J,axiom,
    ! [V: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT,Vj: nat] : vEBT_vebt_pred(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vh,Vi),Vj) = none_nat ).

% vebt_pred.simps(6)
tff(fact_1102_vebt__pred_Osimps_I2_J,axiom,
    ! [Aa: $o,Uw: $o] :
      vEBT_vebt_pred(vEBT_Leaf((Aa),(Uw)),aa_nat_nat(suc,zero_zero_nat)) = $ite((Aa),some_nat(zero_zero_nat),none_nat) ).

% vebt_pred.simps(2)
tff(fact_1103_vebt__pred_Osimps_I3_J,axiom,
    ! [Aa: $o,Ba: $o,Va: nat] :
      vEBT_vebt_pred(vEBT_Leaf((Aa),(Ba)),aa_nat_nat(suc,aa_nat_nat(suc,Va))) = $ite(
        (Ba),
        some_nat(one_one_nat),
        $ite((Aa),some_nat(zero_zero_nat),none_nat) ) ).

% vebt_pred.simps(3)
tff(fact_1104_not__psubset__empty,axiom,
    ! [A3: set_real] : ~ ord_less_set_real(A3,bot_bot_set_real) ).

% not_psubset_empty
tff(fact_1105_not__psubset__empty,axiom,
    ! [A3: set_nat] : ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A3),bot_bot_set_nat) ).

% not_psubset_empty
tff(fact_1106_not__psubset__empty,axiom,
    ! [A3: set_int] : ~ aa_set_int_o(ord_less_set_int(A3),bot_bot_set_int) ).

% not_psubset_empty
tff(fact_1107_subset__iff__psubset__eq,axiom,
    ! [A3: set_int,B4: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A3),B4)
    <=> ( aa_set_int_o(ord_less_set_int(A3),B4)
        | ( A3 = B4 ) ) ) ).

% subset_iff_psubset_eq
tff(fact_1108_subset__psubset__trans,axiom,
    ! [A3: set_int,B4: set_int,C2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A3),B4)
     => ( aa_set_int_o(ord_less_set_int(B4),C2)
       => aa_set_int_o(ord_less_set_int(A3),C2) ) ) ).

% subset_psubset_trans
tff(fact_1109_subset__not__subset__eq,axiom,
    ! [A3: set_int,B4: set_int] :
      ( aa_set_int_o(ord_less_set_int(A3),B4)
    <=> ( aa_set_int_o(ord_less_eq_set_int(A3),B4)
        & ~ aa_set_int_o(ord_less_eq_set_int(B4),A3) ) ) ).

% subset_not_subset_eq
tff(fact_1110_psubset__subset__trans,axiom,
    ! [A3: set_int,B4: set_int,C2: set_int] :
      ( aa_set_int_o(ord_less_set_int(A3),B4)
     => ( aa_set_int_o(ord_less_eq_set_int(B4),C2)
       => aa_set_int_o(ord_less_set_int(A3),C2) ) ) ).

% psubset_subset_trans
tff(fact_1111_psubset__imp__subset,axiom,
    ! [A3: set_int,B4: set_int] :
      ( aa_set_int_o(ord_less_set_int(A3),B4)
     => aa_set_int_o(ord_less_eq_set_int(A3),B4) ) ).

% psubset_imp_subset
tff(fact_1112_psubset__eq,axiom,
    ! [A3: set_int,B4: set_int] :
      ( aa_set_int_o(ord_less_set_int(A3),B4)
    <=> ( aa_set_int_o(ord_less_eq_set_int(A3),B4)
        & ( A3 != B4 ) ) ) ).

% psubset_eq
tff(fact_1113_psubsetE,axiom,
    ! [A3: set_int,B4: set_int] :
      ( aa_set_int_o(ord_less_set_int(A3),B4)
     => ~ ( aa_set_int_o(ord_less_eq_set_int(A3),B4)
         => aa_set_int_o(ord_less_eq_set_int(B4),A3) ) ) ).

% psubsetE
tff(fact_1114_vebt__maxt_Osimps_I1_J,axiom,
    ! [Aa: $o,Ba: $o] :
      vEBT_vebt_maxt(vEBT_Leaf((Aa),(Ba))) = $ite(
        (Ba),
        some_nat(one_one_nat),
        $ite((Aa),some_nat(zero_zero_nat),none_nat) ) ).

% vebt_maxt.simps(1)
tff(fact_1115_vebt__mint_Osimps_I1_J,axiom,
    ! [Aa: $o,Ba: $o] :
      vEBT_vebt_mint(vEBT_Leaf((Aa),(Ba))) = $ite(
        (Aa),
        some_nat(zero_zero_nat),
        $ite((Ba),some_nat(one_one_nat),none_nat) ) ).

% vebt_mint.simps(1)
tff(fact_1116_option_Osize__gen_I1_J,axiom,
    ! [Xb: fun_nat_nat] : size_option_nat(Xb,none_nat) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size_gen(1)
tff(fact_1117_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_1118_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_1119_arg__min__if__finite_I2_J,axiom,
    ! [S2: set_complex,F: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S2)
     => ( ( S2 != bot_bot_set_complex )
       => ~ ? [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),S2)
              & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(F,X3)),aa_complex_real(F,lattic8794016678065449205x_real(F,S2))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1120_arg__min__if__finite_I2_J,axiom,
    ! [S2: set_real,F: fun_real_real] :
      ( finite_finite_real(S2)
     => ( ( S2 != bot_bot_set_real )
       => ~ ? [X3: real] :
              ( member_real(X3,S2)
              & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,X3)),aa_real_real(F,lattic8440615504127631091l_real(F,S2))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1121_arg__min__if__finite_I2_J,axiom,
    ! [S2: set_nat,F: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( ( S2 != bot_bot_set_nat )
       => ~ ? [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),S2)
              & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,X3)),aa_nat_real(F,aa_set_nat_nat(lattic488527866317076247t_real(F),S2))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1122_arg__min__if__finite_I2_J,axiom,
    ! [S2: set_int,F: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,S2)
     => ( ( S2 != bot_bot_set_int )
       => ~ ? [X3: int] :
              ( aa_set_int_o(member_int(X3),S2)
              & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(F,X3)),aa_int_real(F,lattic2675449441010098035t_real(F,S2))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1123_arg__min__if__finite_I2_J,axiom,
    ! [S2: set_complex,F: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S2)
     => ( ( S2 != bot_bot_set_complex )
       => ~ ? [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),S2)
              & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_complex_rat(F,X3)),aa_complex_rat(F,lattic4729654577720512673ex_rat(F,S2))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1124_arg__min__if__finite_I2_J,axiom,
    ! [S2: set_real,F: fun_real_rat] :
      ( finite_finite_real(S2)
     => ( ( S2 != bot_bot_set_real )
       => ~ ? [X3: real] :
              ( member_real(X3,S2)
              & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,X3)),aa_real_rat(F,lattic4420706379359479199al_rat(F,S2))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1125_arg__min__if__finite_I2_J,axiom,
    ! [S2: set_nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( ( S2 != bot_bot_set_nat )
       => ~ ? [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),S2)
              & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,X3)),aa_nat_rat(F,aa_set_nat_nat(lattic6811802900495863747at_rat(F),S2))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1126_arg__min__if__finite_I2_J,axiom,
    ! [S2: set_int,F: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,S2)
     => ( ( S2 != bot_bot_set_int )
       => ~ ? [X3: int] :
              ( aa_set_int_o(member_int(X3),S2)
              & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(F,X3)),aa_int_rat(F,lattic7811156612396918303nt_rat(F,S2))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1127_arg__min__if__finite_I2_J,axiom,
    ! [S2: set_complex,F: fun_complex_num] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S2)
     => ( ( S2 != bot_bot_set_complex )
       => ~ ? [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),S2)
              & aa_num_o(aa_num_fun_num_o(ord_less_num,aa_complex_num(F,X3)),aa_complex_num(F,lattic1922116423962787043ex_num(F,S2))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1128_arg__min__if__finite_I2_J,axiom,
    ! [S2: set_real,F: fun_real_num] :
      ( finite_finite_real(S2)
     => ( ( S2 != bot_bot_set_real )
       => ~ ? [X3: real] :
              ( member_real(X3,S2)
              & aa_num_o(aa_num_fun_num_o(ord_less_num,aa_real_num(F,X3)),aa_real_num(F,lattic1613168225601753569al_num(F,S2))) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_1129_arg__min__least,axiom,
    ! [S2: set_complex,Y: complex,F: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S2)
     => ( ( S2 != bot_bot_set_complex )
       => ( aa_set_complex_o(member_complex(Y),S2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_complex_rat(F,lattic4729654577720512673ex_rat(F,S2))),aa_complex_rat(F,Y)) ) ) ) ).

% arg_min_least
tff(fact_1130_arg__min__least,axiom,
    ! [S2: set_real,Y: real,F: fun_real_rat] :
      ( finite_finite_real(S2)
     => ( ( S2 != bot_bot_set_real )
       => ( member_real(Y,S2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_real_rat(F,lattic4420706379359479199al_rat(F,S2))),aa_real_rat(F,Y)) ) ) ) ).

% arg_min_least
tff(fact_1131_arg__min__least,axiom,
    ! [S2: set_nat,Y: nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( ( S2 != bot_bot_set_nat )
       => ( aa_set_nat_o(member_nat(Y),S2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,aa_set_nat_nat(lattic6811802900495863747at_rat(F),S2))),aa_nat_rat(F,Y)) ) ) ) ).

% arg_min_least
tff(fact_1132_arg__min__least,axiom,
    ! [S2: set_int,Y: int,F: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,S2)
     => ( ( S2 != bot_bot_set_int )
       => ( aa_set_int_o(member_int(Y),S2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,lattic7811156612396918303nt_rat(F,S2))),aa_int_rat(F,Y)) ) ) ) ).

% arg_min_least
tff(fact_1133_arg__min__least,axiom,
    ! [S2: set_complex,Y: complex,F: fun_complex_num] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S2)
     => ( ( S2 != bot_bot_set_complex )
       => ( aa_set_complex_o(member_complex(Y),S2)
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_complex_num(F,lattic1922116423962787043ex_num(F,S2))),aa_complex_num(F,Y)) ) ) ) ).

% arg_min_least
tff(fact_1134_arg__min__least,axiom,
    ! [S2: set_real,Y: real,F: fun_real_num] :
      ( finite_finite_real(S2)
     => ( ( S2 != bot_bot_set_real )
       => ( member_real(Y,S2)
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_real_num(F,lattic1613168225601753569al_num(F,S2))),aa_real_num(F,Y)) ) ) ) ).

% arg_min_least
tff(fact_1135_arg__min__least,axiom,
    ! [S2: set_nat,Y: nat,F: fun_nat_num] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( ( S2 != bot_bot_set_nat )
       => ( aa_set_nat_o(member_nat(Y),S2)
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,aa_set_nat_nat(lattic4004264746738138117at_num(F),S2))),aa_nat_num(F,Y)) ) ) ) ).

% arg_min_least
tff(fact_1136_arg__min__least,axiom,
    ! [S2: set_int,Y: int,F: fun_int_num] :
      ( aa_set_int_o(finite_finite_int,S2)
     => ( ( S2 != bot_bot_set_int )
       => ( aa_set_int_o(member_int(Y),S2)
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_int_num(F,lattic5003618458639192673nt_num(F,S2))),aa_int_num(F,Y)) ) ) ) ).

% arg_min_least
tff(fact_1137_arg__min__least,axiom,
    ! [S2: set_complex,Y: complex,F: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S2)
     => ( ( S2 != bot_bot_set_complex )
       => ( aa_set_complex_o(member_complex(Y),S2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_complex_nat(F,lattic5364784637807008409ex_nat(F,S2))),aa_complex_nat(F,Y)) ) ) ) ).

% arg_min_least
tff(fact_1138_arg__min__least,axiom,
    ! [S2: set_real,Y: real,F: fun_real_nat] :
      ( finite_finite_real(S2)
     => ( ( S2 != bot_bot_set_real )
       => ( member_real(Y,S2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_real_nat(F,lattic5055836439445974935al_nat(F,S2))),aa_real_nat(F,Y)) ) ) ) ).

% arg_min_least
tff(fact_1139_option_Osize_I3_J,axiom,
    size_size_option_nat(none_nat) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(3)
tff(fact_1140_option_Osize_I3_J,axiom,
    size_s170228958280169651at_nat(none_P5556105721700978146at_nat) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(3)
tff(fact_1141_option_Osize_I3_J,axiom,
    size_size_option_num(none_num) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(3)
tff(fact_1142_option_Osize_I4_J,axiom,
    ! [X2: nat] : size_size_option_nat(some_nat(X2)) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(4)
tff(fact_1143_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_1144_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_1145_nat__descend__induct,axiom,
    ! [Na: nat,P: fun_nat_o,M: nat] :
      ( ! [K2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),K2)
         => aa_nat_o(P,K2) )
     => ( ! [K2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Na)
           => ( ! [I4: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K2),I4)
                 => aa_nat_o(P,I4) )
             => aa_nat_o(P,K2) ) )
       => aa_nat_o(P,M) ) ) ).

% nat_descend_induct
tff(fact_1146_subset__emptyI,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] :
      ( ! [X: product_prod_nat_nat] : ~ member8440522571783428010at_nat(X,A3)
     => ord_le3146513528884898305at_nat(A3,bot_bo2099793752762293965at_nat) ) ).

% subset_emptyI
tff(fact_1147_subset__emptyI,axiom,
    ! [A3: set_set_nat] :
      ( ! [X: set_nat] : ~ member_set_nat(X,A3)
     => ord_le6893508408891458716et_nat(A3,bot_bot_set_set_nat) ) ).

% subset_emptyI
tff(fact_1148_subset__emptyI,axiom,
    ! [A3: set_real] :
      ( ! [X: real] : ~ member_real(X,A3)
     => ord_less_eq_set_real(A3,bot_bot_set_real) ) ).

% subset_emptyI
tff(fact_1149_subset__emptyI,axiom,
    ! [A3: set_nat] :
      ( ! [X: nat] : ~ aa_set_nat_o(member_nat(X),A3)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A3),bot_bot_set_nat) ) ).

% subset_emptyI
tff(fact_1150_subset__emptyI,axiom,
    ! [A3: set_int] :
      ( ! [X: int] : ~ aa_set_int_o(member_int(X),A3)
     => aa_set_int_o(ord_less_eq_set_int(A3),bot_bot_set_int) ) ).

% subset_emptyI
tff(fact_1151_size__neq__size__imp__neq,axiom,
    ! [Xb: char,Y: char] :
      ( ( size_size_char(Xb) != size_size_char(Y) )
     => ( Xb != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_1152_size__neq__size__imp__neq,axiom,
    ! [Xb: list_VEBT_VEBT,Y: list_VEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT(Xb) != size_s6755466524823107622T_VEBT(Y) )
     => ( Xb != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_1153_size__neq__size__imp__neq,axiom,
    ! [Xb: list_o,Y: list_o] :
      ( ( size_size_list_o(Xb) != size_size_list_o(Y) )
     => ( Xb != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_1154_size__neq__size__imp__neq,axiom,
    ! [Xb: list_nat,Y: list_nat] :
      ( ( size_size_list_nat(Xb) != size_size_list_nat(Y) )
     => ( Xb != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_1155_size__neq__size__imp__neq,axiom,
    ! [Xb: num,Y: num] :
      ( ( aa_num_nat(size_size_num,Xb) != aa_num_nat(size_size_num,Y) )
     => ( Xb != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_1156_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_1157_vebt__mint_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] : vEBT_vebt_mint(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw)) = none_nat ).

% vebt_mint.simps(2)
tff(fact_1158_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_1159_vebt__maxt_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] : vEBT_vebt_maxt(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw)) = none_nat ).

% vebt_maxt.simps(2)
tff(fact_1160_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_1161_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_1162_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_1163_vebt__pred_Osimps_I4_J,axiom,
    ! [Uy: nat,Uz: list_VEBT_VEBT,Va: vEBT_VEBT,Vb: nat] : vEBT_vebt_pred(vEBT_Node(none_P5556105721700978146at_nat,Uy,Uz,Va),Vb) = none_nat ).

% vebt_pred.simps(4)
tff(fact_1164_VEBT__internal_OminNull_Oelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Y: $o] :
      ( ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Xb)
      <=> (Y) )
     => ( ( ( Xb = vEBT_Leaf($false,$false) )
         => ~ (Y) )
       => ( ( ? [Uv2: $o] : Xb = vEBT_Leaf($true,(Uv2))
           => (Y) )
         => ( ( ? [Uu2: $o] : Xb = vEBT_Leaf((Uu2),$true)
             => (Y) )
           => ( ( ? [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : Xb = vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2,Uy2)
               => ~ (Y) )
             => ~ ( ? [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2,Vc2)
                 => (Y) ) ) ) ) ) ) ).

% VEBT_internal.minNull.elims(1)
tff(fact_1165_vebt__maxt_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Y: option_nat] :
      ( ( vEBT_vebt_maxt(Xb) = Y )
     => ( ! [A2: $o,B2: $o] :
            ( ( Xb = vEBT_Leaf((A2),(B2)) )
           => ( Y != $ite(
                  (B2),
                  some_nat(one_one_nat),
                  $ite((A2),some_nat(zero_zero_nat),none_nat) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : Xb = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2)
           => ( Y != none_nat ) )
         => ~ ! [Mi: nat,Ma: nat] :
                ( ? [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Ux2,Uy2,Uz2)
               => ( Y != some_nat(Ma) ) ) ) ) ) ).

% vebt_maxt.elims
tff(fact_1166_vebt__mint_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Y: option_nat] :
      ( ( vEBT_vebt_mint(Xb) = Y )
     => ( ! [A2: $o,B2: $o] :
            ( ( Xb = vEBT_Leaf((A2),(B2)) )
           => ( Y != $ite(
                  (A2),
                  some_nat(zero_zero_nat),
                  $ite((B2),some_nat(one_one_nat),none_nat) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : Xb = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2)
           => ( Y != none_nat ) )
         => ~ ! [Mi: nat] :
                ( ? [Ma: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Ux2,Uy2,Uz2)
               => ( Y != some_nat(Mi) ) ) ) ) ) ).

% vebt_mint.elims
tff(fact_1167_bot__empty__eq,axiom,
    ! [X3: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(bot_bo5074786513356115630_nat_o,X3)
    <=> member8440522571783428010at_nat(X3,bot_bo2099793752762293965at_nat) ) ).

% bot_empty_eq
tff(fact_1168_bot__empty__eq,axiom,
    ! [X3: set_nat] :
      ( aa_set_nat_o(bot_bo3967814874310145751_nat_o,X3)
    <=> member_set_nat(X3,bot_bot_set_set_nat) ) ).

% bot_empty_eq
tff(fact_1169_bot__empty__eq,axiom,
    ! [X3: real] :
      ( aa_real_o(bot_bot_fun_real_o,X3)
    <=> member_real(X3,bot_bot_set_real) ) ).

% bot_empty_eq
tff(fact_1170_bot__empty__eq,axiom,
    ! [X3: nat] :
      ( aa_nat_o(bot_bot_fun_nat_o,X3)
    <=> aa_set_nat_o(member_nat(X3),bot_bot_set_nat) ) ).

% bot_empty_eq
tff(fact_1171_bot__empty__eq,axiom,
    ! [X3: int] :
      ( aa_int_o(bot_bot_fun_int_o,X3)
    <=> aa_set_int_o(member_int(X3),bot_bot_set_int) ) ).

% bot_empty_eq
tff(fact_1172_Collect__empty__eq__bot,axiom,
    ! [P: fun_list_nat_o] :
      ( ( collect_list_nat(P) = bot_bot_set_list_nat )
    <=> ( P = bot_bo8828579575001852285_nat_o ) ) ).

% Collect_empty_eq_bot
tff(fact_1173_Collect__empty__eq__bot,axiom,
    ! [P: fun_set_nat_o] :
      ( ( collect_set_nat(P) = bot_bot_set_set_nat )
    <=> ( P = bot_bo3967814874310145751_nat_o ) ) ).

% Collect_empty_eq_bot
tff(fact_1174_Collect__empty__eq__bot,axiom,
    ! [P: fun_real_o] :
      ( ( collect_real(P) = bot_bot_set_real )
    <=> ( P = bot_bot_fun_real_o ) ) ).

% Collect_empty_eq_bot
tff(fact_1175_Collect__empty__eq__bot,axiom,
    ! [P: fun_nat_o] :
      ( ( collect_nat(P) = bot_bot_set_nat )
    <=> ( P = bot_bot_fun_nat_o ) ) ).

% Collect_empty_eq_bot
tff(fact_1176_Collect__empty__eq__bot,axiom,
    ! [P: fun_int_o] :
      ( ( collect_int(P) = bot_bot_set_int )
    <=> ( P = bot_bot_fun_int_o ) ) ).

% Collect_empty_eq_bot
tff(fact_1177_list__decode_Ocases,axiom,
    ! [Xb: nat] :
      ( ( Xb != zero_zero_nat )
     => ~ ! [N: nat] : Xb != aa_nat_nat(suc,N) ) ).

% list_decode.cases
tff(fact_1178_exists__least__lemma,axiom,
    ! [P: fun_nat_o] :
      ( ~ aa_nat_o(P,zero_zero_nat)
     => ( ? [X_13: nat] : aa_nat_o(P,X_13)
       => ? [N: nat] :
            ( ~ aa_nat_o(P,N)
            & aa_nat_o(P,aa_nat_nat(suc,N)) ) ) ) ).

% exists_least_lemma
tff(fact_1179_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_1180_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_1181_minf_I8_J,axiom,
    ! [Ta: real] :
    ? [Z3: real] :
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Z3)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ta),X3) ) ).

% minf(8)
tff(fact_1182_minf_I8_J,axiom,
    ! [Ta: rat] :
    ? [Z3: rat] :
    ! [X3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Z3)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ta),X3) ) ).

% minf(8)
tff(fact_1183_minf_I8_J,axiom,
    ! [Ta: num] :
    ? [Z3: num] :
    ! [X3: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Z3)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ta),X3) ) ).

% minf(8)
tff(fact_1184_minf_I8_J,axiom,
    ! [Ta: nat] :
    ? [Z3: nat] :
    ! [X3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Z3)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ta),X3) ) ).

% minf(8)
tff(fact_1185_minf_I8_J,axiom,
    ! [Ta: int] :
    ? [Z3: int] :
    ! [X3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Z3)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ta),X3) ) ).

% minf(8)
tff(fact_1186_minf_I6_J,axiom,
    ! [Ta: real] :
    ? [Z3: real] :
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Z3)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),Ta) ) ).

% minf(6)
tff(fact_1187_minf_I6_J,axiom,
    ! [Ta: rat] :
    ? [Z3: rat] :
    ! [X3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Z3)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Ta) ) ).

% minf(6)
tff(fact_1188_minf_I6_J,axiom,
    ! [Ta: num] :
    ? [Z3: num] :
    ! [X3: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Z3)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Ta) ) ).

% minf(6)
tff(fact_1189_minf_I6_J,axiom,
    ! [Ta: nat] :
    ? [Z3: nat] :
    ! [X3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Z3)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Ta) ) ).

% minf(6)
tff(fact_1190_minf_I6_J,axiom,
    ! [Ta: int] :
    ? [Z3: int] :
    ! [X3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Z3)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),Ta) ) ).

% minf(6)
tff(fact_1191_pinf_I8_J,axiom,
    ! [Ta: real] :
    ? [Z3: real] :
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),X3)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ta),X3) ) ).

% pinf(8)
tff(fact_1192_pinf_I8_J,axiom,
    ! [Ta: rat] :
    ? [Z3: rat] :
    ! [X3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z3),X3)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ta),X3) ) ).

% pinf(8)
tff(fact_1193_pinf_I8_J,axiom,
    ! [Ta: num] :
    ? [Z3: num] :
    ! [X3: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z3),X3)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ta),X3) ) ).

% pinf(8)
tff(fact_1194_pinf_I8_J,axiom,
    ! [Ta: nat] :
    ? [Z3: nat] :
    ! [X3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z3),X3)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ta),X3) ) ).

% pinf(8)
tff(fact_1195_pinf_I8_J,axiom,
    ! [Ta: int] :
    ? [Z3: int] :
    ! [X3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),X3)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ta),X3) ) ).

% pinf(8)
tff(fact_1196_subrelI,axiom,
    ! [R2: set_Pr4811707699266497531nteger,S3: set_Pr4811707699266497531nteger] :
      ( ! [X: code_integer,Y3: code_integer] :
          ( member157494554546826820nteger(produc1086072967326762835nteger(X,Y3),R2)
         => member157494554546826820nteger(produc1086072967326762835nteger(X,Y3),S3) )
     => ord_le3725938330318615451nteger(R2,S3) ) ).

% subrelI
tff(fact_1197_subrelI,axiom,
    ! [R2: set_Pr448751882837621926eger_o,S3: set_Pr448751882837621926eger_o] :
      ( ! [X: code_integer,Y3: $o] :
          ( member1379723562493234055eger_o(produc6677183202524767010eger_o(X,(Y3)),R2)
         => member1379723562493234055eger_o(produc6677183202524767010eger_o(X,(Y3)),S3) )
     => ord_le8980329558974975238eger_o(R2,S3) ) ).

% subrelI
tff(fact_1198_subrelI,axiom,
    ! [R2: set_Pr8693737435421807431at_nat,S3: set_Pr8693737435421807431at_nat] :
      ( ! [X: product_prod_nat_nat,Y3: product_prod_nat_nat] :
          ( member8206827879206165904at_nat(produc6161850002892822231at_nat(X,Y3),R2)
         => member8206827879206165904at_nat(produc6161850002892822231at_nat(X,Y3),S3) )
     => ord_le3000389064537975527at_nat(R2,S3) ) ).

% subrelI
tff(fact_1199_subrelI,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat] :
      ( ! [X: nat,Y3: nat] :
          ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y3),R2)
         => member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X),Y3),S3) )
     => ord_le3146513528884898305at_nat(R2,S3) ) ).

% subrelI
tff(fact_1200_subrelI,axiom,
    ! [R2: set_Pr958786334691620121nt_int,S3: set_Pr958786334691620121nt_int] :
      ( ! [X: int,Y3: int] :
          ( member5262025264175285858nt_int(product_Pair_int_int(X,Y3),R2)
         => member5262025264175285858nt_int(product_Pair_int_int(X,Y3),S3) )
     => ord_le2843351958646193337nt_int(R2,S3) ) ).

% subrelI
tff(fact_1201_pinf_I1_J,axiom,
    ! [P: fun_real_o,P2: fun_real_o,Q: fun_real_o,Q2: fun_real_o] :
      ( ? [Z4: real] :
        ! [X: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z4),X)
         => ( aa_real_o(P,X)
          <=> aa_real_o(P2,X) ) )
     => ( ? [Z4: real] :
          ! [X: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z4),X)
           => ( aa_real_o(Q,X)
            <=> aa_real_o(Q2,X) ) )
       => ? [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(Q,X3) )
            <=> ( aa_real_o(P2,X3)
                & aa_real_o(Q2,X3) ) ) ) ) ) ).

% pinf(1)
tff(fact_1202_pinf_I1_J,axiom,
    ! [P: fun_rat_o,P2: fun_rat_o,Q: fun_rat_o,Q2: fun_rat_o] :
      ( ? [Z4: rat] :
        ! [X: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z4),X)
         => ( aa_rat_o(P,X)
          <=> aa_rat_o(P2,X) ) )
     => ( ? [Z4: rat] :
          ! [X: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z4),X)
           => ( aa_rat_o(Q,X)
            <=> aa_rat_o(Q2,X) ) )
       => ? [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(Q,X3) )
            <=> ( aa_rat_o(P2,X3)
                & aa_rat_o(Q2,X3) ) ) ) ) ) ).

% pinf(1)
tff(fact_1203_pinf_I1_J,axiom,
    ! [P: fun_num_o,P2: fun_num_o,Q: fun_num_o,Q2: fun_num_o] :
      ( ? [Z4: num] :
        ! [X: num] :
          ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z4),X)
         => ( aa_num_o(P,X)
          <=> aa_num_o(P2,X) ) )
     => ( ? [Z4: num] :
          ! [X: num] :
            ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z4),X)
           => ( aa_num_o(Q,X)
            <=> aa_num_o(Q2,X) ) )
       => ? [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(Q,X3) )
            <=> ( aa_num_o(P2,X3)
                & aa_num_o(Q2,X3) ) ) ) ) ) ).

% pinf(1)
tff(fact_1204_pinf_I1_J,axiom,
    ! [P: fun_nat_o,P2: fun_nat_o,Q: fun_nat_o,Q2: fun_nat_o] :
      ( ? [Z4: nat] :
        ! [X: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z4),X)
         => ( aa_nat_o(P,X)
          <=> aa_nat_o(P2,X) ) )
     => ( ? [Z4: nat] :
          ! [X: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z4),X)
           => ( aa_nat_o(Q,X)
            <=> aa_nat_o(Q2,X) ) )
       => ? [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(Q,X3) )
            <=> ( aa_nat_o(P2,X3)
                & aa_nat_o(Q2,X3) ) ) ) ) ) ).

% pinf(1)
tff(fact_1205_pinf_I1_J,axiom,
    ! [P: fun_int_o,P2: fun_int_o,Q: fun_int_o,Q2: fun_int_o] :
      ( ? [Z4: int] :
        ! [X: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z4),X)
         => ( aa_int_o(P,X)
          <=> aa_int_o(P2,X) ) )
     => ( ? [Z4: int] :
          ! [X: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z4),X)
           => ( aa_int_o(Q,X)
            <=> aa_int_o(Q2,X) ) )
       => ? [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(Q,X3) )
            <=> ( aa_int_o(P2,X3)
                & aa_int_o(Q2,X3) ) ) ) ) ) ).

% pinf(1)
tff(fact_1206_pinf_I2_J,axiom,
    ! [P: fun_real_o,P2: fun_real_o,Q: fun_real_o,Q2: fun_real_o] :
      ( ? [Z4: real] :
        ! [X: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z4),X)
         => ( aa_real_o(P,X)
          <=> aa_real_o(P2,X) ) )
     => ( ? [Z4: real] :
          ! [X: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z4),X)
           => ( aa_real_o(Q,X)
            <=> aa_real_o(Q2,X) ) )
       => ? [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(Q,X3) )
            <=> ( aa_real_o(P2,X3)
                | aa_real_o(Q2,X3) ) ) ) ) ) ).

% pinf(2)
tff(fact_1207_pinf_I2_J,axiom,
    ! [P: fun_rat_o,P2: fun_rat_o,Q: fun_rat_o,Q2: fun_rat_o] :
      ( ? [Z4: rat] :
        ! [X: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z4),X)
         => ( aa_rat_o(P,X)
          <=> aa_rat_o(P2,X) ) )
     => ( ? [Z4: rat] :
          ! [X: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z4),X)
           => ( aa_rat_o(Q,X)
            <=> aa_rat_o(Q2,X) ) )
       => ? [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(Q,X3) )
            <=> ( aa_rat_o(P2,X3)
                | aa_rat_o(Q2,X3) ) ) ) ) ) ).

% pinf(2)
tff(fact_1208_pinf_I2_J,axiom,
    ! [P: fun_num_o,P2: fun_num_o,Q: fun_num_o,Q2: fun_num_o] :
      ( ? [Z4: num] :
        ! [X: num] :
          ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z4),X)
         => ( aa_num_o(P,X)
          <=> aa_num_o(P2,X) ) )
     => ( ? [Z4: num] :
          ! [X: num] :
            ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z4),X)
           => ( aa_num_o(Q,X)
            <=> aa_num_o(Q2,X) ) )
       => ? [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(Q,X3) )
            <=> ( aa_num_o(P2,X3)
                | aa_num_o(Q2,X3) ) ) ) ) ) ).

% pinf(2)
tff(fact_1209_pinf_I2_J,axiom,
    ! [P: fun_nat_o,P2: fun_nat_o,Q: fun_nat_o,Q2: fun_nat_o] :
      ( ? [Z4: nat] :
        ! [X: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z4),X)
         => ( aa_nat_o(P,X)
          <=> aa_nat_o(P2,X) ) )
     => ( ? [Z4: nat] :
          ! [X: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z4),X)
           => ( aa_nat_o(Q,X)
            <=> aa_nat_o(Q2,X) ) )
       => ? [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(Q,X3) )
            <=> ( aa_nat_o(P2,X3)
                | aa_nat_o(Q2,X3) ) ) ) ) ) ).

% pinf(2)
tff(fact_1210_pinf_I2_J,axiom,
    ! [P: fun_int_o,P2: fun_int_o,Q: fun_int_o,Q2: fun_int_o] :
      ( ? [Z4: int] :
        ! [X: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z4),X)
         => ( aa_int_o(P,X)
          <=> aa_int_o(P2,X) ) )
     => ( ? [Z4: int] :
          ! [X: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z4),X)
           => ( aa_int_o(Q,X)
            <=> aa_int_o(Q2,X) ) )
       => ? [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(Q,X3) )
            <=> ( aa_int_o(P2,X3)
                | aa_int_o(Q2,X3) ) ) ) ) ) ).

% pinf(2)
tff(fact_1211_pinf_I3_J,axiom,
    ! [Ta: real] :
    ? [Z3: real] :
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),X3)
     => ( X3 != Ta ) ) ).

% pinf(3)
tff(fact_1212_pinf_I3_J,axiom,
    ! [Ta: rat] :
    ? [Z3: rat] :
    ! [X3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z3),X3)
     => ( X3 != Ta ) ) ).

% pinf(3)
tff(fact_1213_pinf_I3_J,axiom,
    ! [Ta: num] :
    ? [Z3: num] :
    ! [X3: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z3),X3)
     => ( X3 != Ta ) ) ).

% pinf(3)
tff(fact_1214_pinf_I3_J,axiom,
    ! [Ta: nat] :
    ? [Z3: nat] :
    ! [X3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z3),X3)
     => ( X3 != Ta ) ) ).

% pinf(3)
tff(fact_1215_pinf_I3_J,axiom,
    ! [Ta: int] :
    ? [Z3: int] :
    ! [X3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),X3)
     => ( X3 != Ta ) ) ).

% pinf(3)
tff(fact_1216_pinf_I4_J,axiom,
    ! [Ta: real] :
    ? [Z3: real] :
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),X3)
     => ( X3 != Ta ) ) ).

% pinf(4)
tff(fact_1217_pinf_I4_J,axiom,
    ! [Ta: rat] :
    ? [Z3: rat] :
    ! [X3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z3),X3)
     => ( X3 != Ta ) ) ).

% pinf(4)
tff(fact_1218_pinf_I4_J,axiom,
    ! [Ta: num] :
    ? [Z3: num] :
    ! [X3: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z3),X3)
     => ( X3 != Ta ) ) ).

% pinf(4)
tff(fact_1219_pinf_I4_J,axiom,
    ! [Ta: nat] :
    ? [Z3: nat] :
    ! [X3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z3),X3)
     => ( X3 != Ta ) ) ).

% pinf(4)
tff(fact_1220_pinf_I4_J,axiom,
    ! [Ta: int] :
    ? [Z3: int] :
    ! [X3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),X3)
     => ( X3 != Ta ) ) ).

% pinf(4)
tff(fact_1221_pinf_I5_J,axiom,
    ! [Ta: real] :
    ? [Z3: real] :
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),X3)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Ta) ) ).

% pinf(5)
tff(fact_1222_pinf_I5_J,axiom,
    ! [Ta: rat] :
    ? [Z3: rat] :
    ! [X3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z3),X3)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Ta) ) ).

% pinf(5)
tff(fact_1223_pinf_I5_J,axiom,
    ! [Ta: num] :
    ? [Z3: num] :
    ! [X3: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z3),X3)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Ta) ) ).

% pinf(5)
tff(fact_1224_pinf_I5_J,axiom,
    ! [Ta: nat] :
    ? [Z3: nat] :
    ! [X3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z3),X3)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Ta) ) ).

% pinf(5)
tff(fact_1225_pinf_I5_J,axiom,
    ! [Ta: int] :
    ? [Z3: int] :
    ! [X3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),X3)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Ta) ) ).

% pinf(5)
tff(fact_1226_pinf_I7_J,axiom,
    ! [Ta: real] :
    ? [Z3: real] :
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),X3)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,Ta),X3) ) ).

% pinf(7)
tff(fact_1227_pinf_I7_J,axiom,
    ! [Ta: rat] :
    ? [Z3: rat] :
    ! [X3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z3),X3)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ta),X3) ) ).

% pinf(7)
tff(fact_1228_pinf_I7_J,axiom,
    ! [Ta: num] :
    ? [Z3: num] :
    ! [X3: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z3),X3)
     => aa_num_o(aa_num_fun_num_o(ord_less_num,Ta),X3) ) ).

% pinf(7)
tff(fact_1229_pinf_I7_J,axiom,
    ! [Ta: nat] :
    ? [Z3: nat] :
    ! [X3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z3),X3)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ta),X3) ) ).

% pinf(7)
tff(fact_1230_pinf_I7_J,axiom,
    ! [Ta: int] :
    ? [Z3: int] :
    ! [X3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),X3)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,Ta),X3) ) ).

% pinf(7)
tff(fact_1231_minf_I1_J,axiom,
    ! [P: fun_real_o,P2: fun_real_o,Q: fun_real_o,Q2: fun_real_o] :
      ( ? [Z4: real] :
        ! [X: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Z4)
         => ( aa_real_o(P,X)
          <=> aa_real_o(P2,X) ) )
     => ( ? [Z4: real] :
          ! [X: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Z4)
           => ( aa_real_o(Q,X)
            <=> aa_real_o(Q2,X) ) )
       => ? [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(Q,X3) )
            <=> ( aa_real_o(P2,X3)
                & aa_real_o(Q2,X3) ) ) ) ) ) ).

% minf(1)
tff(fact_1232_minf_I1_J,axiom,
    ! [P: fun_rat_o,P2: fun_rat_o,Q: fun_rat_o,Q2: fun_rat_o] :
      ( ? [Z4: rat] :
        ! [X: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Z4)
         => ( aa_rat_o(P,X)
          <=> aa_rat_o(P2,X) ) )
     => ( ? [Z4: rat] :
          ! [X: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Z4)
           => ( aa_rat_o(Q,X)
            <=> aa_rat_o(Q2,X) ) )
       => ? [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(Q,X3) )
            <=> ( aa_rat_o(P2,X3)
                & aa_rat_o(Q2,X3) ) ) ) ) ) ).

% minf(1)
tff(fact_1233_minf_I1_J,axiom,
    ! [P: fun_num_o,P2: fun_num_o,Q: fun_num_o,Q2: fun_num_o] :
      ( ? [Z4: num] :
        ! [X: num] :
          ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Z4)
         => ( aa_num_o(P,X)
          <=> aa_num_o(P2,X) ) )
     => ( ? [Z4: num] :
          ! [X: num] :
            ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Z4)
           => ( aa_num_o(Q,X)
            <=> aa_num_o(Q2,X) ) )
       => ? [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(Q,X3) )
            <=> ( aa_num_o(P2,X3)
                & aa_num_o(Q2,X3) ) ) ) ) ) ).

% minf(1)
tff(fact_1234_minf_I1_J,axiom,
    ! [P: fun_nat_o,P2: fun_nat_o,Q: fun_nat_o,Q2: fun_nat_o] :
      ( ? [Z4: nat] :
        ! [X: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Z4)
         => ( aa_nat_o(P,X)
          <=> aa_nat_o(P2,X) ) )
     => ( ? [Z4: nat] :
          ! [X: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Z4)
           => ( aa_nat_o(Q,X)
            <=> aa_nat_o(Q2,X) ) )
       => ? [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(Q,X3) )
            <=> ( aa_nat_o(P2,X3)
                & aa_nat_o(Q2,X3) ) ) ) ) ) ).

% minf(1)
tff(fact_1235_minf_I1_J,axiom,
    ! [P: fun_int_o,P2: fun_int_o,Q: fun_int_o,Q2: fun_int_o] :
      ( ? [Z4: int] :
        ! [X: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Z4)
         => ( aa_int_o(P,X)
          <=> aa_int_o(P2,X) ) )
     => ( ? [Z4: int] :
          ! [X: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Z4)
           => ( aa_int_o(Q,X)
            <=> aa_int_o(Q2,X) ) )
       => ? [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(Q,X3) )
            <=> ( aa_int_o(P2,X3)
                & aa_int_o(Q2,X3) ) ) ) ) ) ).

% minf(1)
tff(fact_1236_minf_I2_J,axiom,
    ! [P: fun_real_o,P2: fun_real_o,Q: fun_real_o,Q2: fun_real_o] :
      ( ? [Z4: real] :
        ! [X: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Z4)
         => ( aa_real_o(P,X)
          <=> aa_real_o(P2,X) ) )
     => ( ? [Z4: real] :
          ! [X: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Z4)
           => ( aa_real_o(Q,X)
            <=> aa_real_o(Q2,X) ) )
       => ? [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(Q,X3) )
            <=> ( aa_real_o(P2,X3)
                | aa_real_o(Q2,X3) ) ) ) ) ) ).

% minf(2)
tff(fact_1237_minf_I2_J,axiom,
    ! [P: fun_rat_o,P2: fun_rat_o,Q: fun_rat_o,Q2: fun_rat_o] :
      ( ? [Z4: rat] :
        ! [X: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Z4)
         => ( aa_rat_o(P,X)
          <=> aa_rat_o(P2,X) ) )
     => ( ? [Z4: rat] :
          ! [X: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X),Z4)
           => ( aa_rat_o(Q,X)
            <=> aa_rat_o(Q2,X) ) )
       => ? [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(Q,X3) )
            <=> ( aa_rat_o(P2,X3)
                | aa_rat_o(Q2,X3) ) ) ) ) ) ).

% minf(2)
tff(fact_1238_minf_I2_J,axiom,
    ! [P: fun_num_o,P2: fun_num_o,Q: fun_num_o,Q2: fun_num_o] :
      ( ? [Z4: num] :
        ! [X: num] :
          ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Z4)
         => ( aa_num_o(P,X)
          <=> aa_num_o(P2,X) ) )
     => ( ? [Z4: num] :
          ! [X: num] :
            ( aa_num_o(aa_num_fun_num_o(ord_less_num,X),Z4)
           => ( aa_num_o(Q,X)
            <=> aa_num_o(Q2,X) ) )
       => ? [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(Q,X3) )
            <=> ( aa_num_o(P2,X3)
                | aa_num_o(Q2,X3) ) ) ) ) ) ).

% minf(2)
tff(fact_1239_minf_I2_J,axiom,
    ! [P: fun_nat_o,P2: fun_nat_o,Q: fun_nat_o,Q2: fun_nat_o] :
      ( ? [Z4: nat] :
        ! [X: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Z4)
         => ( aa_nat_o(P,X)
          <=> aa_nat_o(P2,X) ) )
     => ( ? [Z4: nat] :
          ! [X: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Z4)
           => ( aa_nat_o(Q,X)
            <=> aa_nat_o(Q2,X) ) )
       => ? [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(Q,X3) )
            <=> ( aa_nat_o(P2,X3)
                | aa_nat_o(Q2,X3) ) ) ) ) ) ).

% minf(2)
tff(fact_1240_minf_I2_J,axiom,
    ! [P: fun_int_o,P2: fun_int_o,Q: fun_int_o,Q2: fun_int_o] :
      ( ? [Z4: int] :
        ! [X: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Z4)
         => ( aa_int_o(P,X)
          <=> aa_int_o(P2,X) ) )
     => ( ? [Z4: int] :
          ! [X: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Z4)
           => ( aa_int_o(Q,X)
            <=> aa_int_o(Q2,X) ) )
       => ? [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(Q,X3) )
            <=> ( aa_int_o(P2,X3)
                | aa_int_o(Q2,X3) ) ) ) ) ) ).

% minf(2)
tff(fact_1241_minf_I3_J,axiom,
    ! [Ta: real] :
    ? [Z3: real] :
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Z3)
     => ( X3 != Ta ) ) ).

% minf(3)
tff(fact_1242_minf_I3_J,axiom,
    ! [Ta: rat] :
    ? [Z3: rat] :
    ! [X3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Z3)
     => ( X3 != Ta ) ) ).

% minf(3)
tff(fact_1243_minf_I3_J,axiom,
    ! [Ta: num] :
    ? [Z3: num] :
    ! [X3: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Z3)
     => ( X3 != Ta ) ) ).

% minf(3)
tff(fact_1244_minf_I3_J,axiom,
    ! [Ta: nat] :
    ? [Z3: nat] :
    ! [X3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Z3)
     => ( X3 != Ta ) ) ).

% minf(3)
tff(fact_1245_minf_I3_J,axiom,
    ! [Ta: int] :
    ? [Z3: int] :
    ! [X3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Z3)
     => ( X3 != Ta ) ) ).

% minf(3)
tff(fact_1246_minf_I4_J,axiom,
    ! [Ta: real] :
    ? [Z3: real] :
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Z3)
     => ( X3 != Ta ) ) ).

% minf(4)
tff(fact_1247_minf_I4_J,axiom,
    ! [Ta: rat] :
    ? [Z3: rat] :
    ! [X3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Z3)
     => ( X3 != Ta ) ) ).

% minf(4)
tff(fact_1248_minf_I4_J,axiom,
    ! [Ta: num] :
    ? [Z3: num] :
    ! [X3: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Z3)
     => ( X3 != Ta ) ) ).

% minf(4)
tff(fact_1249_minf_I4_J,axiom,
    ! [Ta: nat] :
    ? [Z3: nat] :
    ! [X3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Z3)
     => ( X3 != Ta ) ) ).

% minf(4)
tff(fact_1250_minf_I4_J,axiom,
    ! [Ta: int] :
    ? [Z3: int] :
    ! [X3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Z3)
     => ( X3 != Ta ) ) ).

% minf(4)
tff(fact_1251_minf_I5_J,axiom,
    ! [Ta: real] :
    ? [Z3: real] :
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Z3)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Ta) ) ).

% minf(5)
tff(fact_1252_minf_I5_J,axiom,
    ! [Ta: rat] :
    ? [Z3: rat] :
    ! [X3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Z3)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Ta) ) ).

% minf(5)
tff(fact_1253_minf_I5_J,axiom,
    ! [Ta: num] :
    ? [Z3: num] :
    ! [X3: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Z3)
     => aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Ta) ) ).

% minf(5)
tff(fact_1254_minf_I5_J,axiom,
    ! [Ta: nat] :
    ? [Z3: nat] :
    ! [X3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Z3)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Ta) ) ).

% minf(5)
tff(fact_1255_minf_I5_J,axiom,
    ! [Ta: int] :
    ? [Z3: int] :
    ! [X3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Z3)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Ta) ) ).

% minf(5)
tff(fact_1256_minf_I7_J,axiom,
    ! [Ta: real] :
    ? [Z3: real] :
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Z3)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Ta),X3) ) ).

% minf(7)
tff(fact_1257_minf_I7_J,axiom,
    ! [Ta: rat] :
    ? [Z3: rat] :
    ! [X3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Z3)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ta),X3) ) ).

% minf(7)
tff(fact_1258_minf_I7_J,axiom,
    ! [Ta: num] :
    ? [Z3: num] :
    ! [X3: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Z3)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Ta),X3) ) ).

% minf(7)
tff(fact_1259_minf_I7_J,axiom,
    ! [Ta: nat] :
    ? [Z3: nat] :
    ! [X3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Z3)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ta),X3) ) ).

% minf(7)
tff(fact_1260_minf_I7_J,axiom,
    ! [Ta: int] :
    ? [Z3: int] :
    ! [X3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Z3)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Ta),X3) ) ).

% minf(7)
tff(fact_1261_vebt__mint_Ocases,axiom,
    ! [Xb: vEBT_VEBT] :
      ( ! [A2: $o,B2: $o] : Xb != vEBT_Leaf((A2),(B2))
     => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : Xb != vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2)
       => ~ ! [Mi: nat,Ma: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : Xb != vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Ux2,Uy2,Uz2) ) ) ).

% vebt_mint.cases
tff(fact_1262_VEBT__internal_Omembermima_Osimps_I3_J,axiom,
    ! [Mi2: nat,Ma2: nat,Va: 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(Mi2),Ma2)),zero_zero_nat,Va,Vb)),Xb)
    <=> ( ( Xb = Mi2 )
        | ( Xb = Ma2 ) ) ) ).

% VEBT_internal.membermima.simps(3)
tff(fact_1263_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_1264_vebt__mint_Osimps_I3_J,axiom,
    ! [Mi2: nat,Ma2: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] : vEBT_vebt_mint(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux,Uy,Uz)) = some_nat(Mi2) ).

% vebt_mint.simps(3)
tff(fact_1265_vebt__maxt_Osimps_I3_J,axiom,
    ! [Mi2: nat,Ma2: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] : vEBT_vebt_maxt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Ux,Uy,Uz)) = some_nat(Ma2) ).

% vebt_maxt.simps(3)
tff(fact_1266_pinf_I6_J,axiom,
    ! [Ta: real] :
    ? [Z3: real] :
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),X3)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),Ta) ) ).

% pinf(6)
tff(fact_1267_pinf_I6_J,axiom,
    ! [Ta: rat] :
    ? [Z3: rat] :
    ! [X3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z3),X3)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Ta) ) ).

% pinf(6)
tff(fact_1268_pinf_I6_J,axiom,
    ! [Ta: num] :
    ? [Z3: num] :
    ! [X3: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z3),X3)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Ta) ) ).

% pinf(6)
tff(fact_1269_pinf_I6_J,axiom,
    ! [Ta: nat] :
    ? [Z3: nat] :
    ! [X3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z3),X3)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Ta) ) ).

% pinf(6)
tff(fact_1270_pinf_I6_J,axiom,
    ! [Ta: int] :
    ? [Z3: int] :
    ! [X3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),X3)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),Ta) ) ).

% pinf(6)
tff(fact_1271_vebt__maxt_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Y: option_nat] :
      ( ( vEBT_vebt_maxt(Xb) = Y )
     => ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_vebt_maxt_rel),Xb)
       => ( ! [A2: $o,B2: $o] :
              ( ( Xb = vEBT_Leaf((A2),(B2)) )
             => ( ( Y = $ite(
                      (B2),
                      some_nat(one_one_nat),
                      $ite((A2),some_nat(zero_zero_nat),none_nat) ) )
               => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_vebt_maxt_rel),vEBT_Leaf((A2),(B2))) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2) )
               => ( ( Y = none_nat )
                 => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_vebt_maxt_rel),vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2)) ) )
           => ~ ! [Mi: nat,Ma: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Ux2,Uy2,Uz2) )
                 => ( ( Y = some_nat(Ma) )
                   => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_vebt_maxt_rel),vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Ux2,Uy2,Uz2)) ) ) ) ) ) ) ).

% vebt_maxt.pelims
tff(fact_1272_vebt__mint_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Y: option_nat] :
      ( ( vEBT_vebt_mint(Xb) = Y )
     => ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_vebt_mint_rel),Xb)
       => ( ! [A2: $o,B2: $o] :
              ( ( Xb = vEBT_Leaf((A2),(B2)) )
             => ( ( Y = $ite(
                      (A2),
                      some_nat(zero_zero_nat),
                      $ite((B2),some_nat(one_one_nat),none_nat) ) )
               => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_vebt_mint_rel),vEBT_Leaf((A2),(B2))) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2) )
               => ( ( Y = none_nat )
                 => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_vebt_mint_rel),vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2)) ) )
           => ~ ! [Mi: nat,Ma: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Ux2,Uy2,Uz2) )
                 => ( ( Y = some_nat(Mi) )
                   => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_vebt_mint_rel),vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Ux2,Uy2,Uz2)) ) ) ) ) ) ) ).

% vebt_mint.pelims
tff(fact_1273_vebt__pred_Ocases,axiom,
    ! [Xb: produc9072475918466114483BT_nat] :
      ( ! [Uu2: $o,Uv2: $o] : Xb != produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(Uv2)),zero_zero_nat)
     => ( ! [A2: $o,Uw2: $o] : Xb != produc738532404422230701BT_nat(vEBT_Leaf((A2),(Uw2)),aa_nat_nat(suc,zero_zero_nat))
       => ( ! [A2: $o,B2: $o,Va2: nat] : Xb != produc738532404422230701BT_nat(vEBT_Leaf((A2),(B2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)))
         => ( ! [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT,Vb2: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2,Va3),Vb2)
           => ( ! [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT,Vf2: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vd2,Ve2),Vf2)
             => ( ! [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT,Vj2: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vh2,Vi2),Vj2)
               => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X: 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,Va2)),TreeList2,Summary2),X) ) ) ) ) ) ) ).

% vebt_pred.cases
tff(fact_1274_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,X: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,Va3,Vb2),X)
         => ( ! [Mi: nat,Ma: nat,V2: nat,TreeList2: list_VEBT_VEBT,Vc2: vEBT_VEBT,X: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,V2),TreeList2,Vc2),X)
           => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT,Vd2: vEBT_VEBT,X: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2,Vd2),X) ) ) ) ) ).

% VEBT_internal.membermima.cases
tff(fact_1275_vebt__member_Ocases,axiom,
    ! [Xb: produc9072475918466114483BT_nat] :
      ( ! [A2: $o,B2: $o,X: nat] : Xb != produc738532404422230701BT_nat(vEBT_Leaf((A2),(B2)),X)
     => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,X: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2),X)
       => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,X: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2),X)
         => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT,X: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2),X)
           => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X: 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,Va2)),TreeList2,Summary2),X) ) ) ) ) ).

% vebt_member.cases
tff(fact_1276_complete__interval,axiom,
    ! [Aa: real,Ba: real,P: fun_real_o] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),Ba)
     => ( aa_real_o(P,Aa)
       => ( ~ aa_real_o(P,Ba)
         => ? [C3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),C3)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C3),Ba)
              & ! [X3: real] :
                  ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),X3)
                    & aa_real_o(aa_real_fun_real_o(ord_less_real,X3),C3) )
                 => aa_real_o(P,X3) )
              & ! [D3: real] :
                  ( ! [X: real] :
                      ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),X)
                        & aa_real_o(aa_real_fun_real_o(ord_less_real,X),D3) )
                     => aa_real_o(P,X) )
                 => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,D3),C3) ) ) ) ) ) ).

% complete_interval
tff(fact_1277_complete__interval,axiom,
    ! [Aa: nat,Ba: nat,P: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),Ba)
     => ( aa_nat_o(P,Aa)
       => ( ~ aa_nat_o(P,Ba)
         => ? [C3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),C3)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C3),Ba)
              & ! [X3: nat] :
                  ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),X3)
                    & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),C3) )
                 => aa_nat_o(P,X3) )
              & ! [D3: nat] :
                  ( ! [X: nat] :
                      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),X)
                        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),D3) )
                     => aa_nat_o(P,X) )
                 => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,D3),C3) ) ) ) ) ) ).

% complete_interval
tff(fact_1278_complete__interval,axiom,
    ! [Aa: int,Ba: int,P: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),Ba)
     => ( aa_int_o(P,Aa)
       => ( ~ aa_int_o(P,Ba)
         => ? [C3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),C3)
              & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C3),Ba)
              & ! [X3: int] :
                  ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),X3)
                    & aa_int_o(aa_int_fun_int_o(ord_less_int,X3),C3) )
                 => aa_int_o(P,X3) )
              & ! [D3: int] :
                  ( ! [X: int] :
                      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),X)
                        & aa_int_o(aa_int_fun_int_o(ord_less_int,X),D3) )
                     => aa_int_o(P,X) )
                 => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,D3),C3) ) ) ) ) ) ).

% complete_interval
tff(fact_1279_verit__comp__simplify1_I3_J,axiom,
    ! [B5: real,A5: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B5),A5)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A5),B5) ) ).

% verit_comp_simplify1(3)
tff(fact_1280_verit__comp__simplify1_I3_J,axiom,
    ! [B5: rat,A5: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B5),A5)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A5),B5) ) ).

% verit_comp_simplify1(3)
tff(fact_1281_verit__comp__simplify1_I3_J,axiom,
    ! [B5: num,A5: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B5),A5)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,A5),B5) ) ).

% verit_comp_simplify1(3)
tff(fact_1282_verit__comp__simplify1_I3_J,axiom,
    ! [B5: nat,A5: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B5),A5)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A5),B5) ) ).

% verit_comp_simplify1(3)
tff(fact_1283_verit__comp__simplify1_I3_J,axiom,
    ! [B5: int,A5: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B5),A5)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A5),B5) ) ).

% verit_comp_simplify1(3)
tff(fact_1284_dbl__inc__simps_I2_J,axiom,
    aa_complex_complex(neg_nu8557863876264182079omplex,zero_zero_complex) = one_one_complex ).

% dbl_inc_simps(2)
tff(fact_1285_dbl__inc__simps_I2_J,axiom,
    aa_real_real(neg_nu8295874005876285629c_real,zero_zero_real) = one_one_real ).

% dbl_inc_simps(2)
tff(fact_1286_dbl__inc__simps_I2_J,axiom,
    aa_rat_rat(neg_nu5219082963157363817nc_rat,zero_zero_rat) = one_one_rat ).

% dbl_inc_simps(2)
tff(fact_1287_dbl__inc__simps_I2_J,axiom,
    aa_int_int(neg_nu5851722552734809277nc_int,zero_zero_int) = one_one_int ).

% dbl_inc_simps(2)
tff(fact_1288_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_1289_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_1290_vebt__insert_Ocases,axiom,
    ! [Xb: produc9072475918466114483BT_nat] :
      ( ! [A2: $o,B2: $o,X: nat] : Xb != produc738532404422230701BT_nat(vEBT_Leaf((A2),(B2)),X)
     => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT,X: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(Info2,zero_zero_nat,Ts,S),X)
       => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT,X: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts,S),X)
         => ( ! [V2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2,Summary2),X)
           => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,X: 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,Va2)),TreeList2,Summary2),X) ) ) ) ) ).

% vebt_insert.cases
tff(fact_1291_not__min__Null__member,axiom,
    ! [Ta: vEBT_VEBT] :
      ( ~ aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Ta)
     => ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Ta),X_1) ) ).

% not_min_Null_member
tff(fact_1292_both__member__options__equiv__member,axiom,
    ! [Ta: vEBT_VEBT,Na: nat,Xb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Na)
     => ( aa_nat_o(vEBT_V8194947554948674370ptions(Ta),Xb)
      <=> aa_nat_o(vEBT_vebt_member(Ta),Xb) ) ) ).

% both_member_options_equiv_member
tff(fact_1293_valid__member__both__member__options,axiom,
    ! [Ta: vEBT_VEBT,Na: nat,Xb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Na)
     => ( aa_nat_o(vEBT_V8194947554948674370ptions(Ta),Xb)
       => aa_nat_o(vEBT_vebt_member(Ta),Xb) ) ) ).

% valid_member_both_member_options
tff(fact_1294_maxbmo,axiom,
    ! [Ta: vEBT_VEBT,Xb: nat] :
      ( ( vEBT_vebt_maxt(Ta) = some_nat(Xb) )
     => aa_nat_o(vEBT_V8194947554948674370ptions(Ta),Xb) ) ).

% maxbmo
tff(fact_1295_VEBT__internal_Ooption__shift_Ocases,axiom,
    ! [Xb: produc7004362771917097900on_nat] :
      ( ! [Uu2: fun_nat_fun_nat_nat,Uv2: option_nat] : Xb != produc6699827698519632094on_nat(Uu2,produc5098337634421038937on_nat(none_nat,Uv2))
     => ( ! [Uw2: fun_nat_fun_nat_nat,V2: nat] : Xb != produc6699827698519632094on_nat(Uw2,produc5098337634421038937on_nat(some_nat(V2),none_nat))
       => ~ ! [F2: fun_nat_fun_nat_nat,A2: nat,B2: nat] : Xb != produc6699827698519632094on_nat(F2,produc5098337634421038937on_nat(some_nat(A2),some_nat(B2))) ) ) ).

% VEBT_internal.option_shift.cases
tff(fact_1296_VEBT__internal_Ooption__shift_Ocases,axiom,
    ! [Xb: produc1260151622200656695at_nat] :
      ( ! [Uu2: fun_Pr2477239776880934807at_nat,Uv2: option4927543243414619207at_nat] : Xb != produc7814092408141220007at_nat(Uu2,produc488173922507101015at_nat(none_P5556105721700978146at_nat,Uv2))
     => ( ! [Uw2: fun_Pr2477239776880934807at_nat,V2: product_prod_nat_nat] : Xb != produc7814092408141220007at_nat(Uw2,produc488173922507101015at_nat(some_P7363390416028606310at_nat(V2),none_P5556105721700978146at_nat))
       => ~ ! [F2: fun_Pr2477239776880934807at_nat,A2: product_prod_nat_nat,B2: product_prod_nat_nat] : Xb != produc7814092408141220007at_nat(F2,produc488173922507101015at_nat(some_P7363390416028606310at_nat(A2),some_P7363390416028606310at_nat(B2))) ) ) ).

% VEBT_internal.option_shift.cases
tff(fact_1297_VEBT__internal_Ooption__shift_Ocases,axiom,
    ! [Xb: produc2703417854250991842on_num] :
      ( ! [Uu2: fun_num_fun_num_num,Uv2: option_num] : Xb != produc6360833703471310868on_num(Uu2,produc8585076106096196333on_num(none_num,Uv2))
     => ( ! [Uw2: fun_num_fun_num_num,V2: num] : Xb != produc6360833703471310868on_num(Uw2,produc8585076106096196333on_num(some_num(V2),none_num))
       => ~ ! [F2: fun_num_fun_num_num,A2: num,B2: num] : Xb != produc6360833703471310868on_num(F2,produc8585076106096196333on_num(some_num(A2),some_num(B2))) ) ) ).

% VEBT_internal.option_shift.cases
tff(fact_1298_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
    ! [Xb: produc1615844007061411358on_nat] :
      ( ! [Uu2: fun_nat_fun_nat_o,Uv2: option_nat] : Xb != produc7967433646104525454on_nat(Uu2,produc5098337634421038937on_nat(none_nat,Uv2))
     => ( ! [Uw2: fun_nat_fun_nat_o,V2: nat] : Xb != produc7967433646104525454on_nat(Uw2,produc5098337634421038937on_nat(some_nat(V2),none_nat))
       => ~ ! [F2: fun_nat_fun_nat_o,X: nat,Y3: nat] : Xb != produc7967433646104525454on_nat(F2,produc5098337634421038937on_nat(some_nat(X),some_nat(Y3))) ) ) ).

% VEBT_internal.option_comp_shift.cases
tff(fact_1299_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
    ! [Xb: produc3739950874144486240at_nat] :
      ( ! [Uu2: fun_Pr2028404973800837934_nat_o,Uv2: option4927543243414619207at_nat] : Xb != produc3653971535690468304at_nat(Uu2,produc488173922507101015at_nat(none_P5556105721700978146at_nat,Uv2))
     => ( ! [Uw2: fun_Pr2028404973800837934_nat_o,V2: product_prod_nat_nat] : Xb != produc3653971535690468304at_nat(Uw2,produc488173922507101015at_nat(some_P7363390416028606310at_nat(V2),none_P5556105721700978146at_nat))
       => ~ ! [F2: fun_Pr2028404973800837934_nat_o,X: product_prod_nat_nat,Y3: product_prod_nat_nat] : Xb != produc3653971535690468304at_nat(F2,produc488173922507101015at_nat(some_P7363390416028606310at_nat(X),some_P7363390416028606310at_nat(Y3))) ) ) ).

% VEBT_internal.option_comp_shift.cases
tff(fact_1300_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
    ! [Xb: produc50614890452615238on_num] :
      ( ! [Uu2: fun_num_fun_num_o,Uv2: option_num] : Xb != produc1140783415258738358on_num(Uu2,produc8585076106096196333on_num(none_num,Uv2))
     => ( ! [Uw2: fun_num_fun_num_o,V2: num] : Xb != produc1140783415258738358on_num(Uw2,produc8585076106096196333on_num(some_num(V2),none_num))
       => ~ ! [F2: fun_num_fun_num_o,X: num,Y3: num] : Xb != produc1140783415258738358on_num(F2,produc8585076106096196333on_num(some_num(X),some_num(Y3))) ) ) ).

% VEBT_internal.option_comp_shift.cases
tff(fact_1301_fold__atLeastAtMost__nat_Ocases,axiom,
    ! [Xb: produc9217583074051984497at_nat] :
      ~ ! [F2: fun_nat_fun_nat_nat,A2: nat,B2: nat,Acc: nat] : Xb != produc2818902965508115497at_nat(F2,produc487386426758144856at_nat(A2,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B2),Acc))) ).

% fold_atLeastAtMost_nat.cases
tff(fact_1302_VEBT__internal_Ovalid_H_Ocases,axiom,
    ! [Xb: produc9072475918466114483BT_nat] :
      ( ! [Uu2: $o,Uv2: $o,D4: nat] : Xb != produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(Uv2)),D4)
     => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT,Deg3: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(Mima,Deg2,TreeList2,Summary2),Deg3) ) ).

% VEBT_internal.valid'.cases
tff(fact_1303_verit__comp__simplify1_I2_J,axiom,
    ! [Aa: set_int] : aa_set_int_o(ord_less_eq_set_int(Aa),Aa) ).

% verit_comp_simplify1(2)
tff(fact_1304_verit__comp__simplify1_I2_J,axiom,
    ! [Aa: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),Aa) ).

% verit_comp_simplify1(2)
tff(fact_1305_verit__comp__simplify1_I2_J,axiom,
    ! [Aa: num] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Aa),Aa) ).

% verit_comp_simplify1(2)
tff(fact_1306_verit__comp__simplify1_I2_J,axiom,
    ! [Aa: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),Aa) ).

% verit_comp_simplify1(2)
tff(fact_1307_verit__comp__simplify1_I2_J,axiom,
    ! [Aa: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),Aa) ).

% verit_comp_simplify1(2)
tff(fact_1308_verit__la__disequality,axiom,
    ! [Aa: rat,Ba: rat] :
      ( ( Aa = Ba )
      | ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),Ba)
      | ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ba),Aa) ) ).

% verit_la_disequality
tff(fact_1309_verit__la__disequality,axiom,
    ! [Aa: num,Ba: num] :
      ( ( Aa = Ba )
      | ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Aa),Ba)
      | ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ba),Aa) ) ).

% verit_la_disequality
tff(fact_1310_verit__la__disequality,axiom,
    ! [Aa: nat,Ba: nat] :
      ( ( Aa = Ba )
      | ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),Ba)
      | ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ba),Aa) ) ).

% verit_la_disequality
tff(fact_1311_verit__la__disequality,axiom,
    ! [Aa: int,Ba: int] :
      ( ( Aa = Ba )
      | ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),Ba)
      | ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ba),Aa) ) ).

% verit_la_disequality
tff(fact_1312_verit__comp__simplify1_I1_J,axiom,
    ! [Aa: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),Aa) ).

% verit_comp_simplify1(1)
tff(fact_1313_verit__comp__simplify1_I1_J,axiom,
    ! [Aa: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),Aa) ).

% verit_comp_simplify1(1)
tff(fact_1314_verit__comp__simplify1_I1_J,axiom,
    ! [Aa: num] : ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Aa),Aa) ).

% verit_comp_simplify1(1)
tff(fact_1315_verit__comp__simplify1_I1_J,axiom,
    ! [Aa: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),Aa) ).

% verit_comp_simplify1(1)
tff(fact_1316_verit__comp__simplify1_I1_J,axiom,
    ! [Aa: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),Aa) ).

% verit_comp_simplify1(1)
tff(fact_1317_ex__gt__or__lt,axiom,
    ! [Aa: real] :
    ? [B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),B2)
      | aa_real_o(aa_real_fun_real_o(ord_less_real,B2),Aa) ) ).

% ex_gt_or_lt
tff(fact_1318_VEBT__internal_Onaive__member_Ocases,axiom,
    ! [Xb: produc9072475918466114483BT_nat] :
      ( ! [A2: $o,B2: $o,X: nat] : Xb != produc738532404422230701BT_nat(vEBT_Leaf((A2),(B2)),X)
     => ( ! [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,TreeList2: list_VEBT_VEBT,S: vEBT_VEBT,X: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList2,S),X) ) ) ).

% VEBT_internal.naive_member.cases
tff(fact_1319_mi__eq__ma__no__ch,axiom,
    ! [Mi2: nat,Ma2: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( aa_nat_o(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Deg,TreeList,Summary)),Deg)
     => ( ( Mi2 = Ma2 )
       => ( ! [X3: vEBT_VEBT] :
              ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeList))
             => ~ ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X3),X_13) )
          & ~ ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary),X_13) ) ) ) ).

% mi_eq_ma_no_ch
tff(fact_1320_VEBT__internal_OminNull_Opelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Y: $o] :
      ( ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Xb)
      <=> (Y) )
     => ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),Xb)
       => ( ( ( Xb = vEBT_Leaf($false,$false) )
           => ( (Y)
             => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf($false,$false)) ) )
         => ( ! [Uv2: $o] :
                ( ( Xb = vEBT_Leaf($true,(Uv2)) )
               => ( ~ (Y)
                 => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf($true,(Uv2))) ) )
           => ( ! [Uu2: $o] :
                  ( ( Xb = vEBT_Leaf((Uu2),$true) )
                 => ( ~ (Y)
                   => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf((Uu2),$true)) ) )
             => ( ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2,Uy2) )
                   => ( (Y)
                     => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2,Uy2)) ) )
               => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2,Vc2) )
                     => ( ~ (Y)
                       => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2,Vc2)) ) ) ) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(1)
tff(fact_1321_VEBT__internal_OminNull_Opelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Xb)
     => ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),Xb)
       => ( ( ( Xb = vEBT_Leaf($false,$false) )
           => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf($false,$false)) )
         => ~ ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2,Uy2) )
               => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux2,Uy2)) ) ) ) ) ).

% VEBT_internal.minNull.pelims(2)
tff(fact_1322_VEBT__internal_OminNull_Opelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT] :
      ( ~ aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Xb)
     => ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),Xb)
       => ( ! [Uv2: $o] :
              ( ( Xb = vEBT_Leaf($true,(Uv2)) )
             => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Leaf($true,(Uv2))) )
         => ( ! [Uu2: $o] :
                ( ( Xb = vEBT_Leaf((Uu2),$true) )
               => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),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) )
                 => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),vEBT_Node(some_P7363390416028606310at_nat(Uz2),Va3,Vb2,Vc2)) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(3)
tff(fact_1323_vebt__insert_Osimps_I4_J,axiom,
    ! [V: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Xb: nat] : vEBT_vebt_insert(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V)),TreeList,Summary),Xb) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xb),Xb)),aa_nat_nat(suc,aa_nat_nat(suc,V)),TreeList,Summary) ).

% vebt_insert.simps(4)
tff(fact_1324_pair__lessI2,axiom,
    ! [Aa: nat,Ba: nat,S3: nat,Ta: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),Ba)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,S3),Ta)
       => member8206827879206165904at_nat(produc6161850002892822231at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Aa),S3),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Ba),Ta)),fun_pair_less) ) ) ).

% pair_lessI2
tff(fact_1325_pair__less__iff1,axiom,
    ! [Xb: nat,Y: nat,Z: nat] :
      ( member8206827879206165904at_nat(produc6161850002892822231at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xb),Y),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xb),Z)),fun_pair_less)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Z) ) ).

% pair_less_iff1
tff(fact_1326_set__encode__empty,axiom,
    aa_set_nat_nat(nat_set_encode,bot_bot_set_nat) = zero_zero_nat ).

% set_encode_empty
tff(fact_1327_VEBT__internal_Ooption__shift_Oelims,axiom,
    ! [Xb: fun_Pr2477239776880934807at_nat,Xaa: option4927543243414619207at_nat,Xba: option4927543243414619207at_nat,Y: option4927543243414619207at_nat] :
      ( ( vEBT_V1502963449132264192at_nat(Xb,Xaa,Xba) = Y )
     => ( ( ( Xaa = none_P5556105721700978146at_nat )
         => ( Y != none_P5556105721700978146at_nat ) )
       => ( ( ? [V2: product_prod_nat_nat] : Xaa = some_P7363390416028606310at_nat(V2)
           => ( ( Xba = none_P5556105721700978146at_nat )
             => ( Y != none_P5556105721700978146at_nat ) ) )
         => ~ ! [A2: product_prod_nat_nat] :
                ( ( Xaa = some_P7363390416028606310at_nat(A2) )
               => ! [B2: product_prod_nat_nat] :
                    ( ( Xba = some_P7363390416028606310at_nat(B2) )
                   => ( Y != some_P7363390416028606310at_nat(aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(Xb,A2),B2)) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
tff(fact_1328_VEBT__internal_Ooption__shift_Oelims,axiom,
    ! [Xb: fun_num_fun_num_num,Xaa: option_num,Xba: option_num,Y: option_num] :
      ( ( vEBT_V819420779217536731ft_num(Xb,Xaa,Xba) = Y )
     => ( ( ( Xaa = none_num )
         => ( Y != none_num ) )
       => ( ( ? [V2: num] : Xaa = some_num(V2)
           => ( ( Xba = none_num )
             => ( Y != none_num ) ) )
         => ~ ! [A2: num] :
                ( ( Xaa = some_num(A2) )
               => ! [B2: num] :
                    ( ( Xba = some_num(B2) )
                   => ( Y != some_num(aa_num_num(aa_num_fun_num_num(Xb,A2),B2)) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
tff(fact_1329_VEBT__internal_Ooption__shift_Oelims,axiom,
    ! [Xb: fun_nat_fun_nat_nat,Xaa: option_nat,Xba: option_nat,Y: option_nat] :
      ( ( aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_V4262088993061758097ft_nat(Xb),Xaa),Xba) = Y )
     => ( ( ( Xaa = none_nat )
         => ( Y != none_nat ) )
       => ( ( ? [V2: nat] : Xaa = some_nat(V2)
           => ( ( Xba = none_nat )
             => ( Y != none_nat ) ) )
         => ~ ! [A2: nat] :
                ( ( Xaa = some_nat(A2) )
               => ! [B2: nat] :
                    ( ( Xba = some_nat(B2) )
                   => ( Y != some_nat(aa_nat_nat(aa_nat_fun_nat_nat(Xb,A2),B2)) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
tff(fact_1330_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
    ! [Uw: fun_Pr2477239776880934807at_nat,V: product_prod_nat_nat] : vEBT_V1502963449132264192at_nat(Uw,some_P7363390416028606310at_nat(V),none_P5556105721700978146at_nat) = none_P5556105721700978146at_nat ).

% VEBT_internal.option_shift.simps(2)
tff(fact_1331_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
    ! [Uw: fun_num_fun_num_num,V: num] : vEBT_V819420779217536731ft_num(Uw,some_num(V),none_num) = none_num ).

% VEBT_internal.option_shift.simps(2)
tff(fact_1332_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
    ! [Uw: fun_nat_fun_nat_nat,V: nat] : aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_V4262088993061758097ft_nat(Uw),some_nat(V)),none_nat) = none_nat ).

% VEBT_internal.option_shift.simps(2)
tff(fact_1333_set__encode__eq,axiom,
    ! [A3: set_nat,B4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( ( aa_set_nat_nat(nat_set_encode,A3) = aa_set_nat_nat(nat_set_encode,B4) )
        <=> ( A3 = B4 ) ) ) ) ).

% set_encode_eq
tff(fact_1334_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
    ! [F: fun_Pr2477239776880934807at_nat,Aa: product_prod_nat_nat,Ba: product_prod_nat_nat] : vEBT_V1502963449132264192at_nat(F,some_P7363390416028606310at_nat(Aa),some_P7363390416028606310at_nat(Ba)) = some_P7363390416028606310at_nat(aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(F,Aa),Ba)) ).

% VEBT_internal.option_shift.simps(3)
tff(fact_1335_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
    ! [F: fun_num_fun_num_num,Aa: num,Ba: num] : vEBT_V819420779217536731ft_num(F,some_num(Aa),some_num(Ba)) = some_num(aa_num_num(aa_num_fun_num_num(F,Aa),Ba)) ).

% VEBT_internal.option_shift.simps(3)
tff(fact_1336_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
    ! [F: fun_nat_fun_nat_nat,Aa: nat,Ba: nat] : aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_V4262088993061758097ft_nat(F),some_nat(Aa)),some_nat(Ba)) = some_nat(aa_nat_nat(aa_nat_fun_nat_nat(F,Aa),Ba)) ).

% VEBT_internal.option_shift.simps(3)
tff(fact_1337_vebt__insert_Osimps_I2_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT,Xb: nat] : vEBT_vebt_insert(vEBT_Node(Info,zero_zero_nat,Ts2,S3),Xb) = vEBT_Node(Info,zero_zero_nat,Ts2,S3) ).

% vebt_insert.simps(2)
tff(fact_1338_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_1339_pair__lessI1,axiom,
    ! [Aa: nat,Ba: nat,S3: nat,Ta: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),Ba)
     => member8206827879206165904at_nat(produc6161850002892822231at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Aa),S3),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Ba),Ta)),fun_pair_less) ) ).

% pair_lessI1
tff(fact_1340_vebt__insert_Osimps_I3_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT,Xb: nat] : vEBT_vebt_insert(vEBT_Node(Info,aa_nat_nat(suc,zero_zero_nat),Ts2,S3),Xb) = vEBT_Node(Info,aa_nat_nat(suc,zero_zero_nat),Ts2,S3) ).

% vebt_insert.simps(3)
tff(fact_1341_vebt__insert_Osimps_I1_J,axiom,
    ! [Aa: $o,Ba: $o,Xb: nat] :
      vEBT_vebt_insert(vEBT_Leaf((Aa),(Ba)),Xb) = $ite(
        Xb = zero_zero_nat,
        vEBT_Leaf($true,(Ba)),
        $ite(Xb = one_one_nat,vEBT_Leaf((Aa),$true),vEBT_Leaf((Aa),(Ba))) ) ).

% vebt_insert.simps(1)
tff(fact_1342_List_Ofinite__set,axiom,
    ! [Xs: list_VEBT_VEBT] : finite5795047828879050333T_VEBT(set_VEBT_VEBT2(Xs)) ).

% List.finite_set
tff(fact_1343_List_Ofinite__set,axiom,
    ! [Xs: list_nat] : aa_set_nat_o(finite_finite_nat,set_nat2(Xs)) ).

% List.finite_set
tff(fact_1344_List_Ofinite__set,axiom,
    ! [Xs: list_int] : aa_set_int_o(finite_finite_int,set_int2(Xs)) ).

% List.finite_set
tff(fact_1345_List_Ofinite__set,axiom,
    ! [Xs: list_complex] : aa_set_complex_o(finite3207457112153483333omplex,set_complex2(Xs)) ).

% List.finite_set
tff(fact_1346_length__pos__if__in__set,axiom,
    ! [Xb: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( member8440522571783428010at_nat(Xb,set_Pr5648618587558075414at_nat(Xs))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),size_s5460976970255530739at_nat(Xs)) ) ).

% length_pos_if_in_set
tff(fact_1347_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_1348_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_1349_length__pos__if__in__set,axiom,
    ! [Xb: int,Xs: list_int] :
      ( aa_set_int_o(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_1350_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_1351_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_1352_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_1353_pair__leqI2,axiom,
    ! [Aa: nat,Ba: nat,S3: nat,Ta: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),Ba)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,S3),Ta)
       => member8206827879206165904at_nat(produc6161850002892822231at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Aa),S3),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Ba),Ta)),fun_pair_leq) ) ) ).

% pair_leqI2
tff(fact_1354_pair__leqI1,axiom,
    ! [Aa: nat,Ba: nat,S3: nat,Ta: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),Ba)
     => member8206827879206165904at_nat(produc6161850002892822231at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Aa),S3),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Ba),Ta)),fun_pair_leq) ) ).

% pair_leqI1
tff(fact_1355_subset__code_I1_J,axiom,
    ! [Xs: list_P6011104703257516679at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ord_le3146513528884898305at_nat(set_Pr5648618587558075414at_nat(Xs),B4)
    <=> ! [X4: product_prod_nat_nat] :
          ( member8440522571783428010at_nat(X4,set_Pr5648618587558075414at_nat(Xs))
         => member8440522571783428010at_nat(X4,B4) ) ) ).

% subset_code(1)
tff(fact_1356_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_1357_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_1358_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_1359_subset__code_I1_J,axiom,
    ! [Xs: list_nat,B4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_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_1360_subset__code_I1_J,axiom,
    ! [Xs: list_int,B4: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(set_int2(Xs)),B4)
    <=> ! [X4: int] :
          ( aa_set_int_o(member_int(X4),set_int2(Xs))
         => aa_set_int_o(member_int(X4),B4) ) ) ).

% subset_code(1)
tff(fact_1361_finite__list,axiom,
    ! [A3: set_VEBT_VEBT] :
      ( finite5795047828879050333T_VEBT(A3)
     => ? [Xs2: list_VEBT_VEBT] : set_VEBT_VEBT2(Xs2) = A3 ) ).

% finite_list
tff(fact_1362_finite__list,axiom,
    ! [A3: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ? [Xs2: list_nat] : set_nat2(Xs2) = A3 ) ).

% finite_list
tff(fact_1363_finite__list,axiom,
    ! [A3: set_int] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ? [Xs2: list_int] : set_int2(Xs2) = A3 ) ).

% finite_list
tff(fact_1364_finite__list,axiom,
    ! [A3: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A3)
     => ? [Xs2: list_complex] : set_complex2(Xs2) = A3 ) ).

% finite_list
tff(fact_1365_set__encode__inverse,axiom,
    ! [A3: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( nat_set_decode(aa_set_nat_nat(nat_set_encode,A3)) = A3 ) ) ).

% set_encode_inverse
tff(fact_1366_frac__eq,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(archim2898591450579166408c_real,Xb) = Xb )
    <=> ( 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) ) ) ).

% frac_eq
tff(fact_1367_frac__eq,axiom,
    ! [Xb: rat] :
      ( ( aa_rat_rat(archimedean_frac_rat,Xb) = Xb )
    <=> ( 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),one_one_rat) ) ) ).

% frac_eq
tff(fact_1368_Suc__diff__1,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat)) = Na ) ) ).

% Suc_diff_1
tff(fact_1369_option_Osize__gen_I2_J,axiom,
    ! [Xb: fun_nat_nat,X2: nat] : size_option_nat(Xb,some_nat(X2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(Xb,X2)),aa_nat_nat(suc,zero_zero_nat)) ).

% option.size_gen(2)
tff(fact_1370_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_1371_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_1372_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_1373_add__right__cancel,axiom,
    ! [Ba: real,Aa: real,C: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Ba),Aa) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),Aa) )
    <=> ( Ba = C ) ) ).

% add_right_cancel
tff(fact_1374_add__right__cancel,axiom,
    ! [Ba: rat,Aa: rat,C: rat] :
      ( ( aa_rat_rat(plus_plus_rat(Ba),Aa) = aa_rat_rat(plus_plus_rat(C),Aa) )
    <=> ( Ba = C ) ) ).

% add_right_cancel
tff(fact_1375_add__right__cancel,axiom,
    ! [Ba: nat,Aa: nat,C: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ba),Aa) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),Aa) )
    <=> ( Ba = C ) ) ).

% add_right_cancel
tff(fact_1376_add__right__cancel,axiom,
    ! [Ba: int,Aa: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,Ba),Aa) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),Aa) )
    <=> ( Ba = C ) ) ).

% add_right_cancel
tff(fact_1377_add__left__cancel,axiom,
    ! [Aa: real,Ba: real,C: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),Ba) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),C) )
    <=> ( Ba = C ) ) ).

% add_left_cancel
tff(fact_1378_add__left__cancel,axiom,
    ! [Aa: rat,Ba: rat,C: rat] :
      ( ( aa_rat_rat(plus_plus_rat(Aa),Ba) = aa_rat_rat(plus_plus_rat(Aa),C) )
    <=> ( Ba = C ) ) ).

% add_left_cancel
tff(fact_1379_add__left__cancel,axiom,
    ! [Aa: nat,Ba: nat,C: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),Ba) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),C) )
    <=> ( Ba = C ) ) ).

% add_left_cancel
tff(fact_1380_add__left__cancel,axiom,
    ! [Aa: int,Ba: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),Ba) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),C) )
    <=> ( Ba = C ) ) ).

% add_left_cancel
tff(fact_1381_add__le__cancel__right,axiom,
    ! [Aa: real,C: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Ba),C))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),Ba) ) ).

% add_le_cancel_right
tff(fact_1382_add__le__cancel__right,axiom,
    ! [Aa: rat,C: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(Aa),C)),aa_rat_rat(plus_plus_rat(Ba),C))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),Ba) ) ).

% add_le_cancel_right
tff(fact_1383_add__le__cancel__right,axiom,
    ! [Aa: nat,C: nat,Ba: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ba),C))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),Ba) ) ).

% add_le_cancel_right
tff(fact_1384_add__le__cancel__right,axiom,
    ! [Aa: int,C: int,Ba: 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),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,Ba),C))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),Ba) ) ).

% add_le_cancel_right
tff(fact_1385_add__le__cancel__left,axiom,
    ! [C: real,Aa: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),Aa)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),Ba))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),Ba) ) ).

% add_le_cancel_left
tff(fact_1386_add__le__cancel__left,axiom,
    ! [C: rat,Aa: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(C),Aa)),aa_rat_rat(plus_plus_rat(C),Ba))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),Ba) ) ).

% add_le_cancel_left
tff(fact_1387_add__le__cancel__left,axiom,
    ! [C: nat,Aa: nat,Ba: 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),Aa)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),Ba))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),Ba) ) ).

% add_le_cancel_left
tff(fact_1388_add__le__cancel__left,axiom,
    ! [C: int,Aa: int,Ba: 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),Aa)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),Ba))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),Ba) ) ).

% add_le_cancel_left
tff(fact_1389_add__0,axiom,
    ! [Aa: literal] : plus_plus_literal(zero_zero_literal,Aa) = Aa ).

% add_0
tff(fact_1390_add__0,axiom,
    ! [Aa: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,zero_zero_real),Aa) = Aa ).

% add_0
tff(fact_1391_add__0,axiom,
    ! [Aa: rat] : aa_rat_rat(plus_plus_rat(zero_zero_rat),Aa) = Aa ).

% add_0
tff(fact_1392_add__0,axiom,
    ! [Aa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,zero_zero_nat),Aa) = Aa ).

% add_0
tff(fact_1393_add__0,axiom,
    ! [Aa: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,zero_zero_int),Aa) = Aa ).

% add_0
tff(fact_1394_zero__eq__add__iff__both__eq__0,axiom,
    ! [Xb: nat,Y: nat] :
      ( ( zero_zero_nat = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xb),Y) )
    <=> ( ( Xb = zero_zero_nat )
        & ( Y = zero_zero_nat ) ) ) ).

% zero_eq_add_iff_both_eq_0
tff(fact_1395_add__eq__0__iff__both__eq__0,axiom,
    ! [Xb: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xb),Y) = zero_zero_nat )
    <=> ( ( Xb = zero_zero_nat )
        & ( Y = zero_zero_nat ) ) ) ).

% add_eq_0_iff_both_eq_0
tff(fact_1396_add__cancel__right__right,axiom,
    ! [Aa: real,Ba: real] :
      ( ( Aa = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),Ba) )
    <=> ( Ba = zero_zero_real ) ) ).

% add_cancel_right_right
tff(fact_1397_add__cancel__right__right,axiom,
    ! [Aa: rat,Ba: rat] :
      ( ( Aa = aa_rat_rat(plus_plus_rat(Aa),Ba) )
    <=> ( Ba = zero_zero_rat ) ) ).

% add_cancel_right_right
tff(fact_1398_add__cancel__right__right,axiom,
    ! [Aa: nat,Ba: nat] :
      ( ( Aa = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),Ba) )
    <=> ( Ba = zero_zero_nat ) ) ).

% add_cancel_right_right
tff(fact_1399_add__cancel__right__right,axiom,
    ! [Aa: int,Ba: int] :
      ( ( Aa = aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),Ba) )
    <=> ( Ba = zero_zero_int ) ) ).

% add_cancel_right_right
tff(fact_1400_add__cancel__right__left,axiom,
    ! [Aa: real,Ba: real] :
      ( ( Aa = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Ba),Aa) )
    <=> ( Ba = zero_zero_real ) ) ).

% add_cancel_right_left
tff(fact_1401_add__cancel__right__left,axiom,
    ! [Aa: rat,Ba: rat] :
      ( ( Aa = aa_rat_rat(plus_plus_rat(Ba),Aa) )
    <=> ( Ba = zero_zero_rat ) ) ).

% add_cancel_right_left
tff(fact_1402_add__cancel__right__left,axiom,
    ! [Aa: nat,Ba: nat] :
      ( ( Aa = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ba),Aa) )
    <=> ( Ba = zero_zero_nat ) ) ).

% add_cancel_right_left
tff(fact_1403_add__cancel__right__left,axiom,
    ! [Aa: int,Ba: int] :
      ( ( Aa = aa_int_int(aa_int_fun_int_int(plus_plus_int,Ba),Aa) )
    <=> ( Ba = zero_zero_int ) ) ).

% add_cancel_right_left
tff(fact_1404_add__cancel__left__right,axiom,
    ! [Aa: real,Ba: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),Ba) = Aa )
    <=> ( Ba = zero_zero_real ) ) ).

% add_cancel_left_right
tff(fact_1405_add__cancel__left__right,axiom,
    ! [Aa: rat,Ba: rat] :
      ( ( aa_rat_rat(plus_plus_rat(Aa),Ba) = Aa )
    <=> ( Ba = zero_zero_rat ) ) ).

% add_cancel_left_right
tff(fact_1406_add__cancel__left__right,axiom,
    ! [Aa: nat,Ba: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),Ba) = Aa )
    <=> ( Ba = zero_zero_nat ) ) ).

% add_cancel_left_right
tff(fact_1407_add__cancel__left__right,axiom,
    ! [Aa: int,Ba: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),Ba) = Aa )
    <=> ( Ba = zero_zero_int ) ) ).

% add_cancel_left_right
tff(fact_1408_add__cancel__left__left,axiom,
    ! [Ba: real,Aa: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Ba),Aa) = Aa )
    <=> ( Ba = zero_zero_real ) ) ).

% add_cancel_left_left
tff(fact_1409_add__cancel__left__left,axiom,
    ! [Ba: rat,Aa: rat] :
      ( ( aa_rat_rat(plus_plus_rat(Ba),Aa) = Aa )
    <=> ( Ba = zero_zero_rat ) ) ).

% add_cancel_left_left
tff(fact_1410_add__cancel__left__left,axiom,
    ! [Ba: nat,Aa: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ba),Aa) = Aa )
    <=> ( Ba = zero_zero_nat ) ) ).

% add_cancel_left_left
tff(fact_1411_add__cancel__left__left,axiom,
    ! [Ba: int,Aa: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,Ba),Aa) = Aa )
    <=> ( Ba = zero_zero_int ) ) ).

% add_cancel_left_left
tff(fact_1412_double__zero__sym,axiom,
    ! [Aa: real] :
      ( ( zero_zero_real = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),Aa) )
    <=> ( Aa = zero_zero_real ) ) ).

% double_zero_sym
tff(fact_1413_double__zero__sym,axiom,
    ! [Aa: rat] :
      ( ( zero_zero_rat = aa_rat_rat(plus_plus_rat(Aa),Aa) )
    <=> ( Aa = zero_zero_rat ) ) ).

% double_zero_sym
tff(fact_1414_double__zero__sym,axiom,
    ! [Aa: int] :
      ( ( zero_zero_int = aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),Aa) )
    <=> ( Aa = zero_zero_int ) ) ).

% double_zero_sym
tff(fact_1415_add_Oright__neutral,axiom,
    ! [Aa: literal] : plus_plus_literal(Aa,zero_zero_literal) = Aa ).

% add.right_neutral
tff(fact_1416_add_Oright__neutral,axiom,
    ! [Aa: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),zero_zero_real) = Aa ).

% add.right_neutral
tff(fact_1417_add_Oright__neutral,axiom,
    ! [Aa: rat] : aa_rat_rat(plus_plus_rat(Aa),zero_zero_rat) = Aa ).

% add.right_neutral
tff(fact_1418_add_Oright__neutral,axiom,
    ! [Aa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),zero_zero_nat) = Aa ).

% add.right_neutral
tff(fact_1419_add_Oright__neutral,axiom,
    ! [Aa: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),zero_zero_int) = Aa ).

% add.right_neutral
tff(fact_1420_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [Aa: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Aa),Aa) = zero_zero_real ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_1421_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [Aa: rat] : aa_rat_rat(minus_minus_rat(Aa),Aa) = zero_zero_rat ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_1422_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [Aa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Aa),Aa) = zero_zero_nat ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_1423_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [Aa: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,Aa),Aa) = zero_zero_int ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_1424_diff__zero,axiom,
    ! [Aa: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Aa),zero_zero_real) = Aa ).

% diff_zero
tff(fact_1425_diff__zero,axiom,
    ! [Aa: rat] : aa_rat_rat(minus_minus_rat(Aa),zero_zero_rat) = Aa ).

% diff_zero
tff(fact_1426_diff__zero,axiom,
    ! [Aa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Aa),zero_zero_nat) = Aa ).

% diff_zero
tff(fact_1427_diff__zero,axiom,
    ! [Aa: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,Aa),zero_zero_int) = Aa ).

% diff_zero
tff(fact_1428_zero__diff,axiom,
    ! [Aa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,zero_zero_nat),Aa) = zero_zero_nat ).

% zero_diff
tff(fact_1429_diff__0__right,axiom,
    ! [Aa: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Aa),zero_zero_real) = Aa ).

% diff_0_right
tff(fact_1430_diff__0__right,axiom,
    ! [Aa: rat] : aa_rat_rat(minus_minus_rat(Aa),zero_zero_rat) = Aa ).

% diff_0_right
tff(fact_1431_diff__0__right,axiom,
    ! [Aa: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,Aa),zero_zero_int) = Aa ).

% diff_0_right
tff(fact_1432_diff__self,axiom,
    ! [Aa: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Aa),Aa) = zero_zero_real ).

% diff_self
tff(fact_1433_diff__self,axiom,
    ! [Aa: rat] : aa_rat_rat(minus_minus_rat(Aa),Aa) = zero_zero_rat ).

% diff_self
tff(fact_1434_diff__self,axiom,
    ! [Aa: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,Aa),Aa) = zero_zero_int ).

% diff_self
tff(fact_1435_add__less__cancel__right,axiom,
    ! [Aa: real,C: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Ba),C))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),Ba) ) ).

% add_less_cancel_right
tff(fact_1436_add__less__cancel__right,axiom,
    ! [Aa: rat,C: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(Aa),C)),aa_rat_rat(plus_plus_rat(Ba),C))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),Ba) ) ).

% add_less_cancel_right
tff(fact_1437_add__less__cancel__right,axiom,
    ! [Aa: nat,C: nat,Ba: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ba),C))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),Ba) ) ).

% add_less_cancel_right
tff(fact_1438_add__less__cancel__right,axiom,
    ! [Aa: int,C: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,Ba),C))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),Ba) ) ).

% add_less_cancel_right
tff(fact_1439_add__less__cancel__left,axiom,
    ! [C: real,Aa: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),Aa)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),Ba))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),Ba) ) ).

% add_less_cancel_left
tff(fact_1440_add__less__cancel__left,axiom,
    ! [C: rat,Aa: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(C),Aa)),aa_rat_rat(plus_plus_rat(C),Ba))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),Ba) ) ).

% add_less_cancel_left
tff(fact_1441_add__less__cancel__left,axiom,
    ! [C: nat,Aa: nat,Ba: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),Aa)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),Ba))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),Ba) ) ).

% add_less_cancel_left
tff(fact_1442_add__less__cancel__left,axiom,
    ! [C: int,Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),Aa)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),Ba))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),Ba) ) ).

% add_less_cancel_left
tff(fact_1443_add__diff__cancel__right_H,axiom,
    ! [Aa: real,Ba: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),Ba)),Ba) = Aa ).

% add_diff_cancel_right'
tff(fact_1444_add__diff__cancel__right_H,axiom,
    ! [Aa: rat,Ba: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(Aa),Ba)),Ba) = Aa ).

% add_diff_cancel_right'
tff(fact_1445_add__diff__cancel__right_H,axiom,
    ! [Aa: nat,Ba: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),Ba)),Ba) = Aa ).

% add_diff_cancel_right'
tff(fact_1446_add__diff__cancel__right_H,axiom,
    ! [Aa: int,Ba: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),Ba)),Ba) = Aa ).

% add_diff_cancel_right'
tff(fact_1447_add__diff__cancel__right,axiom,
    ! [Aa: real,C: real,Ba: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Ba),C)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Aa),Ba) ).

% add_diff_cancel_right
tff(fact_1448_add__diff__cancel__right,axiom,
    ! [Aa: rat,C: rat,Ba: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(Aa),C)),aa_rat_rat(plus_plus_rat(Ba),C)) = aa_rat_rat(minus_minus_rat(Aa),Ba) ).

% add_diff_cancel_right
tff(fact_1449_add__diff__cancel__right,axiom,
    ! [Aa: nat,C: nat,Ba: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ba),C)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Aa),Ba) ).

% add_diff_cancel_right
tff(fact_1450_add__diff__cancel__right,axiom,
    ! [Aa: int,C: int,Ba: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,Ba),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,Aa),Ba) ).

% add_diff_cancel_right
tff(fact_1451_add__diff__cancel__left_H,axiom,
    ! [Aa: real,Ba: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),Ba)),Aa) = Ba ).

% add_diff_cancel_left'
tff(fact_1452_add__diff__cancel__left_H,axiom,
    ! [Aa: rat,Ba: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(Aa),Ba)),Aa) = Ba ).

% add_diff_cancel_left'
tff(fact_1453_add__diff__cancel__left_H,axiom,
    ! [Aa: nat,Ba: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),Ba)),Aa) = Ba ).

% add_diff_cancel_left'
tff(fact_1454_add__diff__cancel__left_H,axiom,
    ! [Aa: int,Ba: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),Ba)),Aa) = Ba ).

% add_diff_cancel_left'
tff(fact_1455_add__diff__cancel__left,axiom,
    ! [C: real,Aa: real,Ba: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),Aa)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),Ba)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Aa),Ba) ).

% add_diff_cancel_left
tff(fact_1456_add__diff__cancel__left,axiom,
    ! [C: rat,Aa: rat,Ba: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(C),Aa)),aa_rat_rat(plus_plus_rat(C),Ba)) = aa_rat_rat(minus_minus_rat(Aa),Ba) ).

% add_diff_cancel_left
tff(fact_1457_add__diff__cancel__left,axiom,
    ! [C: nat,Aa: nat,Ba: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),Aa)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),Ba)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Aa),Ba) ).

% add_diff_cancel_left
tff(fact_1458_add__diff__cancel__left,axiom,
    ! [C: int,Aa: int,Ba: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),Aa)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),Ba)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,Aa),Ba) ).

% add_diff_cancel_left
tff(fact_1459_diff__add__cancel,axiom,
    ! [Aa: real,Ba: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Aa),Ba)),Ba) = Aa ).

% diff_add_cancel
tff(fact_1460_diff__add__cancel,axiom,
    ! [Aa: rat,Ba: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(minus_minus_rat(Aa),Ba)),Ba) = Aa ).

% diff_add_cancel
tff(fact_1461_diff__add__cancel,axiom,
    ! [Aa: int,Ba: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Aa),Ba)),Ba) = Aa ).

% diff_add_cancel
tff(fact_1462_add__diff__cancel,axiom,
    ! [Aa: real,Ba: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),Ba)),Ba) = Aa ).

% add_diff_cancel
tff(fact_1463_add__diff__cancel,axiom,
    ! [Aa: rat,Ba: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(Aa),Ba)),Ba) = Aa ).

% add_diff_cancel
tff(fact_1464_add__diff__cancel,axiom,
    ! [Aa: int,Ba: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),Ba)),Ba) = Aa ).

% add_diff_cancel
tff(fact_1465_add__Suc__right,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(suc,Na)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na)) ).

% add_Suc_right
tff(fact_1466_add__is__0,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na) = zero_zero_nat )
    <=> ( ( M = zero_zero_nat )
        & ( Na = zero_zero_nat ) ) ) ).

% add_is_0
tff(fact_1467_Nat_Oadd__0__right,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),zero_zero_nat) = M ).

% Nat.add_0_right
tff(fact_1468_diff__Suc__Suc,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,Na)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na) ).

% diff_Suc_Suc
tff(fact_1469_Suc__diff__diff,axiom,
    ! [M: nat,Na: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,M)),Na)),aa_nat_nat(suc,K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na)),K) ).

% Suc_diff_diff
tff(fact_1470_nat__add__left__cancel__less,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ).

% nat_add_left_cancel_less
tff(fact_1471_diff__0__eq__0,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,zero_zero_nat),Na) = zero_zero_nat ).

% diff_0_eq_0
tff(fact_1472_diff__self__eq__0,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),M) = zero_zero_nat ).

% diff_self_eq_0
tff(fact_1473_nat__add__left__cancel__le,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ).

% nat_add_left_cancel_le
tff(fact_1474_diff__diff__cancel,axiom,
    ! [I: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),I)) = I ) ) ).

% diff_diff_cancel
tff(fact_1475_diff__diff__left,axiom,
    ! [I: nat,J: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),J)),K) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K)) ).

% diff_diff_left
tff(fact_1476_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),Aa))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Aa) ) ).

% zero_le_double_add_iff_zero_le_single_add
tff(fact_1477_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(Aa),Aa))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Aa) ) ).

% zero_le_double_add_iff_zero_le_single_add
tff(fact_1478_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [Aa: 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),Aa))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Aa) ) ).

% zero_le_double_add_iff_zero_le_single_add
tff(fact_1479_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),Aa)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),zero_zero_real) ) ).

% double_add_le_zero_iff_single_add_le_zero
tff(fact_1480_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(Aa),Aa)),zero_zero_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),zero_zero_rat) ) ).

% double_add_le_zero_iff_single_add_le_zero
tff(fact_1481_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [Aa: 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),Aa)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),zero_zero_int) ) ).

% double_add_le_zero_iff_single_add_le_zero
tff(fact_1482_le__add__same__cancel2,axiom,
    ! [Aa: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Ba),Aa))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Ba) ) ).

% le_add_same_cancel2
tff(fact_1483_le__add__same__cancel2,axiom,
    ! [Aa: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),aa_rat_rat(plus_plus_rat(Ba),Aa))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Ba) ) ).

% le_add_same_cancel2
tff(fact_1484_le__add__same__cancel2,axiom,
    ! [Aa: nat,Ba: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ba),Aa))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Ba) ) ).

% le_add_same_cancel2
tff(fact_1485_le__add__same__cancel2,axiom,
    ! [Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),aa_int_int(aa_int_fun_int_int(plus_plus_int,Ba),Aa))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Ba) ) ).

% le_add_same_cancel2
tff(fact_1486_le__add__same__cancel1,axiom,
    ! [Aa: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),Ba))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Ba) ) ).

% le_add_same_cancel1
tff(fact_1487_le__add__same__cancel1,axiom,
    ! [Aa: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),aa_rat_rat(plus_plus_rat(Aa),Ba))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Ba) ) ).

% le_add_same_cancel1
tff(fact_1488_le__add__same__cancel1,axiom,
    ! [Aa: nat,Ba: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),Ba))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Ba) ) ).

% le_add_same_cancel1
tff(fact_1489_le__add__same__cancel1,axiom,
    ! [Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),Ba))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Ba) ) ).

% le_add_same_cancel1
tff(fact_1490_add__le__same__cancel2,axiom,
    ! [Aa: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),Ba)),Ba)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),zero_zero_real) ) ).

% add_le_same_cancel2
tff(fact_1491_add__le__same__cancel2,axiom,
    ! [Aa: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(Aa),Ba)),Ba)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),zero_zero_rat) ) ).

% add_le_same_cancel2
tff(fact_1492_add__le__same__cancel2,axiom,
    ! [Aa: nat,Ba: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),Ba)),Ba)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),zero_zero_nat) ) ).

% add_le_same_cancel2
tff(fact_1493_add__le__same__cancel2,axiom,
    ! [Aa: int,Ba: 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),Ba)),Ba)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),zero_zero_int) ) ).

% add_le_same_cancel2
tff(fact_1494_add__le__same__cancel1,axiom,
    ! [Ba: real,Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Ba),Aa)),Ba)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),zero_zero_real) ) ).

% add_le_same_cancel1
tff(fact_1495_add__le__same__cancel1,axiom,
    ! [Ba: rat,Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(Ba),Aa)),Ba)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),zero_zero_rat) ) ).

% add_le_same_cancel1
tff(fact_1496_add__le__same__cancel1,axiom,
    ! [Ba: nat,Aa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ba),Aa)),Ba)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),zero_zero_nat) ) ).

% add_le_same_cancel1
tff(fact_1497_add__le__same__cancel1,axiom,
    ! [Ba: int,Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Ba),Aa)),Ba)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),zero_zero_int) ) ).

% add_le_same_cancel1
tff(fact_1498_diff__ge__0__iff__ge,axiom,
    ! [Aa: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Aa),Ba))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ba),Aa) ) ).

% diff_ge_0_iff_ge
tff(fact_1499_diff__ge__0__iff__ge,axiom,
    ! [Aa: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(minus_minus_rat(Aa),Ba))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ba),Aa) ) ).

% diff_ge_0_iff_ge
tff(fact_1500_diff__ge__0__iff__ge,axiom,
    ! [Aa: int,Ba: 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,Aa),Ba))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ba),Aa) ) ).

% diff_ge_0_iff_ge
tff(fact_1501_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),Aa))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Aa) ) ).

% zero_less_double_add_iff_zero_less_single_add
tff(fact_1502_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(Aa),Aa))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Aa) ) ).

% zero_less_double_add_iff_zero_less_single_add
tff(fact_1503_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [Aa: 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),Aa))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Aa) ) ).

% zero_less_double_add_iff_zero_less_single_add
tff(fact_1504_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),Aa)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),zero_zero_real) ) ).

% double_add_less_zero_iff_single_add_less_zero
tff(fact_1505_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(Aa),Aa)),zero_zero_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),zero_zero_rat) ) ).

% double_add_less_zero_iff_single_add_less_zero
tff(fact_1506_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),Aa)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),zero_zero_int) ) ).

% double_add_less_zero_iff_single_add_less_zero
tff(fact_1507_less__add__same__cancel2,axiom,
    ! [Aa: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Ba),Aa))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Ba) ) ).

% less_add_same_cancel2
tff(fact_1508_less__add__same__cancel2,axiom,
    ! [Aa: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),aa_rat_rat(plus_plus_rat(Ba),Aa))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Ba) ) ).

% less_add_same_cancel2
tff(fact_1509_less__add__same__cancel2,axiom,
    ! [Aa: nat,Ba: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ba),Aa))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Ba) ) ).

% less_add_same_cancel2
tff(fact_1510_less__add__same__cancel2,axiom,
    ! [Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),aa_int_int(aa_int_fun_int_int(plus_plus_int,Ba),Aa))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Ba) ) ).

% less_add_same_cancel2
tff(fact_1511_less__add__same__cancel1,axiom,
    ! [Aa: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),Ba))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Ba) ) ).

% less_add_same_cancel1
tff(fact_1512_less__add__same__cancel1,axiom,
    ! [Aa: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),aa_rat_rat(plus_plus_rat(Aa),Ba))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Ba) ) ).

% less_add_same_cancel1
tff(fact_1513_less__add__same__cancel1,axiom,
    ! [Aa: nat,Ba: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),Ba))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Ba) ) ).

% less_add_same_cancel1
tff(fact_1514_less__add__same__cancel1,axiom,
    ! [Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),Ba))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Ba) ) ).

% less_add_same_cancel1
tff(fact_1515_add__less__same__cancel2,axiom,
    ! [Aa: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),Ba)),Ba)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),zero_zero_real) ) ).

% add_less_same_cancel2
tff(fact_1516_add__less__same__cancel2,axiom,
    ! [Aa: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(Aa),Ba)),Ba)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),zero_zero_rat) ) ).

% add_less_same_cancel2
tff(fact_1517_add__less__same__cancel2,axiom,
    ! [Aa: nat,Ba: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),Ba)),Ba)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),zero_zero_nat) ) ).

% add_less_same_cancel2
tff(fact_1518_add__less__same__cancel2,axiom,
    ! [Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),Ba)),Ba)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),zero_zero_int) ) ).

% add_less_same_cancel2
tff(fact_1519_add__less__same__cancel1,axiom,
    ! [Ba: real,Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Ba),Aa)),Ba)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),zero_zero_real) ) ).

% add_less_same_cancel1
tff(fact_1520_add__less__same__cancel1,axiom,
    ! [Ba: rat,Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(Ba),Aa)),Ba)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),zero_zero_rat) ) ).

% add_less_same_cancel1
tff(fact_1521_add__less__same__cancel1,axiom,
    ! [Ba: nat,Aa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ba),Aa)),Ba)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),zero_zero_nat) ) ).

% add_less_same_cancel1
tff(fact_1522_add__less__same__cancel1,axiom,
    ! [Ba: int,Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Ba),Aa)),Ba)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),zero_zero_int) ) ).

% add_less_same_cancel1
tff(fact_1523_diff__gt__0__iff__gt,axiom,
    ! [Aa: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Aa),Ba))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Ba),Aa) ) ).

% diff_gt_0_iff_gt
tff(fact_1524_diff__gt__0__iff__gt,axiom,
    ! [Aa: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(minus_minus_rat(Aa),Ba))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ba),Aa) ) ).

% diff_gt_0_iff_gt
tff(fact_1525_diff__gt__0__iff__gt,axiom,
    ! [Aa: int,Ba: 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,Aa),Ba))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Ba),Aa) ) ).

% diff_gt_0_iff_gt
tff(fact_1526_le__add__diff__inverse,axiom,
    ! [Ba: real,Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ba),Aa)
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Ba),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Aa),Ba)) = Aa ) ) ).

% le_add_diff_inverse
tff(fact_1527_le__add__diff__inverse,axiom,
    ! [Ba: rat,Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ba),Aa)
     => ( aa_rat_rat(plus_plus_rat(Ba),aa_rat_rat(minus_minus_rat(Aa),Ba)) = Aa ) ) ).

% le_add_diff_inverse
tff(fact_1528_le__add__diff__inverse,axiom,
    ! [Ba: nat,Aa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ba),Aa)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ba),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Aa),Ba)) = Aa ) ) ).

% le_add_diff_inverse
tff(fact_1529_le__add__diff__inverse,axiom,
    ! [Ba: int,Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ba),Aa)
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,Ba),aa_int_int(aa_int_fun_int_int(minus_minus_int,Aa),Ba)) = Aa ) ) ).

% le_add_diff_inverse
tff(fact_1530_le__add__diff__inverse2,axiom,
    ! [Ba: real,Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ba),Aa)
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Aa),Ba)),Ba) = Aa ) ) ).

% le_add_diff_inverse2
tff(fact_1531_le__add__diff__inverse2,axiom,
    ! [Ba: rat,Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ba),Aa)
     => ( aa_rat_rat(plus_plus_rat(aa_rat_rat(minus_minus_rat(Aa),Ba)),Ba) = Aa ) ) ).

% le_add_diff_inverse2
tff(fact_1532_le__add__diff__inverse2,axiom,
    ! [Ba: nat,Aa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ba),Aa)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Aa),Ba)),Ba) = Aa ) ) ).

% le_add_diff_inverse2
tff(fact_1533_le__add__diff__inverse2,axiom,
    ! [Ba: int,Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ba),Aa)
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Aa),Ba)),Ba) = Aa ) ) ).

% le_add_diff_inverse2
tff(fact_1534_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_1535_diff__numeral__special_I9_J,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),one_one_real) = zero_zero_real ).

% diff_numeral_special(9)
tff(fact_1536_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_1537_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_1538_diff__add__zero,axiom,
    ! [Aa: nat,Ba: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),Ba)) = zero_zero_nat ).

% diff_add_zero
tff(fact_1539_add__gr__0,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ) ).

% add_gr_0
tff(fact_1540_zero__less__diff,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),M))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ).

% zero_less_diff
tff(fact_1541_diff__is__0__eq_H,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na) = zero_zero_nat ) ) ).

% diff_is_0_eq'
tff(fact_1542_diff__is__0__eq,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na) = zero_zero_nat )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ).

% diff_is_0_eq
tff(fact_1543_Nat_Odiff__diff__right,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),J) ) ) ).

% Nat.diff_diff_right
tff(fact_1544_Nat_Oadd__diff__assoc2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)),I) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),I)),K) ) ) ).

% Nat.add_diff_assoc2
tff(fact_1545_Nat_Oadd__diff__assoc,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J)),K) ) ) ).

% Nat.add_diff_assoc
tff(fact_1546_diff__Suc__1,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,Na)),one_one_nat) = Na ).

% diff_Suc_1
tff(fact_1547_set__decode__zero,axiom,
    nat_set_decode(zero_zero_nat) = bot_bot_set_nat ).

% set_decode_zero
tff(fact_1548_Suc__pred,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(suc,zero_zero_nat))) = Na ) ) ).

% Suc_pred
tff(fact_1549_diff__Suc__diff__eq1,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(suc,J)) ) ) ).

% diff_Suc_diff_eq1
tff(fact_1550_diff__Suc__diff__eq2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K))),I) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,J)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),I)) ) ) ).

% diff_Suc_diff_eq2
tff(fact_1551_diff__less__eq,axiom,
    ! [Aa: real,Ba: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Aa),Ba)),C)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),Ba)) ) ).

% diff_less_eq
tff(fact_1552_diff__less__eq,axiom,
    ! [Aa: rat,Ba: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(Aa),Ba)),C)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),aa_rat_rat(plus_plus_rat(C),Ba)) ) ).

% diff_less_eq
tff(fact_1553_diff__less__eq,axiom,
    ! [Aa: int,Ba: 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,Aa),Ba)),C)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),Ba)) ) ).

% diff_less_eq
tff(fact_1554_less__diff__eq,axiom,
    ! [Aa: real,C: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),Ba))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),Ba)),C) ) ).

% less_diff_eq
tff(fact_1555_less__diff__eq,axiom,
    ! [Aa: rat,C: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),aa_rat_rat(minus_minus_rat(C),Ba))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(Aa),Ba)),C) ) ).

% less_diff_eq
tff(fact_1556_less__diff__eq,axiom,
    ! [Aa: int,C: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),Ba))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),Ba)),C) ) ).

% less_diff_eq
tff(fact_1557_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [Aa: real,Ba: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),Ba)
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Ba),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Aa),Ba)) = Aa ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_1558_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [Aa: rat,Ba: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),Ba)
     => ( aa_rat_rat(plus_plus_rat(Ba),aa_rat_rat(minus_minus_rat(Aa),Ba)) = Aa ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_1559_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [Aa: nat,Ba: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),Ba)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ba),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Aa),Ba)) = Aa ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_1560_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [Aa: int,Ba: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),Ba)
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,Ba),aa_int_int(aa_int_fun_int_int(minus_minus_int,Aa),Ba)) = Aa ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_1561_diff__add__inverse2,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na)),Na) = M ).

% diff_add_inverse2
tff(fact_1562_diff__add__inverse,axiom,
    ! [Na: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),M)),Na) = M ).

% diff_add_inverse
tff(fact_1563_diff__commute,axiom,
    ! [I: nat,J: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),J)),K) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),K)),J) ).

% diff_commute
tff(fact_1564_diff__cancel2,axiom,
    ! [M: nat,K: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na) ).

% diff_cancel2
tff(fact_1565_Nat_Odiff__cancel,axiom,
    ! [K: nat,M: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na) ).

% Nat.diff_cancel
tff(fact_1566_diff__diff__eq,axiom,
    ! [Aa: real,Ba: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Aa),Ba)),C) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Aa),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Ba),C)) ).

% diff_diff_eq
tff(fact_1567_diff__diff__eq,axiom,
    ! [Aa: rat,Ba: rat,C: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(minus_minus_rat(Aa),Ba)),C) = aa_rat_rat(minus_minus_rat(Aa),aa_rat_rat(plus_plus_rat(Ba),C)) ).

% diff_diff_eq
tff(fact_1568_diff__diff__eq,axiom,
    ! [Aa: nat,Ba: 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,Aa),Ba)),C) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ba),C)) ).

% diff_diff_eq
tff(fact_1569_diff__diff__eq,axiom,
    ! [Aa: int,Ba: 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,Aa),Ba)),C) = aa_int_int(aa_int_fun_int_int(minus_minus_int,Aa),aa_int_int(aa_int_fun_int_int(plus_plus_int,Ba),C)) ).

% diff_diff_eq
tff(fact_1570_diff__right__commute,axiom,
    ! [Aa: real,C: real,Ba: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Aa),C)),Ba) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Aa),Ba)),C) ).

% diff_right_commute
tff(fact_1571_diff__right__commute,axiom,
    ! [Aa: rat,C: rat,Ba: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(minus_minus_rat(Aa),C)),Ba) = aa_rat_rat(minus_minus_rat(aa_rat_rat(minus_minus_rat(Aa),Ba)),C) ).

% diff_right_commute
tff(fact_1572_diff__right__commute,axiom,
    ! [Aa: nat,C: nat,Ba: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Aa),C)),Ba) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Aa),Ba)),C) ).

% diff_right_commute
tff(fact_1573_diff__right__commute,axiom,
    ! [Aa: int,C: int,Ba: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Aa),C)),Ba) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Aa),Ba)),C) ).

% diff_right_commute
tff(fact_1574_add__implies__diff,axiom,
    ! [C: real,Ba: real,Aa: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),Ba) = Aa )
     => ( C = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Aa),Ba) ) ) ).

% add_implies_diff
tff(fact_1575_add__implies__diff,axiom,
    ! [C: rat,Ba: rat,Aa: rat] :
      ( ( aa_rat_rat(plus_plus_rat(C),Ba) = Aa )
     => ( C = aa_rat_rat(minus_minus_rat(Aa),Ba) ) ) ).

% add_implies_diff
tff(fact_1576_add__implies__diff,axiom,
    ! [C: nat,Ba: nat,Aa: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),Ba) = Aa )
     => ( C = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Aa),Ba) ) ) ).

% add_implies_diff
tff(fact_1577_add__implies__diff,axiom,
    ! [C: int,Ba: int,Aa: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,C),Ba) = Aa )
     => ( C = aa_int_int(aa_int_fun_int_int(minus_minus_int,Aa),Ba) ) ) ).

% add_implies_diff
tff(fact_1578_add__right__imp__eq,axiom,
    ! [Ba: real,Aa: real,C: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Ba),Aa) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),Aa) )
     => ( Ba = C ) ) ).

% add_right_imp_eq
tff(fact_1579_add__right__imp__eq,axiom,
    ! [Ba: rat,Aa: rat,C: rat] :
      ( ( aa_rat_rat(plus_plus_rat(Ba),Aa) = aa_rat_rat(plus_plus_rat(C),Aa) )
     => ( Ba = C ) ) ).

% add_right_imp_eq
tff(fact_1580_add__right__imp__eq,axiom,
    ! [Ba: nat,Aa: nat,C: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ba),Aa) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),Aa) )
     => ( Ba = C ) ) ).

% add_right_imp_eq
tff(fact_1581_add__right__imp__eq,axiom,
    ! [Ba: int,Aa: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,Ba),Aa) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),Aa) )
     => ( Ba = C ) ) ).

% add_right_imp_eq
tff(fact_1582_diff__add__eq__diff__diff__swap,axiom,
    ! [Aa: real,Ba: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Aa),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Ba),C)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Aa),C)),Ba) ).

% diff_add_eq_diff_diff_swap
tff(fact_1583_diff__add__eq__diff__diff__swap,axiom,
    ! [Aa: rat,Ba: rat,C: rat] : aa_rat_rat(minus_minus_rat(Aa),aa_rat_rat(plus_plus_rat(Ba),C)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(minus_minus_rat(Aa),C)),Ba) ).

% diff_add_eq_diff_diff_swap
tff(fact_1584_diff__add__eq__diff__diff__swap,axiom,
    ! [Aa: int,Ba: int,C: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,Aa),aa_int_int(aa_int_fun_int_int(plus_plus_int,Ba),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Aa),C)),Ba) ).

% diff_add_eq_diff_diff_swap
tff(fact_1585_add__left__imp__eq,axiom,
    ! [Aa: real,Ba: real,C: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),Ba) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),C) )
     => ( Ba = C ) ) ).

% add_left_imp_eq
tff(fact_1586_add__left__imp__eq,axiom,
    ! [Aa: rat,Ba: rat,C: rat] :
      ( ( aa_rat_rat(plus_plus_rat(Aa),Ba) = aa_rat_rat(plus_plus_rat(Aa),C) )
     => ( Ba = C ) ) ).

% add_left_imp_eq
tff(fact_1587_add__left__imp__eq,axiom,
    ! [Aa: nat,Ba: nat,C: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),Ba) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),C) )
     => ( Ba = C ) ) ).

% add_left_imp_eq
tff(fact_1588_add__left__imp__eq,axiom,
    ! [Aa: int,Ba: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),Ba) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),C) )
     => ( Ba = C ) ) ).

% add_left_imp_eq
tff(fact_1589_add_Oleft__commute,axiom,
    ! [Ba: real,Aa: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Ba),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),C)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Ba),C)) ).

% add.left_commute
tff(fact_1590_add_Oleft__commute,axiom,
    ! [Ba: rat,Aa: rat,C: rat] : aa_rat_rat(plus_plus_rat(Ba),aa_rat_rat(plus_plus_rat(Aa),C)) = aa_rat_rat(plus_plus_rat(Aa),aa_rat_rat(plus_plus_rat(Ba),C)) ).

% add.left_commute
tff(fact_1591_add_Oleft__commute,axiom,
    ! [Ba: nat,Aa: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ba),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),C)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ba),C)) ).

% add.left_commute
tff(fact_1592_add_Oleft__commute,axiom,
    ! [Ba: int,Aa: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,Ba),aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),aa_int_int(aa_int_fun_int_int(plus_plus_int,Ba),C)) ).

% add.left_commute
tff(fact_1593_add_Ocommute,axiom,
    ! [Aa: real,Ba: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),Ba) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Ba),Aa) ).

% add.commute
tff(fact_1594_add_Ocommute,axiom,
    ! [Aa: rat,Ba: rat] : aa_rat_rat(plus_plus_rat(Aa),Ba) = aa_rat_rat(plus_plus_rat(Ba),Aa) ).

% add.commute
tff(fact_1595_add_Ocommute,axiom,
    ! [Aa: nat,Ba: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),Ba) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ba),Aa) ).

% add.commute
tff(fact_1596_add_Ocommute,axiom,
    ! [Aa: int,Ba: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),Ba) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Ba),Aa) ).

% add.commute
tff(fact_1597_add_Oright__cancel,axiom,
    ! [Ba: real,Aa: real,C: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Ba),Aa) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),Aa) )
    <=> ( Ba = C ) ) ).

% add.right_cancel
tff(fact_1598_add_Oright__cancel,axiom,
    ! [Ba: rat,Aa: rat,C: rat] :
      ( ( aa_rat_rat(plus_plus_rat(Ba),Aa) = aa_rat_rat(plus_plus_rat(C),Aa) )
    <=> ( Ba = C ) ) ).

% add.right_cancel
tff(fact_1599_add_Oright__cancel,axiom,
    ! [Ba: int,Aa: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,Ba),Aa) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),Aa) )
    <=> ( Ba = C ) ) ).

% add.right_cancel
tff(fact_1600_diff__eq__diff__eq,axiom,
    ! [Aa: real,Ba: real,C: real,D: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Aa),Ba) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),D) )
     => ( ( Aa = Ba )
      <=> ( C = D ) ) ) ).

% diff_eq_diff_eq
tff(fact_1601_diff__eq__diff__eq,axiom,
    ! [Aa: rat,Ba: rat,C: rat,D: rat] :
      ( ( aa_rat_rat(minus_minus_rat(Aa),Ba) = aa_rat_rat(minus_minus_rat(C),D) )
     => ( ( Aa = Ba )
      <=> ( C = D ) ) ) ).

% diff_eq_diff_eq
tff(fact_1602_diff__eq__diff__eq,axiom,
    ! [Aa: int,Ba: int,C: int,D: int] :
      ( ( aa_int_int(aa_int_fun_int_int(minus_minus_int,Aa),Ba) = aa_int_int(aa_int_fun_int_int(minus_minus_int,C),D) )
     => ( ( Aa = Ba )
      <=> ( C = D ) ) ) ).

% diff_eq_diff_eq
tff(fact_1603_add_Oleft__cancel,axiom,
    ! [Aa: real,Ba: real,C: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),Ba) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),C) )
    <=> ( Ba = C ) ) ).

% add.left_cancel
tff(fact_1604_add_Oleft__cancel,axiom,
    ! [Aa: rat,Ba: rat,C: rat] :
      ( ( aa_rat_rat(plus_plus_rat(Aa),Ba) = aa_rat_rat(plus_plus_rat(Aa),C) )
    <=> ( Ba = C ) ) ).

% add.left_cancel
tff(fact_1605_add_Oleft__cancel,axiom,
    ! [Aa: int,Ba: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),Ba) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),C) )
    <=> ( Ba = C ) ) ).

% add.left_cancel
tff(fact_1606_diff__add__eq,axiom,
    ! [Aa: real,Ba: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Aa),Ba)),C) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),C)),Ba) ).

% diff_add_eq
tff(fact_1607_diff__add__eq,axiom,
    ! [Aa: rat,Ba: rat,C: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(minus_minus_rat(Aa),Ba)),C) = aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(Aa),C)),Ba) ).

% diff_add_eq
tff(fact_1608_diff__add__eq,axiom,
    ! [Aa: int,Ba: 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,Aa),Ba)),C) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),C)),Ba) ).

% diff_add_eq
tff(fact_1609_add_Oassoc,axiom,
    ! [Aa: real,Ba: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),Ba)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Ba),C)) ).

% add.assoc
tff(fact_1610_add_Oassoc,axiom,
    ! [Aa: rat,Ba: rat,C: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(Aa),Ba)),C) = aa_rat_rat(plus_plus_rat(Aa),aa_rat_rat(plus_plus_rat(Ba),C)) ).

% add.assoc
tff(fact_1611_add_Oassoc,axiom,
    ! [Aa: nat,Ba: 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),Ba)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ba),C)) ).

% add.assoc
tff(fact_1612_add_Oassoc,axiom,
    ! [Aa: int,Ba: 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),Ba)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),aa_int_int(aa_int_fun_int_int(plus_plus_int,Ba),C)) ).

% add.assoc
tff(fact_1613_diff__diff__eq2,axiom,
    ! [Aa: real,Ba: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Aa),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Ba),C)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),C)),Ba) ).

% diff_diff_eq2
tff(fact_1614_diff__diff__eq2,axiom,
    ! [Aa: rat,Ba: rat,C: rat] : aa_rat_rat(minus_minus_rat(Aa),aa_rat_rat(minus_minus_rat(Ba),C)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(Aa),C)),Ba) ).

% diff_diff_eq2
tff(fact_1615_diff__diff__eq2,axiom,
    ! [Aa: int,Ba: int,C: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,Aa),aa_int_int(aa_int_fun_int_int(minus_minus_int,Ba),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),C)),Ba) ).

% diff_diff_eq2
tff(fact_1616_add__diff__eq,axiom,
    ! [Aa: real,Ba: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Ba),C)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),Ba)),C) ).

% add_diff_eq
tff(fact_1617_add__diff__eq,axiom,
    ! [Aa: rat,Ba: rat,C: rat] : aa_rat_rat(plus_plus_rat(Aa),aa_rat_rat(minus_minus_rat(Ba),C)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(Aa),Ba)),C) ).

% add_diff_eq
tff(fact_1618_add__diff__eq,axiom,
    ! [Aa: int,Ba: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),aa_int_int(aa_int_fun_int_int(minus_minus_int,Ba),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),Ba)),C) ).

% add_diff_eq
tff(fact_1619_eq__diff__eq,axiom,
    ! [Aa: real,C: real,Ba: real] :
      ( ( Aa = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),Ba) )
    <=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),Ba) = C ) ) ).

% eq_diff_eq
tff(fact_1620_eq__diff__eq,axiom,
    ! [Aa: rat,C: rat,Ba: rat] :
      ( ( Aa = aa_rat_rat(minus_minus_rat(C),Ba) )
    <=> ( aa_rat_rat(plus_plus_rat(Aa),Ba) = C ) ) ).

% eq_diff_eq
tff(fact_1621_eq__diff__eq,axiom,
    ! [Aa: int,C: int,Ba: int] :
      ( ( Aa = aa_int_int(aa_int_fun_int_int(minus_minus_int,C),Ba) )
    <=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),Ba) = C ) ) ).

% eq_diff_eq
tff(fact_1622_diff__eq__eq,axiom,
    ! [Aa: real,Ba: real,C: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Aa),Ba) = C )
    <=> ( Aa = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),Ba) ) ) ).

% diff_eq_eq
tff(fact_1623_diff__eq__eq,axiom,
    ! [Aa: rat,Ba: rat,C: rat] :
      ( ( aa_rat_rat(minus_minus_rat(Aa),Ba) = C )
    <=> ( Aa = aa_rat_rat(plus_plus_rat(C),Ba) ) ) ).

% diff_eq_eq
tff(fact_1624_diff__eq__eq,axiom,
    ! [Aa: int,Ba: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(minus_minus_int,Aa),Ba) = C )
    <=> ( Aa = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),Ba) ) ) ).

% diff_eq_eq
tff(fact_1625_group__cancel_Osub1,axiom,
    ! [A3: real,K: real,Aa: real,Ba: real] :
      ( ( A3 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),Aa) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A3),Ba) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Aa),Ba)) ) ) ).

% group_cancel.sub1
tff(fact_1626_group__cancel_Osub1,axiom,
    ! [A3: rat,K: rat,Aa: rat,Ba: rat] :
      ( ( A3 = aa_rat_rat(plus_plus_rat(K),Aa) )
     => ( aa_rat_rat(minus_minus_rat(A3),Ba) = aa_rat_rat(plus_plus_rat(K),aa_rat_rat(minus_minus_rat(Aa),Ba)) ) ) ).

% group_cancel.sub1
tff(fact_1627_group__cancel_Osub1,axiom,
    ! [A3: int,K: int,Aa: int,Ba: int] :
      ( ( A3 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),Aa) )
     => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A3),Ba) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_int_int(aa_int_fun_int_int(minus_minus_int,Aa),Ba)) ) ) ).

% group_cancel.sub1
tff(fact_1628_group__cancel_Oadd2,axiom,
    ! [B4: real,K: real,Ba: real,Aa: real] :
      ( ( B4 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),Ba) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),B4) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),Ba)) ) ) ).

% group_cancel.add2
tff(fact_1629_group__cancel_Oadd2,axiom,
    ! [B4: rat,K: rat,Ba: rat,Aa: rat] :
      ( ( B4 = aa_rat_rat(plus_plus_rat(K),Ba) )
     => ( aa_rat_rat(plus_plus_rat(Aa),B4) = aa_rat_rat(plus_plus_rat(K),aa_rat_rat(plus_plus_rat(Aa),Ba)) ) ) ).

% group_cancel.add2
tff(fact_1630_group__cancel_Oadd2,axiom,
    ! [B4: nat,K: nat,Ba: nat,Aa: nat] :
      ( ( B4 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Ba) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),B4) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),Ba)) ) ) ).

% group_cancel.add2
tff(fact_1631_group__cancel_Oadd2,axiom,
    ! [B4: int,K: int,Ba: int,Aa: int] :
      ( ( B4 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),Ba) )
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),B4) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),Ba)) ) ) ).

% group_cancel.add2
tff(fact_1632_group__cancel_Oadd1,axiom,
    ! [A3: real,K: real,Aa: real,Ba: real] :
      ( ( A3 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),Aa) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A3),Ba) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),Ba)) ) ) ).

% group_cancel.add1
tff(fact_1633_group__cancel_Oadd1,axiom,
    ! [A3: rat,K: rat,Aa: rat,Ba: rat] :
      ( ( A3 = aa_rat_rat(plus_plus_rat(K),Aa) )
     => ( aa_rat_rat(plus_plus_rat(A3),Ba) = aa_rat_rat(plus_plus_rat(K),aa_rat_rat(plus_plus_rat(Aa),Ba)) ) ) ).

% group_cancel.add1
tff(fact_1634_group__cancel_Oadd1,axiom,
    ! [A3: nat,K: nat,Aa: nat,Ba: nat] :
      ( ( A3 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Aa) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A3),Ba) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),Ba)) ) ) ).

% group_cancel.add1
tff(fact_1635_group__cancel_Oadd1,axiom,
    ! [A3: int,K: int,Aa: int,Ba: int] :
      ( ( A3 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),Aa) )
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A3),Ba) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),Ba)) ) ) ).

% group_cancel.add1
tff(fact_1636_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( ( I = J )
        & ( K = L ) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_1637_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( I = J )
        & ( K = L ) )
     => ( aa_rat_rat(plus_plus_rat(I),K) = aa_rat_rat(plus_plus_rat(J),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_1638_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( I = J )
        & ( K = L ) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_1639_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( I = J )
        & ( K = L ) )
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K) = aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_1640_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [Aa: real,Ba: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),Ba)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Ba),C)) ).

% ab_semigroup_add_class.add_ac(1)
tff(fact_1641_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [Aa: rat,Ba: rat,C: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(Aa),Ba)),C) = aa_rat_rat(plus_plus_rat(Aa),aa_rat_rat(plus_plus_rat(Ba),C)) ).

% ab_semigroup_add_class.add_ac(1)
tff(fact_1642_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [Aa: nat,Ba: 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),Ba)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ba),C)) ).

% ab_semigroup_add_class.add_ac(1)
tff(fact_1643_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [Aa: int,Ba: 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),Ba)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),aa_int_int(aa_int_fun_int_int(plus_plus_int,Ba),C)) ).

% ab_semigroup_add_class.add_ac(1)
tff(fact_1644_is__num__normalize_I1_J,axiom,
    ! [Aa: real,Ba: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),Ba)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Ba),C)) ).

% is_num_normalize(1)
tff(fact_1645_is__num__normalize_I1_J,axiom,
    ! [Aa: rat,Ba: rat,C: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(Aa),Ba)),C) = aa_rat_rat(plus_plus_rat(Aa),aa_rat_rat(plus_plus_rat(Ba),C)) ).

% is_num_normalize(1)
tff(fact_1646_is__num__normalize_I1_J,axiom,
    ! [Aa: int,Ba: 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),Ba)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),aa_int_int(aa_int_fun_int_int(plus_plus_int,Ba),C)) ).

% is_num_normalize(1)
tff(fact_1647_diff__le__eq,axiom,
    ! [Aa: real,Ba: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Aa),Ba)),C)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),Ba)) ) ).

% diff_le_eq
tff(fact_1648_diff__le__eq,axiom,
    ! [Aa: rat,Ba: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(Aa),Ba)),C)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),aa_rat_rat(plus_plus_rat(C),Ba)) ) ).

% diff_le_eq
tff(fact_1649_diff__le__eq,axiom,
    ! [Aa: int,Ba: 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,Aa),Ba)),C)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),Ba)) ) ).

% diff_le_eq
tff(fact_1650_le__diff__eq,axiom,
    ! [Aa: real,C: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),Ba))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),Ba)),C) ) ).

% le_diff_eq
tff(fact_1651_le__diff__eq,axiom,
    ! [Aa: rat,C: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),aa_rat_rat(minus_minus_rat(C),Ba))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(Aa),Ba)),C) ) ).

% le_diff_eq
tff(fact_1652_le__diff__eq,axiom,
    ! [Aa: int,C: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),Ba))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),Ba)),C) ) ).

% le_diff_eq
tff(fact_1653_diff__add,axiom,
    ! [Aa: nat,Ba: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),Ba)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ba),Aa)),Aa) = Ba ) ) ).

% diff_add
tff(fact_1654_le__add__diff,axiom,
    ! [Aa: nat,Ba: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),Ba)
     => 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,Ba),C)),Aa)) ) ).

% le_add_diff
tff(fact_1655_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
    ! [Aa: nat,Ba: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),Ba)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ba),Aa))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),Aa)),Ba) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
tff(fact_1656_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
    ! [Aa: nat,Ba: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),Ba)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ba),Aa)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),Ba)),Aa) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
tff(fact_1657_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
    ! [Aa: nat,Ba: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),Ba)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),Ba)),Aa) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ba),Aa)) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
tff(fact_1658_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
    ! [Aa: nat,Ba: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),Ba)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ba),Aa)),C) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ba),C)),Aa) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
tff(fact_1659_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
    ! [Aa: nat,Ba: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),Ba)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ba),C)),Aa) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ba),Aa)),C) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
tff(fact_1660_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
    ! [Aa: nat,Ba: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),Ba)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ba),Aa)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),Aa)),Ba) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_diff_right
tff(fact_1661_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
    ! [Aa: nat,Ba: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),Ba)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ba),Aa)) = Ba ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
tff(fact_1662_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
    ! [Aa: nat,Ba: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),Ba)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),Ba)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ba),Aa) = C )
        <=> ( Ba = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),Aa) ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
tff(fact_1663_add__le__add__imp__diff__le,axiom,
    ! [I: real,K: real,Na: real,J: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Na),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),K))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),Na)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Na),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),K))
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Na),K)),J) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_1664_add__le__add__imp__diff__le,axiom,
    ! [I: rat,K: rat,Na: rat,J: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(I),K)),Na)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Na),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(I),K)),Na)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Na),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(Na),K)),J) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_1665_add__le__add__imp__diff__le,axiom,
    ! [I: nat,K: nat,Na: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),Na)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K))
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K)),J) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_1666_add__le__add__imp__diff__le,axiom,
    ! [I: int,K: int,Na: int,J: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),Na)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Na),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),K))
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),Na)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Na),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),K))
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Na),K)),J) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_1667_add__le__imp__le__diff,axiom,
    ! [I: real,K: real,Na: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),Na)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,I),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Na),K)) ) ).

% add_le_imp_le_diff
tff(fact_1668_add__le__imp__le__diff,axiom,
    ! [I: rat,K: rat,Na: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(I),K)),Na)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,I),aa_rat_rat(minus_minus_rat(Na),K)) ) ).

% add_le_imp_le_diff
tff(fact_1669_add__le__imp__le__diff,axiom,
    ! [I: nat,K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K)) ) ).

% add_le_imp_le_diff
tff(fact_1670_add__le__imp__le__diff,axiom,
    ! [I: int,K: int,Na: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),Na)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),aa_int_int(aa_int_fun_int_int(minus_minus_int,Na),K)) ) ).

% add_le_imp_le_diff
tff(fact_1671_diff__add__0,axiom,
    ! [Na: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),M)) = zero_zero_nat ).

% diff_add_0
tff(fact_1672_add__diff__inverse__nat,axiom,
    ! [M: nat,Na: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na)) = M ) ) ).

% add_diff_inverse_nat
tff(fact_1673_less__diff__conv,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),J) ) ).

% less_diff_conv
tff(fact_1674_Nat_Ole__imp__diff__is__add,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I) = K )
      <=> ( J = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),I) ) ) ) ).

% Nat.le_imp_diff_is_add
tff(fact_1675_Nat_Odiff__add__assoc2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),I)),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)),I) ) ) ).

% Nat.diff_add_assoc2
tff(fact_1676_Nat_Odiff__add__assoc,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J)),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)) ) ) ).

% Nat.diff_add_assoc
tff(fact_1677_Nat_Ole__diff__conv2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),J) ) ) ).

% Nat.le_diff_conv2
tff(fact_1678_le__diff__conv,axiom,
    ! [J: nat,K: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)),I)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)) ) ).

% le_diff_conv
tff(fact_1679_frac__1__eq,axiom,
    ! [Xb: real] : aa_real_real(archim2898591450579166408c_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),one_one_real)) = aa_real_real(archim2898591450579166408c_real,Xb) ).

% frac_1_eq
tff(fact_1680_frac__1__eq,axiom,
    ! [Xb: rat] : aa_rat_rat(archimedean_frac_rat,aa_rat_rat(plus_plus_rat(Xb),one_one_rat)) = aa_rat_rat(archimedean_frac_rat,Xb) ).

% frac_1_eq
tff(fact_1681_frac__add,axiom,
    ! [Xb: real,Y: real] :
      aa_real_real(archim2898591450579166408c_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y)) = $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(archim2898591450579166408c_real,Xb)),aa_real_real(archim2898591450579166408c_real,Y))),one_one_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(archim2898591450579166408c_real,Xb)),aa_real_real(archim2898591450579166408c_real,Y)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(archim2898591450579166408c_real,Xb)),aa_real_real(archim2898591450579166408c_real,Y))),one_one_real)) ).

% frac_add
tff(fact_1682_frac__add,axiom,
    ! [Xb: rat,Y: rat] :
      aa_rat_rat(archimedean_frac_rat,aa_rat_rat(plus_plus_rat(Xb),Y)) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(archimedean_frac_rat,Xb)),aa_rat_rat(archimedean_frac_rat,Y))),one_one_rat),aa_rat_rat(plus_plus_rat(aa_rat_rat(archimedean_frac_rat,Xb)),aa_rat_rat(archimedean_frac_rat,Y)),aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(archimedean_frac_rat,Xb)),aa_rat_rat(archimedean_frac_rat,Y))),one_one_rat)) ).

% frac_add
tff(fact_1683_nat__diff__split__asm,axiom,
    ! [P: fun_nat_o,Aa: nat,Ba: nat] :
      ( aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Aa),Ba))
    <=> ~ ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),Ba)
            & ~ aa_nat_o(P,zero_zero_nat) )
          | ? [D5: nat] :
              ( ( Aa = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ba),D5) )
              & ~ aa_nat_o(P,D5) ) ) ) ).

% nat_diff_split_asm
tff(fact_1684_nat__diff__split,axiom,
    ! [P: fun_nat_o,Aa: nat,Ba: nat] :
      ( aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Aa),Ba))
    <=> ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),Ba)
         => aa_nat_o(P,zero_zero_nat) )
        & ! [D5: nat] :
            ( ( Aa = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ba),D5) )
           => aa_nat_o(P,D5) ) ) ) ).

% nat_diff_split
tff(fact_1685_less__diff__conv2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)),I)
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)) ) ) ).

% less_diff_conv2
tff(fact_1686_diff__eq__diff__less__eq,axiom,
    ! [Aa: real,Ba: real,C: real,D: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Aa),Ba) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),D) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),Ba)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),D) ) ) ).

% diff_eq_diff_less_eq
tff(fact_1687_diff__eq__diff__less__eq,axiom,
    ! [Aa: rat,Ba: rat,C: rat,D: rat] :
      ( ( aa_rat_rat(minus_minus_rat(Aa),Ba) = aa_rat_rat(minus_minus_rat(C),D) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),Ba)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),D) ) ) ).

% diff_eq_diff_less_eq
tff(fact_1688_diff__eq__diff__less__eq,axiom,
    ! [Aa: int,Ba: int,C: int,D: int] :
      ( ( aa_int_int(aa_int_fun_int_int(minus_minus_int,Aa),Ba) = 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,Aa),Ba)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),D) ) ) ).

% diff_eq_diff_less_eq
tff(fact_1689_diff__right__mono,axiom,
    ! [Aa: real,Ba: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),Ba)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Aa),C)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Ba),C)) ) ).

% diff_right_mono
tff(fact_1690_diff__right__mono,axiom,
    ! [Aa: rat,Ba: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),Ba)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(Aa),C)),aa_rat_rat(minus_minus_rat(Ba),C)) ) ).

% diff_right_mono
tff(fact_1691_diff__right__mono,axiom,
    ! [Aa: int,Ba: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),Ba)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Aa),C)),aa_int_int(aa_int_fun_int_int(minus_minus_int,Ba),C)) ) ).

% diff_right_mono
tff(fact_1692_diff__left__mono,axiom,
    ! [Ba: real,Aa: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ba),Aa)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),Aa)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),Ba)) ) ).

% diff_left_mono
tff(fact_1693_diff__left__mono,axiom,
    ! [Ba: rat,Aa: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ba),Aa)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(C),Aa)),aa_rat_rat(minus_minus_rat(C),Ba)) ) ).

% diff_left_mono
tff(fact_1694_diff__left__mono,axiom,
    ! [Ba: int,Aa: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ba),Aa)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,C),Aa)),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),Ba)) ) ).

% diff_left_mono
tff(fact_1695_diff__mono,axiom,
    ! [Aa: real,Ba: real,D: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),Ba)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,D),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Aa),C)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Ba),D)) ) ) ).

% diff_mono
tff(fact_1696_diff__mono,axiom,
    ! [Aa: rat,Ba: rat,D: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),Ba)
     => ( 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(Aa),C)),aa_rat_rat(minus_minus_rat(Ba),D)) ) ) ).

% diff_mono
tff(fact_1697_diff__mono,axiom,
    ! [Aa: int,Ba: int,D: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),Ba)
     => ( 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,Aa),C)),aa_int_int(aa_int_fun_int_int(minus_minus_int,Ba),D)) ) ) ).

% diff_mono
tff(fact_1698_eq__iff__diff__eq__0,axiom,
    ! [Aa: real,Ba: real] :
      ( ( Aa = Ba )
    <=> ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Aa),Ba) = zero_zero_real ) ) ).

% eq_iff_diff_eq_0
tff(fact_1699_eq__iff__diff__eq__0,axiom,
    ! [Aa: rat,Ba: rat] :
      ( ( Aa = Ba )
    <=> ( aa_rat_rat(minus_minus_rat(Aa),Ba) = zero_zero_rat ) ) ).

% eq_iff_diff_eq_0
tff(fact_1700_eq__iff__diff__eq__0,axiom,
    ! [Aa: int,Ba: int] :
      ( ( Aa = Ba )
    <=> ( aa_int_int(aa_int_fun_int_int(minus_minus_int,Aa),Ba) = zero_zero_int ) ) ).

% eq_iff_diff_eq_0
tff(fact_1701_diff__strict__right__mono,axiom,
    ! [Aa: real,Ba: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),Ba)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Aa),C)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Ba),C)) ) ).

% diff_strict_right_mono
tff(fact_1702_diff__strict__right__mono,axiom,
    ! [Aa: rat,Ba: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),Ba)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(Aa),C)),aa_rat_rat(minus_minus_rat(Ba),C)) ) ).

% diff_strict_right_mono
tff(fact_1703_diff__strict__right__mono,axiom,
    ! [Aa: int,Ba: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),Ba)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Aa),C)),aa_int_int(aa_int_fun_int_int(minus_minus_int,Ba),C)) ) ).

% diff_strict_right_mono
tff(fact_1704_diff__strict__left__mono,axiom,
    ! [Ba: real,Aa: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Ba),Aa)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),Aa)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),Ba)) ) ).

% diff_strict_left_mono
tff(fact_1705_diff__strict__left__mono,axiom,
    ! [Ba: rat,Aa: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ba),Aa)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(C),Aa)),aa_rat_rat(minus_minus_rat(C),Ba)) ) ).

% diff_strict_left_mono
tff(fact_1706_diff__strict__left__mono,axiom,
    ! [Ba: int,Aa: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Ba),Aa)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,C),Aa)),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),Ba)) ) ).

% diff_strict_left_mono
tff(fact_1707_diff__eq__diff__less,axiom,
    ! [Aa: real,Ba: real,C: real,D: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Aa),Ba) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),D) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),Ba)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,C),D) ) ) ).

% diff_eq_diff_less
tff(fact_1708_diff__eq__diff__less,axiom,
    ! [Aa: rat,Ba: rat,C: rat,D: rat] :
      ( ( aa_rat_rat(minus_minus_rat(Aa),Ba) = aa_rat_rat(minus_minus_rat(C),D) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),Ba)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),D) ) ) ).

% diff_eq_diff_less
tff(fact_1709_diff__eq__diff__less,axiom,
    ! [Aa: int,Ba: int,C: int,D: int] :
      ( ( aa_int_int(aa_int_fun_int_int(minus_minus_int,Aa),Ba) = aa_int_int(aa_int_fun_int_int(minus_minus_int,C),D) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),Ba)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,C),D) ) ) ).

% diff_eq_diff_less
tff(fact_1710_diff__strict__mono,axiom,
    ! [Aa: real,Ba: real,D: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),Ba)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,D),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Aa),C)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Ba),D)) ) ) ).

% diff_strict_mono
tff(fact_1711_diff__strict__mono,axiom,
    ! [Aa: rat,Ba: rat,D: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),Ba)
     => ( 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(Aa),C)),aa_rat_rat(minus_minus_rat(Ba),D)) ) ) ).

% diff_strict_mono
tff(fact_1712_diff__strict__mono,axiom,
    ! [Aa: int,Ba: int,D: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),Ba)
     => ( 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,Aa),C)),aa_int_int(aa_int_fun_int_int(minus_minus_int,Ba),D)) ) ) ).

% diff_strict_mono
tff(fact_1713_add__le__imp__le__right,axiom,
    ! [Aa: real,C: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Ba),C))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),Ba) ) ).

% add_le_imp_le_right
tff(fact_1714_add__le__imp__le__right,axiom,
    ! [Aa: rat,C: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(Aa),C)),aa_rat_rat(plus_plus_rat(Ba),C))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),Ba) ) ).

% add_le_imp_le_right
tff(fact_1715_add__le__imp__le__right,axiom,
    ! [Aa: nat,C: nat,Ba: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ba),C))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),Ba) ) ).

% add_le_imp_le_right
tff(fact_1716_add__le__imp__le__right,axiom,
    ! [Aa: int,C: int,Ba: 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),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,Ba),C))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),Ba) ) ).

% add_le_imp_le_right
tff(fact_1717_add__le__imp__le__left,axiom,
    ! [C: real,Aa: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),Aa)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),Ba))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),Ba) ) ).

% add_le_imp_le_left
tff(fact_1718_add__le__imp__le__left,axiom,
    ! [C: rat,Aa: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(C),Aa)),aa_rat_rat(plus_plus_rat(C),Ba))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),Ba) ) ).

% add_le_imp_le_left
tff(fact_1719_add__le__imp__le__left,axiom,
    ! [C: nat,Aa: nat,Ba: 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),Aa)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),Ba))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),Ba) ) ).

% add_le_imp_le_left
tff(fact_1720_add__le__imp__le__left,axiom,
    ! [C: int,Aa: int,Ba: 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),Aa)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),Ba))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),Ba) ) ).

% add_le_imp_le_left
tff(fact_1721_le__iff__add,axiom,
    ! [Aa: nat,Ba: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),Ba)
    <=> ? [C4: nat] : Ba = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),C4) ) ).

% le_iff_add
tff(fact_1722_add__right__mono,axiom,
    ! [Aa: real,Ba: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),Ba)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Ba),C)) ) ).

% add_right_mono
tff(fact_1723_add__right__mono,axiom,
    ! [Aa: rat,Ba: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),Ba)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(Aa),C)),aa_rat_rat(plus_plus_rat(Ba),C)) ) ).

% add_right_mono
tff(fact_1724_add__right__mono,axiom,
    ! [Aa: nat,Ba: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),Ba)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ba),C)) ) ).

% add_right_mono
tff(fact_1725_add__right__mono,axiom,
    ! [Aa: int,Ba: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),Ba)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,Ba),C)) ) ).

% add_right_mono
tff(fact_1726_less__eqE,axiom,
    ! [Aa: nat,Ba: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),Ba)
     => ~ ! [C3: nat] : Ba != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),C3) ) ).

% less_eqE
tff(fact_1727_add__left__mono,axiom,
    ! [Aa: real,Ba: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),Ba)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),Aa)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),Ba)) ) ).

% add_left_mono
tff(fact_1728_add__left__mono,axiom,
    ! [Aa: rat,Ba: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),Ba)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(C),Aa)),aa_rat_rat(plus_plus_rat(C),Ba)) ) ).

% add_left_mono
tff(fact_1729_add__left__mono,axiom,
    ! [Aa: nat,Ba: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),Ba)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),Aa)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),Ba)) ) ).

% add_left_mono
tff(fact_1730_add__left__mono,axiom,
    ! [Aa: int,Ba: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),Ba)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),Aa)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),Ba)) ) ).

% add_left_mono
tff(fact_1731_add__mono,axiom,
    ! [Aa: real,Ba: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),Ba)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),D)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Ba),D)) ) ) ).

% add_mono
tff(fact_1732_add__mono,axiom,
    ! [Aa: rat,Ba: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),Ba)
     => ( 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(Aa),C)),aa_rat_rat(plus_plus_rat(Ba),D)) ) ) ).

% add_mono
tff(fact_1733_add__mono,axiom,
    ! [Aa: nat,Ba: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),Ba)
     => ( 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,Aa),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ba),D)) ) ) ).

% add_mono
tff(fact_1734_add__mono,axiom,
    ! [Aa: int,Ba: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),Ba)
     => ( 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,Aa),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,Ba),D)) ) ) ).

% add_mono
tff(fact_1735_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,I),J)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,K),L) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L)) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_1736_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,I),J)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,K),L) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(I),K)),aa_rat_rat(plus_plus_rat(J),L)) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_1737_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),L) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_1738_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),L) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_1739_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( ( I = J )
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,K),L) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L)) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_1740_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( I = J )
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,K),L) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(I),K)),aa_rat_rat(plus_plus_rat(J),L)) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_1741_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( I = J )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),L) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_1742_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( I = J )
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),L) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_1743_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,I),J)
        & ( K = L ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L)) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_1744_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,I),J)
        & ( K = L ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(I),K)),aa_rat_rat(plus_plus_rat(J),L)) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_1745_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
        & ( K = L ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_1746_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J)
        & ( K = L ) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_1747_verit__sum__simplify,axiom,
    ! [Aa: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),zero_zero_real) = Aa ).

% verit_sum_simplify
tff(fact_1748_verit__sum__simplify,axiom,
    ! [Aa: rat] : aa_rat_rat(plus_plus_rat(Aa),zero_zero_rat) = Aa ).

% verit_sum_simplify
tff(fact_1749_verit__sum__simplify,axiom,
    ! [Aa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),zero_zero_nat) = Aa ).

% verit_sum_simplify
tff(fact_1750_verit__sum__simplify,axiom,
    ! [Aa: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),zero_zero_int) = Aa ).

% verit_sum_simplify
tff(fact_1751_add_Ogroup__left__neutral,axiom,
    ! [Aa: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,zero_zero_real),Aa) = Aa ).

% add.group_left_neutral
tff(fact_1752_add_Ogroup__left__neutral,axiom,
    ! [Aa: rat] : aa_rat_rat(plus_plus_rat(zero_zero_rat),Aa) = Aa ).

% add.group_left_neutral
tff(fact_1753_add_Ogroup__left__neutral,axiom,
    ! [Aa: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,zero_zero_int),Aa) = Aa ).

% add.group_left_neutral
tff(fact_1754_add_Ocomm__neutral,axiom,
    ! [Aa: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),zero_zero_real) = Aa ).

% add.comm_neutral
tff(fact_1755_add_Ocomm__neutral,axiom,
    ! [Aa: rat] : aa_rat_rat(plus_plus_rat(Aa),zero_zero_rat) = Aa ).

% add.comm_neutral
tff(fact_1756_add_Ocomm__neutral,axiom,
    ! [Aa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),zero_zero_nat) = Aa ).

% add.comm_neutral
tff(fact_1757_add_Ocomm__neutral,axiom,
    ! [Aa: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),zero_zero_int) = Aa ).

% add.comm_neutral
tff(fact_1758_comm__monoid__add__class_Oadd__0,axiom,
    ! [Aa: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,zero_zero_real),Aa) = Aa ).

% comm_monoid_add_class.add_0
tff(fact_1759_comm__monoid__add__class_Oadd__0,axiom,
    ! [Aa: rat] : aa_rat_rat(plus_plus_rat(zero_zero_rat),Aa) = Aa ).

% comm_monoid_add_class.add_0
tff(fact_1760_comm__monoid__add__class_Oadd__0,axiom,
    ! [Aa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,zero_zero_nat),Aa) = Aa ).

% comm_monoid_add_class.add_0
tff(fact_1761_comm__monoid__add__class_Oadd__0,axiom,
    ! [Aa: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,zero_zero_int),Aa) = Aa ).

% comm_monoid_add_class.add_0
tff(fact_1762_add__less__imp__less__right,axiom,
    ! [Aa: real,C: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Ba),C))
     => aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),Ba) ) ).

% add_less_imp_less_right
tff(fact_1763_add__less__imp__less__right,axiom,
    ! [Aa: rat,C: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(Aa),C)),aa_rat_rat(plus_plus_rat(Ba),C))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),Ba) ) ).

% add_less_imp_less_right
tff(fact_1764_add__less__imp__less__right,axiom,
    ! [Aa: nat,C: nat,Ba: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ba),C))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),Ba) ) ).

% add_less_imp_less_right
tff(fact_1765_add__less__imp__less__right,axiom,
    ! [Aa: int,C: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,Ba),C))
     => aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),Ba) ) ).

% add_less_imp_less_right
tff(fact_1766_add__less__imp__less__left,axiom,
    ! [C: real,Aa: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),Aa)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),Ba))
     => aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),Ba) ) ).

% add_less_imp_less_left
tff(fact_1767_add__less__imp__less__left,axiom,
    ! [C: rat,Aa: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(C),Aa)),aa_rat_rat(plus_plus_rat(C),Ba))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),Ba) ) ).

% add_less_imp_less_left
tff(fact_1768_add__less__imp__less__left,axiom,
    ! [C: nat,Aa: nat,Ba: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),Aa)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),Ba))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),Ba) ) ).

% add_less_imp_less_left
tff(fact_1769_add__less__imp__less__left,axiom,
    ! [C: int,Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),Aa)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),Ba))
     => aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),Ba) ) ).

% add_less_imp_less_left
tff(fact_1770_add__strict__right__mono,axiom,
    ! [Aa: real,Ba: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),Ba)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Ba),C)) ) ).

% add_strict_right_mono
tff(fact_1771_add__strict__right__mono,axiom,
    ! [Aa: rat,Ba: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),Ba)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(Aa),C)),aa_rat_rat(plus_plus_rat(Ba),C)) ) ).

% add_strict_right_mono
tff(fact_1772_add__strict__right__mono,axiom,
    ! [Aa: nat,Ba: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),Ba)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ba),C)) ) ).

% add_strict_right_mono
tff(fact_1773_add__strict__right__mono,axiom,
    ! [Aa: int,Ba: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),Ba)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,Ba),C)) ) ).

% add_strict_right_mono
tff(fact_1774_add__strict__left__mono,axiom,
    ! [Aa: real,Ba: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),Ba)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),Aa)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),Ba)) ) ).

% add_strict_left_mono
tff(fact_1775_add__strict__left__mono,axiom,
    ! [Aa: rat,Ba: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),Ba)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(C),Aa)),aa_rat_rat(plus_plus_rat(C),Ba)) ) ).

% add_strict_left_mono
tff(fact_1776_add__strict__left__mono,axiom,
    ! [Aa: nat,Ba: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),Ba)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),Aa)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),Ba)) ) ).

% add_strict_left_mono
tff(fact_1777_add__strict__left__mono,axiom,
    ! [Aa: int,Ba: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),Ba)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),Aa)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),Ba)) ) ).

% add_strict_left_mono
tff(fact_1778_add__strict__mono,axiom,
    ! [Aa: real,Ba: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),Ba)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),D)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Ba),D)) ) ) ).

% add_strict_mono
tff(fact_1779_add__strict__mono,axiom,
    ! [Aa: rat,Ba: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),Ba)
     => ( 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(Aa),C)),aa_rat_rat(plus_plus_rat(Ba),D)) ) ) ).

% add_strict_mono
tff(fact_1780_add__strict__mono,axiom,
    ! [Aa: nat,Ba: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),Ba)
     => ( 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,Aa),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ba),D)) ) ) ).

% add_strict_mono
tff(fact_1781_add__strict__mono,axiom,
    ! [Aa: int,Ba: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),Ba)
     => ( 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,Aa),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,Ba),D)) ) ) ).

% add_strict_mono
tff(fact_1782_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,I),J)
        & ( K = L ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L)) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_1783_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,I),J)
        & ( K = L ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(I),K)),aa_rat_rat(plus_plus_rat(J),L)) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_1784_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
        & ( K = L ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_1785_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,I),J)
        & ( K = L ) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_1786_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( ( I = J )
        & aa_real_o(aa_real_fun_real_o(ord_less_real,K),L) )
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L)) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_1787_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( I = J )
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,K),L) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(I),K)),aa_rat_rat(plus_plus_rat(J),L)) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_1788_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( I = J )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),L) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_1789_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( I = J )
        & aa_int_o(aa_int_fun_int_o(ord_less_int,K),L) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_1790_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,I),J)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,K),L) )
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L)) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_1791_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,I),J)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,K),L) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(I),K)),aa_rat_rat(plus_plus_rat(J),L)) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_1792_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),L) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_1793_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,I),J)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,K),L) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_1794_zero__induct__lemma,axiom,
    ! [P: fun_nat_o,K: nat,I: nat] :
      ( aa_nat_o(P,K)
     => ( ! [N: nat] :
            ( aa_nat_o(P,aa_nat_nat(suc,N))
           => aa_nat_o(P,N) )
       => aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),I)) ) ) ).

% zero_induct_lemma
tff(fact_1795_minus__nat_Odiff__0,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),zero_zero_nat) = M ).

% minus_nat.diff_0
tff(fact_1796_diffs0__imp__equal,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na) = zero_zero_nat )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),M) = zero_zero_nat )
       => ( M = Na ) ) ) ).

% diffs0_imp_equal
tff(fact_1797_diff__less__mono2,axiom,
    ! [M: nat,Na: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),L)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),Na)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),M)) ) ) ).

% diff_less_mono2
tff(fact_1798_less__imp__diff__less,axiom,
    ! [J: nat,K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),K)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),Na)),K) ) ).

% less_imp_diff_less
tff(fact_1799_nat__arith_Osuc1,axiom,
    ! [A3: nat,K: nat,Aa: nat] :
      ( ( A3 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Aa) )
     => ( aa_nat_nat(suc,A3) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_nat_nat(suc,Aa)) ) ) ).

% nat_arith.suc1
tff(fact_1800_add__Suc,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(suc,M)),Na) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na)) ).

% add_Suc
tff(fact_1801_add__Suc__shift,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(suc,M)),Na) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(suc,Na)) ).

% add_Suc_shift
tff(fact_1802_eq__diff__iff,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K) )
        <=> ( M = Na ) ) ) ) ).

% eq_diff_iff
tff(fact_1803_le__diff__iff,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ) ) ).

% le_diff_iff
tff(fact_1804_Nat_Odiff__diff__eq,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na) ) ) ) ).

% Nat.diff_diff_eq
tff(fact_1805_diff__le__mono,axiom,
    ! [M: nat,Na: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),L)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),L)) ) ).

% diff_le_mono
tff(fact_1806_diff__le__self,axiom,
    ! [M: nat,Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na)),M) ).

% diff_le_self
tff(fact_1807_le__diff__iff_H,axiom,
    ! [Aa: nat,C: nat,Ba: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),C)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ba),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),Aa)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,C),Ba))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ba),Aa) ) ) ) ).

% le_diff_iff'
tff(fact_1808_diff__le__mono2,axiom,
    ! [M: nat,Na: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),Na)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),M)) ) ).

% diff_le_mono2
tff(fact_1809_plus__nat_Oadd__0,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,zero_zero_nat),Na) = Na ).

% plus_nat.add_0
tff(fact_1810_add__eq__self__zero,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na) = M )
     => ( Na = zero_zero_nat ) ) ).

% add_eq_self_zero
tff(fact_1811_add__lessD1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J)),K)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),K) ) ).

% add_lessD1
tff(fact_1812_add__less__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),L)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ) ).

% add_less_mono
tff(fact_1813_not__add__less1,axiom,
    ! [I: nat,J: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J)),I) ).

% not_add_less1
tff(fact_1814_not__add__less2,axiom,
    ! [J: nat,I: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),I)),I) ).

% not_add_less2
tff(fact_1815_add__less__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K)) ) ).

% add_less_mono1
tff(fact_1816_trans__less__add1,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),M)) ) ).

% trans_less_add1
tff(fact_1817_trans__less__add2,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),J)) ) ).

% trans_less_add2
tff(fact_1818_less__add__eq__less,axiom,
    ! [K: nat,L: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),L)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),L) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Na) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ) ).

% less_add_eq_less
tff(fact_1819_add__leE,axiom,
    ! [M: nat,K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),Na)
     => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
         => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na) ) ) ).

% add_leE
tff(fact_1820_le__add1,axiom,
    ! [Na: nat,M: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),M)) ).

% le_add1
tff(fact_1821_le__add2,axiom,
    ! [Na: nat,M: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na)) ).

% le_add2
tff(fact_1822_add__leD1,axiom,
    ! [M: nat,K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ).

% add_leD1
tff(fact_1823_add__leD2,axiom,
    ! [M: nat,K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na) ) ).

% add_leD2
tff(fact_1824_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_1825_add__le__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),L)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ) ).

% add_le_mono
tff(fact_1826_add__le__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K)) ) ).

% add_le_mono1
tff(fact_1827_trans__le__add1,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),M)) ) ).

% trans_le_add1
tff(fact_1828_trans__le__add2,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),J)) ) ).

% trans_le_add2
tff(fact_1829_nat__le__iff__add,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
    <=> ? [K3: nat] : Na = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K3) ) ).

% nat_le_iff_add
tff(fact_1830_add__eq__if,axiom,
    ! [M: nat,Na: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na) = $ite(M = zero_zero_nat,Na,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)),Na))) ).

% add_eq_if
tff(fact_1831_subset__decode__imp__le,axiom,
    ! [M: nat,Na: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,nat_set_decode(M)),nat_set_decode(Na))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ).

% subset_decode_imp_le
tff(fact_1832_le__iff__diff__le__0,axiom,
    ! [Aa: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),Ba)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Aa),Ba)),zero_zero_real) ) ).

% le_iff_diff_le_0
tff(fact_1833_le__iff__diff__le__0,axiom,
    ! [Aa: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),Ba)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(Aa),Ba)),zero_zero_rat) ) ).

% le_iff_diff_le_0
tff(fact_1834_le__iff__diff__le__0,axiom,
    ! [Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),Ba)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Aa),Ba)),zero_zero_int) ) ).

% le_iff_diff_le_0
tff(fact_1835_less__iff__diff__less__0,axiom,
    ! [Aa: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),Ba)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Aa),Ba)),zero_zero_real) ) ).

% less_iff_diff_less_0
tff(fact_1836_less__iff__diff__less__0,axiom,
    ! [Aa: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),Ba)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(Aa),Ba)),zero_zero_rat) ) ).

% less_iff_diff_less_0
tff(fact_1837_less__iff__diff__less__0,axiom,
    ! [Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),Ba)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Aa),Ba)),zero_zero_int) ) ).

% less_iff_diff_less_0
tff(fact_1838_finite__set__decode,axiom,
    ! [Na: nat] : aa_set_nat_o(finite_finite_nat,nat_set_decode(Na)) ).

% finite_set_decode
tff(fact_1839_add__nonpos__eq__0__iff,axiom,
    ! [Xb: real,Y: 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,Y),zero_zero_real)
       => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y) = zero_zero_real )
        <=> ( ( Xb = zero_zero_real )
            & ( Y = zero_zero_real ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_1840_add__nonpos__eq__0__iff,axiom,
    ! [Xb: rat,Y: 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,Y),zero_zero_rat)
       => ( ( aa_rat_rat(plus_plus_rat(Xb),Y) = zero_zero_rat )
        <=> ( ( Xb = zero_zero_rat )
            & ( Y = zero_zero_rat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_1841_add__nonpos__eq__0__iff,axiom,
    ! [Xb: nat,Y: 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,Y),zero_zero_nat)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xb),Y) = zero_zero_nat )
        <=> ( ( Xb = zero_zero_nat )
            & ( Y = zero_zero_nat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_1842_add__nonpos__eq__0__iff,axiom,
    ! [Xb: int,Y: 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,Y),zero_zero_int)
       => ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Y) = zero_zero_int )
        <=> ( ( Xb = zero_zero_int )
            & ( Y = zero_zero_int ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_1843_add__nonneg__eq__0__iff,axiom,
    ! [Xb: real,Y: 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),Y)
       => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y) = zero_zero_real )
        <=> ( ( Xb = zero_zero_real )
            & ( Y = zero_zero_real ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_1844_add__nonneg__eq__0__iff,axiom,
    ! [Xb: rat,Y: 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),Y)
       => ( ( aa_rat_rat(plus_plus_rat(Xb),Y) = zero_zero_rat )
        <=> ( ( Xb = zero_zero_rat )
            & ( Y = zero_zero_rat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_1845_add__nonneg__eq__0__iff,axiom,
    ! [Xb: nat,Y: 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),Y)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xb),Y) = zero_zero_nat )
        <=> ( ( Xb = zero_zero_nat )
            & ( Y = zero_zero_nat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_1846_add__nonneg__eq__0__iff,axiom,
    ! [Xb: int,Y: 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),Y)
       => ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Y) = zero_zero_int )
        <=> ( ( Xb = zero_zero_int )
            & ( Y = zero_zero_int ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_1847_add__nonpos__nonpos,axiom,
    ! [Aa: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ba),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),Ba)),zero_zero_real) ) ) ).

% add_nonpos_nonpos
tff(fact_1848_add__nonpos__nonpos,axiom,
    ! [Aa: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ba),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(Aa),Ba)),zero_zero_rat) ) ) ).

% add_nonpos_nonpos
tff(fact_1849_add__nonpos__nonpos,axiom,
    ! [Aa: nat,Ba: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ba),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,Aa),Ba)),zero_zero_nat) ) ) ).

% add_nonpos_nonpos
tff(fact_1850_add__nonpos__nonpos,axiom,
    ! [Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ba),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,Aa),Ba)),zero_zero_int) ) ) ).

% add_nonpos_nonpos
tff(fact_1851_add__nonneg__nonneg,axiom,
    ! [Aa: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Aa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Ba)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),Ba)) ) ) ).

% add_nonneg_nonneg
tff(fact_1852_add__nonneg__nonneg,axiom,
    ! [Aa: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Aa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Ba)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(Aa),Ba)) ) ) ).

% add_nonneg_nonneg
tff(fact_1853_add__nonneg__nonneg,axiom,
    ! [Aa: nat,Ba: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Aa)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Ba)
       => 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,Aa),Ba)) ) ) ).

% add_nonneg_nonneg
tff(fact_1854_add__nonneg__nonneg,axiom,
    ! [Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Aa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Ba)
       => 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),Ba)) ) ) ).

% add_nonneg_nonneg
tff(fact_1855_add__increasing2,axiom,
    ! [C: real,Ba: real,Aa: 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,Ba),Aa)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ba),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),C)) ) ) ).

% add_increasing2
tff(fact_1856_add__increasing2,axiom,
    ! [C: rat,Ba: rat,Aa: 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,Ba),Aa)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ba),aa_rat_rat(plus_plus_rat(Aa),C)) ) ) ).

% add_increasing2
tff(fact_1857_add__increasing2,axiom,
    ! [C: nat,Ba: nat,Aa: 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,Ba),Aa)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ba),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),C)) ) ) ).

% add_increasing2
tff(fact_1858_add__increasing2,axiom,
    ! [C: int,Ba: int,Aa: 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,Ba),Aa)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ba),aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),C)) ) ) ).

% add_increasing2
tff(fact_1859_add__decreasing2,axiom,
    ! [C: real,Aa: real,Ba: 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,Aa),Ba)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),C)),Ba) ) ) ).

% add_decreasing2
tff(fact_1860_add__decreasing2,axiom,
    ! [C: rat,Aa: rat,Ba: 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,Aa),Ba)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(Aa),C)),Ba) ) ) ).

% add_decreasing2
tff(fact_1861_add__decreasing2,axiom,
    ! [C: nat,Aa: nat,Ba: 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,Aa),Ba)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),C)),Ba) ) ) ).

% add_decreasing2
tff(fact_1862_add__decreasing2,axiom,
    ! [C: int,Aa: int,Ba: 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,Aa),Ba)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),C)),Ba) ) ) ).

% add_decreasing2
tff(fact_1863_add__increasing,axiom,
    ! [Aa: real,Ba: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Aa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ba),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ba),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),C)) ) ) ).

% add_increasing
tff(fact_1864_add__increasing,axiom,
    ! [Aa: rat,Ba: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Aa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ba),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ba),aa_rat_rat(plus_plus_rat(Aa),C)) ) ) ).

% add_increasing
tff(fact_1865_add__increasing,axiom,
    ! [Aa: nat,Ba: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Aa)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ba),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ba),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),C)) ) ) ).

% add_increasing
tff(fact_1866_add__increasing,axiom,
    ! [Aa: int,Ba: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Aa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ba),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ba),aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),C)) ) ) ).

% add_increasing
tff(fact_1867_add__decreasing,axiom,
    ! [Aa: real,C: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),Ba)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),C)),Ba) ) ) ).

% add_decreasing
tff(fact_1868_add__decreasing,axiom,
    ! [Aa: rat,C: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),Ba)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(Aa),C)),Ba) ) ) ).

% add_decreasing
tff(fact_1869_add__decreasing,axiom,
    ! [Aa: nat,C: nat,Ba: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),Ba)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),C)),Ba) ) ) ).

% add_decreasing
tff(fact_1870_add__decreasing,axiom,
    ! [Aa: int,C: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),Ba)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),C)),Ba) ) ) ).

% add_decreasing
tff(fact_1871_add__less__le__mono,axiom,
    ! [Aa: real,Ba: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),Ba)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),D)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Ba),D)) ) ) ).

% add_less_le_mono
tff(fact_1872_add__less__le__mono,axiom,
    ! [Aa: rat,Ba: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),Ba)
     => ( 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(Aa),C)),aa_rat_rat(plus_plus_rat(Ba),D)) ) ) ).

% add_less_le_mono
tff(fact_1873_add__less__le__mono,axiom,
    ! [Aa: nat,Ba: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),Ba)
     => ( 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,Aa),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ba),D)) ) ) ).

% add_less_le_mono
tff(fact_1874_add__less__le__mono,axiom,
    ! [Aa: int,Ba: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),Ba)
     => ( 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,Aa),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,Ba),D)) ) ) ).

% add_less_le_mono
tff(fact_1875_add__le__less__mono,axiom,
    ! [Aa: real,Ba: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),Ba)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),D)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Ba),D)) ) ) ).

% add_le_less_mono
tff(fact_1876_add__le__less__mono,axiom,
    ! [Aa: rat,Ba: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),Ba)
     => ( 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(Aa),C)),aa_rat_rat(plus_plus_rat(Ba),D)) ) ) ).

% add_le_less_mono
tff(fact_1877_add__le__less__mono,axiom,
    ! [Aa: nat,Ba: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),Ba)
     => ( 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,Aa),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ba),D)) ) ) ).

% add_le_less_mono
tff(fact_1878_add__le__less__mono,axiom,
    ! [Aa: int,Ba: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),Ba)
     => ( 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,Aa),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,Ba),D)) ) ) ).

% add_le_less_mono
tff(fact_1879_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,I),J)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,K),L) )
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L)) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_1880_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,I),J)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,K),L) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(I),K)),aa_rat_rat(plus_plus_rat(J),L)) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_1881_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),L) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_1882_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,I),J)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),L) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_1883_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,I),J)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,K),L) )
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L)) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_1884_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,I),J)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,K),L) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(I),K)),aa_rat_rat(plus_plus_rat(J),L)) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_1885_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),L) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_1886_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,K),L) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_1887_add__less__zeroD,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y)),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,Y),zero_zero_real) ) ) ).

% add_less_zeroD
tff(fact_1888_add__less__zeroD,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(Xb),Y)),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,Y),zero_zero_rat) ) ) ).

% add_less_zeroD
tff(fact_1889_add__less__zeroD,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Y)),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,Y),zero_zero_int) ) ) ).

% add_less_zeroD
tff(fact_1890_pos__add__strict,axiom,
    ! [Aa: real,Ba: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Aa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Ba),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Ba),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),C)) ) ) ).

% pos_add_strict
tff(fact_1891_pos__add__strict,axiom,
    ! [Aa: rat,Ba: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Aa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ba),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ba),aa_rat_rat(plus_plus_rat(Aa),C)) ) ) ).

% pos_add_strict
tff(fact_1892_pos__add__strict,axiom,
    ! [Aa: nat,Ba: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Aa)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ba),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ba),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),C)) ) ) ).

% pos_add_strict
tff(fact_1893_pos__add__strict,axiom,
    ! [Aa: int,Ba: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Aa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Ba),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,Ba),aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),C)) ) ) ).

% pos_add_strict
tff(fact_1894_canonically__ordered__monoid__add__class_OlessE,axiom,
    ! [Aa: nat,Ba: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),Ba)
     => ~ ! [C3: nat] :
            ( ( Ba = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),C3) )
           => ( C3 = zero_zero_nat ) ) ) ).

% canonically_ordered_monoid_add_class.lessE
tff(fact_1895_add__pos__pos,axiom,
    ! [Aa: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Aa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Ba)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),Ba)) ) ) ).

% add_pos_pos
tff(fact_1896_add__pos__pos,axiom,
    ! [Aa: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Aa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Ba)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(Aa),Ba)) ) ) ).

% add_pos_pos
tff(fact_1897_add__pos__pos,axiom,
    ! [Aa: nat,Ba: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Aa)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Ba)
       => 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,Aa),Ba)) ) ) ).

% add_pos_pos
tff(fact_1898_add__pos__pos,axiom,
    ! [Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Aa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Ba)
       => 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),Ba)) ) ) ).

% add_pos_pos
tff(fact_1899_add__neg__neg,axiom,
    ! [Aa: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Ba),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),Ba)),zero_zero_real) ) ) ).

% add_neg_neg
tff(fact_1900_add__neg__neg,axiom,
    ! [Aa: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ba),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(Aa),Ba)),zero_zero_rat) ) ) ).

% add_neg_neg
tff(fact_1901_add__neg__neg,axiom,
    ! [Aa: nat,Ba: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ba),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,Aa),Ba)),zero_zero_nat) ) ) ).

% add_neg_neg
tff(fact_1902_add__neg__neg,axiom,
    ! [Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Ba),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,Aa),Ba)),zero_zero_int) ) ) ).

% add_neg_neg
tff(fact_1903_add__mono1,axiom,
    ! [Aa: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),Ba)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),one_one_real)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Ba),one_one_real)) ) ).

% add_mono1
tff(fact_1904_add__mono1,axiom,
    ! [Aa: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),Ba)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(Aa),one_one_rat)),aa_rat_rat(plus_plus_rat(Ba),one_one_rat)) ) ).

% add_mono1
tff(fact_1905_add__mono1,axiom,
    ! [Aa: nat,Ba: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),Ba)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ba),one_one_nat)) ) ).

% add_mono1
tff(fact_1906_add__mono1,axiom,
    ! [Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),Ba)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),one_one_int)),aa_int_int(aa_int_fun_int_int(plus_plus_int,Ba),one_one_int)) ) ).

% add_mono1
tff(fact_1907_less__add__one,axiom,
    ! [Aa: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),one_one_real)) ).

% less_add_one
tff(fact_1908_less__add__one,axiom,
    ! [Aa: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),aa_rat_rat(plus_plus_rat(Aa),one_one_rat)) ).

% less_add_one
tff(fact_1909_less__add__one,axiom,
    ! [Aa: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),one_one_nat)) ).

% less_add_one
tff(fact_1910_less__add__one,axiom,
    ! [Aa: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),one_one_int)) ).

% less_add_one
tff(fact_1911_Suc__diff__Suc,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M)
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(suc,Na))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na) ) ) ).

% Suc_diff_Suc
tff(fact_1912_diff__less__Suc,axiom,
    ! [M: nat,Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na)),aa_nat_nat(suc,M)) ).

% diff_less_Suc
tff(fact_1913_diff__less,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na)),M) ) ) ).

% diff_less
tff(fact_1914_Suc__diff__le,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,M)),Na) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na)) ) ) ).

% Suc_diff_le
tff(fact_1915_less__diff__iff,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ) ) ).

% less_diff_iff
tff(fact_1916_diff__less__mono,axiom,
    ! [Aa: nat,Ba: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),Ba)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),Aa)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Aa),C)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ba),C)) ) ) ).

% diff_less_mono
tff(fact_1917_one__is__add,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa_nat_nat(suc,zero_zero_nat) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na) )
    <=> ( ( ( M = aa_nat_nat(suc,zero_zero_nat) )
          & ( Na = zero_zero_nat ) )
        | ( ( M = zero_zero_nat )
          & ( Na = aa_nat_nat(suc,zero_zero_nat) ) ) ) ) ).

% one_is_add
tff(fact_1918_add__is__1,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ( ( ( M = aa_nat_nat(suc,zero_zero_nat) )
          & ( Na = zero_zero_nat ) )
        | ( ( M = zero_zero_nat )
          & ( Na = aa_nat_nat(suc,zero_zero_nat) ) ) ) ) ).

% add_is_1
tff(fact_1919_less__natE,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
     => ~ ! [Q3: nat] : Na != aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Q3)) ) ).

% less_natE
tff(fact_1920_less__add__Suc1,axiom,
    ! [I: nat,M: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),M))) ).

% less_add_Suc1
tff(fact_1921_less__add__Suc2,axiom,
    ! [I: nat,M: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),I))) ).

% less_add_Suc2
tff(fact_1922_less__iff__Suc__add,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
    <=> ? [K3: nat] : Na = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K3)) ) ).

% less_iff_Suc_add
tff(fact_1923_less__imp__Suc__add,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
     => ? [K2: nat] : Na = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K2)) ) ).

% less_imp_Suc_add
tff(fact_1924_less__imp__add__positive,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),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,I),K2) = J ) ) ) ).

% less_imp_add_positive
tff(fact_1925_diff__Suc__eq__diff__pred,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(suc,Na)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)),Na) ).

% diff_Suc_eq_diff_pred
tff(fact_1926_mono__nat__linear__lb,axiom,
    ! [F: fun_nat_nat,M: nat,K: nat] :
      ( ! [M3: nat,N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M3),N)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,M3)),aa_nat_nat(F,N)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(F,M)),K)),aa_nat_nat(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K))) ) ).

% mono_nat_linear_lb
tff(fact_1927_Suc__eq__plus1__left,axiom,
    ! [Na: nat] : aa_nat_nat(suc,Na) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),Na) ).

% Suc_eq_plus1_left
tff(fact_1928_plus__1__eq__Suc,axiom,
    aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat) = suc ).

% plus_1_eq_Suc
tff(fact_1929_Suc__eq__plus1,axiom,
    ! [Na: nat] : aa_nat_nat(suc,Na) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),one_one_nat) ).

% Suc_eq_plus1
tff(fact_1930_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_1931_dbl__inc__def,axiom,
    ! [Xb: real] : aa_real_real(neg_nu8295874005876285629c_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Xb)),one_one_real) ).

% dbl_inc_def
tff(fact_1932_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_1933_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_1934_frac__ge__0,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(archim2898591450579166408c_real,Xb)) ).

% frac_ge_0
tff(fact_1935_frac__ge__0,axiom,
    ! [Xb: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(archimedean_frac_rat,Xb)) ).

% frac_ge_0
tff(fact_1936_frac__lt__1,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(archim2898591450579166408c_real,Xb)),one_one_real) ).

% frac_lt_1
tff(fact_1937_frac__lt__1,axiom,
    ! [Xb: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(archimedean_frac_rat,Xb)),one_one_rat) ).

% frac_lt_1
tff(fact_1938_add__neg__nonpos,axiom,
    ! [Aa: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ba),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),Ba)),zero_zero_real) ) ) ).

% add_neg_nonpos
tff(fact_1939_add__neg__nonpos,axiom,
    ! [Aa: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ba),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(Aa),Ba)),zero_zero_rat) ) ) ).

% add_neg_nonpos
tff(fact_1940_add__neg__nonpos,axiom,
    ! [Aa: nat,Ba: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ba),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,Aa),Ba)),zero_zero_nat) ) ) ).

% add_neg_nonpos
tff(fact_1941_add__neg__nonpos,axiom,
    ! [Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ba),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,Aa),Ba)),zero_zero_int) ) ) ).

% add_neg_nonpos
tff(fact_1942_add__nonneg__pos,axiom,
    ! [Aa: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Aa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Ba)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),Ba)) ) ) ).

% add_nonneg_pos
tff(fact_1943_add__nonneg__pos,axiom,
    ! [Aa: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Aa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Ba)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(Aa),Ba)) ) ) ).

% add_nonneg_pos
tff(fact_1944_add__nonneg__pos,axiom,
    ! [Aa: nat,Ba: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Aa)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Ba)
       => 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,Aa),Ba)) ) ) ).

% add_nonneg_pos
tff(fact_1945_add__nonneg__pos,axiom,
    ! [Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Aa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Ba)
       => 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),Ba)) ) ) ).

% add_nonneg_pos
tff(fact_1946_add__nonpos__neg,axiom,
    ! [Aa: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Ba),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),Ba)),zero_zero_real) ) ) ).

% add_nonpos_neg
tff(fact_1947_add__nonpos__neg,axiom,
    ! [Aa: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ba),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(Aa),Ba)),zero_zero_rat) ) ) ).

% add_nonpos_neg
tff(fact_1948_add__nonpos__neg,axiom,
    ! [Aa: nat,Ba: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ba),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,Aa),Ba)),zero_zero_nat) ) ) ).

% add_nonpos_neg
tff(fact_1949_add__nonpos__neg,axiom,
    ! [Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Ba),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,Aa),Ba)),zero_zero_int) ) ) ).

% add_nonpos_neg
tff(fact_1950_add__pos__nonneg,axiom,
    ! [Aa: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Aa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Ba)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),Ba)) ) ) ).

% add_pos_nonneg
tff(fact_1951_add__pos__nonneg,axiom,
    ! [Aa: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Aa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Ba)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(Aa),Ba)) ) ) ).

% add_pos_nonneg
tff(fact_1952_add__pos__nonneg,axiom,
    ! [Aa: nat,Ba: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Aa)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Ba)
       => 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,Aa),Ba)) ) ) ).

% add_pos_nonneg
tff(fact_1953_add__pos__nonneg,axiom,
    ! [Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Aa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Ba)
       => 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),Ba)) ) ) ).

% add_pos_nonneg
tff(fact_1954_add__strict__increasing,axiom,
    ! [Aa: real,Ba: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Aa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ba),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Ba),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),C)) ) ) ).

% add_strict_increasing
tff(fact_1955_add__strict__increasing,axiom,
    ! [Aa: rat,Ba: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Aa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ba),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ba),aa_rat_rat(plus_plus_rat(Aa),C)) ) ) ).

% add_strict_increasing
tff(fact_1956_add__strict__increasing,axiom,
    ! [Aa: nat,Ba: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Aa)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ba),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ba),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),C)) ) ) ).

% add_strict_increasing
tff(fact_1957_add__strict__increasing,axiom,
    ! [Aa: int,Ba: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Aa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ba),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,Ba),aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),C)) ) ) ).

% add_strict_increasing
tff(fact_1958_add__strict__increasing2,axiom,
    ! [Aa: real,Ba: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Aa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Ba),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Ba),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),C)) ) ) ).

% add_strict_increasing2
tff(fact_1959_add__strict__increasing2,axiom,
    ! [Aa: rat,Ba: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Aa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ba),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ba),aa_rat_rat(plus_plus_rat(Aa),C)) ) ) ).

% add_strict_increasing2
tff(fact_1960_add__strict__increasing2,axiom,
    ! [Aa: nat,Ba: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Aa)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ba),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ba),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),C)) ) ) ).

% add_strict_increasing2
tff(fact_1961_add__strict__increasing2,axiom,
    ! [Aa: int,Ba: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Aa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Ba),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,Ba),aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),C)) ) ) ).

% add_strict_increasing2
tff(fact_1962_zero__less__two,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),one_one_real)) ).

% zero_less_two
tff(fact_1963_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_1964_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_1965_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_1966_diff__Suc__less,axiom,
    ! [Na: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(suc,I))),Na) ) ).

% diff_Suc_less
tff(fact_1967_finite__maxlen,axiom,
    ! [M7: set_list_VEBT_VEBT] :
      ( finite3004134309566078307T_VEBT(M7)
     => ? [N: nat] :
        ! [X3: list_VEBT_VEBT] :
          ( member2936631157270082147T_VEBT(X3,M7)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_s6755466524823107622T_VEBT(X3)),N) ) ) ).

% finite_maxlen
tff(fact_1968_finite__maxlen,axiom,
    ! [M7: set_list_o] :
      ( finite_finite_list_o(M7)
     => ? [N: nat] :
        ! [X3: list_o] :
          ( member_list_o(X3,M7)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_size_list_o(X3)),N) ) ) ).

% finite_maxlen
tff(fact_1969_finite__maxlen,axiom,
    ! [M7: set_list_nat] :
      ( finite8100373058378681591st_nat(M7)
     => ? [N: nat] :
        ! [X3: list_nat] :
          ( member_list_nat(X3,M7)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_size_list_nat(X3)),N) ) ) ).

% finite_maxlen
tff(fact_1970_length__induct,axiom,
    ! [P: fun_list_VEBT_VEBT_o,Xs: list_VEBT_VEBT] :
      ( ! [Xs2: list_VEBT_VEBT] :
          ( ! [Ys: list_VEBT_VEBT] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_s6755466524823107622T_VEBT(Ys)),size_s6755466524823107622T_VEBT(Xs2))
             => aa_list_VEBT_VEBT_o(P,Ys) )
         => aa_list_VEBT_VEBT_o(P,Xs2) )
     => aa_list_VEBT_VEBT_o(P,Xs) ) ).

% length_induct
tff(fact_1971_length__induct,axiom,
    ! [P: fun_list_o_o,Xs: list_o] :
      ( ! [Xs2: list_o] :
          ( ! [Ys: list_o] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_size_list_o(Ys)),size_size_list_o(Xs2))
             => aa_list_o_o(P,Ys) )
         => aa_list_o_o(P,Xs2) )
     => aa_list_o_o(P,Xs) ) ).

% length_induct
tff(fact_1972_length__induct,axiom,
    ! [P: fun_list_nat_o,Xs: list_nat] :
      ( ! [Xs2: list_nat] :
          ( ! [Ys: list_nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_size_list_nat(Ys)),size_size_list_nat(Xs2))
             => aa_list_nat_o(P,Ys) )
         => aa_list_nat_o(P,Xs2) )
     => aa_list_nat_o(P,Xs) ) ).

% length_induct
tff(fact_1973_Suc__diff__eq__diff__pred,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,M)),Na) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat)) ) ) ).

% Suc_diff_eq_diff_pred
tff(fact_1974_Suc__pred_H,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( Na = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat)) ) ) ).

% Suc_pred'
tff(fact_1975_add__shift,axiom,
    ! [Xb: nat,Y: nat,Z: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xb),Y) = Z )
    <=> ( aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,some_nat(Xb)),some_nat(Y)) = some_nat(Z) ) ) ).

% add_shift
tff(fact_1976_add__def,axiom,
    vEBT_VEBT_add = vEBT_V4262088993061758097ft_nat(plus_plus_nat) ).

% add_def
tff(fact_1977_double__eq__0__iff,axiom,
    ! [Aa: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),Aa) = zero_zero_real )
    <=> ( Aa = zero_zero_real ) ) ).

% double_eq_0_iff
tff(fact_1978_double__eq__0__iff,axiom,
    ! [Aa: rat] :
      ( ( aa_rat_rat(plus_plus_rat(Aa),Aa) = zero_zero_rat )
    <=> ( Aa = zero_zero_rat ) ) ).

% double_eq_0_iff
tff(fact_1979_double__eq__0__iff,axiom,
    ! [Aa: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),Aa) = zero_zero_int )
    <=> ( Aa = zero_zero_int ) ) ).

% double_eq_0_iff
tff(fact_1980_discrete,axiom,
    ! [Aa: nat,Ba: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),Ba)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),one_one_nat)),Ba) ) ).

% discrete
tff(fact_1981_discrete,axiom,
    ! [Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),Ba)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),one_one_int)),Ba) ) ).

% discrete
tff(fact_1982_field__le__epsilon,axiom,
    ! [Xb: real,Y: real] :
      ( ! [E: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Y),E)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y) ) ).

% field_le_epsilon
tff(fact_1983_field__le__epsilon,axiom,
    ! [Xb: rat,Y: rat] :
      ( ! [E: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),E)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),aa_rat_rat(plus_plus_rat(Y),E)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y) ) ).

% field_le_epsilon
tff(fact_1984_diff__shunt__var,axiom,
    ! [Xb: set_real,Y: set_real] :
      ( ( minus_minus_set_real(Xb,Y) = bot_bot_set_real )
    <=> ord_less_eq_set_real(Xb,Y) ) ).

% diff_shunt_var
tff(fact_1985_diff__shunt__var,axiom,
    ! [Xb: set_nat,Y: set_nat] :
      ( ( minus_minus_set_nat(Xb,Y) = bot_bot_set_nat )
    <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Xb),Y) ) ).

% diff_shunt_var
tff(fact_1986_diff__shunt__var,axiom,
    ! [Xb: set_int,Y: set_int] :
      ( ( minus_minus_set_int(Xb,Y) = bot_bot_set_int )
    <=> aa_set_int_o(ord_less_eq_set_int(Xb),Y) ) ).

% diff_shunt_var
tff(fact_1987_prod__decode__aux_Oelims,axiom,
    ! [Xb: nat,Xaa: nat,Y: product_prod_nat_nat] :
      ( ( nat_prod_decode_aux(Xb,Xaa) = Y )
     => ( Y = $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_1988_prod__decode__aux_Osimps,axiom,
    ! [K: nat,M: nat] :
      nat_prod_decode_aux(K,M) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),K),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),M)),nat_prod_decode_aux(aa_nat_nat(suc,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(suc,K)))) ).

% prod_decode_aux.simps
tff(fact_1989_inthall,axiom,
    ! [Xs: list_P6011104703257516679at_nat,P: fun_Pr4796943543331071514_nat_o,Na: nat] :
      ( ! [X: product_prod_nat_nat] :
          ( member8440522571783428010at_nat(X,set_Pr5648618587558075414at_nat(Xs))
         => aa_Pro6150637604445964947_nat_o(P,X) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_s5460976970255530739at_nat(Xs))
       => aa_Pro6150637604445964947_nat_o(P,nth_Pr7617993195940197384at_nat(Xs,Na)) ) ) ).

% inthall
tff(fact_1990_inthall,axiom,
    ! [Xs: list_real,P: fun_real_o,Na: nat] :
      ( ! [X: real] :
          ( member_real(X,set_real2(Xs))
         => aa_real_o(P,X) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_size_list_real(Xs))
       => aa_real_o(P,aa_nat_real(nth_real(Xs),Na)) ) ) ).

% inthall
tff(fact_1991_inthall,axiom,
    ! [Xs: list_set_nat,P: fun_set_nat_o,Na: nat] :
      ( ! [X: set_nat] :
          ( member_set_nat(X,set_set_nat2(Xs))
         => aa_set_nat_o(P,X) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_s3254054031482475050et_nat(Xs))
       => aa_set_nat_o(P,nth_set_nat(Xs,Na)) ) ) ).

% inthall
tff(fact_1992_inthall,axiom,
    ! [Xs: list_int,P: fun_int_o,Na: nat] :
      ( ! [X: int] :
          ( aa_set_int_o(member_int(X),set_int2(Xs))
         => aa_int_o(P,X) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_size_list_int(Xs))
       => aa_int_o(P,aa_nat_int(nth_int(Xs),Na)) ) ) ).

% inthall
tff(fact_1993_inthall,axiom,
    ! [Xs: list_VEBT_VEBT,P: fun_VEBT_VEBT_o,Na: nat] :
      ( ! [X: vEBT_VEBT] :
          ( member_VEBT_VEBT(X,set_VEBT_VEBT2(Xs))
         => aa_VEBT_VEBT_o(P,X) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_s6755466524823107622T_VEBT(Xs))
       => aa_VEBT_VEBT_o(P,nth_VEBT_VEBT(Xs,Na)) ) ) ).

% inthall
tff(fact_1994_inthall,axiom,
    ! [Xs: list_o,P: fun_o_o,Na: nat] :
      ( ! [X: $o] :
          ( member_o((X),set_o2(Xs))
         => aa_o_o(P,(X)) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_size_list_o(Xs))
       => aa_o_o(P,aa_nat_o(nth_o(Xs),Na)) ) ) ).

% inthall
tff(fact_1995_inthall,axiom,
    ! [Xs: list_nat,P: fun_nat_o,Na: nat] :
      ( ! [X: nat] :
          ( aa_set_nat_o(member_nat(X),set_nat2(Xs))
         => aa_nat_o(P,X) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_size_list_nat(Xs))
       => aa_nat_o(P,aa_nat_nat(nth_nat(Xs),Na)) ) ) ).

% inthall
tff(fact_1996_Diff__cancel,axiom,
    ! [A3: set_real] : minus_minus_set_real(A3,A3) = bot_bot_set_real ).

% Diff_cancel
tff(fact_1997_Diff__cancel,axiom,
    ! [A3: set_int] : minus_minus_set_int(A3,A3) = bot_bot_set_int ).

% Diff_cancel
tff(fact_1998_Diff__cancel,axiom,
    ! [A3: set_nat] : minus_minus_set_nat(A3,A3) = bot_bot_set_nat ).

% Diff_cancel
tff(fact_1999_empty__Diff,axiom,
    ! [A3: set_real] : minus_minus_set_real(bot_bot_set_real,A3) = bot_bot_set_real ).

% empty_Diff
tff(fact_2000_empty__Diff,axiom,
    ! [A3: set_int] : minus_minus_set_int(bot_bot_set_int,A3) = bot_bot_set_int ).

% empty_Diff
tff(fact_2001_empty__Diff,axiom,
    ! [A3: set_nat] : minus_minus_set_nat(bot_bot_set_nat,A3) = bot_bot_set_nat ).

% empty_Diff
tff(fact_2002_Diff__empty,axiom,
    ! [A3: set_real] : minus_minus_set_real(A3,bot_bot_set_real) = A3 ).

% Diff_empty
tff(fact_2003_Diff__empty,axiom,
    ! [A3: set_int] : minus_minus_set_int(A3,bot_bot_set_int) = A3 ).

% Diff_empty
tff(fact_2004_Diff__empty,axiom,
    ! [A3: set_nat] : minus_minus_set_nat(A3,bot_bot_set_nat) = A3 ).

% Diff_empty
tff(fact_2005_finite__Diff2,axiom,
    ! [B4: set_int,A3: set_int] :
      ( aa_set_int_o(finite_finite_int,B4)
     => ( aa_set_int_o(finite_finite_int,minus_minus_set_int(A3,B4))
      <=> aa_set_int_o(finite_finite_int,A3) ) ) ).

% finite_Diff2
tff(fact_2006_finite__Diff2,axiom,
    ! [B4: set_complex,A3: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B4)
     => ( aa_set_complex_o(finite3207457112153483333omplex,minus_811609699411566653omplex(A3,B4))
      <=> aa_set_complex_o(finite3207457112153483333omplex,A3) ) ) ).

% finite_Diff2
tff(fact_2007_finite__Diff2,axiom,
    ! [B4: set_nat,A3: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,B4)
     => ( aa_set_nat_o(finite_finite_nat,minus_minus_set_nat(A3,B4))
      <=> aa_set_nat_o(finite_finite_nat,A3) ) ) ).

% finite_Diff2
tff(fact_2008_finite__Diff,axiom,
    ! [A3: set_int,B4: set_int] :
      ( aa_set_int_o(finite_finite_int,A3)
     => aa_set_int_o(finite_finite_int,minus_minus_set_int(A3,B4)) ) ).

% finite_Diff
tff(fact_2009_finite__Diff,axiom,
    ! [A3: set_complex,B4: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A3)
     => aa_set_complex_o(finite3207457112153483333omplex,minus_811609699411566653omplex(A3,B4)) ) ).

% finite_Diff
tff(fact_2010_finite__Diff,axiom,
    ! [A3: set_nat,B4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => aa_set_nat_o(finite_finite_nat,minus_minus_set_nat(A3,B4)) ) ).

% finite_Diff
tff(fact_2011_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_2012_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(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A3),B4) ) ).

% Diff_eq_empty_iff
tff(fact_2013_Diff__eq__empty__iff,axiom,
    ! [A3: set_int,B4: set_int] :
      ( ( minus_minus_set_int(A3,B4) = bot_bot_set_int )
    <=> aa_set_int_o(ord_less_eq_set_int(A3),B4) ) ).

% Diff_eq_empty_iff
tff(fact_2014_Diff__infinite__finite,axiom,
    ! [T: set_int,S2: set_int] :
      ( aa_set_int_o(finite_finite_int,T)
     => ( ~ aa_set_int_o(finite_finite_int,S2)
       => ~ aa_set_int_o(finite_finite_int,minus_minus_set_int(S2,T)) ) ) ).

% Diff_infinite_finite
tff(fact_2015_Diff__infinite__finite,axiom,
    ! [T: set_complex,S2: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T)
     => ( ~ aa_set_complex_o(finite3207457112153483333omplex,S2)
       => ~ aa_set_complex_o(finite3207457112153483333omplex,minus_811609699411566653omplex(S2,T)) ) ) ).

% Diff_infinite_finite
tff(fact_2016_Diff__infinite__finite,axiom,
    ! [T: set_nat,S2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,T)
     => ( ~ aa_set_nat_o(finite_finite_nat,S2)
       => ~ aa_set_nat_o(finite_finite_nat,minus_minus_set_nat(S2,T)) ) ) ).

% Diff_infinite_finite
tff(fact_2017_double__diff,axiom,
    ! [A3: set_nat,B4: set_nat,C2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A3),B4)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),C2)
       => ( minus_minus_set_nat(B4,minus_minus_set_nat(C2,A3)) = A3 ) ) ) ).

% double_diff
tff(fact_2018_double__diff,axiom,
    ! [A3: set_int,B4: set_int,C2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A3),B4)
     => ( aa_set_int_o(ord_less_eq_set_int(B4),C2)
       => ( minus_minus_set_int(B4,minus_minus_set_int(C2,A3)) = A3 ) ) ) ).

% double_diff
tff(fact_2019_Diff__subset,axiom,
    ! [A3: set_nat,B4: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,minus_minus_set_nat(A3,B4)),A3) ).

% Diff_subset
tff(fact_2020_Diff__subset,axiom,
    ! [A3: set_int,B4: set_int] : aa_set_int_o(ord_less_eq_set_int(minus_minus_set_int(A3,B4)),A3) ).

% Diff_subset
tff(fact_2021_Diff__mono,axiom,
    ! [A3: set_nat,C2: set_nat,D6: set_nat,B4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A3),C2)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,D6),B4)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,minus_minus_set_nat(A3,B4)),minus_minus_set_nat(C2,D6)) ) ) ).

% Diff_mono
tff(fact_2022_Diff__mono,axiom,
    ! [A3: set_int,C2: set_int,D6: set_int,B4: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A3),C2)
     => ( aa_set_int_o(ord_less_eq_set_int(D6),B4)
       => aa_set_int_o(ord_less_eq_set_int(minus_minus_set_int(A3,B4)),minus_minus_set_int(C2,D6)) ) ) ).

% Diff_mono
tff(fact_2023_psubset__imp__ex__mem,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ord_le7866589430770878221at_nat(A3,B4)
     => ? [B2: product_prod_nat_nat] : member8440522571783428010at_nat(B2,minus_1356011639430497352at_nat(B4,A3)) ) ).

% psubset_imp_ex_mem
tff(fact_2024_psubset__imp__ex__mem,axiom,
    ! [A3: set_real,B4: set_real] :
      ( ord_less_set_real(A3,B4)
     => ? [B2: real] : member_real(B2,minus_minus_set_real(B4,A3)) ) ).

% psubset_imp_ex_mem
tff(fact_2025_psubset__imp__ex__mem,axiom,
    ! [A3: set_set_nat,B4: set_set_nat] :
      ( ord_less_set_set_nat(A3,B4)
     => ? [B2: set_nat] : member_set_nat(B2,minus_2163939370556025621et_nat(B4,A3)) ) ).

% psubset_imp_ex_mem
tff(fact_2026_psubset__imp__ex__mem,axiom,
    ! [A3: set_int,B4: set_int] :
      ( aa_set_int_o(ord_less_set_int(A3),B4)
     => ? [B2: int] : aa_set_int_o(member_int(B2),minus_minus_set_int(B4,A3)) ) ).

% psubset_imp_ex_mem
tff(fact_2027_psubset__imp__ex__mem,axiom,
    ! [A3: set_nat,B4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A3),B4)
     => ? [B2: nat] : aa_set_nat_o(member_nat(B2),minus_minus_set_nat(B4,A3)) ) ).

% psubset_imp_ex_mem
tff(fact_2028_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_int,Ys2: list_int] :
      ( ( Xs = Ys2 )
    <=> ( ( size_size_list_int(Xs) = size_size_list_int(Ys2) )
        & ! [I3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_int(Xs))
           => ( aa_nat_int(nth_int(Xs),I3) = aa_nat_int(nth_int(Ys2),I3) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_2029_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_VEBT_VEBT,Ys2: list_VEBT_VEBT] :
      ( ( Xs = Ys2 )
    <=> ( ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Ys2) )
        & ! [I3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_s6755466524823107622T_VEBT(Xs))
           => ( nth_VEBT_VEBT(Xs,I3) = nth_VEBT_VEBT(Ys2,I3) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_2030_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_o,Ys2: list_o] :
      ( ( Xs = Ys2 )
    <=> ( ( size_size_list_o(Xs) = size_size_list_o(Ys2) )
        & ! [I3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_o(Xs))
           => ( aa_nat_o(nth_o(Xs),I3)
            <=> aa_nat_o(nth_o(Ys2),I3) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_2031_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_nat,Ys2: list_nat] :
      ( ( Xs = Ys2 )
    <=> ( ( size_size_list_nat(Xs) = size_size_list_nat(Ys2) )
        & ! [I3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_nat(Xs))
           => ( aa_nat_nat(nth_nat(Xs),I3) = aa_nat_nat(nth_nat(Ys2),I3) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_2032_Skolem__list__nth,axiom,
    ! [K: nat,P: fun_nat_fun_int_o] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),K)
         => ? [X_12: int] : aa_int_o(aa_nat_fun_int_o(P,I3),X_12) )
    <=> ? [Xs3: list_int] :
          ( ( size_size_list_int(Xs3) = K )
          & ! [I3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),K)
             => aa_int_o(aa_nat_fun_int_o(P,I3),aa_nat_int(nth_int(Xs3),I3)) ) ) ) ).

% Skolem_list_nth
tff(fact_2033_Skolem__list__nth,axiom,
    ! [K: nat,P: fun_na6887940067898170868VEBT_o] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),K)
         => ? [X_12: vEBT_VEBT] : aa_VEBT_VEBT_o(aa_nat1925708948643195573VEBT_o(P,I3),X_12) )
    <=> ? [Xs3: list_VEBT_VEBT] :
          ( ( size_s6755466524823107622T_VEBT(Xs3) = K )
          & ! [I3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),K)
             => aa_VEBT_VEBT_o(aa_nat1925708948643195573VEBT_o(P,I3),nth_VEBT_VEBT(Xs3,I3)) ) ) ) ).

% Skolem_list_nth
tff(fact_2034_Skolem__list__nth,axiom,
    ! [K: nat,P: fun_nat_fun_o_o] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),K)
         => ? [X_12: $o] : aa_o_o(aa_nat_fun_o_o(P,I3),(X_12)) )
    <=> ? [Xs3: list_o] :
          ( ( size_size_list_o(Xs3) = K )
          & ! [I3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),K)
             => aa_o_o(aa_nat_fun_o_o(P,I3),aa_nat_o(nth_o(Xs3),I3)) ) ) ) ).

% Skolem_list_nth
tff(fact_2035_Skolem__list__nth,axiom,
    ! [K: nat,P: fun_nat_fun_nat_o] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),K)
         => ? [X_12: nat] : aa_nat_o(aa_nat_fun_nat_o(P,I3),X_12) )
    <=> ? [Xs3: list_nat] :
          ( ( size_size_list_nat(Xs3) = K )
          & ! [I3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),K)
             => aa_nat_o(aa_nat_fun_nat_o(P,I3),aa_nat_nat(nth_nat(Xs3),I3)) ) ) ) ).

% Skolem_list_nth
tff(fact_2036_nth__equalityI,axiom,
    ! [Xs: list_int,Ys2: list_int] :
      ( ( size_size_list_int(Xs) = size_size_list_int(Ys2) )
     => ( ! [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(Ys2),I2) ) )
       => ( Xs = Ys2 ) ) ) ).

% nth_equalityI
tff(fact_2037_nth__equalityI,axiom,
    ! [Xs: list_VEBT_VEBT,Ys2: list_VEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Ys2) )
     => ( ! [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(Ys2,I2) ) )
       => ( Xs = Ys2 ) ) ) ).

% nth_equalityI
tff(fact_2038_nth__equalityI,axiom,
    ! [Xs: list_o,Ys2: list_o] :
      ( ( size_size_list_o(Xs) = size_size_list_o(Ys2) )
     => ( ! [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(Ys2),I2) ) )
       => ( Xs = Ys2 ) ) ) ).

% nth_equalityI
tff(fact_2039_nth__equalityI,axiom,
    ! [Xs: list_nat,Ys2: list_nat] :
      ( ( size_size_list_nat(Xs) = size_size_list_nat(Ys2) )
     => ( ! [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(Ys2),I2) ) )
       => ( Xs = Ys2 ) ) ) ).

% nth_equalityI
tff(fact_2040_all__set__conv__all__nth,axiom,
    ! [Xs: list_int,P: fun_int_o] :
      ( ! [X4: int] :
          ( aa_set_int_o(member_int(X4),set_int2(Xs))
         => aa_int_o(P,X4) )
    <=> ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_int(Xs))
         => aa_int_o(P,aa_nat_int(nth_int(Xs),I3)) ) ) ).

% all_set_conv_all_nth
tff(fact_2041_all__set__conv__all__nth,axiom,
    ! [Xs: list_VEBT_VEBT,P: fun_VEBT_VEBT_o] :
      ( ! [X4: vEBT_VEBT] :
          ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(Xs))
         => aa_VEBT_VEBT_o(P,X4) )
    <=> ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_s6755466524823107622T_VEBT(Xs))
         => aa_VEBT_VEBT_o(P,nth_VEBT_VEBT(Xs,I3)) ) ) ).

% all_set_conv_all_nth
tff(fact_2042_all__set__conv__all__nth,axiom,
    ! [Xs: list_o,P: fun_o_o] :
      ( ! [X4: $o] :
          ( member_o((X4),set_o2(Xs))
         => aa_o_o(P,(X4)) )
    <=> ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_o(Xs))
         => aa_o_o(P,aa_nat_o(nth_o(Xs),I3)) ) ) ).

% all_set_conv_all_nth
tff(fact_2043_all__set__conv__all__nth,axiom,
    ! [Xs: list_nat,P: fun_nat_o] :
      ( ! [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),set_nat2(Xs))
         => aa_nat_o(P,X4) )
    <=> ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_nat(Xs))
         => aa_nat_o(P,aa_nat_nat(nth_nat(Xs),I3)) ) ) ).

% all_set_conv_all_nth
tff(fact_2044_all__nth__imp__all__set,axiom,
    ! [Xs: list_P6011104703257516679at_nat,P: fun_Pr4796943543331071514_nat_o,Xb: product_prod_nat_nat] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_s5460976970255530739at_nat(Xs))
         => aa_Pro6150637604445964947_nat_o(P,nth_Pr7617993195940197384at_nat(Xs,I2)) )
     => ( member8440522571783428010at_nat(Xb,set_Pr5648618587558075414at_nat(Xs))
       => aa_Pro6150637604445964947_nat_o(P,Xb) ) ) ).

% all_nth_imp_all_set
tff(fact_2045_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_2046_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,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_2047_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)) )
     => ( aa_set_int_o(member_int(Xb),set_int2(Xs))
       => aa_int_o(P,Xb) ) ) ).

% all_nth_imp_all_set
tff(fact_2048_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_2049_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_2050_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_2051_in__set__conv__nth,axiom,
    ! [Xb: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( member8440522571783428010at_nat(Xb,set_Pr5648618587558075414at_nat(Xs))
    <=> ? [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_s5460976970255530739at_nat(Xs))
          & ( nth_Pr7617993195940197384at_nat(Xs,I3) = Xb ) ) ) ).

% in_set_conv_nth
tff(fact_2052_in__set__conv__nth,axiom,
    ! [Xb: real,Xs: list_real] :
      ( member_real(Xb,set_real2(Xs))
    <=> ? [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_real(Xs))
          & ( aa_nat_real(nth_real(Xs),I3) = Xb ) ) ) ).

% in_set_conv_nth
tff(fact_2053_in__set__conv__nth,axiom,
    ! [Xb: set_nat,Xs: list_set_nat] :
      ( member_set_nat(Xb,set_set_nat2(Xs))
    <=> ? [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_s3254054031482475050et_nat(Xs))
          & ( nth_set_nat(Xs,I3) = Xb ) ) ) ).

% in_set_conv_nth
tff(fact_2054_in__set__conv__nth,axiom,
    ! [Xb: int,Xs: list_int] :
      ( aa_set_int_o(member_int(Xb),set_int2(Xs))
    <=> ? [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_int(Xs))
          & ( aa_nat_int(nth_int(Xs),I3) = Xb ) ) ) ).

% in_set_conv_nth
tff(fact_2055_in__set__conv__nth,axiom,
    ! [Xb: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( member_VEBT_VEBT(Xb,set_VEBT_VEBT2(Xs))
    <=> ? [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_s6755466524823107622T_VEBT(Xs))
          & ( nth_VEBT_VEBT(Xs,I3) = Xb ) ) ) ).

% in_set_conv_nth
tff(fact_2056_in__set__conv__nth,axiom,
    ! [Xb: $o,Xs: list_o] :
      ( member_o((Xb),set_o2(Xs))
    <=> ? [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_o(Xs))
          & ( aa_nat_o(nth_o(Xs),I3)
          <=> (Xb) ) ) ) ).

% in_set_conv_nth
tff(fact_2057_in__set__conv__nth,axiom,
    ! [Xb: nat,Xs: list_nat] :
      ( aa_set_nat_o(member_nat(Xb),set_nat2(Xs))
    <=> ? [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_nat(Xs))
          & ( aa_nat_nat(nth_nat(Xs),I3) = Xb ) ) ) ).

% in_set_conv_nth
tff(fact_2058_list__ball__nth,axiom,
    ! [Na: nat,Xs: list_int,P: fun_int_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_size_list_int(Xs))
     => ( ! [X: int] :
            ( aa_set_int_o(member_int(X),set_int2(Xs))
           => aa_int_o(P,X) )
       => aa_int_o(P,aa_nat_int(nth_int(Xs),Na)) ) ) ).

% list_ball_nth
tff(fact_2059_list__ball__nth,axiom,
    ! [Na: nat,Xs: list_VEBT_VEBT,P: fun_VEBT_VEBT_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_s6755466524823107622T_VEBT(Xs))
     => ( ! [X: vEBT_VEBT] :
            ( member_VEBT_VEBT(X,set_VEBT_VEBT2(Xs))
           => aa_VEBT_VEBT_o(P,X) )
       => aa_VEBT_VEBT_o(P,nth_VEBT_VEBT(Xs,Na)) ) ) ).

% list_ball_nth
tff(fact_2060_list__ball__nth,axiom,
    ! [Na: nat,Xs: list_o,P: fun_o_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_size_list_o(Xs))
     => ( ! [X: $o] :
            ( member_o((X),set_o2(Xs))
           => aa_o_o(P,(X)) )
       => aa_o_o(P,aa_nat_o(nth_o(Xs),Na)) ) ) ).

% list_ball_nth
tff(fact_2061_list__ball__nth,axiom,
    ! [Na: nat,Xs: list_nat,P: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_size_list_nat(Xs))
     => ( ! [X: nat] :
            ( aa_set_nat_o(member_nat(X),set_nat2(Xs))
           => aa_nat_o(P,X) )
       => aa_nat_o(P,aa_nat_nat(nth_nat(Xs),Na)) ) ) ).

% list_ball_nth
tff(fact_2062_nth__mem,axiom,
    ! [Na: nat,Xs: list_P6011104703257516679at_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_s5460976970255530739at_nat(Xs))
     => member8440522571783428010at_nat(nth_Pr7617993195940197384at_nat(Xs,Na),set_Pr5648618587558075414at_nat(Xs)) ) ).

% nth_mem
tff(fact_2063_nth__mem,axiom,
    ! [Na: nat,Xs: list_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_size_list_real(Xs))
     => member_real(aa_nat_real(nth_real(Xs),Na),set_real2(Xs)) ) ).

% nth_mem
tff(fact_2064_nth__mem,axiom,
    ! [Na: nat,Xs: list_set_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_s3254054031482475050et_nat(Xs))
     => member_set_nat(nth_set_nat(Xs,Na),set_set_nat2(Xs)) ) ).

% nth_mem
tff(fact_2065_nth__mem,axiom,
    ! [Na: nat,Xs: list_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_size_list_int(Xs))
     => aa_set_int_o(member_int(aa_nat_int(nth_int(Xs),Na)),set_int2(Xs)) ) ).

% nth_mem
tff(fact_2066_nth__mem,axiom,
    ! [Na: nat,Xs: list_VEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_s6755466524823107622T_VEBT(Xs))
     => member_VEBT_VEBT(nth_VEBT_VEBT(Xs,Na),set_VEBT_VEBT2(Xs)) ) ).

% nth_mem
tff(fact_2067_nth__mem,axiom,
    ! [Na: nat,Xs: list_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_size_list_o(Xs))
     => member_o(aa_nat_o(nth_o(Xs),Na),set_o2(Xs)) ) ).

% nth_mem
tff(fact_2068_nth__mem,axiom,
    ! [Na: nat,Xs: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_size_list_nat(Xs))
     => aa_set_nat_o(member_nat(aa_nat_nat(nth_nat(Xs),Na)),set_nat2(Xs)) ) ).

% nth_mem
tff(fact_2069_linordered__field__no__ub,axiom,
    ! [X3: real] :
    ? [X_1: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,X3),X_1) ).

% linordered_field_no_ub
tff(fact_2070_linordered__field__no__ub,axiom,
    ! [X3: rat] :
    ? [X_1: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),X_1) ).

% linordered_field_no_ub
tff(fact_2071_linordered__field__no__lb,axiom,
    ! [X3: real] :
    ? [Y3: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,Y3),X3) ).

% linordered_field_no_lb
tff(fact_2072_linordered__field__no__lb,axiom,
    ! [X3: rat] :
    ? [Y3: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y3),X3) ).

% linordered_field_no_lb
tff(fact_2073_divides__aux__eq,axiom,
    ! [Q4: code_integer,R2: code_integer] :
      ( unique5706413561485394159nteger(produc1086072967326762835nteger(Q4,R2))
    <=> ( R2 = zero_z3403309356797280102nteger ) ) ).

% divides_aux_eq
tff(fact_2074_divides__aux__eq,axiom,
    ! [Q4: nat,R2: nat] :
      ( aa_Pro6150637604445964947_nat_o(unique6322359934112328802ux_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Q4),R2))
    <=> ( R2 = zero_zero_nat ) ) ).

% divides_aux_eq
tff(fact_2075_divides__aux__eq,axiom,
    ! [Q4: int,R2: int] :
      ( aa_Pro4592068669816069595_int_o(unique6319869463603278526ux_int,product_Pair_int_int(Q4,R2))
    <=> ( R2 = zero_zero_int ) ) ).

% divides_aux_eq
tff(fact_2076_nth__enumerate__eq,axiom,
    ! [M: nat,Xs: list_int,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),size_size_list_int(Xs))
     => ( nth_Pr3440142176431000676at_int(enumerate_int(Na,Xs),M) = product_Pair_nat_int(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),M),aa_nat_int(nth_int(Xs),M)) ) ) ).

% nth_enumerate_eq
tff(fact_2077_nth__enumerate__eq,axiom,
    ! [M: nat,Xs: list_VEBT_VEBT,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),size_s6755466524823107622T_VEBT(Xs))
     => ( nth_Pr744662078594809490T_VEBT(enumerate_VEBT_VEBT(Na,Xs),M) = produc599794634098209291T_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),M),nth_VEBT_VEBT(Xs,M)) ) ) ).

% nth_enumerate_eq
tff(fact_2078_nth__enumerate__eq,axiom,
    ! [M: nat,Xs: list_o,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),size_size_list_o(Xs))
     => ( nth_Pr112076138515278198_nat_o(enumerate_o(Na,Xs),M) = product_Pair_nat_o(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),M),aa_nat_o(nth_o(Xs),M)) ) ) ).

% nth_enumerate_eq
tff(fact_2079_nth__enumerate__eq,axiom,
    ! [M: nat,Xs: list_nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),size_size_list_nat(Xs))
     => ( nth_Pr7617993195940197384at_nat(enumerate_nat(Na,Xs),M) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),M)),aa_nat_nat(nth_nat(Xs),M)) ) ) ).

% nth_enumerate_eq
tff(fact_2080_Euclid__induct,axiom,
    ! [P: fun_nat_fun_nat_o,Aa: nat,Ba: nat] :
      ( ! [A2: nat,B2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(P,A2),B2)
        <=> aa_nat_o(aa_nat_fun_nat_o(P,B2),A2) )
     => ( ! [A2: nat] : aa_nat_o(aa_nat_fun_nat_o(P,A2),zero_zero_nat)
       => ( ! [A2: nat,B2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(P,A2),B2)
             => aa_nat_o(aa_nat_fun_nat_o(P,A2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)) )
         => aa_nat_o(aa_nat_fun_nat_o(P,Aa),Ba) ) ) ) ).

% Euclid_induct
tff(fact_2081_triangle__Suc,axiom,
    ! [Na: nat] : aa_nat_nat(nat_triangle,aa_nat_nat(suc,Na)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(nat_triangle,Na)),aa_nat_nat(suc,Na)) ).

% triangle_Suc
tff(fact_2082_add__0__iff,axiom,
    ! [Ba: real,Aa: real] :
      ( ( Ba = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Ba),Aa) )
    <=> ( Aa = zero_zero_real ) ) ).

% add_0_iff
tff(fact_2083_add__0__iff,axiom,
    ! [Ba: rat,Aa: rat] :
      ( ( Ba = aa_rat_rat(plus_plus_rat(Ba),Aa) )
    <=> ( Aa = zero_zero_rat ) ) ).

% add_0_iff
tff(fact_2084_add__0__iff,axiom,
    ! [Ba: nat,Aa: nat] :
      ( ( Ba = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ba),Aa) )
    <=> ( Aa = zero_zero_nat ) ) ).

% add_0_iff
tff(fact_2085_add__0__iff,axiom,
    ! [Ba: int,Aa: int] :
      ( ( Ba = aa_int_int(aa_int_fun_int_int(plus_plus_int,Ba),Aa) )
    <=> ( Aa = zero_zero_int ) ) ).

% add_0_iff
tff(fact_2086_find__Some__iff2,axiom,
    ! [Xb: int,P: fun_int_o,Xs: list_int] :
      ( ( some_int(Xb) = find_int(P,Xs) )
    <=> ? [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_int(Xs))
          & aa_int_o(P,aa_nat_int(nth_int(Xs),I3))
          & ( Xb = aa_nat_int(nth_int(Xs),I3) )
          & ! [J3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),I3)
             => ~ aa_int_o(P,aa_nat_int(nth_int(Xs),J3)) ) ) ) ).

% find_Some_iff2
tff(fact_2087_find__Some__iff2,axiom,
    ! [Xb: product_prod_nat_nat,P: fun_Pr4796943543331071514_nat_o,Xs: list_P6011104703257516679at_nat] :
      ( ( some_P7363390416028606310at_nat(Xb) = find_P8199882355184865565at_nat(P,Xs) )
    <=> ? [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_s5460976970255530739at_nat(Xs))
          & aa_Pro6150637604445964947_nat_o(P,nth_Pr7617993195940197384at_nat(Xs,I3))
          & ( Xb = nth_Pr7617993195940197384at_nat(Xs,I3) )
          & ! [J3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),I3)
             => ~ aa_Pro6150637604445964947_nat_o(P,nth_Pr7617993195940197384at_nat(Xs,J3)) ) ) ) ).

% find_Some_iff2
tff(fact_2088_find__Some__iff2,axiom,
    ! [Xb: num,P: fun_num_o,Xs: list_num] :
      ( ( some_num(Xb) = find_num(P,Xs) )
    <=> ? [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_num(Xs))
          & aa_num_o(P,aa_nat_num(nth_num(Xs),I3))
          & ( Xb = aa_nat_num(nth_num(Xs),I3) )
          & ! [J3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),I3)
             => ~ aa_num_o(P,aa_nat_num(nth_num(Xs),J3)) ) ) ) ).

% find_Some_iff2
tff(fact_2089_find__Some__iff2,axiom,
    ! [Xb: vEBT_VEBT,P: fun_VEBT_VEBT_o,Xs: list_VEBT_VEBT] :
      ( ( some_VEBT_VEBT(Xb) = find_VEBT_VEBT(P,Xs) )
    <=> ? [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_s6755466524823107622T_VEBT(Xs))
          & aa_VEBT_VEBT_o(P,nth_VEBT_VEBT(Xs,I3))
          & ( Xb = nth_VEBT_VEBT(Xs,I3) )
          & ! [J3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),I3)
             => ~ aa_VEBT_VEBT_o(P,nth_VEBT_VEBT(Xs,J3)) ) ) ) ).

% find_Some_iff2
tff(fact_2090_find__Some__iff2,axiom,
    ! [Xb: $o,P: fun_o_o,Xs: list_o] :
      ( ( some_o((Xb)) = find_o(P,Xs) )
    <=> ? [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_o(Xs))
          & aa_o_o(P,aa_nat_o(nth_o(Xs),I3))
          & ( (Xb)
          <=> aa_nat_o(nth_o(Xs),I3) )
          & ! [J3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),I3)
             => ~ aa_o_o(P,aa_nat_o(nth_o(Xs),J3)) ) ) ) ).

% find_Some_iff2
tff(fact_2091_find__Some__iff2,axiom,
    ! [Xb: nat,P: fun_nat_o,Xs: list_nat] :
      ( ( some_nat(Xb) = find_nat(P,Xs) )
    <=> ? [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_nat(Xs))
          & aa_nat_o(P,aa_nat_nat(nth_nat(Xs),I3))
          & ( Xb = aa_nat_nat(nth_nat(Xs),I3) )
          & ! [J3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),I3)
             => ~ aa_nat_o(P,aa_nat_nat(nth_nat(Xs),J3)) ) ) ) ).

% find_Some_iff2
tff(fact_2092_find__Some__iff,axiom,
    ! [P: fun_int_o,Xs: list_int,Xb: int] :
      ( ( find_int(P,Xs) = some_int(Xb) )
    <=> ? [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_int(Xs))
          & aa_int_o(P,aa_nat_int(nth_int(Xs),I3))
          & ( Xb = aa_nat_int(nth_int(Xs),I3) )
          & ! [J3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),I3)
             => ~ aa_int_o(P,aa_nat_int(nth_int(Xs),J3)) ) ) ) ).

% find_Some_iff
tff(fact_2093_find__Some__iff,axiom,
    ! [P: fun_Pr4796943543331071514_nat_o,Xs: list_P6011104703257516679at_nat,Xb: product_prod_nat_nat] :
      ( ( find_P8199882355184865565at_nat(P,Xs) = some_P7363390416028606310at_nat(Xb) )
    <=> ? [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_s5460976970255530739at_nat(Xs))
          & aa_Pro6150637604445964947_nat_o(P,nth_Pr7617993195940197384at_nat(Xs,I3))
          & ( Xb = nth_Pr7617993195940197384at_nat(Xs,I3) )
          & ! [J3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),I3)
             => ~ aa_Pro6150637604445964947_nat_o(P,nth_Pr7617993195940197384at_nat(Xs,J3)) ) ) ) ).

% find_Some_iff
tff(fact_2094_find__Some__iff,axiom,
    ! [P: fun_num_o,Xs: list_num,Xb: num] :
      ( ( find_num(P,Xs) = some_num(Xb) )
    <=> ? [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_num(Xs))
          & aa_num_o(P,aa_nat_num(nth_num(Xs),I3))
          & ( Xb = aa_nat_num(nth_num(Xs),I3) )
          & ! [J3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),I3)
             => ~ aa_num_o(P,aa_nat_num(nth_num(Xs),J3)) ) ) ) ).

% find_Some_iff
tff(fact_2095_find__Some__iff,axiom,
    ! [P: fun_VEBT_VEBT_o,Xs: list_VEBT_VEBT,Xb: vEBT_VEBT] :
      ( ( find_VEBT_VEBT(P,Xs) = some_VEBT_VEBT(Xb) )
    <=> ? [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_s6755466524823107622T_VEBT(Xs))
          & aa_VEBT_VEBT_o(P,nth_VEBT_VEBT(Xs,I3))
          & ( Xb = nth_VEBT_VEBT(Xs,I3) )
          & ! [J3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),I3)
             => ~ aa_VEBT_VEBT_o(P,nth_VEBT_VEBT(Xs,J3)) ) ) ) ).

% find_Some_iff
tff(fact_2096_find__Some__iff,axiom,
    ! [P: fun_o_o,Xs: list_o,Xb: $o] :
      ( ( find_o(P,Xs) = some_o((Xb)) )
    <=> ? [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_o(Xs))
          & aa_o_o(P,aa_nat_o(nth_o(Xs),I3))
          & ( (Xb)
          <=> aa_nat_o(nth_o(Xs),I3) )
          & ! [J3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),I3)
             => ~ aa_o_o(P,aa_nat_o(nth_o(Xs),J3)) ) ) ) ).

% find_Some_iff
tff(fact_2097_find__Some__iff,axiom,
    ! [P: fun_nat_o,Xs: list_nat,Xb: nat] :
      ( ( find_nat(P,Xs) = some_nat(Xb) )
    <=> ? [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_nat(Xs))
          & aa_nat_o(P,aa_nat_nat(nth_nat(Xs),I3))
          & ( Xb = aa_nat_nat(nth_nat(Xs),I3) )
          & ! [J3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),I3)
             => ~ aa_nat_o(P,aa_nat_nat(nth_nat(Xs),J3)) ) ) ) ).

% find_Some_iff
tff(fact_2098_frac__unique__iff,axiom,
    ! [Xb: real,Aa: real] :
      ( ( aa_real_real(archim2898591450579166408c_real,Xb) = Aa )
    <=> ( member_real(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Aa),ring_1_Ints_real)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Aa)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),one_one_real) ) ) ).

% frac_unique_iff
tff(fact_2099_frac__unique__iff,axiom,
    ! [Xb: rat,Aa: rat] :
      ( ( aa_rat_rat(archimedean_frac_rat,Xb) = Aa )
    <=> ( member_rat(aa_rat_rat(minus_minus_rat(Xb),Aa),ring_1_Ints_rat)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Aa)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),one_one_rat) ) ) ).

% frac_unique_iff
tff(fact_2100_nth__Cons__pos,axiom,
    ! [Na: nat,Xb: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( nth_VEBT_VEBT(cons_VEBT_VEBT(Xb,Xs),Na) = nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat)) ) ) ).

% nth_Cons_pos
tff(fact_2101_nth__Cons__pos,axiom,
    ! [Na: nat,Xb: int,Xs: list_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_int(nth_int(cons_int(Xb,Xs)),Na) = aa_nat_int(nth_int(Xs),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat)) ) ) ).

% nth_Cons_pos
tff(fact_2102_nth__Cons__pos,axiom,
    ! [Na: nat,Xb: nat,Xs: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_nat(nth_nat(cons_nat(Xb,Xs)),Na) = aa_nat_nat(nth_nat(Xs),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat)) ) ) ).

% nth_Cons_pos
tff(fact_2103_rotate1__length01,axiom,
    ! [Xs: list_VEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_s6755466524823107622T_VEBT(Xs)),one_one_nat)
     => ( rotate1_VEBT_VEBT(Xs) = Xs ) ) ).

% rotate1_length01
tff(fact_2104_rotate1__length01,axiom,
    ! [Xs: list_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_o(Xs)),one_one_nat)
     => ( rotate1_o(Xs) = Xs ) ) ).

% rotate1_length01
tff(fact_2105_rotate1__length01,axiom,
    ! [Xs: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_nat(Xs)),one_one_nat)
     => ( rotate1_nat(Xs) = Xs ) ) ).

% rotate1_length01
tff(fact_2106_triangle__0,axiom,
    aa_nat_nat(nat_triangle,zero_zero_nat) = zero_zero_nat ).

% triangle_0
tff(fact_2107_nth__Cons__Suc,axiom,
    ! [Xb: vEBT_VEBT,Xs: list_VEBT_VEBT,Na: nat] : nth_VEBT_VEBT(cons_VEBT_VEBT(Xb,Xs),aa_nat_nat(suc,Na)) = nth_VEBT_VEBT(Xs,Na) ).

% nth_Cons_Suc
tff(fact_2108_nth__Cons__Suc,axiom,
    ! [Xb: int,Xs: list_int,Na: nat] : aa_nat_int(nth_int(cons_int(Xb,Xs)),aa_nat_nat(suc,Na)) = aa_nat_int(nth_int(Xs),Na) ).

% nth_Cons_Suc
tff(fact_2109_nth__Cons__Suc,axiom,
    ! [Xb: nat,Xs: list_nat,Na: nat] : aa_nat_nat(nth_nat(cons_nat(Xb,Xs)),aa_nat_nat(suc,Na)) = aa_nat_nat(nth_nat(Xs),Na) ).

% nth_Cons_Suc
tff(fact_2110_nth__Cons__0,axiom,
    ! [Xb: vEBT_VEBT,Xs: list_VEBT_VEBT] : nth_VEBT_VEBT(cons_VEBT_VEBT(Xb,Xs),zero_zero_nat) = Xb ).

% nth_Cons_0
tff(fact_2111_nth__Cons__0,axiom,
    ! [Xb: int,Xs: list_int] : aa_nat_int(nth_int(cons_int(Xb,Xs)),zero_zero_nat) = Xb ).

% nth_Cons_0
tff(fact_2112_nth__Cons__0,axiom,
    ! [Xb: nat,Xs: list_nat] : aa_nat_nat(nth_nat(cons_nat(Xb,Xs)),zero_zero_nat) = Xb ).

% nth_Cons_0
tff(fact_2113_frac__eq__0__iff,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(archim2898591450579166408c_real,Xb) = zero_zero_real )
    <=> member_real(Xb,ring_1_Ints_real) ) ).

% frac_eq_0_iff
tff(fact_2114_frac__eq__0__iff,axiom,
    ! [Xb: rat] :
      ( ( aa_rat_rat(archimedean_frac_rat,Xb) = zero_zero_rat )
    <=> member_rat(Xb,ring_1_Ints_rat) ) ).

% frac_eq_0_iff
tff(fact_2115_frac__gt__0__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(archim2898591450579166408c_real,Xb))
    <=> ~ member_real(Xb,ring_1_Ints_real) ) ).

% frac_gt_0_iff
tff(fact_2116_frac__gt__0__iff,axiom,
    ! [Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(archimedean_frac_rat,Xb))
    <=> ~ member_rat(Xb,ring_1_Ints_rat) ) ).

% frac_gt_0_iff
tff(fact_2117_enumerate__simps_I2_J,axiom,
    ! [Na: nat,Xb: int,Xs: list_int] : enumerate_int(Na,cons_int(Xb,Xs)) = cons_P2335045147070616083at_int(product_Pair_nat_int(Na,Xb),enumerate_int(aa_nat_nat(suc,Na),Xs)) ).

% enumerate_simps(2)
tff(fact_2118_enumerate__simps_I2_J,axiom,
    ! [Na: nat,Xb: nat,Xs: list_nat] : enumerate_nat(Na,cons_nat(Xb,Xs)) = cons_P6512896166579812791at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Na),Xb),enumerate_nat(aa_nat_nat(suc,Na),Xs)) ).

% enumerate_simps(2)
tff(fact_2119_find_Osimps_I2_J,axiom,
    ! [P: fun_int_o,Xb: int,Xs: list_int] :
      find_int(P,cons_int(Xb,Xs)) = $ite(aa_int_o(P,Xb),some_int(Xb),find_int(P,Xs)) ).

% find.simps(2)
tff(fact_2120_find_Osimps_I2_J,axiom,
    ! [P: fun_nat_o,Xb: nat,Xs: list_nat] :
      find_nat(P,cons_nat(Xb,Xs)) = $ite(aa_nat_o(P,Xb),some_nat(Xb),find_nat(P,Xs)) ).

% find.simps(2)
tff(fact_2121_find_Osimps_I2_J,axiom,
    ! [P: fun_Pr4796943543331071514_nat_o,Xb: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      find_P8199882355184865565at_nat(P,cons_P6512896166579812791at_nat(Xb,Xs)) = $ite(aa_Pro6150637604445964947_nat_o(P,Xb),some_P7363390416028606310at_nat(Xb),find_P8199882355184865565at_nat(P,Xs)) ).

% find.simps(2)
tff(fact_2122_find_Osimps_I2_J,axiom,
    ! [P: fun_num_o,Xb: num,Xs: list_num] :
      find_num(P,cons_num(Xb,Xs)) = $ite(aa_num_o(P,Xb),some_num(Xb),find_num(P,Xs)) ).

% find.simps(2)
tff(fact_2123_Ints__0,axiom,
    member_real(zero_zero_real,ring_1_Ints_real) ).

% Ints_0
tff(fact_2124_Ints__0,axiom,
    member_rat(zero_zero_rat,ring_1_Ints_rat) ).

% Ints_0
tff(fact_2125_Ints__0,axiom,
    aa_set_int_o(member_int(zero_zero_int),ring_1_Ints_int) ).

% Ints_0
tff(fact_2126_Ints__1,axiom,
    aa_set_complex_o(member_complex(one_one_complex),ring_1_Ints_complex) ).

% Ints_1
tff(fact_2127_Ints__1,axiom,
    member_real(one_one_real,ring_1_Ints_real) ).

% Ints_1
tff(fact_2128_Ints__1,axiom,
    member_rat(one_one_rat,ring_1_Ints_rat) ).

% Ints_1
tff(fact_2129_Ints__1,axiom,
    aa_set_int_o(member_int(one_one_int),ring_1_Ints_int) ).

% Ints_1
tff(fact_2130_fact__in__Ints,axiom,
    ! [Na: nat] : aa_set_int_o(member_int(aa_nat_int(semiri1406184849735516958ct_int,Na)),ring_1_Ints_int) ).

% fact_in_Ints
tff(fact_2131_fact__in__Ints,axiom,
    ! [Na: nat] : member_real(aa_nat_real(semiri2265585572941072030t_real,Na),ring_1_Ints_real) ).

% fact_in_Ints
tff(fact_2132_Suc__length__conv,axiom,
    ! [Na: nat,Xs: list_int] :
      ( ( aa_nat_nat(suc,Na) = size_size_list_int(Xs) )
    <=> ? [Y5: int,Ys3: list_int] :
          ( ( Xs = cons_int(Y5,Ys3) )
          & ( size_size_list_int(Ys3) = Na ) ) ) ).

% Suc_length_conv
tff(fact_2133_Suc__length__conv,axiom,
    ! [Na: nat,Xs: list_VEBT_VEBT] :
      ( ( aa_nat_nat(suc,Na) = size_s6755466524823107622T_VEBT(Xs) )
    <=> ? [Y5: vEBT_VEBT,Ys3: list_VEBT_VEBT] :
          ( ( Xs = cons_VEBT_VEBT(Y5,Ys3) )
          & ( size_s6755466524823107622T_VEBT(Ys3) = Na ) ) ) ).

% Suc_length_conv
tff(fact_2134_Suc__length__conv,axiom,
    ! [Na: nat,Xs: list_o] :
      ( ( aa_nat_nat(suc,Na) = size_size_list_o(Xs) )
    <=> ? [Y5: $o,Ys3: list_o] :
          ( ( Xs = cons_o((Y5),Ys3) )
          & ( size_size_list_o(Ys3) = Na ) ) ) ).

% Suc_length_conv
tff(fact_2135_Suc__length__conv,axiom,
    ! [Na: nat,Xs: list_nat] :
      ( ( aa_nat_nat(suc,Na) = size_size_list_nat(Xs) )
    <=> ? [Y5: nat,Ys3: list_nat] :
          ( ( Xs = cons_nat(Y5,Ys3) )
          & ( size_size_list_nat(Ys3) = Na ) ) ) ).

% Suc_length_conv
tff(fact_2136_length__Suc__conv,axiom,
    ! [Xs: list_int,Na: nat] :
      ( ( size_size_list_int(Xs) = aa_nat_nat(suc,Na) )
    <=> ? [Y5: int,Ys3: list_int] :
          ( ( Xs = cons_int(Y5,Ys3) )
          & ( size_size_list_int(Ys3) = Na ) ) ) ).

% length_Suc_conv
tff(fact_2137_length__Suc__conv,axiom,
    ! [Xs: list_VEBT_VEBT,Na: nat] :
      ( ( size_s6755466524823107622T_VEBT(Xs) = aa_nat_nat(suc,Na) )
    <=> ? [Y5: vEBT_VEBT,Ys3: list_VEBT_VEBT] :
          ( ( Xs = cons_VEBT_VEBT(Y5,Ys3) )
          & ( size_s6755466524823107622T_VEBT(Ys3) = Na ) ) ) ).

% length_Suc_conv
tff(fact_2138_length__Suc__conv,axiom,
    ! [Xs: list_o,Na: nat] :
      ( ( size_size_list_o(Xs) = aa_nat_nat(suc,Na) )
    <=> ? [Y5: $o,Ys3: list_o] :
          ( ( Xs = cons_o((Y5),Ys3) )
          & ( size_size_list_o(Ys3) = Na ) ) ) ).

% length_Suc_conv
tff(fact_2139_length__Suc__conv,axiom,
    ! [Xs: list_nat,Na: nat] :
      ( ( size_size_list_nat(Xs) = aa_nat_nat(suc,Na) )
    <=> ? [Y5: nat,Ys3: list_nat] :
          ( ( Xs = cons_nat(Y5,Ys3) )
          & ( size_size_list_nat(Ys3) = Na ) ) ) ).

% length_Suc_conv
tff(fact_2140_set__subset__Cons,axiom,
    ! [Xs: list_VEBT_VEBT,Xb: vEBT_VEBT] : ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(Xs),set_VEBT_VEBT2(cons_VEBT_VEBT(Xb,Xs))) ).

% set_subset_Cons
tff(fact_2141_set__subset__Cons,axiom,
    ! [Xs: list_nat,Xb: nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,set_nat2(Xs)),set_nat2(cons_nat(Xb,Xs))) ).

% set_subset_Cons
tff(fact_2142_set__subset__Cons,axiom,
    ! [Xs: list_int,Xb: int] : aa_set_int_o(ord_less_eq_set_int(set_int2(Xs)),set_int2(cons_int(Xb,Xs))) ).

% set_subset_Cons
tff(fact_2143_impossible__Cons,axiom,
    ! [Xs: list_int,Ys2: list_int,Xb: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_int(Xs)),size_size_list_int(Ys2))
     => ( Xs != cons_int(Xb,Ys2) ) ) ).

% impossible_Cons
tff(fact_2144_impossible__Cons,axiom,
    ! [Xs: list_VEBT_VEBT,Ys2: list_VEBT_VEBT,Xb: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_s6755466524823107622T_VEBT(Xs)),size_s6755466524823107622T_VEBT(Ys2))
     => ( Xs != cons_VEBT_VEBT(Xb,Ys2) ) ) ).

% impossible_Cons
tff(fact_2145_impossible__Cons,axiom,
    ! [Xs: list_o,Ys2: list_o,Xb: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_o(Xs)),size_size_list_o(Ys2))
     => ( Xs != cons_o((Xb),Ys2) ) ) ).

% impossible_Cons
tff(fact_2146_impossible__Cons,axiom,
    ! [Xs: list_nat,Ys2: list_nat,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_nat(Xs)),size_size_list_nat(Ys2))
     => ( Xs != cons_nat(Xb,Ys2) ) ) ).

% impossible_Cons
tff(fact_2147_Ints__double__eq__0__iff,axiom,
    ! [Aa: real] :
      ( member_real(Aa,ring_1_Ints_real)
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),Aa) = zero_zero_real )
      <=> ( Aa = zero_zero_real ) ) ) ).

% Ints_double_eq_0_iff
tff(fact_2148_Ints__double__eq__0__iff,axiom,
    ! [Aa: rat] :
      ( member_rat(Aa,ring_1_Ints_rat)
     => ( ( aa_rat_rat(plus_plus_rat(Aa),Aa) = zero_zero_rat )
      <=> ( Aa = zero_zero_rat ) ) ) ).

% Ints_double_eq_0_iff
tff(fact_2149_Ints__double__eq__0__iff,axiom,
    ! [Aa: int] :
      ( aa_set_int_o(member_int(Aa),ring_1_Ints_int)
     => ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),Aa) = zero_zero_int )
      <=> ( Aa = zero_zero_int ) ) ) ).

% Ints_double_eq_0_iff
tff(fact_2150_Suc__le__length__iff,axiom,
    ! [Na: nat,Xs: list_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Na)),size_size_list_int(Xs))
    <=> ? [X4: int,Ys3: list_int] :
          ( ( Xs = cons_int(X4,Ys3) )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),size_size_list_int(Ys3)) ) ) ).

% Suc_le_length_iff
tff(fact_2151_Suc__le__length__iff,axiom,
    ! [Na: nat,Xs: list_VEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Na)),size_s6755466524823107622T_VEBT(Xs))
    <=> ? [X4: vEBT_VEBT,Ys3: list_VEBT_VEBT] :
          ( ( Xs = cons_VEBT_VEBT(X4,Ys3) )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),size_s6755466524823107622T_VEBT(Ys3)) ) ) ).

% Suc_le_length_iff
tff(fact_2152_Suc__le__length__iff,axiom,
    ! [Na: nat,Xs: list_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Na)),size_size_list_o(Xs))
    <=> ? [X4: $o,Ys3: list_o] :
          ( ( Xs = cons_o((X4),Ys3) )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),size_size_list_o(Ys3)) ) ) ).

% Suc_le_length_iff
tff(fact_2153_Suc__le__length__iff,axiom,
    ! [Na: nat,Xs: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Na)),size_size_list_nat(Xs))
    <=> ? [X4: nat,Ys3: list_nat] :
          ( ( Xs = cons_nat(X4,Ys3) )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),size_size_list_nat(Ys3)) ) ) ).

% Suc_le_length_iff
tff(fact_2154_Ints__odd__nonzero,axiom,
    ! [Aa: complex] :
      ( aa_set_complex_o(member_complex(Aa),ring_1_Ints_complex)
     => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(plus_plus_complex(one_one_complex),Aa)),Aa) != zero_zero_complex ) ) ).

% Ints_odd_nonzero
tff(fact_2155_Ints__odd__nonzero,axiom,
    ! [Aa: real] :
      ( member_real(Aa,ring_1_Ints_real)
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Aa)),Aa) != zero_zero_real ) ) ).

% Ints_odd_nonzero
tff(fact_2156_Ints__odd__nonzero,axiom,
    ! [Aa: rat] :
      ( member_rat(Aa,ring_1_Ints_rat)
     => ( aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(one_one_rat),Aa)),Aa) != zero_zero_rat ) ) ).

% Ints_odd_nonzero
tff(fact_2157_Ints__odd__nonzero,axiom,
    ! [Aa: int] :
      ( aa_set_int_o(member_int(Aa),ring_1_Ints_int)
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),Aa)),Aa) != zero_zero_int ) ) ).

% Ints_odd_nonzero
tff(fact_2158_list_Osize_I4_J,axiom,
    ! [X21: int,X22: list_int] : size_size_list_int(cons_int(X21,X22)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,size_size_list_int(X22)),aa_nat_nat(suc,zero_zero_nat)) ).

% list.size(4)
tff(fact_2159_list_Osize_I4_J,axiom,
    ! [X21: vEBT_VEBT,X22: list_VEBT_VEBT] : size_s6755466524823107622T_VEBT(cons_VEBT_VEBT(X21,X22)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,size_s6755466524823107622T_VEBT(X22)),aa_nat_nat(suc,zero_zero_nat)) ).

% list.size(4)
tff(fact_2160_list_Osize_I4_J,axiom,
    ! [X21: $o,X22: list_o] : size_size_list_o(cons_o((X21),X22)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,size_size_list_o(X22)),aa_nat_nat(suc,zero_zero_nat)) ).

% list.size(4)
tff(fact_2161_list_Osize_I4_J,axiom,
    ! [X21: nat,X22: list_nat] : size_size_list_nat(cons_nat(X21,X22)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,size_size_list_nat(X22)),aa_nat_nat(suc,zero_zero_nat)) ).

% list.size(4)
tff(fact_2162_Ints__odd__less__0,axiom,
    ! [Aa: real] :
      ( member_real(Aa,ring_1_Ints_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Aa)),Aa)),zero_zero_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),zero_zero_real) ) ) ).

% Ints_odd_less_0
tff(fact_2163_Ints__odd__less__0,axiom,
    ! [Aa: rat] :
      ( member_rat(Aa,ring_1_Ints_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(one_one_rat),Aa)),Aa)),zero_zero_rat)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),zero_zero_rat) ) ) ).

% Ints_odd_less_0
tff(fact_2164_Ints__odd__less__0,axiom,
    ! [Aa: int] :
      ( aa_set_int_o(member_int(Aa),ring_1_Ints_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),Aa)),Aa)),zero_zero_int)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),zero_zero_int) ) ) ).

% Ints_odd_less_0
tff(fact_2165_nth__Cons_H,axiom,
    ! [Xb: vEBT_VEBT,Xs: list_VEBT_VEBT,Na: nat] :
      nth_VEBT_VEBT(cons_VEBT_VEBT(Xb,Xs),Na) = $ite(Na = zero_zero_nat,Xb,nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat))) ).

% nth_Cons'
tff(fact_2166_nth__Cons_H,axiom,
    ! [Xb: int,Xs: list_int,Na: nat] :
      aa_nat_int(nth_int(cons_int(Xb,Xs)),Na) = $ite(Na = zero_zero_nat,Xb,aa_nat_int(nth_int(Xs),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat))) ).

% nth_Cons'
tff(fact_2167_nth__Cons_H,axiom,
    ! [Xb: nat,Xs: list_nat,Na: nat] :
      aa_nat_nat(nth_nat(cons_nat(Xb,Xs)),Na) = $ite(Na = zero_zero_nat,Xb,aa_nat_nat(nth_nat(Xs),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat))) ).

% nth_Cons'
tff(fact_2168_nth__equal__first__eq,axiom,
    ! [Xb: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat,Na: nat] :
      ( ~ member8440522571783428010at_nat(Xb,set_Pr5648618587558075414at_nat(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),size_s5460976970255530739at_nat(Xs))
       => ( ( nth_Pr7617993195940197384at_nat(cons_P6512896166579812791at_nat(Xb,Xs),Na) = Xb )
        <=> ( Na = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
tff(fact_2169_nth__equal__first__eq,axiom,
    ! [Xb: real,Xs: list_real,Na: nat] :
      ( ~ member_real(Xb,set_real2(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),size_size_list_real(Xs))
       => ( ( aa_nat_real(nth_real(cons_real(Xb,Xs)),Na) = Xb )
        <=> ( Na = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
tff(fact_2170_nth__equal__first__eq,axiom,
    ! [Xb: set_nat,Xs: list_set_nat,Na: nat] :
      ( ~ member_set_nat(Xb,set_set_nat2(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),size_s3254054031482475050et_nat(Xs))
       => ( ( nth_set_nat(cons_set_nat(Xb,Xs),Na) = Xb )
        <=> ( Na = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
tff(fact_2171_nth__equal__first__eq,axiom,
    ! [Xb: int,Xs: list_int,Na: nat] :
      ( ~ aa_set_int_o(member_int(Xb),set_int2(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),size_size_list_int(Xs))
       => ( ( aa_nat_int(nth_int(cons_int(Xb,Xs)),Na) = Xb )
        <=> ( Na = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
tff(fact_2172_nth__equal__first__eq,axiom,
    ! [Xb: vEBT_VEBT,Xs: list_VEBT_VEBT,Na: nat] :
      ( ~ member_VEBT_VEBT(Xb,set_VEBT_VEBT2(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),size_s6755466524823107622T_VEBT(Xs))
       => ( ( nth_VEBT_VEBT(cons_VEBT_VEBT(Xb,Xs),Na) = Xb )
        <=> ( Na = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
tff(fact_2173_nth__equal__first__eq,axiom,
    ! [Xb: $o,Xs: list_o,Na: nat] :
      ( ~ member_o((Xb),set_o2(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),size_size_list_o(Xs))
       => ( ( aa_nat_o(nth_o(cons_o((Xb),Xs)),Na)
          <=> (Xb) )
        <=> ( Na = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
tff(fact_2174_nth__equal__first__eq,axiom,
    ! [Xb: nat,Xs: list_nat,Na: nat] :
      ( ~ aa_set_nat_o(member_nat(Xb),set_nat2(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),size_size_list_nat(Xs))
       => ( ( aa_nat_nat(nth_nat(cons_nat(Xb,Xs)),Na) = Xb )
        <=> ( Na = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
tff(fact_2175_nth__non__equal__first__eq,axiom,
    ! [Xb: vEBT_VEBT,Y: vEBT_VEBT,Xs: list_VEBT_VEBT,Na: nat] :
      ( ( Xb != Y )
     => ( ( nth_VEBT_VEBT(cons_VEBT_VEBT(Xb,Xs),Na) = Y )
      <=> ( ( nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat)) = Y )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ) ) ).

% nth_non_equal_first_eq
tff(fact_2176_nth__non__equal__first__eq,axiom,
    ! [Xb: int,Y: int,Xs: list_int,Na: nat] :
      ( ( Xb != Y )
     => ( ( aa_nat_int(nth_int(cons_int(Xb,Xs)),Na) = Y )
      <=> ( ( aa_nat_int(nth_int(Xs),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat)) = Y )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ) ) ).

% nth_non_equal_first_eq
tff(fact_2177_nth__non__equal__first__eq,axiom,
    ! [Xb: nat,Y: nat,Xs: list_nat,Na: nat] :
      ( ( Xb != Y )
     => ( ( aa_nat_nat(nth_nat(cons_nat(Xb,Xs)),Na) = Y )
      <=> ( ( aa_nat_nat(nth_nat(Xs),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat)) = Y )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ) ) ).

% nth_non_equal_first_eq
tff(fact_2178_length__Cons,axiom,
    ! [Xb: int,Xs: list_int] : size_size_list_int(cons_int(Xb,Xs)) = aa_nat_nat(suc,size_size_list_int(Xs)) ).

% length_Cons
tff(fact_2179_length__Cons,axiom,
    ! [Xb: vEBT_VEBT,Xs: list_VEBT_VEBT] : size_s6755466524823107622T_VEBT(cons_VEBT_VEBT(Xb,Xs)) = aa_nat_nat(suc,size_s6755466524823107622T_VEBT(Xs)) ).

% length_Cons
tff(fact_2180_length__Cons,axiom,
    ! [Xb: $o,Xs: list_o] : size_size_list_o(cons_o((Xb),Xs)) = aa_nat_nat(suc,size_size_list_o(Xs)) ).

% length_Cons
tff(fact_2181_length__Cons,axiom,
    ! [Xb: nat,Xs: list_nat] : size_size_list_nat(cons_nat(Xb,Xs)) = aa_nat_nat(suc,size_size_list_nat(Xs)) ).

% length_Cons
tff(fact_2182_nth__zip,axiom,
    ! [I: nat,Xs: list_Code_integer,Ys2: list_Code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s3445333598471063425nteger(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s3445333598471063425nteger(Ys2))
       => ( nth_Pr2304437835452373666nteger(zip_Co3543743374963494515nteger(Xs,Ys2),I) = produc1086072967326762835nteger(aa_nat_Code_integer(nth_Code_integer(Xs),I),aa_nat_Code_integer(nth_Code_integer(Ys2),I)) ) ) ) ).

% nth_zip
tff(fact_2183_nth__zip,axiom,
    ! [I: nat,Xs: list_int,Ys2: list_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_int(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_int(Ys2))
       => ( nth_Pr4439495888332055232nt_int(zip_int_int(Xs,Ys2),I) = product_Pair_int_int(aa_nat_int(nth_int(Xs),I),aa_nat_int(nth_int(Ys2),I)) ) ) ) ).

% nth_zip
tff(fact_2184_nth__zip,axiom,
    ! [I: nat,Xs: list_int,Ys2: list_VEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_int(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(Ys2))
       => ( nth_Pr3474266648193625910T_VEBT(zip_int_VEBT_VEBT(Xs,Ys2),I) = produc3329399203697025711T_VEBT(aa_nat_int(nth_int(Xs),I),nth_VEBT_VEBT(Ys2,I)) ) ) ) ).

% nth_zip
tff(fact_2185_nth__zip,axiom,
    ! [I: nat,Xs: list_int,Ys2: list_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_int(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_o(Ys2))
       => ( nth_Pr7514405829937366042_int_o(zip_int_o(Xs,Ys2),I) = product_Pair_int_o(aa_nat_int(nth_int(Xs),I),aa_nat_o(nth_o(Ys2),I)) ) ) ) ).

% nth_zip
tff(fact_2186_nth__zip,axiom,
    ! [I: nat,Xs: list_Code_integer,Ys2: list_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s3445333598471063425nteger(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_o(Ys2))
       => ( nth_Pr8522763379788166057eger_o(zip_Code_integer_o(Xs,Ys2),I) = produc6677183202524767010eger_o(aa_nat_Code_integer(nth_Code_integer(Xs),I),aa_nat_o(nth_o(Ys2),I)) ) ) ) ).

% nth_zip
tff(fact_2187_nth__zip,axiom,
    ! [I: nat,Xs: list_int,Ys2: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_int(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_nat(Ys2))
       => ( nth_Pr8617346907841251940nt_nat(zip_int_nat(Xs,Ys2),I) = product_Pair_int_nat(aa_nat_int(nth_int(Xs),I),aa_nat_nat(nth_nat(Ys2),I)) ) ) ) ).

% nth_zip
tff(fact_2188_nth__zip,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,Ys2: list_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_int(Ys2))
       => ( nth_Pr6837108013167703752BT_int(zip_VEBT_VEBT_int(Xs,Ys2),I) = produc736041933913180425BT_int(nth_VEBT_VEBT(Xs,I),aa_nat_int(nth_int(Ys2),I)) ) ) ) ).

% nth_zip
tff(fact_2189_nth__zip,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,Ys2: list_VEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(Ys2))
       => ( nth_Pr4953567300277697838T_VEBT(zip_VE537291747668921783T_VEBT(Xs,Ys2),I) = produc537772716801021591T_VEBT(nth_VEBT_VEBT(Xs,I),nth_VEBT_VEBT(Ys2,I)) ) ) ) ).

% nth_zip
tff(fact_2190_nth__zip,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,Ys2: list_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_o(Ys2))
       => ( nth_Pr4606735188037164562VEBT_o(zip_VEBT_VEBT_o(Xs,Ys2),I) = produc8721562602347293563VEBT_o(nth_VEBT_VEBT(Xs,I),aa_nat_o(nth_o(Ys2),I)) ) ) ) ).

% nth_zip
tff(fact_2191_nth__zip,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,Ys2: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_nat(Ys2))
       => ( nth_Pr1791586995822124652BT_nat(zip_VEBT_VEBT_nat(Xs,Ys2),I) = produc738532404422230701BT_nat(nth_VEBT_VEBT(Xs,I),aa_nat_nat(nth_nat(Ys2),I)) ) ) ) ).

% nth_zip
tff(fact_2192_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_2193_dbl__dec__def,axiom,
    ! [Xb: real] : aa_real_real(neg_nu6075765906172075777c_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Xb)),one_one_real) ).

% dbl_dec_def
tff(fact_2194_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_2195_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_2196_count__list_Osimps_I2_J,axiom,
    ! [Xb: int,Xs: list_int,Y: int] :
      aa_int_nat(count_list_int(cons_int(Xb,Xs)),Y) = $ite(Xb = Y,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_int_nat(count_list_int(Xs),Y)),one_one_nat),aa_int_nat(count_list_int(Xs),Y)) ).

% count_list.simps(2)
tff(fact_2197_count__list_Osimps_I2_J,axiom,
    ! [Xb: nat,Xs: list_nat,Y: nat] :
      aa_nat_nat(count_list_nat(cons_nat(Xb,Xs)),Y) = $ite(Xb = Y,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(count_list_nat(Xs),Y)),one_one_nat),aa_nat_nat(count_list_nat(Xs),Y)) ).

% count_list.simps(2)
tff(fact_2198_count__notin,axiom,
    ! [Xb: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( ~ member8440522571783428010at_nat(Xb,set_Pr5648618587558075414at_nat(Xs))
     => ( aa_Pro4243173770044397973at_nat(count_4203492906077236349at_nat(Xs),Xb) = zero_zero_nat ) ) ).

% count_notin
tff(fact_2199_count__notin,axiom,
    ! [Xb: real,Xs: list_real] :
      ( ~ member_real(Xb,set_real2(Xs))
     => ( aa_real_nat(count_list_real(Xs),Xb) = zero_zero_nat ) ) ).

% count_notin
tff(fact_2200_count__notin,axiom,
    ! [Xb: set_nat,Xs: list_set_nat] :
      ( ~ member_set_nat(Xb,set_set_nat2(Xs))
     => ( aa_set_nat_nat(count_list_set_nat(Xs),Xb) = zero_zero_nat ) ) ).

% count_notin
tff(fact_2201_count__notin,axiom,
    ! [Xb: int,Xs: list_int] :
      ( ~ aa_set_int_o(member_int(Xb),set_int2(Xs))
     => ( aa_int_nat(count_list_int(Xs),Xb) = zero_zero_nat ) ) ).

% count_notin
tff(fact_2202_count__notin,axiom,
    ! [Xb: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( ~ member_VEBT_VEBT(Xb,set_VEBT_VEBT2(Xs))
     => ( aa_VEBT_VEBT_nat(count_list_VEBT_VEBT(Xs),Xb) = zero_zero_nat ) ) ).

% count_notin
tff(fact_2203_count__notin,axiom,
    ! [Xb: nat,Xs: list_nat] :
      ( ~ aa_set_nat_o(member_nat(Xb),set_nat2(Xs))
     => ( aa_nat_nat(count_list_nat(Xs),Xb) = zero_zero_nat ) ) ).

% count_notin
tff(fact_2204_Cons__lenlex__iff,axiom,
    ! [M: code_integer,Ms: list_Code_integer,Na: code_integer,Ns: list_Code_integer,R2: set_Pr4811707699266497531nteger] :
      ( member749217712838834276nteger(produc750622340256944499nteger(cons_Code_integer(M,Ms),cons_Code_integer(Na,Ns)),lenlex_Code_integer(R2))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_s3445333598471063425nteger(Ms)),size_s3445333598471063425nteger(Ns))
        | ( ( size_s3445333598471063425nteger(Ms) = size_s3445333598471063425nteger(Ns) )
          & member157494554546826820nteger(produc1086072967326762835nteger(M,Na),R2) )
        | ( ( M = Na )
          & member749217712838834276nteger(produc750622340256944499nteger(Ms,Ns),lenlex_Code_integer(R2)) ) ) ) ).

% Cons_lenlex_iff
tff(fact_2205_Cons__lenlex__iff,axiom,
    ! [M: product_prod_nat_nat,Ms: list_P6011104703257516679at_nat,Na: product_prod_nat_nat,Ns: list_P6011104703257516679at_nat,R2: set_Pr8693737435421807431at_nat] :
      ( member6693912407220327184at_nat(produc5943733680697469783at_nat(cons_P6512896166579812791at_nat(M,Ms),cons_P6512896166579812791at_nat(Na,Ns)),lenlex325483962726685836at_nat(R2))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_s5460976970255530739at_nat(Ms)),size_s5460976970255530739at_nat(Ns))
        | ( ( size_s5460976970255530739at_nat(Ms) = size_s5460976970255530739at_nat(Ns) )
          & member8206827879206165904at_nat(produc6161850002892822231at_nat(M,Na),R2) )
        | ( ( M = Na )
          & member6693912407220327184at_nat(produc5943733680697469783at_nat(Ms,Ns),lenlex325483962726685836at_nat(R2)) ) ) ) ).

% Cons_lenlex_iff
tff(fact_2206_Cons__lenlex__iff,axiom,
    ! [M: int,Ms: list_int,Na: int,Ns: list_int,R2: set_Pr958786334691620121nt_int] :
      ( member6698963635872716290st_int(produc364263696895485585st_int(cons_int(M,Ms),cons_int(Na,Ns)),lenlex_int(R2))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_size_list_int(Ms)),size_size_list_int(Ns))
        | ( ( size_size_list_int(Ms) = size_size_list_int(Ns) )
          & member5262025264175285858nt_int(product_Pair_int_int(M,Na),R2) )
        | ( ( M = Na )
          & member6698963635872716290st_int(produc364263696895485585st_int(Ms,Ns),lenlex_int(R2)) ) ) ) ).

% Cons_lenlex_iff
tff(fact_2207_Cons__lenlex__iff,axiom,
    ! [M: vEBT_VEBT,Ms: list_VEBT_VEBT,Na: vEBT_VEBT,Ns: list_VEBT_VEBT,R2: set_Pr6192946355708809607T_VEBT] :
      ( member4439316823752958928T_VEBT(produc3897820843166775703T_VEBT(cons_VEBT_VEBT(M,Ms),cons_VEBT_VEBT(Na,Ns)),lenlex_VEBT_VEBT(R2))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_s6755466524823107622T_VEBT(Ms)),size_s6755466524823107622T_VEBT(Ns))
        | ( ( size_s6755466524823107622T_VEBT(Ms) = size_s6755466524823107622T_VEBT(Ns) )
          & member568628332442017744T_VEBT(produc537772716801021591T_VEBT(M,Na),R2) )
        | ( ( M = Na )
          & member4439316823752958928T_VEBT(produc3897820843166775703T_VEBT(Ms,Ns),lenlex_VEBT_VEBT(R2)) ) ) ) ).

% Cons_lenlex_iff
tff(fact_2208_Cons__lenlex__iff,axiom,
    ! [M: $o,Ms: list_o,Na: $o,Ns: list_o,R2: set_Product_prod_o_o] :
      ( member4159035015898711888list_o(produc8435520187683070743list_o(cons_o((M),Ms),cons_o((Na),Ns)),lenlex_o(R2))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_size_list_o(Ms)),size_size_list_o(Ns))
        | ( ( size_size_list_o(Ms) = size_size_list_o(Ns) )
          & member7466972457876170832od_o_o(product_Pair_o_o((M),(Na)),R2) )
        | ( ( (M)
          <=> (Na) )
          & member4159035015898711888list_o(produc8435520187683070743list_o(Ms,Ns),lenlex_o(R2)) ) ) ) ).

% Cons_lenlex_iff
tff(fact_2209_Cons__lenlex__iff,axiom,
    ! [M: nat,Ms: list_nat,Na: nat,Ns: list_nat,R2: set_Pr1261947904930325089at_nat] :
      ( member7340969449405702474st_nat(produc2694037385005941721st_nat(cons_nat(M,Ms),cons_nat(Na,Ns)),lenlex_nat(R2))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_size_list_nat(Ms)),size_size_list_nat(Ns))
        | ( ( size_size_list_nat(Ms) = size_size_list_nat(Ns) )
          & member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(M),Na),R2) )
        | ( ( M = Na )
          & member7340969449405702474st_nat(produc2694037385005941721st_nat(Ms,Ns),lenlex_nat(R2)) ) ) ) ).

% Cons_lenlex_iff
tff(fact_2210_in__measures_I2_J,axiom,
    ! [Xb: code_integer,Y: code_integer,F: fun_Code_integer_nat,Fs: list_f8684324736568799034er_nat] :
      ( member157494554546826820nteger(produc1086072967326762835nteger(Xb,Y),measur8870801148506250077nteger(cons_f3120178438656451060er_nat(F,Fs)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_Code_integer_nat(F,Xb)),aa_Code_integer_nat(F,Y))
        | ( ( aa_Code_integer_nat(F,Xb) = aa_Code_integer_nat(F,Y) )
          & member157494554546826820nteger(produc1086072967326762835nteger(Xb,Y),measur8870801148506250077nteger(Fs)) ) ) ) ).

% in_measures(2)
tff(fact_2211_in__measures_I2_J,axiom,
    ! [Xb: product_prod_nat_nat,Y: product_prod_nat_nat,F: fun_Pr2602117396471556196at_nat,Fs: list_f8256639991604547178at_nat] :
      ( member8206827879206165904at_nat(produc6161850002892822231at_nat(Xb,Y),measur2679027848233739777at_nat(cons_f462676766793977114at_nat(F,Fs)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_Pro4243173770044397973at_nat(F,Xb)),aa_Pro4243173770044397973at_nat(F,Y))
        | ( ( aa_Pro4243173770044397973at_nat(F,Xb) = aa_Pro4243173770044397973at_nat(F,Y) )
          & member8206827879206165904at_nat(produc6161850002892822231at_nat(Xb,Y),measur2679027848233739777at_nat(Fs)) ) ) ) ).

% in_measures(2)
tff(fact_2212_in__measures_I2_J,axiom,
    ! [Xb: nat,Y: nat,F: fun_nat_nat,Fs: list_fun_nat_nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xb),Y),measures_nat(cons_fun_nat_nat(F,Fs)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,Xb)),aa_nat_nat(F,Y))
        | ( ( aa_nat_nat(F,Xb) = aa_nat_nat(F,Y) )
          & member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xb),Y),measures_nat(Fs)) ) ) ) ).

% in_measures(2)
tff(fact_2213_in__measures_I2_J,axiom,
    ! [Xb: int,Y: int,F: fun_int_nat,Fs: list_fun_int_nat] :
      ( member5262025264175285858nt_int(product_Pair_int_int(Xb,Y),measures_int(cons_fun_int_nat(F,Fs)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(F,Xb)),aa_int_nat(F,Y))
        | ( ( aa_int_nat(F,Xb) = aa_int_nat(F,Y) )
          & member5262025264175285858nt_int(product_Pair_int_int(Xb,Y),measures_int(Fs)) ) ) ) ).

% in_measures(2)
tff(fact_2214_intind,axiom,
    ! [I: nat,Na: nat,P: fun_int_o,Xb: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),Na)
     => ( aa_int_o(P,Xb)
       => aa_int_o(P,aa_nat_int(nth_int(replicate_int(Na,Xb)),I)) ) ) ).

% intind
tff(fact_2215_intind,axiom,
    ! [I: nat,Na: nat,P: fun_nat_o,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),Na)
     => ( aa_nat_o(P,Xb)
       => aa_nat_o(P,aa_nat_nat(nth_nat(replicate_nat(Na,Xb)),I)) ) ) ).

% intind
tff(fact_2216_intind,axiom,
    ! [I: nat,Na: nat,P: fun_VEBT_VEBT_o,Xb: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),Na)
     => ( aa_VEBT_VEBT_o(P,Xb)
       => aa_VEBT_VEBT_o(P,nth_VEBT_VEBT(replicate_VEBT_VEBT(Na,Xb),I)) ) ) ).

% intind
tff(fact_2217_frac__neg,axiom,
    ! [Xb: real] :
      aa_real_real(archim2898591450579166408c_real,aa_real_real(uminus_uminus_real,Xb)) = $ite(member_real(Xb,ring_1_Ints_real),zero_zero_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_real_real(archim2898591450579166408c_real,Xb))) ).

% frac_neg
tff(fact_2218_frac__neg,axiom,
    ! [Xb: rat] :
      aa_rat_rat(archimedean_frac_rat,aa_rat_rat(uminus_uminus_rat,Xb)) = $ite(member_rat(Xb,ring_1_Ints_rat),zero_zero_rat,aa_rat_rat(minus_minus_rat(one_one_rat),aa_rat_rat(archimedean_frac_rat,Xb))) ).

% frac_neg
tff(fact_2219_nth__Cons__numeral,axiom,
    ! [Xb: vEBT_VEBT,Xs: list_VEBT_VEBT,V: num] : nth_VEBT_VEBT(cons_VEBT_VEBT(Xb,Xs),aa_num_nat(numeral_numeral_nat,V)) = nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_num_nat(numeral_numeral_nat,V)),one_one_nat)) ).

% nth_Cons_numeral
tff(fact_2220_nth__Cons__numeral,axiom,
    ! [Xb: int,Xs: list_int,V: num] : aa_nat_int(nth_int(cons_int(Xb,Xs)),aa_num_nat(numeral_numeral_nat,V)) = aa_nat_int(nth_int(Xs),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_num_nat(numeral_numeral_nat,V)),one_one_nat)) ).

% nth_Cons_numeral
tff(fact_2221_nth__Cons__numeral,axiom,
    ! [Xb: nat,Xs: list_nat,V: num] : aa_nat_nat(nth_nat(cons_nat(Xb,Xs)),aa_num_nat(numeral_numeral_nat,V)) = aa_nat_nat(nth_nat(Xs),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_num_nat(numeral_numeral_nat,V)),one_one_nat)) ).

% nth_Cons_numeral
tff(fact_2222_numeral__eq__iff,axiom,
    ! [M: num,Na: num] :
      ( ( aa_num_nat(numeral_numeral_nat,M) = aa_num_nat(numeral_numeral_nat,Na) )
    <=> ( M = Na ) ) ).

% numeral_eq_iff
tff(fact_2223_numeral__eq__iff,axiom,
    ! [M: num,Na: num] :
      ( ( aa_num_real(numeral_numeral_real,M) = aa_num_real(numeral_numeral_real,Na) )
    <=> ( M = Na ) ) ).

% numeral_eq_iff
tff(fact_2224_numeral__eq__iff,axiom,
    ! [M: num,Na: num] :
      ( ( aa_num_int(numeral_numeral_int,M) = aa_num_int(numeral_numeral_int,Na) )
    <=> ( M = Na ) ) ).

% numeral_eq_iff
tff(fact_2225_numeral__eq__iff,axiom,
    ! [M: num,Na: num] :
      ( ( numera1916890842035813515d_enat(M) = numera1916890842035813515d_enat(Na) )
    <=> ( M = Na ) ) ).

% numeral_eq_iff
tff(fact_2226_numeral__eq__iff,axiom,
    ! [M: num,Na: num] :
      ( ( aa_num_Code_integer(numera6620942414471956472nteger,M) = aa_num_Code_integer(numera6620942414471956472nteger,Na) )
    <=> ( M = Na ) ) ).

% numeral_eq_iff
tff(fact_2227_add_Oinverse__inverse,axiom,
    ! [Aa: real] : aa_real_real(uminus_uminus_real,aa_real_real(uminus_uminus_real,Aa)) = Aa ).

% add.inverse_inverse
tff(fact_2228_add_Oinverse__inverse,axiom,
    ! [Aa: int] : aa_int_int(uminus_uminus_int,aa_int_int(uminus_uminus_int,Aa)) = Aa ).

% add.inverse_inverse
tff(fact_2229_add_Oinverse__inverse,axiom,
    ! [Aa: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa)) = Aa ).

% add.inverse_inverse
tff(fact_2230_add_Oinverse__inverse,axiom,
    ! [Aa: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(uminus_uminus_rat,Aa)) = Aa ).

% add.inverse_inverse
tff(fact_2231_neg__equal__iff__equal,axiom,
    ! [Aa: real,Ba: real] :
      ( ( aa_real_real(uminus_uminus_real,Aa) = aa_real_real(uminus_uminus_real,Ba) )
    <=> ( Aa = Ba ) ) ).

% neg_equal_iff_equal
tff(fact_2232_neg__equal__iff__equal,axiom,
    ! [Aa: int,Ba: int] :
      ( ( aa_int_int(uminus_uminus_int,Aa) = aa_int_int(uminus_uminus_int,Ba) )
    <=> ( Aa = Ba ) ) ).

% neg_equal_iff_equal
tff(fact_2233_neg__equal__iff__equal,axiom,
    ! [Aa: code_integer,Ba: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Ba) )
    <=> ( Aa = Ba ) ) ).

% neg_equal_iff_equal
tff(fact_2234_neg__equal__iff__equal,axiom,
    ! [Aa: rat,Ba: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,Aa) = aa_rat_rat(uminus_uminus_rat,Ba) )
    <=> ( Aa = Ba ) ) ).

% neg_equal_iff_equal
tff(fact_2235_Compl__subset__Compl__iff,axiom,
    ! [A3: set_int,B4: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(uminus1532241313380277803et_int(A3)),uminus1532241313380277803et_int(B4))
    <=> aa_set_int_o(ord_less_eq_set_int(B4),A3) ) ).

% Compl_subset_Compl_iff
tff(fact_2236_Compl__anti__mono,axiom,
    ! [A3: set_int,B4: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A3),B4)
     => aa_set_int_o(ord_less_eq_set_int(uminus1532241313380277803et_int(B4)),uminus1532241313380277803et_int(A3)) ) ).

% Compl_anti_mono
tff(fact_2237_numeral__le__iff,axiom,
    ! [M: num,Na: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,M)),aa_num_real(numeral_numeral_real,Na))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,M),Na) ) ).

% numeral_le_iff
tff(fact_2238_numeral__le__iff,axiom,
    ! [M: num,Na: num] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(numera1916890842035813515d_enat(M)),numera1916890842035813515d_enat(Na))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,M),Na) ) ).

% numeral_le_iff
tff(fact_2239_numeral__le__iff,axiom,
    ! [M: num,Na: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)),aa_num_Code_integer(numera6620942414471956472nteger,Na))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,M),Na) ) ).

% numeral_le_iff
tff(fact_2240_numeral__le__iff,axiom,
    ! [M: num,Na: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,M)),aa_num_rat(numeral_numeral_rat,Na))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,M),Na) ) ).

% numeral_le_iff
tff(fact_2241_numeral__le__iff,axiom,
    ! [M: num,Na: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,Na))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,M),Na) ) ).

% numeral_le_iff
tff(fact_2242_numeral__le__iff,axiom,
    ! [M: num,Na: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,Na))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,M),Na) ) ).

% numeral_le_iff
tff(fact_2243_numeral__less__iff,axiom,
    ! [M: num,Na: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,M)),aa_num_rat(numeral_numeral_rat,Na))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,M),Na) ) ).

% numeral_less_iff
tff(fact_2244_numeral__less__iff,axiom,
    ! [M: num,Na: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,Na))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,M),Na) ) ).

% numeral_less_iff
tff(fact_2245_numeral__less__iff,axiom,
    ! [M: num,Na: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,M)),aa_num_real(numeral_numeral_real,Na))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,M),Na) ) ).

% numeral_less_iff
tff(fact_2246_numeral__less__iff,axiom,
    ! [M: num,Na: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,Na))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,M),Na) ) ).

% numeral_less_iff
tff(fact_2247_numeral__less__iff,axiom,
    ! [M: num,Na: num] :
      ( aa_Extended_enat_o(ord_le72135733267957522d_enat(numera1916890842035813515d_enat(M)),numera1916890842035813515d_enat(Na))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,M),Na) ) ).

% numeral_less_iff
tff(fact_2248_numeral__less__iff,axiom,
    ! [M: num,Na: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)),aa_num_Code_integer(numera6620942414471956472nteger,Na))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,M),Na) ) ).

% numeral_less_iff
tff(fact_2249_compl__le__compl__iff,axiom,
    ! [Xb: set_int,Y: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(uminus1532241313380277803et_int(Xb)),uminus1532241313380277803et_int(Y))
    <=> aa_set_int_o(ord_less_eq_set_int(Y),Xb) ) ).

% compl_le_compl_iff
tff(fact_2250_neg__le__iff__le,axiom,
    ! [Ba: real,Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,Ba)),aa_real_real(uminus_uminus_real,Aa))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),Ba) ) ).

% neg_le_iff_le
tff(fact_2251_neg__le__iff__le,axiom,
    ! [Ba: code_integer,Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Ba)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Aa),Ba) ) ).

% neg_le_iff_le
tff(fact_2252_neg__le__iff__le,axiom,
    ! [Ba: rat,Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,Ba)),aa_rat_rat(uminus_uminus_rat,Aa))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),Ba) ) ).

% neg_le_iff_le
tff(fact_2253_neg__le__iff__le,axiom,
    ! [Ba: int,Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,Ba)),aa_int_int(uminus_uminus_int,Aa))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),Ba) ) ).

% neg_le_iff_le
tff(fact_2254_add_Oinverse__neutral,axiom,
    aa_real_real(uminus_uminus_real,zero_zero_real) = zero_zero_real ).

% add.inverse_neutral
tff(fact_2255_add_Oinverse__neutral,axiom,
    aa_int_int(uminus_uminus_int,zero_zero_int) = zero_zero_int ).

% add.inverse_neutral
tff(fact_2256_add_Oinverse__neutral,axiom,
    aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% add.inverse_neutral
tff(fact_2257_add_Oinverse__neutral,axiom,
    aa_rat_rat(uminus_uminus_rat,zero_zero_rat) = zero_zero_rat ).

% add.inverse_neutral
tff(fact_2258_neg__0__equal__iff__equal,axiom,
    ! [Aa: real] :
      ( ( zero_zero_real = aa_real_real(uminus_uminus_real,Aa) )
    <=> ( zero_zero_real = Aa ) ) ).

% neg_0_equal_iff_equal
tff(fact_2259_neg__0__equal__iff__equal,axiom,
    ! [Aa: int] :
      ( ( zero_zero_int = aa_int_int(uminus_uminus_int,Aa) )
    <=> ( zero_zero_int = Aa ) ) ).

% neg_0_equal_iff_equal
tff(fact_2260_neg__0__equal__iff__equal,axiom,
    ! [Aa: code_integer] :
      ( ( zero_z3403309356797280102nteger = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa) )
    <=> ( zero_z3403309356797280102nteger = Aa ) ) ).

% neg_0_equal_iff_equal
tff(fact_2261_neg__0__equal__iff__equal,axiom,
    ! [Aa: rat] :
      ( ( zero_zero_rat = aa_rat_rat(uminus_uminus_rat,Aa) )
    <=> ( zero_zero_rat = Aa ) ) ).

% neg_0_equal_iff_equal
tff(fact_2262_neg__equal__0__iff__equal,axiom,
    ! [Aa: real] :
      ( ( aa_real_real(uminus_uminus_real,Aa) = zero_zero_real )
    <=> ( Aa = zero_zero_real ) ) ).

% neg_equal_0_iff_equal
tff(fact_2263_neg__equal__0__iff__equal,axiom,
    ! [Aa: int] :
      ( ( aa_int_int(uminus_uminus_int,Aa) = zero_zero_int )
    <=> ( Aa = zero_zero_int ) ) ).

% neg_equal_0_iff_equal
tff(fact_2264_neg__equal__0__iff__equal,axiom,
    ! [Aa: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa) = zero_z3403309356797280102nteger )
    <=> ( Aa = zero_z3403309356797280102nteger ) ) ).

% neg_equal_0_iff_equal
tff(fact_2265_neg__equal__0__iff__equal,axiom,
    ! [Aa: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,Aa) = zero_zero_rat )
    <=> ( Aa = zero_zero_rat ) ) ).

% neg_equal_0_iff_equal
tff(fact_2266_equal__neg__zero,axiom,
    ! [Aa: real] :
      ( ( Aa = aa_real_real(uminus_uminus_real,Aa) )
    <=> ( Aa = zero_zero_real ) ) ).

% equal_neg_zero
tff(fact_2267_equal__neg__zero,axiom,
    ! [Aa: int] :
      ( ( Aa = aa_int_int(uminus_uminus_int,Aa) )
    <=> ( Aa = zero_zero_int ) ) ).

% equal_neg_zero
tff(fact_2268_equal__neg__zero,axiom,
    ! [Aa: code_integer] :
      ( ( Aa = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa) )
    <=> ( Aa = zero_z3403309356797280102nteger ) ) ).

% equal_neg_zero
tff(fact_2269_equal__neg__zero,axiom,
    ! [Aa: rat] :
      ( ( Aa = aa_rat_rat(uminus_uminus_rat,Aa) )
    <=> ( Aa = zero_zero_rat ) ) ).

% equal_neg_zero
tff(fact_2270_neg__equal__zero,axiom,
    ! [Aa: real] :
      ( ( aa_real_real(uminus_uminus_real,Aa) = Aa )
    <=> ( Aa = zero_zero_real ) ) ).

% neg_equal_zero
tff(fact_2271_neg__equal__zero,axiom,
    ! [Aa: int] :
      ( ( aa_int_int(uminus_uminus_int,Aa) = Aa )
    <=> ( Aa = zero_zero_int ) ) ).

% neg_equal_zero
tff(fact_2272_neg__equal__zero,axiom,
    ! [Aa: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa) = Aa )
    <=> ( Aa = zero_z3403309356797280102nteger ) ) ).

% neg_equal_zero
tff(fact_2273_neg__equal__zero,axiom,
    ! [Aa: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,Aa) = Aa )
    <=> ( Aa = zero_zero_rat ) ) ).

% neg_equal_zero
tff(fact_2274_numeral__plus__numeral,axiom,
    ! [M: num,Na: num] : aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,M)),aa_num_rat(numeral_numeral_rat,Na)) = aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),Na)) ).

% numeral_plus_numeral
tff(fact_2275_numeral__plus__numeral,axiom,
    ! [M: num,Na: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,Na)) = aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),Na)) ).

% numeral_plus_numeral
tff(fact_2276_numeral__plus__numeral,axiom,
    ! [M: num,Na: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,M)),aa_num_real(numeral_numeral_real,Na)) = aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),Na)) ).

% numeral_plus_numeral
tff(fact_2277_numeral__plus__numeral,axiom,
    ! [M: num,Na: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,Na)) = aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),Na)) ).

% numeral_plus_numeral
tff(fact_2278_numeral__plus__numeral,axiom,
    ! [M: num,Na: num] : plus_p3455044024723400733d_enat(numera1916890842035813515d_enat(M),numera1916890842035813515d_enat(Na)) = numera1916890842035813515d_enat(aa_num_num(aa_num_fun_num_num(plus_plus_num,M),Na)) ).

% numeral_plus_numeral
tff(fact_2279_numeral__plus__numeral,axiom,
    ! [M: num,Na: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)),aa_num_Code_integer(numera6620942414471956472nteger,Na)) = aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),Na)) ).

% numeral_plus_numeral
tff(fact_2280_add__numeral__left,axiom,
    ! [V: num,W2: 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,W2)),Z)) = aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W2))),Z) ).

% add_numeral_left
tff(fact_2281_add__numeral__left,axiom,
    ! [V: num,W2: 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,W2)),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W2))),Z) ).

% add_numeral_left
tff(fact_2282_add__numeral__left,axiom,
    ! [V: num,W2: num,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,V)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,W2)),Z)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W2))),Z) ).

% add_numeral_left
tff(fact_2283_add__numeral__left,axiom,
    ! [V: num,W2: 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,W2)),Z)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W2))),Z) ).

% add_numeral_left
tff(fact_2284_add__numeral__left,axiom,
    ! [V: num,W2: num,Z: extended_enat] : plus_p3455044024723400733d_enat(numera1916890842035813515d_enat(V),plus_p3455044024723400733d_enat(numera1916890842035813515d_enat(W2),Z)) = plus_p3455044024723400733d_enat(numera1916890842035813515d_enat(aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W2)),Z) ).

% add_numeral_left
tff(fact_2285_add__numeral__left,axiom,
    ! [V: num,W2: num,Z: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_num_Code_integer(numera6620942414471956472nteger,V)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_num_Code_integer(numera6620942414471956472nteger,W2)),Z)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(aa_num_fun_num_num(plus_plus_num,V),W2))),Z) ).

% add_numeral_left
tff(fact_2286_neg__less__iff__less,axiom,
    ! [Ba: real,Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,Ba)),aa_real_real(uminus_uminus_real,Aa))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),Ba) ) ).

% neg_less_iff_less
tff(fact_2287_neg__less__iff__less,axiom,
    ! [Ba: int,Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,Ba)),aa_int_int(uminus_uminus_int,Aa))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),Ba) ) ).

% neg_less_iff_less
tff(fact_2288_neg__less__iff__less,axiom,
    ! [Ba: code_integer,Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Ba)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Aa),Ba) ) ).

% neg_less_iff_less
tff(fact_2289_neg__less__iff__less,axiom,
    ! [Ba: rat,Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,Ba)),aa_rat_rat(uminus_uminus_rat,Aa))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),Ba) ) ).

% neg_less_iff_less
tff(fact_2290_neg__numeral__eq__iff,axiom,
    ! [M: num,Na: num] :
      ( ( aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M)) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Na)) )
    <=> ( M = Na ) ) ).

% neg_numeral_eq_iff
tff(fact_2291_neg__numeral__eq__iff,axiom,
    ! [M: num,Na: num] :
      ( ( aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Na)) )
    <=> ( M = Na ) ) ).

% neg_numeral_eq_iff
tff(fact_2292_neg__numeral__eq__iff,axiom,
    ! [M: num,Na: num] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Na)) )
    <=> ( M = Na ) ) ).

% neg_numeral_eq_iff
tff(fact_2293_neg__numeral__eq__iff,axiom,
    ! [M: num,Na: num] :
      ( ( aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M)) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Na)) )
    <=> ( M = Na ) ) ).

% neg_numeral_eq_iff
tff(fact_2294_add__minus__cancel,axiom,
    ! [Aa: real,Ba: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,Aa)),Ba)) = Ba ).

% add_minus_cancel
tff(fact_2295_add__minus__cancel,axiom,
    ! [Aa: int,Ba: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,Aa)),Ba)) = Ba ).

% add_minus_cancel
tff(fact_2296_add__minus__cancel,axiom,
    ! [Aa: code_integer,Ba: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Aa),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa)),Ba)) = Ba ).

% add_minus_cancel
tff(fact_2297_add__minus__cancel,axiom,
    ! [Aa: rat,Ba: rat] : aa_rat_rat(plus_plus_rat(Aa),aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,Aa)),Ba)) = Ba ).

% add_minus_cancel
tff(fact_2298_minus__add__cancel,axiom,
    ! [Aa: real,Ba: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,Aa)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),Ba)) = Ba ).

% minus_add_cancel
tff(fact_2299_minus__add__cancel,axiom,
    ! [Aa: int,Ba: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,Aa)),aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),Ba)) = Ba ).

% minus_add_cancel
tff(fact_2300_minus__add__cancel,axiom,
    ! [Aa: code_integer,Ba: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Aa),Ba)) = Ba ).

% minus_add_cancel
tff(fact_2301_minus__add__cancel,axiom,
    ! [Aa: rat,Ba: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,Aa)),aa_rat_rat(plus_plus_rat(Aa),Ba)) = Ba ).

% minus_add_cancel
tff(fact_2302_minus__add__distrib,axiom,
    ! [Aa: real,Ba: real] : aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),Ba)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,Aa)),aa_real_real(uminus_uminus_real,Ba)) ).

% minus_add_distrib
tff(fact_2303_minus__add__distrib,axiom,
    ! [Aa: int,Ba: int] : aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),Ba)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,Aa)),aa_int_int(uminus_uminus_int,Ba)) ).

% minus_add_distrib
tff(fact_2304_minus__add__distrib,axiom,
    ! [Aa: code_integer,Ba: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Aa),Ba)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Ba)) ).

% minus_add_distrib
tff(fact_2305_minus__add__distrib,axiom,
    ! [Aa: rat,Ba: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(plus_plus_rat(Aa),Ba)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,Aa)),aa_rat_rat(uminus_uminus_rat,Ba)) ).

% minus_add_distrib
tff(fact_2306_minus__diff__eq,axiom,
    ! [Aa: real,Ba: real] : aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Aa),Ba)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Ba),Aa) ).

% minus_diff_eq
tff(fact_2307_minus__diff__eq,axiom,
    ! [Aa: int,Ba: int] : aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Aa),Ba)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,Ba),Aa) ).

% minus_diff_eq
tff(fact_2308_minus__diff__eq,axiom,
    ! [Aa: code_integer,Ba: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,Aa),Ba)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,Ba),Aa) ).

% minus_diff_eq
tff(fact_2309_minus__diff__eq,axiom,
    ! [Aa: rat,Ba: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(minus_minus_rat(Aa),Ba)) = aa_rat_rat(minus_minus_rat(Ba),Aa) ).

% minus_diff_eq
tff(fact_2310_replicate__eq__replicate,axiom,
    ! [M: nat,Xb: vEBT_VEBT,Na: nat,Y: vEBT_VEBT] :
      ( ( replicate_VEBT_VEBT(M,Xb) = replicate_VEBT_VEBT(Na,Y) )
    <=> ( ( M = Na )
        & ( ( M != zero_zero_nat )
         => ( Xb = Y ) ) ) ) ).

% replicate_eq_replicate
tff(fact_2311_dbl__dec__simps_I3_J,axiom,
    aa_complex_complex(neg_nu6511756317524482435omplex,one_one_complex) = one_one_complex ).

% dbl_dec_simps(3)
tff(fact_2312_dbl__dec__simps_I3_J,axiom,
    aa_real_real(neg_nu6075765906172075777c_real,one_one_real) = one_one_real ).

% dbl_dec_simps(3)
tff(fact_2313_dbl__dec__simps_I3_J,axiom,
    aa_rat_rat(neg_nu3179335615603231917ec_rat,one_one_rat) = one_one_rat ).

% dbl_dec_simps(3)
tff(fact_2314_dbl__dec__simps_I3_J,axiom,
    aa_int_int(neg_nu3811975205180677377ec_int,one_one_int) = one_one_int ).

% dbl_dec_simps(3)
tff(fact_2315_neg__0__le__iff__le,axiom,
    ! [Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(uminus_uminus_real,Aa))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),zero_zero_real) ) ).

% neg_0_le_iff_le
tff(fact_2316_neg__0__le__iff__le,axiom,
    ! [Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Aa),zero_z3403309356797280102nteger) ) ).

% neg_0_le_iff_le
tff(fact_2317_neg__0__le__iff__le,axiom,
    ! [Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,Aa))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),zero_zero_rat) ) ).

% neg_0_le_iff_le
tff(fact_2318_neg__0__le__iff__le,axiom,
    ! [Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(uminus_uminus_int,Aa))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),zero_zero_int) ) ).

% neg_0_le_iff_le
tff(fact_2319_neg__le__0__iff__le,axiom,
    ! [Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,Aa)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Aa) ) ).

% neg_le_0_iff_le
tff(fact_2320_neg__le__0__iff__le,axiom,
    ! [Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa)),zero_z3403309356797280102nteger)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Aa) ) ).

% neg_le_0_iff_le
tff(fact_2321_neg__le__0__iff__le,axiom,
    ! [Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,Aa)),zero_zero_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Aa) ) ).

% neg_le_0_iff_le
tff(fact_2322_neg__le__0__iff__le,axiom,
    ! [Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,Aa)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Aa) ) ).

% neg_le_0_iff_le
tff(fact_2323_less__eq__neg__nonpos,axiom,
    ! [Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),aa_real_real(uminus_uminus_real,Aa))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),zero_zero_real) ) ).

% less_eq_neg_nonpos
tff(fact_2324_less__eq__neg__nonpos,axiom,
    ! [Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Aa),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Aa),zero_z3403309356797280102nteger) ) ).

% less_eq_neg_nonpos
tff(fact_2325_less__eq__neg__nonpos,axiom,
    ! [Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),aa_rat_rat(uminus_uminus_rat,Aa))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),zero_zero_rat) ) ).

% less_eq_neg_nonpos
tff(fact_2326_less__eq__neg__nonpos,axiom,
    ! [Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),aa_int_int(uminus_uminus_int,Aa))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),zero_zero_int) ) ).

% less_eq_neg_nonpos
tff(fact_2327_neg__less__eq__nonneg,axiom,
    ! [Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,Aa)),Aa)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Aa) ) ).

% neg_less_eq_nonneg
tff(fact_2328_neg__less__eq__nonneg,axiom,
    ! [Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa)),Aa)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Aa) ) ).

% neg_less_eq_nonneg
tff(fact_2329_neg__less__eq__nonneg,axiom,
    ! [Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,Aa)),Aa)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Aa) ) ).

% neg_less_eq_nonneg
tff(fact_2330_neg__less__eq__nonneg,axiom,
    ! [Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,Aa)),Aa)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Aa) ) ).

% neg_less_eq_nonneg
tff(fact_2331_less__neg__neg,axiom,
    ! [Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),aa_real_real(uminus_uminus_real,Aa))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),zero_zero_real) ) ).

% less_neg_neg
tff(fact_2332_less__neg__neg,axiom,
    ! [Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),aa_int_int(uminus_uminus_int,Aa))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),zero_zero_int) ) ).

% less_neg_neg
tff(fact_2333_less__neg__neg,axiom,
    ! [Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Aa),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Aa),zero_z3403309356797280102nteger) ) ).

% less_neg_neg
tff(fact_2334_less__neg__neg,axiom,
    ! [Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),aa_rat_rat(uminus_uminus_rat,Aa))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),zero_zero_rat) ) ).

% less_neg_neg
tff(fact_2335_neg__less__pos,axiom,
    ! [Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,Aa)),Aa)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Aa) ) ).

% neg_less_pos
tff(fact_2336_neg__less__pos,axiom,
    ! [Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,Aa)),Aa)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Aa) ) ).

% neg_less_pos
tff(fact_2337_neg__less__pos,axiom,
    ! [Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa)),Aa)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),Aa) ) ).

% neg_less_pos
tff(fact_2338_neg__less__pos,axiom,
    ! [Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,Aa)),Aa)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Aa) ) ).

% neg_less_pos
tff(fact_2339_neg__0__less__iff__less,axiom,
    ! [Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(uminus_uminus_real,Aa))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),zero_zero_real) ) ).

% neg_0_less_iff_less
tff(fact_2340_neg__0__less__iff__less,axiom,
    ! [Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(uminus_uminus_int,Aa))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),zero_zero_int) ) ).

% neg_0_less_iff_less
tff(fact_2341_neg__0__less__iff__less,axiom,
    ! [Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Aa),zero_z3403309356797280102nteger) ) ).

% neg_0_less_iff_less
tff(fact_2342_neg__0__less__iff__less,axiom,
    ! [Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,Aa))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),zero_zero_rat) ) ).

% neg_0_less_iff_less
tff(fact_2343_neg__less__0__iff__less,axiom,
    ! [Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,Aa)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Aa) ) ).

% neg_less_0_iff_less
tff(fact_2344_neg__less__0__iff__less,axiom,
    ! [Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,Aa)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Aa) ) ).

% neg_less_0_iff_less
tff(fact_2345_neg__less__0__iff__less,axiom,
    ! [Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa)),zero_z3403309356797280102nteger)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),Aa) ) ).

% neg_less_0_iff_less
tff(fact_2346_neg__less__0__iff__less,axiom,
    ! [Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,Aa)),zero_zero_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Aa) ) ).

% neg_less_0_iff_less
tff(fact_2347_neg__numeral__le__iff,axiom,
    ! [M: num,Na: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M))),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Na)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Na),M) ) ).

% neg_numeral_le_iff
tff(fact_2348_neg__numeral__le__iff,axiom,
    ! [M: num,Na: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Na)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Na),M) ) ).

% neg_numeral_le_iff
tff(fact_2349_neg__numeral__le__iff,axiom,
    ! [M: num,Na: 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,M))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Na)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Na),M) ) ).

% neg_numeral_le_iff
tff(fact_2350_neg__numeral__le__iff,axiom,
    ! [M: num,Na: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Na)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Na),M) ) ).

% neg_numeral_le_iff
tff(fact_2351_neg__numeral__less__iff,axiom,
    ! [M: num,Na: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M))),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Na)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Na),M) ) ).

% neg_numeral_less_iff
tff(fact_2352_neg__numeral__less__iff,axiom,
    ! [M: num,Na: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Na)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Na),M) ) ).

% neg_numeral_less_iff
tff(fact_2353_neg__numeral__less__iff,axiom,
    ! [M: num,Na: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Na)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Na),M) ) ).

% neg_numeral_less_iff
tff(fact_2354_neg__numeral__less__iff,axiom,
    ! [M: num,Na: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Na)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Na),M) ) ).

% neg_numeral_less_iff
tff(fact_2355_add_Oright__inverse,axiom,
    ! [Aa: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),aa_real_real(uminus_uminus_real,Aa)) = zero_zero_real ).

% add.right_inverse
tff(fact_2356_add_Oright__inverse,axiom,
    ! [Aa: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),aa_int_int(uminus_uminus_int,Aa)) = zero_zero_int ).

% add.right_inverse
tff(fact_2357_add_Oright__inverse,axiom,
    ! [Aa: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Aa),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa)) = zero_z3403309356797280102nteger ).

% add.right_inverse
tff(fact_2358_add_Oright__inverse,axiom,
    ! [Aa: rat] : aa_rat_rat(plus_plus_rat(Aa),aa_rat_rat(uminus_uminus_rat,Aa)) = zero_zero_rat ).

% add.right_inverse
tff(fact_2359_ab__left__minus,axiom,
    ! [Aa: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,Aa)),Aa) = zero_zero_real ).

% ab_left_minus
tff(fact_2360_ab__left__minus,axiom,
    ! [Aa: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,Aa)),Aa) = zero_zero_int ).

% ab_left_minus
tff(fact_2361_ab__left__minus,axiom,
    ! [Aa: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa)),Aa) = zero_z3403309356797280102nteger ).

% ab_left_minus
tff(fact_2362_ab__left__minus,axiom,
    ! [Aa: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,Aa)),Aa) = zero_zero_rat ).

% ab_left_minus
tff(fact_2363_diff__0,axiom,
    ! [Aa: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,zero_zero_real),Aa) = aa_real_real(uminus_uminus_real,Aa) ).

% diff_0
tff(fact_2364_diff__0,axiom,
    ! [Aa: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,zero_zero_int),Aa) = aa_int_int(uminus_uminus_int,Aa) ).

% diff_0
tff(fact_2365_diff__0,axiom,
    ! [Aa: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,zero_z3403309356797280102nteger),Aa) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa) ).

% diff_0
tff(fact_2366_diff__0,axiom,
    ! [Aa: rat] : aa_rat_rat(minus_minus_rat(zero_zero_rat),Aa) = aa_rat_rat(uminus_uminus_rat,Aa) ).

% diff_0
tff(fact_2367_verit__minus__simplify_I3_J,axiom,
    ! [Ba: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,zero_zero_real),Ba) = aa_real_real(uminus_uminus_real,Ba) ).

% verit_minus_simplify(3)
tff(fact_2368_verit__minus__simplify_I3_J,axiom,
    ! [Ba: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,zero_zero_int),Ba) = aa_int_int(uminus_uminus_int,Ba) ).

% verit_minus_simplify(3)
tff(fact_2369_verit__minus__simplify_I3_J,axiom,
    ! [Ba: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,zero_z3403309356797280102nteger),Ba) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Ba) ).

% verit_minus_simplify(3)
tff(fact_2370_verit__minus__simplify_I3_J,axiom,
    ! [Ba: rat] : aa_rat_rat(minus_minus_rat(zero_zero_rat),Ba) = aa_rat_rat(uminus_uminus_rat,Ba) ).

% verit_minus_simplify(3)
tff(fact_2371_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,Na: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M))),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Na))) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,M)),aa_num_real(numeral_numeral_real,Na))) ).

% add_neg_numeral_simps(3)
tff(fact_2372_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,Na: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Na))) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,Na))) ).

% add_neg_numeral_simps(3)
tff(fact_2373_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,Na: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Na))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)),aa_num_Code_integer(numera6620942414471956472nteger,Na))) ).

% add_neg_numeral_simps(3)
tff(fact_2374_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,Na: num] : aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Na))) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,M)),aa_num_rat(numeral_numeral_rat,Na))) ).

% add_neg_numeral_simps(3)
tff(fact_2375_diff__numeral__simps_I2_J,axiom,
    ! [M: num,Na: num] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_num_real(numeral_numeral_real,M)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Na))) = aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),Na)) ).

% diff_numeral_simps(2)
tff(fact_2376_diff__numeral__simps_I2_J,axiom,
    ! [M: num,Na: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Na))) = aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),Na)) ).

% diff_numeral_simps(2)
tff(fact_2377_diff__numeral__simps_I2_J,axiom,
    ! [M: num,Na: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Na))) = aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),Na)) ).

% diff_numeral_simps(2)
tff(fact_2378_diff__numeral__simps_I2_J,axiom,
    ! [M: num,Na: num] : aa_rat_rat(minus_minus_rat(aa_num_rat(numeral_numeral_rat,M)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Na))) = aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),Na)) ).

% diff_numeral_simps(2)
tff(fact_2379_diff__numeral__simps_I3_J,axiom,
    ! [M: num,Na: num] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M))),aa_num_real(numeral_numeral_real,Na)) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),Na))) ).

% diff_numeral_simps(3)
tff(fact_2380_diff__numeral__simps_I3_J,axiom,
    ! [M: num,Na: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,Na)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),Na))) ).

% diff_numeral_simps(3)
tff(fact_2381_diff__numeral__simps_I3_J,axiom,
    ! [M: num,Na: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M))),aa_num_Code_integer(numera6620942414471956472nteger,Na)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),Na))) ).

% diff_numeral_simps(3)
tff(fact_2382_diff__numeral__simps_I3_J,axiom,
    ! [M: num,Na: num] : aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))),aa_num_rat(numeral_numeral_rat,Na)) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),Na))) ).

% diff_numeral_simps(3)
tff(fact_2383_diff__minus__eq__add,axiom,
    ! [Aa: real,Ba: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Aa),aa_real_real(uminus_uminus_real,Ba)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),Ba) ).

% diff_minus_eq_add
tff(fact_2384_diff__minus__eq__add,axiom,
    ! [Aa: int,Ba: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,Aa),aa_int_int(uminus_uminus_int,Ba)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),Ba) ).

% diff_minus_eq_add
tff(fact_2385_diff__minus__eq__add,axiom,
    ! [Aa: code_integer,Ba: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,Aa),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Ba)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Aa),Ba) ).

% diff_minus_eq_add
tff(fact_2386_diff__minus__eq__add,axiom,
    ! [Aa: rat,Ba: rat] : aa_rat_rat(minus_minus_rat(Aa),aa_rat_rat(uminus_uminus_rat,Ba)) = aa_rat_rat(plus_plus_rat(Aa),Ba) ).

% diff_minus_eq_add
tff(fact_2387_uminus__add__conv__diff,axiom,
    ! [Aa: real,Ba: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,Aa)),Ba) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Ba),Aa) ).

% uminus_add_conv_diff
tff(fact_2388_uminus__add__conv__diff,axiom,
    ! [Aa: int,Ba: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,Aa)),Ba) = aa_int_int(aa_int_fun_int_int(minus_minus_int,Ba),Aa) ).

% uminus_add_conv_diff
tff(fact_2389_uminus__add__conv__diff,axiom,
    ! [Aa: code_integer,Ba: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa)),Ba) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,Ba),Aa) ).

% uminus_add_conv_diff
tff(fact_2390_uminus__add__conv__diff,axiom,
    ! [Aa: rat,Ba: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,Aa)),Ba) = aa_rat_rat(minus_minus_rat(Ba),Aa) ).

% uminus_add_conv_diff
tff(fact_2391_euclidean__size__numeral,axiom,
    ! [K: num] : aa_Code_integer_nat(euclid6377331345833325938nteger,aa_num_Code_integer(numera6620942414471956472nteger,K)) = aa_num_nat(numeral_numeral_nat,K) ).

% euclidean_size_numeral
tff(fact_2392_euclidean__size__numeral,axiom,
    ! [K: num] : aa_int_nat(euclid4774559944035922753ze_int,aa_num_int(numeral_numeral_int,K)) = aa_num_nat(numeral_numeral_nat,K) ).

% euclidean_size_numeral
tff(fact_2393_euclidean__size__numeral,axiom,
    ! [K: num] : aa_nat_nat(euclid4777050414544973029ze_nat,aa_num_nat(numeral_numeral_nat,K)) = aa_num_nat(numeral_numeral_nat,K) ).

% euclidean_size_numeral
tff(fact_2394_Ball__set__replicate,axiom,
    ! [Na: nat,Aa: nat,P: fun_nat_o] :
      ( ! [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),set_nat2(replicate_nat(Na,Aa)))
         => aa_nat_o(P,X4) )
    <=> ( aa_nat_o(P,Aa)
        | ( Na = zero_zero_nat ) ) ) ).

% Ball_set_replicate
tff(fact_2395_Ball__set__replicate,axiom,
    ! [Na: nat,Aa: vEBT_VEBT,P: fun_VEBT_VEBT_o] :
      ( ! [X4: vEBT_VEBT] :
          ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(replicate_VEBT_VEBT(Na,Aa)))
         => aa_VEBT_VEBT_o(P,X4) )
    <=> ( aa_VEBT_VEBT_o(P,Aa)
        | ( Na = zero_zero_nat ) ) ) ).

% Ball_set_replicate
tff(fact_2396_Bex__set__replicate,axiom,
    ! [Na: nat,Aa: nat,P: fun_nat_o] :
      ( ? [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),set_nat2(replicate_nat(Na,Aa)))
          & aa_nat_o(P,X4) )
    <=> ( aa_nat_o(P,Aa)
        & ( Na != zero_zero_nat ) ) ) ).

% Bex_set_replicate
tff(fact_2397_Bex__set__replicate,axiom,
    ! [Na: nat,Aa: vEBT_VEBT,P: fun_VEBT_VEBT_o] :
      ( ? [X4: vEBT_VEBT] :
          ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(replicate_VEBT_VEBT(Na,Aa)))
          & aa_VEBT_VEBT_o(P,X4) )
    <=> ( aa_VEBT_VEBT_o(P,Aa)
        & ( Na != zero_zero_nat ) ) ) ).

% Bex_set_replicate
tff(fact_2398_in__set__replicate,axiom,
    ! [Xb: product_prod_nat_nat,Na: nat,Y: product_prod_nat_nat] :
      ( member8440522571783428010at_nat(Xb,set_Pr5648618587558075414at_nat(replic4235873036481779905at_nat(Na,Y)))
    <=> ( ( Xb = Y )
        & ( Na != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_2399_in__set__replicate,axiom,
    ! [Xb: real,Na: nat,Y: real] :
      ( member_real(Xb,set_real2(replicate_real(Na,Y)))
    <=> ( ( Xb = Y )
        & ( Na != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_2400_in__set__replicate,axiom,
    ! [Xb: set_nat,Na: nat,Y: set_nat] :
      ( member_set_nat(Xb,set_set_nat2(replicate_set_nat(Na,Y)))
    <=> ( ( Xb = Y )
        & ( Na != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_2401_in__set__replicate,axiom,
    ! [Xb: int,Na: nat,Y: int] :
      ( aa_set_int_o(member_int(Xb),set_int2(replicate_int(Na,Y)))
    <=> ( ( Xb = Y )
        & ( Na != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_2402_in__set__replicate,axiom,
    ! [Xb: nat,Na: nat,Y: nat] :
      ( aa_set_nat_o(member_nat(Xb),set_nat2(replicate_nat(Na,Y)))
    <=> ( ( Xb = Y )
        & ( Na != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_2403_in__set__replicate,axiom,
    ! [Xb: vEBT_VEBT,Na: nat,Y: vEBT_VEBT] :
      ( member_VEBT_VEBT(Xb,set_VEBT_VEBT2(replicate_VEBT_VEBT(Na,Y)))
    <=> ( ( Xb = Y )
        & ( Na != zero_zero_nat ) ) ) ).

% in_set_replicate
tff(fact_2404_nth__replicate,axiom,
    ! [I: nat,Na: nat,Xb: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),Na)
     => ( aa_nat_int(nth_int(replicate_int(Na,Xb)),I) = Xb ) ) ).

% nth_replicate
tff(fact_2405_nth__replicate,axiom,
    ! [I: nat,Na: nat,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),Na)
     => ( aa_nat_nat(nth_nat(replicate_nat(Na,Xb)),I) = Xb ) ) ).

% nth_replicate
tff(fact_2406_nth__replicate,axiom,
    ! [I: nat,Na: nat,Xb: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),Na)
     => ( nth_VEBT_VEBT(replicate_VEBT_VEBT(Na,Xb),I) = Xb ) ) ).

% nth_replicate
tff(fact_2407_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_2408_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_2409_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_2410_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_2411_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_2412_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_2413_add__neg__numeral__special_I8_J,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,one_one_real)),one_one_real) = zero_zero_real ).

% add_neg_numeral_special(8)
tff(fact_2414_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_2415_add__neg__numeral__special_I8_J,axiom,
    aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),one_one_Code_integer) = zero_z3403309356797280102nteger ).

% add_neg_numeral_special(8)
tff(fact_2416_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_2417_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_2418_add__neg__numeral__special_I7_J,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(uminus_uminus_real,one_one_real)) = zero_zero_real ).

% add_neg_numeral_special(7)
tff(fact_2419_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_2420_add__neg__numeral__special_I7_J,axiom,
    aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = zero_z3403309356797280102nteger ).

% add_neg_numeral_special(7)
tff(fact_2421_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_2422_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_2423_diff__numeral__special_I12_J,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(uminus_uminus_real,one_one_real)) = zero_zero_real ).

% diff_numeral_special(12)
tff(fact_2424_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_2425_diff__numeral__special_I12_J,axiom,
    aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = zero_z3403309356797280102nteger ).

% diff_numeral_special(12)
tff(fact_2426_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_2427_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_2428_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_2429_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_2430_dbl__dec__simps_I2_J,axiom,
    neg_nu7757733837767384882nteger(zero_z3403309356797280102nteger) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% dbl_dec_simps(2)
tff(fact_2431_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_2432_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_2433_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_2434_dbl__dec__simps_I1_J,axiom,
    ! [K: num] : neg_nu7757733837767384882nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,K))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,neg_nu5831290666863070958nteger(aa_num_Code_integer(numera6620942414471956472nteger,K))) ).

% dbl_dec_simps(1)
tff(fact_2435_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_2436_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_2437_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_2438_dbl__inc__simps_I1_J,axiom,
    ! [K: num] : neg_nu5831290666863070958nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,K))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,neg_nu7757733837767384882nteger(aa_num_Code_integer(numera6620942414471956472nteger,K))) ).

% dbl_inc_simps(1)
tff(fact_2439_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_2440_neg__numeral__neq__numeral,axiom,
    ! [M: num,Na: num] : aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M)) != aa_num_real(numeral_numeral_real,Na) ).

% neg_numeral_neq_numeral
tff(fact_2441_neg__numeral__neq__numeral,axiom,
    ! [M: num,Na: num] : aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)) != aa_num_int(numeral_numeral_int,Na) ).

% neg_numeral_neq_numeral
tff(fact_2442_neg__numeral__neq__numeral,axiom,
    ! [M: num,Na: num] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)) != aa_num_Code_integer(numera6620942414471956472nteger,Na) ).

% neg_numeral_neq_numeral
tff(fact_2443_neg__numeral__neq__numeral,axiom,
    ! [M: num,Na: num] : aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M)) != aa_num_rat(numeral_numeral_rat,Na) ).

% neg_numeral_neq_numeral
tff(fact_2444_numeral__neq__neg__numeral,axiom,
    ! [M: num,Na: num] : aa_num_real(numeral_numeral_real,M) != aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Na)) ).

% numeral_neq_neg_numeral
tff(fact_2445_numeral__neq__neg__numeral,axiom,
    ! [M: num,Na: num] : aa_num_int(numeral_numeral_int,M) != aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Na)) ).

% numeral_neq_neg_numeral
tff(fact_2446_numeral__neq__neg__numeral,axiom,
    ! [M: num,Na: num] : aa_num_Code_integer(numera6620942414471956472nteger,M) != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Na)) ).

% numeral_neq_neg_numeral
tff(fact_2447_numeral__neq__neg__numeral,axiom,
    ! [M: num,Na: num] : aa_num_rat(numeral_numeral_rat,M) != aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Na)) ).

% numeral_neq_neg_numeral
tff(fact_2448_equation__minus__iff,axiom,
    ! [Aa: real,Ba: real] :
      ( ( Aa = aa_real_real(uminus_uminus_real,Ba) )
    <=> ( Ba = aa_real_real(uminus_uminus_real,Aa) ) ) ).

% equation_minus_iff
tff(fact_2449_equation__minus__iff,axiom,
    ! [Aa: int,Ba: int] :
      ( ( Aa = aa_int_int(uminus_uminus_int,Ba) )
    <=> ( Ba = aa_int_int(uminus_uminus_int,Aa) ) ) ).

% equation_minus_iff
tff(fact_2450_equation__minus__iff,axiom,
    ! [Aa: code_integer,Ba: code_integer] :
      ( ( Aa = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Ba) )
    <=> ( Ba = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa) ) ) ).

% equation_minus_iff
tff(fact_2451_equation__minus__iff,axiom,
    ! [Aa: rat,Ba: rat] :
      ( ( Aa = aa_rat_rat(uminus_uminus_rat,Ba) )
    <=> ( Ba = aa_rat_rat(uminus_uminus_rat,Aa) ) ) ).

% equation_minus_iff
tff(fact_2452_minus__equation__iff,axiom,
    ! [Aa: real,Ba: real] :
      ( ( aa_real_real(uminus_uminus_real,Aa) = Ba )
    <=> ( aa_real_real(uminus_uminus_real,Ba) = Aa ) ) ).

% minus_equation_iff
tff(fact_2453_minus__equation__iff,axiom,
    ! [Aa: int,Ba: int] :
      ( ( aa_int_int(uminus_uminus_int,Aa) = Ba )
    <=> ( aa_int_int(uminus_uminus_int,Ba) = Aa ) ) ).

% minus_equation_iff
tff(fact_2454_minus__equation__iff,axiom,
    ! [Aa: code_integer,Ba: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa) = Ba )
    <=> ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Ba) = Aa ) ) ).

% minus_equation_iff
tff(fact_2455_minus__equation__iff,axiom,
    ! [Aa: rat,Ba: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,Aa) = Ba )
    <=> ( aa_rat_rat(uminus_uminus_rat,Ba) = Aa ) ) ).

% minus_equation_iff
tff(fact_2456_neg__numeral__less__numeral,axiom,
    ! [M: num,Na: num] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M))),aa_num_real(numeral_numeral_real,Na)) ).

% neg_numeral_less_numeral
tff(fact_2457_neg__numeral__less__numeral,axiom,
    ! [M: num,Na: num] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,Na)) ).

% neg_numeral_less_numeral
tff(fact_2458_neg__numeral__less__numeral,axiom,
    ! [M: num,Na: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M))),aa_num_Code_integer(numera6620942414471956472nteger,Na)) ).

% neg_numeral_less_numeral
tff(fact_2459_neg__numeral__less__numeral,axiom,
    ! [M: num,Na: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))),aa_num_rat(numeral_numeral_rat,Na)) ).

% neg_numeral_less_numeral
tff(fact_2460_not__numeral__less__neg__numeral,axiom,
    ! [M: num,Na: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,M)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Na))) ).

% not_numeral_less_neg_numeral
tff(fact_2461_not__numeral__less__neg__numeral,axiom,
    ! [M: num,Na: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Na))) ).

% not_numeral_less_neg_numeral
tff(fact_2462_not__numeral__less__neg__numeral,axiom,
    ! [M: num,Na: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Na))) ).

% not_numeral_less_neg_numeral
tff(fact_2463_not__numeral__less__neg__numeral,axiom,
    ! [M: num,Na: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,M)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Na))) ).

% not_numeral_less_neg_numeral
tff(fact_2464_zero__neq__neg__numeral,axiom,
    ! [Na: num] : zero_zero_real != aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Na)) ).

% zero_neq_neg_numeral
tff(fact_2465_zero__neq__neg__numeral,axiom,
    ! [Na: num] : zero_zero_int != aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Na)) ).

% zero_neq_neg_numeral
tff(fact_2466_zero__neq__neg__numeral,axiom,
    ! [Na: num] : zero_z3403309356797280102nteger != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Na)) ).

% zero_neq_neg_numeral
tff(fact_2467_zero__neq__neg__numeral,axiom,
    ! [Na: num] : zero_zero_rat != aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Na)) ).

% zero_neq_neg_numeral
tff(fact_2468_one__neq__neg__numeral,axiom,
    ! [Na: num] : one_one_complex != aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Na)) ).

% one_neq_neg_numeral
tff(fact_2469_one__neq__neg__numeral,axiom,
    ! [Na: num] : one_one_real != aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Na)) ).

% one_neq_neg_numeral
tff(fact_2470_one__neq__neg__numeral,axiom,
    ! [Na: num] : one_one_int != aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Na)) ).

% one_neq_neg_numeral
tff(fact_2471_one__neq__neg__numeral,axiom,
    ! [Na: num] : one_one_Code_integer != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Na)) ).

% one_neq_neg_numeral
tff(fact_2472_one__neq__neg__numeral,axiom,
    ! [Na: num] : one_one_rat != aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Na)) ).

% one_neq_neg_numeral
tff(fact_2473_numeral__neq__neg__one,axiom,
    ! [Na: num] : numera6690914467698888265omplex(Na) != aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% numeral_neq_neg_one
tff(fact_2474_numeral__neq__neg__one,axiom,
    ! [Na: num] : aa_num_real(numeral_numeral_real,Na) != aa_real_real(uminus_uminus_real,one_one_real) ).

% numeral_neq_neg_one
tff(fact_2475_numeral__neq__neg__one,axiom,
    ! [Na: num] : aa_num_int(numeral_numeral_int,Na) != aa_int_int(uminus_uminus_int,one_one_int) ).

% numeral_neq_neg_one
tff(fact_2476_numeral__neq__neg__one,axiom,
    ! [Na: num] : aa_num_Code_integer(numera6620942414471956472nteger,Na) != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% numeral_neq_neg_one
tff(fact_2477_numeral__neq__neg__one,axiom,
    ! [Na: num] : aa_num_rat(numeral_numeral_rat,Na) != aa_rat_rat(uminus_uminus_rat,one_one_rat) ).

% numeral_neq_neg_one
tff(fact_2478_not__numeral__le__neg__numeral,axiom,
    ! [M: num,Na: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,M)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Na))) ).

% not_numeral_le_neg_numeral
tff(fact_2479_not__numeral__le__neg__numeral,axiom,
    ! [M: num,Na: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Na))) ).

% not_numeral_le_neg_numeral
tff(fact_2480_not__numeral__le__neg__numeral,axiom,
    ! [M: num,Na: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,M)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Na))) ).

% not_numeral_le_neg_numeral
tff(fact_2481_not__numeral__le__neg__numeral,axiom,
    ! [M: num,Na: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Na))) ).

% not_numeral_le_neg_numeral
tff(fact_2482_neg__numeral__le__numeral,axiom,
    ! [M: num,Na: num] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M))),aa_num_real(numeral_numeral_real,Na)) ).

% neg_numeral_le_numeral
tff(fact_2483_neg__numeral__le__numeral,axiom,
    ! [M: num,Na: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M))),aa_num_Code_integer(numera6620942414471956472nteger,Na)) ).

% neg_numeral_le_numeral
tff(fact_2484_neg__numeral__le__numeral,axiom,
    ! [M: num,Na: 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,M))),aa_num_rat(numeral_numeral_rat,Na)) ).

% neg_numeral_le_numeral
tff(fact_2485_neg__numeral__le__numeral,axiom,
    ! [M: num,Na: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,Na)) ).

% neg_numeral_le_numeral
tff(fact_2486_neg__numeral__le__zero,axiom,
    ! [Na: num] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Na))),zero_zero_real) ).

% neg_numeral_le_zero
tff(fact_2487_neg__numeral__le__zero,axiom,
    ! [Na: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Na))),zero_z3403309356797280102nteger) ).

% neg_numeral_le_zero
tff(fact_2488_neg__numeral__le__zero,axiom,
    ! [Na: 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,Na))),zero_zero_rat) ).

% neg_numeral_le_zero
tff(fact_2489_neg__numeral__le__zero,axiom,
    ! [Na: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Na))),zero_zero_int) ).

% neg_numeral_le_zero
tff(fact_2490_not__zero__le__neg__numeral,axiom,
    ! [Na: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Na))) ).

% not_zero_le_neg_numeral
tff(fact_2491_not__zero__le__neg__numeral,axiom,
    ! [Na: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Na))) ).

% not_zero_le_neg_numeral
tff(fact_2492_not__zero__le__neg__numeral,axiom,
    ! [Na: 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,Na))) ).

% not_zero_le_neg_numeral
tff(fact_2493_not__zero__le__neg__numeral,axiom,
    ! [Na: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Na))) ).

% not_zero_le_neg_numeral
tff(fact_2494_neg__numeral__less__zero,axiom,
    ! [Na: num] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Na))),zero_zero_real) ).

% neg_numeral_less_zero
tff(fact_2495_neg__numeral__less__zero,axiom,
    ! [Na: num] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Na))),zero_zero_int) ).

% neg_numeral_less_zero
tff(fact_2496_neg__numeral__less__zero,axiom,
    ! [Na: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Na))),zero_z3403309356797280102nteger) ).

% neg_numeral_less_zero
tff(fact_2497_neg__numeral__less__zero,axiom,
    ! [Na: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Na))),zero_zero_rat) ).

% neg_numeral_less_zero
tff(fact_2498_not__zero__less__neg__numeral,axiom,
    ! [Na: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Na))) ).

% not_zero_less_neg_numeral
tff(fact_2499_not__zero__less__neg__numeral,axiom,
    ! [Na: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Na))) ).

% not_zero_less_neg_numeral
tff(fact_2500_not__zero__less__neg__numeral,axiom,
    ! [Na: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Na))) ).

% not_zero_less_neg_numeral
tff(fact_2501_not__zero__less__neg__numeral,axiom,
    ! [Na: 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,Na))) ).

% not_zero_less_neg_numeral
tff(fact_2502_neg__numeral__le__one,axiom,
    ! [M: num] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M))),one_one_real) ).

% neg_numeral_le_one
tff(fact_2503_neg__numeral__le__one,axiom,
    ! [M: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M))),one_one_Code_integer) ).

% neg_numeral_le_one
tff(fact_2504_neg__numeral__le__one,axiom,
    ! [M: 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,M))),one_one_rat) ).

% neg_numeral_le_one
tff(fact_2505_neg__numeral__le__one,axiom,
    ! [M: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),one_one_int) ).

% neg_numeral_le_one
tff(fact_2506_neg__one__le__numeral,axiom,
    ! [M: num] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_num_real(numeral_numeral_real,M)) ).

% neg_one_le_numeral
tff(fact_2507_neg__one__le__numeral,axiom,
    ! [M: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_num_Code_integer(numera6620942414471956472nteger,M)) ).

% neg_one_le_numeral
tff(fact_2508_neg__one__le__numeral,axiom,
    ! [M: 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,M)) ).

% neg_one_le_numeral
tff(fact_2509_neg__one__le__numeral,axiom,
    ! [M: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,M)) ).

% neg_one_le_numeral
tff(fact_2510_neg__numeral__le__neg__one,axiom,
    ! [M: num] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M))),aa_real_real(uminus_uminus_real,one_one_real)) ).

% neg_numeral_le_neg_one
tff(fact_2511_neg__numeral__le__neg__one,axiom,
    ! [M: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).

% neg_numeral_le_neg_one
tff(fact_2512_neg__numeral__le__neg__one,axiom,
    ! [M: 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,M))),aa_rat_rat(uminus_uminus_rat,one_one_rat)) ).

% neg_numeral_le_neg_one
tff(fact_2513_neg__numeral__le__neg__one,axiom,
    ! [M: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,one_one_int)) ).

% neg_numeral_le_neg_one
tff(fact_2514_not__numeral__le__neg__one,axiom,
    ! [M: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,M)),aa_real_real(uminus_uminus_real,one_one_real)) ).

% not_numeral_le_neg_one
tff(fact_2515_not__numeral__le__neg__one,axiom,
    ! [M: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).

% not_numeral_le_neg_one
tff(fact_2516_not__numeral__le__neg__one,axiom,
    ! [M: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,M)),aa_rat_rat(uminus_uminus_rat,one_one_rat)) ).

% not_numeral_le_neg_one
tff(fact_2517_not__numeral__le__neg__one,axiom,
    ! [M: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,one_one_int)) ).

% not_numeral_le_neg_one
tff(fact_2518_not__one__le__neg__numeral,axiom,
    ! [M: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M))) ).

% not_one_le_neg_numeral
tff(fact_2519_not__one__le__neg__numeral,axiom,
    ! [M: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M))) ).

% not_one_le_neg_numeral
tff(fact_2520_not__one__le__neg__numeral,axiom,
    ! [M: 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,M))) ).

% not_one_le_neg_numeral
tff(fact_2521_not__one__le__neg__numeral,axiom,
    ! [M: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))) ).

% not_one_le_neg_numeral
tff(fact_2522_neg__numeral__less__one,axiom,
    ! [M: num] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M))),one_one_real) ).

% neg_numeral_less_one
tff(fact_2523_neg__numeral__less__one,axiom,
    ! [M: num] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),one_one_int) ).

% neg_numeral_less_one
tff(fact_2524_neg__numeral__less__one,axiom,
    ! [M: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M))),one_one_Code_integer) ).

% neg_numeral_less_one
tff(fact_2525_neg__numeral__less__one,axiom,
    ! [M: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))),one_one_rat) ).

% neg_numeral_less_one
tff(fact_2526_neg__one__less__numeral,axiom,
    ! [M: num] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_num_real(numeral_numeral_real,M)) ).

% neg_one_less_numeral
tff(fact_2527_neg__one__less__numeral,axiom,
    ! [M: num] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,M)) ).

% neg_one_less_numeral
tff(fact_2528_neg__one__less__numeral,axiom,
    ! [M: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_num_Code_integer(numera6620942414471956472nteger,M)) ).

% neg_one_less_numeral
tff(fact_2529_neg__one__less__numeral,axiom,
    ! [M: 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,M)) ).

% neg_one_less_numeral
tff(fact_2530_not__numeral__less__neg__one,axiom,
    ! [M: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,M)),aa_real_real(uminus_uminus_real,one_one_real)) ).

% not_numeral_less_neg_one
tff(fact_2531_not__numeral__less__neg__one,axiom,
    ! [M: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,one_one_int)) ).

% not_numeral_less_neg_one
tff(fact_2532_not__numeral__less__neg__one,axiom,
    ! [M: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).

% not_numeral_less_neg_one
tff(fact_2533_not__numeral__less__neg__one,axiom,
    ! [M: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,M)),aa_rat_rat(uminus_uminus_rat,one_one_rat)) ).

% not_numeral_less_neg_one
tff(fact_2534_not__one__less__neg__numeral,axiom,
    ! [M: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M))) ).

% not_one_less_neg_numeral
tff(fact_2535_not__one__less__neg__numeral,axiom,
    ! [M: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))) ).

% not_one_less_neg_numeral
tff(fact_2536_not__one__less__neg__numeral,axiom,
    ! [M: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M))) ).

% not_one_less_neg_numeral
tff(fact_2537_not__one__less__neg__numeral,axiom,
    ! [M: 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,M))) ).

% not_one_less_neg_numeral
tff(fact_2538_not__neg__one__less__neg__numeral,axiom,
    ! [M: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M))) ).

% not_neg_one_less_neg_numeral
tff(fact_2539_not__neg__one__less__neg__numeral,axiom,
    ! [M: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))) ).

% not_neg_one_less_neg_numeral
tff(fact_2540_not__neg__one__less__neg__numeral,axiom,
    ! [M: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M))) ).

% not_neg_one_less_neg_numeral
tff(fact_2541_not__neg__one__less__neg__numeral,axiom,
    ! [M: 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,M))) ).

% not_neg_one_less_neg_numeral
tff(fact_2542_compl__le__swap2,axiom,
    ! [Y: set_int,Xb: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(uminus1532241313380277803et_int(Y)),Xb)
     => aa_set_int_o(ord_less_eq_set_int(uminus1532241313380277803et_int(Xb)),Y) ) ).

% compl_le_swap2
tff(fact_2543_compl__le__swap1,axiom,
    ! [Y: set_int,Xb: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(Y),uminus1532241313380277803et_int(Xb))
     => aa_set_int_o(ord_less_eq_set_int(Xb),uminus1532241313380277803et_int(Y)) ) ).

% compl_le_swap1
tff(fact_2544_compl__mono,axiom,
    ! [Xb: set_int,Y: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(Xb),Y)
     => aa_set_int_o(ord_less_eq_set_int(uminus1532241313380277803et_int(Y)),uminus1532241313380277803et_int(Xb)) ) ).

% compl_mono
tff(fact_2545_le__minus__iff,axiom,
    ! [Aa: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),aa_real_real(uminus_uminus_real,Ba))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ba),aa_real_real(uminus_uminus_real,Aa)) ) ).

% le_minus_iff
tff(fact_2546_le__minus__iff,axiom,
    ! [Aa: code_integer,Ba: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Aa),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Ba))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Ba),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa)) ) ).

% le_minus_iff
tff(fact_2547_le__minus__iff,axiom,
    ! [Aa: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),aa_rat_rat(uminus_uminus_rat,Ba))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ba),aa_rat_rat(uminus_uminus_rat,Aa)) ) ).

% le_minus_iff
tff(fact_2548_le__minus__iff,axiom,
    ! [Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),aa_int_int(uminus_uminus_int,Ba))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ba),aa_int_int(uminus_uminus_int,Aa)) ) ).

% le_minus_iff
tff(fact_2549_minus__le__iff,axiom,
    ! [Aa: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,Aa)),Ba)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,Ba)),Aa) ) ).

% minus_le_iff
tff(fact_2550_minus__le__iff,axiom,
    ! [Aa: code_integer,Ba: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa)),Ba)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Ba)),Aa) ) ).

% minus_le_iff
tff(fact_2551_minus__le__iff,axiom,
    ! [Aa: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,Aa)),Ba)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,Ba)),Aa) ) ).

% minus_le_iff
tff(fact_2552_minus__le__iff,axiom,
    ! [Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,Aa)),Ba)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,Ba)),Aa) ) ).

% minus_le_iff
tff(fact_2553_le__imp__neg__le,axiom,
    ! [Aa: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),Ba)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,Ba)),aa_real_real(uminus_uminus_real,Aa)) ) ).

% le_imp_neg_le
tff(fact_2554_le__imp__neg__le,axiom,
    ! [Aa: code_integer,Ba: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Aa),Ba)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Ba)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa)) ) ).

% le_imp_neg_le
tff(fact_2555_le__imp__neg__le,axiom,
    ! [Aa: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),Ba)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,Ba)),aa_rat_rat(uminus_uminus_rat,Aa)) ) ).

% le_imp_neg_le
tff(fact_2556_le__imp__neg__le,axiom,
    ! [Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),Ba)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,Ba)),aa_int_int(uminus_uminus_int,Aa)) ) ).

% le_imp_neg_le
tff(fact_2557_less__minus__iff,axiom,
    ! [Aa: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),aa_real_real(uminus_uminus_real,Ba))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Ba),aa_real_real(uminus_uminus_real,Aa)) ) ).

% less_minus_iff
tff(fact_2558_less__minus__iff,axiom,
    ! [Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),aa_int_int(uminus_uminus_int,Ba))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Ba),aa_int_int(uminus_uminus_int,Aa)) ) ).

% less_minus_iff
tff(fact_2559_less__minus__iff,axiom,
    ! [Aa: code_integer,Ba: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Aa),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Ba))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Ba),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa)) ) ).

% less_minus_iff
tff(fact_2560_less__minus__iff,axiom,
    ! [Aa: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),aa_rat_rat(uminus_uminus_rat,Ba))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ba),aa_rat_rat(uminus_uminus_rat,Aa)) ) ).

% less_minus_iff
tff(fact_2561_minus__less__iff,axiom,
    ! [Aa: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,Aa)),Ba)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,Ba)),Aa) ) ).

% minus_less_iff
tff(fact_2562_minus__less__iff,axiom,
    ! [Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,Aa)),Ba)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,Ba)),Aa) ) ).

% minus_less_iff
tff(fact_2563_minus__less__iff,axiom,
    ! [Aa: code_integer,Ba: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa)),Ba)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Ba)),Aa) ) ).

% minus_less_iff
tff(fact_2564_minus__less__iff,axiom,
    ! [Aa: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,Aa)),Ba)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,Ba)),Aa) ) ).

% minus_less_iff
tff(fact_2565_verit__negate__coefficient_I2_J,axiom,
    ! [Aa: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),Ba)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,Ba)),aa_real_real(uminus_uminus_real,Aa)) ) ).

% verit_negate_coefficient(2)
tff(fact_2566_verit__negate__coefficient_I2_J,axiom,
    ! [Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),Ba)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,Ba)),aa_int_int(uminus_uminus_int,Aa)) ) ).

% verit_negate_coefficient(2)
tff(fact_2567_verit__negate__coefficient_I2_J,axiom,
    ! [Aa: code_integer,Ba: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Aa),Ba)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Ba)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa)) ) ).

% verit_negate_coefficient(2)
tff(fact_2568_verit__negate__coefficient_I2_J,axiom,
    ! [Aa: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),Ba)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,Ba)),aa_rat_rat(uminus_uminus_rat,Aa)) ) ).

% verit_negate_coefficient(2)
tff(fact_2569_zero__neq__numeral,axiom,
    ! [Na: num] : zero_zero_rat != aa_num_rat(numeral_numeral_rat,Na) ).

% zero_neq_numeral
tff(fact_2570_zero__neq__numeral,axiom,
    ! [Na: num] : zero_zero_nat != aa_num_nat(numeral_numeral_nat,Na) ).

% zero_neq_numeral
tff(fact_2571_zero__neq__numeral,axiom,
    ! [Na: num] : zero_zero_real != aa_num_real(numeral_numeral_real,Na) ).

% zero_neq_numeral
tff(fact_2572_zero__neq__numeral,axiom,
    ! [Na: num] : zero_zero_int != aa_num_int(numeral_numeral_int,Na) ).

% zero_neq_numeral
tff(fact_2573_zero__neq__numeral,axiom,
    ! [Na: num] : zero_z5237406670263579293d_enat != numera1916890842035813515d_enat(Na) ).

% zero_neq_numeral
tff(fact_2574_zero__neq__numeral,axiom,
    ! [Na: num] : zero_z3403309356797280102nteger != aa_num_Code_integer(numera6620942414471956472nteger,Na) ).

% zero_neq_numeral
tff(fact_2575_one__neq__neg__one,axiom,
    one_one_complex != aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% one_neq_neg_one
tff(fact_2576_one__neq__neg__one,axiom,
    one_one_real != aa_real_real(uminus_uminus_real,one_one_real) ).

% one_neq_neg_one
tff(fact_2577_one__neq__neg__one,axiom,
    one_one_int != aa_int_int(uminus_uminus_int,one_one_int) ).

% one_neq_neg_one
tff(fact_2578_one__neq__neg__one,axiom,
    one_one_Code_integer != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% one_neq_neg_one
tff(fact_2579_one__neq__neg__one,axiom,
    one_one_rat != aa_rat_rat(uminus_uminus_rat,one_one_rat) ).

% one_neq_neg_one
tff(fact_2580_is__num__normalize_I8_J,axiom,
    ! [Aa: real,Ba: real] : aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),Ba)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,Ba)),aa_real_real(uminus_uminus_real,Aa)) ).

% is_num_normalize(8)
tff(fact_2581_is__num__normalize_I8_J,axiom,
    ! [Aa: int,Ba: int] : aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),Ba)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,Ba)),aa_int_int(uminus_uminus_int,Aa)) ).

% is_num_normalize(8)
tff(fact_2582_is__num__normalize_I8_J,axiom,
    ! [Aa: code_integer,Ba: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Aa),Ba)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Ba)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa)) ).

% is_num_normalize(8)
tff(fact_2583_is__num__normalize_I8_J,axiom,
    ! [Aa: rat,Ba: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(plus_plus_rat(Aa),Ba)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,Ba)),aa_rat_rat(uminus_uminus_rat,Aa)) ).

% is_num_normalize(8)
tff(fact_2584_group__cancel_Oneg1,axiom,
    ! [A3: real,K: real,Aa: real] :
      ( ( A3 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),Aa) )
     => ( aa_real_real(uminus_uminus_real,A3) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,K)),aa_real_real(uminus_uminus_real,Aa)) ) ) ).

% group_cancel.neg1
tff(fact_2585_group__cancel_Oneg1,axiom,
    ! [A3: int,K: int,Aa: int] :
      ( ( A3 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),Aa) )
     => ( 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,Aa)) ) ) ).

% group_cancel.neg1
tff(fact_2586_group__cancel_Oneg1,axiom,
    ! [A3: code_integer,K: code_integer,Aa: code_integer] :
      ( ( A3 = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,K),Aa) )
     => ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A3) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,K)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa)) ) ) ).

% group_cancel.neg1
tff(fact_2587_group__cancel_Oneg1,axiom,
    ! [A3: rat,K: rat,Aa: rat] :
      ( ( A3 = aa_rat_rat(plus_plus_rat(K),Aa) )
     => ( 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,Aa)) ) ) ).

% group_cancel.neg1
tff(fact_2588_add_Oinverse__distrib__swap,axiom,
    ! [Aa: real,Ba: real] : aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),Ba)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,Ba)),aa_real_real(uminus_uminus_real,Aa)) ).

% add.inverse_distrib_swap
tff(fact_2589_add_Oinverse__distrib__swap,axiom,
    ! [Aa: int,Ba: int] : aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),Ba)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,Ba)),aa_int_int(uminus_uminus_int,Aa)) ).

% add.inverse_distrib_swap
tff(fact_2590_add_Oinverse__distrib__swap,axiom,
    ! [Aa: code_integer,Ba: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Aa),Ba)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Ba)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa)) ).

% add.inverse_distrib_swap
tff(fact_2591_add_Oinverse__distrib__swap,axiom,
    ! [Aa: rat,Ba: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(plus_plus_rat(Aa),Ba)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,Ba)),aa_rat_rat(uminus_uminus_rat,Aa)) ).

% add.inverse_distrib_swap
tff(fact_2592_minus__diff__commute,axiom,
    ! [Ba: real,Aa: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,Ba)),Aa) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,Aa)),Ba) ).

% minus_diff_commute
tff(fact_2593_minus__diff__commute,axiom,
    ! [Ba: int,Aa: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,Ba)),Aa) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,Aa)),Ba) ).

% minus_diff_commute
tff(fact_2594_minus__diff__commute,axiom,
    ! [Ba: code_integer,Aa: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Ba)),Aa) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa)),Ba) ).

% minus_diff_commute
tff(fact_2595_minus__diff__commute,axiom,
    ! [Ba: rat,Aa: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,Ba)),Aa) = aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,Aa)),Ba) ).

% minus_diff_commute
tff(fact_2596_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_2597_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_2598_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_2599_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_2600_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_2601_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_2602_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_2603_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_2604_zero__neq__neg__one,axiom,
    zero_zero_complex != aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% zero_neq_neg_one
tff(fact_2605_zero__neq__neg__one,axiom,
    zero_zero_real != aa_real_real(uminus_uminus_real,one_one_real) ).

% zero_neq_neg_one
tff(fact_2606_zero__neq__neg__one,axiom,
    zero_zero_int != aa_int_int(uminus_uminus_int,one_one_int) ).

% zero_neq_neg_one
tff(fact_2607_zero__neq__neg__one,axiom,
    zero_z3403309356797280102nteger != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% zero_neq_neg_one
tff(fact_2608_zero__neq__neg__one,axiom,
    zero_zero_rat != aa_rat_rat(uminus_uminus_rat,one_one_rat) ).

% zero_neq_neg_one
tff(fact_2609_neg__eq__iff__add__eq__0,axiom,
    ! [Aa: real,Ba: real] :
      ( ( aa_real_real(uminus_uminus_real,Aa) = Ba )
    <=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),Ba) = zero_zero_real ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_2610_neg__eq__iff__add__eq__0,axiom,
    ! [Aa: int,Ba: int] :
      ( ( aa_int_int(uminus_uminus_int,Aa) = Ba )
    <=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),Ba) = zero_zero_int ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_2611_neg__eq__iff__add__eq__0,axiom,
    ! [Aa: code_integer,Ba: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa) = Ba )
    <=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Aa),Ba) = zero_z3403309356797280102nteger ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_2612_neg__eq__iff__add__eq__0,axiom,
    ! [Aa: rat,Ba: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,Aa) = Ba )
    <=> ( aa_rat_rat(plus_plus_rat(Aa),Ba) = zero_zero_rat ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_2613_eq__neg__iff__add__eq__0,axiom,
    ! [Aa: real,Ba: real] :
      ( ( Aa = aa_real_real(uminus_uminus_real,Ba) )
    <=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),Ba) = zero_zero_real ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_2614_eq__neg__iff__add__eq__0,axiom,
    ! [Aa: int,Ba: int] :
      ( ( Aa = aa_int_int(uminus_uminus_int,Ba) )
    <=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),Ba) = zero_zero_int ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_2615_eq__neg__iff__add__eq__0,axiom,
    ! [Aa: code_integer,Ba: code_integer] :
      ( ( Aa = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Ba) )
    <=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Aa),Ba) = zero_z3403309356797280102nteger ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_2616_eq__neg__iff__add__eq__0,axiom,
    ! [Aa: rat,Ba: rat] :
      ( ( Aa = aa_rat_rat(uminus_uminus_rat,Ba) )
    <=> ( aa_rat_rat(plus_plus_rat(Aa),Ba) = zero_zero_rat ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_2617_add_Oinverse__unique,axiom,
    ! [Aa: real,Ba: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),Ba) = zero_zero_real )
     => ( aa_real_real(uminus_uminus_real,Aa) = Ba ) ) ).

% add.inverse_unique
tff(fact_2618_add_Oinverse__unique,axiom,
    ! [Aa: int,Ba: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),Ba) = zero_zero_int )
     => ( aa_int_int(uminus_uminus_int,Aa) = Ba ) ) ).

% add.inverse_unique
tff(fact_2619_add_Oinverse__unique,axiom,
    ! [Aa: code_integer,Ba: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Aa),Ba) = zero_z3403309356797280102nteger )
     => ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa) = Ba ) ) ).

% add.inverse_unique
tff(fact_2620_add_Oinverse__unique,axiom,
    ! [Aa: rat,Ba: rat] :
      ( ( aa_rat_rat(plus_plus_rat(Aa),Ba) = zero_zero_rat )
     => ( aa_rat_rat(uminus_uminus_rat,Aa) = Ba ) ) ).

% add.inverse_unique
tff(fact_2621_ab__group__add__class_Oab__left__minus,axiom,
    ! [Aa: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,Aa)),Aa) = zero_zero_real ).

% ab_group_add_class.ab_left_minus
tff(fact_2622_ab__group__add__class_Oab__left__minus,axiom,
    ! [Aa: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,Aa)),Aa) = zero_zero_int ).

% ab_group_add_class.ab_left_minus
tff(fact_2623_ab__group__add__class_Oab__left__minus,axiom,
    ! [Aa: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa)),Aa) = zero_z3403309356797280102nteger ).

% ab_group_add_class.ab_left_minus
tff(fact_2624_ab__group__add__class_Oab__left__minus,axiom,
    ! [Aa: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,Aa)),Aa) = zero_zero_rat ).

% ab_group_add_class.ab_left_minus
tff(fact_2625_add__eq__0__iff,axiom,
    ! [Aa: real,Ba: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),Ba) = zero_zero_real )
    <=> ( Ba = aa_real_real(uminus_uminus_real,Aa) ) ) ).

% add_eq_0_iff
tff(fact_2626_add__eq__0__iff,axiom,
    ! [Aa: int,Ba: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),Ba) = zero_zero_int )
    <=> ( Ba = aa_int_int(uminus_uminus_int,Aa) ) ) ).

% add_eq_0_iff
tff(fact_2627_add__eq__0__iff,axiom,
    ! [Aa: code_integer,Ba: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Aa),Ba) = zero_z3403309356797280102nteger )
    <=> ( Ba = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa) ) ) ).

% add_eq_0_iff
tff(fact_2628_add__eq__0__iff,axiom,
    ! [Aa: rat,Ba: rat] :
      ( ( aa_rat_rat(plus_plus_rat(Aa),Ba) = zero_zero_rat )
    <=> ( Ba = aa_rat_rat(uminus_uminus_rat,Aa) ) ) ).

% add_eq_0_iff
tff(fact_2629_zero__le__numeral,axiom,
    ! [Na: num] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_num_real(numeral_numeral_real,Na)) ).

% zero_le_numeral
tff(fact_2630_zero__le__numeral,axiom,
    ! [Na: num] : aa_Extended_enat_o(ord_le2932123472753598470d_enat(zero_z5237406670263579293d_enat),numera1916890842035813515d_enat(Na)) ).

% zero_le_numeral
tff(fact_2631_zero__le__numeral,axiom,
    ! [Na: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_num_Code_integer(numera6620942414471956472nteger,Na)) ).

% zero_le_numeral
tff(fact_2632_zero__le__numeral,axiom,
    ! [Na: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_num_rat(numeral_numeral_rat,Na)) ).

% zero_le_numeral
tff(fact_2633_zero__le__numeral,axiom,
    ! [Na: num] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,Na)) ).

% zero_le_numeral
tff(fact_2634_zero__le__numeral,axiom,
    ! [Na: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_num_int(numeral_numeral_int,Na)) ).

% zero_le_numeral
tff(fact_2635_not__numeral__le__zero,axiom,
    ! [Na: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,Na)),zero_zero_real) ).

% not_numeral_le_zero
tff(fact_2636_not__numeral__le__zero,axiom,
    ! [Na: num] : ~ aa_Extended_enat_o(ord_le2932123472753598470d_enat(numera1916890842035813515d_enat(Na)),zero_z5237406670263579293d_enat) ).

% not_numeral_le_zero
tff(fact_2637_not__numeral__le__zero,axiom,
    ! [Na: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_num_Code_integer(numera6620942414471956472nteger,Na)),zero_z3403309356797280102nteger) ).

% not_numeral_le_zero
tff(fact_2638_not__numeral__le__zero,axiom,
    ! [Na: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,Na)),zero_zero_rat) ).

% not_numeral_le_zero
tff(fact_2639_not__numeral__le__zero,axiom,
    ! [Na: num] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,Na)),zero_zero_nat) ).

% not_numeral_le_zero
tff(fact_2640_not__numeral__le__zero,axiom,
    ! [Na: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Na)),zero_zero_int) ).

% not_numeral_le_zero
tff(fact_2641_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_2642_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_2643_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_2644_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_2645_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_2646_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_2647_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_2648_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_2649_zero__less__numeral,axiom,
    ! [Na: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_num_rat(numeral_numeral_rat,Na)) ).

% zero_less_numeral
tff(fact_2650_zero__less__numeral,axiom,
    ! [Na: num] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,Na)) ).

% zero_less_numeral
tff(fact_2651_zero__less__numeral,axiom,
    ! [Na: num] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_num_real(numeral_numeral_real,Na)) ).

% zero_less_numeral
tff(fact_2652_zero__less__numeral,axiom,
    ! [Na: num] : aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_num_int(numeral_numeral_int,Na)) ).

% zero_less_numeral
tff(fact_2653_zero__less__numeral,axiom,
    ! [Na: num] : aa_Extended_enat_o(ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat),numera1916890842035813515d_enat(Na)) ).

% zero_less_numeral
tff(fact_2654_zero__less__numeral,axiom,
    ! [Na: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_num_Code_integer(numera6620942414471956472nteger,Na)) ).

% zero_less_numeral
tff(fact_2655_not__numeral__less__zero,axiom,
    ! [Na: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,Na)),zero_zero_rat) ).

% not_numeral_less_zero
tff(fact_2656_not__numeral__less__zero,axiom,
    ! [Na: num] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,Na)),zero_zero_nat) ).

% not_numeral_less_zero
tff(fact_2657_not__numeral__less__zero,axiom,
    ! [Na: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,Na)),zero_zero_real) ).

% not_numeral_less_zero
tff(fact_2658_not__numeral__less__zero,axiom,
    ! [Na: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Na)),zero_zero_int) ).

% not_numeral_less_zero
tff(fact_2659_not__numeral__less__zero,axiom,
    ! [Na: num] : ~ aa_Extended_enat_o(ord_le72135733267957522d_enat(numera1916890842035813515d_enat(Na)),zero_z5237406670263579293d_enat) ).

% not_numeral_less_zero
tff(fact_2660_not__numeral__less__zero,axiom,
    ! [Na: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_num_Code_integer(numera6620942414471956472nteger,Na)),zero_z3403309356797280102nteger) ).

% not_numeral_less_zero
tff(fact_2661_replicate__Suc,axiom,
    ! [Na: nat,Xb: int] : replicate_int(aa_nat_nat(suc,Na),Xb) = cons_int(Xb,replicate_int(Na,Xb)) ).

% replicate_Suc
tff(fact_2662_replicate__Suc,axiom,
    ! [Na: nat,Xb: nat] : replicate_nat(aa_nat_nat(suc,Na),Xb) = cons_nat(Xb,replicate_nat(Na,Xb)) ).

% replicate_Suc
tff(fact_2663_replicate__Suc,axiom,
    ! [Na: nat,Xb: vEBT_VEBT] : replicate_VEBT_VEBT(aa_nat_nat(suc,Na),Xb) = cons_VEBT_VEBT(Xb,replicate_VEBT_VEBT(Na,Xb)) ).

% replicate_Suc
tff(fact_2664_one__le__numeral,axiom,
    ! [Na: num] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_num_real(numeral_numeral_real,Na)) ).

% one_le_numeral
tff(fact_2665_one__le__numeral,axiom,
    ! [Na: num] : aa_Extended_enat_o(ord_le2932123472753598470d_enat(one_on7984719198319812577d_enat),numera1916890842035813515d_enat(Na)) ).

% one_le_numeral
tff(fact_2666_one__le__numeral,axiom,
    ! [Na: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),aa_num_Code_integer(numera6620942414471956472nteger,Na)) ).

% one_le_numeral
tff(fact_2667_one__le__numeral,axiom,
    ! [Na: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_num_rat(numeral_numeral_rat,Na)) ).

% one_le_numeral
tff(fact_2668_one__le__numeral,axiom,
    ! [Na: num] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,Na)) ).

% one_le_numeral
tff(fact_2669_one__le__numeral,axiom,
    ! [Na: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_num_int(numeral_numeral_int,Na)) ).

% one_le_numeral
tff(fact_2670_group__cancel_Osub2,axiom,
    ! [B4: real,K: real,Ba: real,Aa: real] :
      ( ( B4 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),Ba) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Aa),B4) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,K)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Aa),Ba)) ) ) ).

% group_cancel.sub2
tff(fact_2671_group__cancel_Osub2,axiom,
    ! [B4: int,K: int,Ba: int,Aa: int] :
      ( ( B4 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),Ba) )
     => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,Aa),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,Aa),Ba)) ) ) ).

% group_cancel.sub2
tff(fact_2672_group__cancel_Osub2,axiom,
    ! [B4: code_integer,K: code_integer,Ba: code_integer,Aa: code_integer] :
      ( ( B4 = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,K),Ba) )
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,Aa),B4) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,K)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,Aa),Ba)) ) ) ).

% group_cancel.sub2
tff(fact_2673_group__cancel_Osub2,axiom,
    ! [B4: rat,K: rat,Ba: rat,Aa: rat] :
      ( ( B4 = aa_rat_rat(plus_plus_rat(K),Ba) )
     => ( aa_rat_rat(minus_minus_rat(Aa),B4) = aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,K)),aa_rat_rat(minus_minus_rat(Aa),Ba)) ) ) ).

% group_cancel.sub2
tff(fact_2674_diff__conv__add__uminus,axiom,
    ! [Aa: real,Ba: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Aa),Ba) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),aa_real_real(uminus_uminus_real,Ba)) ).

% diff_conv_add_uminus
tff(fact_2675_diff__conv__add__uminus,axiom,
    ! [Aa: int,Ba: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,Aa),Ba) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),aa_int_int(uminus_uminus_int,Ba)) ).

% diff_conv_add_uminus
tff(fact_2676_diff__conv__add__uminus,axiom,
    ! [Aa: code_integer,Ba: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,Aa),Ba) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Aa),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Ba)) ).

% diff_conv_add_uminus
tff(fact_2677_diff__conv__add__uminus,axiom,
    ! [Aa: rat,Ba: rat] : aa_rat_rat(minus_minus_rat(Aa),Ba) = aa_rat_rat(plus_plus_rat(Aa),aa_rat_rat(uminus_uminus_rat,Ba)) ).

% diff_conv_add_uminus
tff(fact_2678_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [Aa: real,Ba: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Aa),Ba) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),aa_real_real(uminus_uminus_real,Ba)) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_2679_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [Aa: int,Ba: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,Aa),Ba) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),aa_int_int(uminus_uminus_int,Ba)) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_2680_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [Aa: code_integer,Ba: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,Aa),Ba) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Aa),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Ba)) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_2681_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [Aa: rat,Ba: rat] : aa_rat_rat(minus_minus_rat(Aa),Ba) = aa_rat_rat(plus_plus_rat(Aa),aa_rat_rat(uminus_uminus_rat,Ba)) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_2682_not__numeral__less__one,axiom,
    ! [Na: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,Na)),one_one_rat) ).

% not_numeral_less_one
tff(fact_2683_not__numeral__less__one,axiom,
    ! [Na: num] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,Na)),one_one_nat) ).

% not_numeral_less_one
tff(fact_2684_not__numeral__less__one,axiom,
    ! [Na: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,Na)),one_one_real) ).

% not_numeral_less_one
tff(fact_2685_not__numeral__less__one,axiom,
    ! [Na: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Na)),one_one_int) ).

% not_numeral_less_one
tff(fact_2686_not__numeral__less__one,axiom,
    ! [Na: num] : ~ aa_Extended_enat_o(ord_le72135733267957522d_enat(numera1916890842035813515d_enat(Na)),one_on7984719198319812577d_enat) ).

% not_numeral_less_one
tff(fact_2687_not__numeral__less__one,axiom,
    ! [Na: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_num_Code_integer(numera6620942414471956472nteger,Na)),one_one_Code_integer) ).

% not_numeral_less_one
tff(fact_2688_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_2689_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_2690_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_2691_one__plus__numeral__commute,axiom,
    ! [Xb: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_num_real(numeral_numeral_real,Xb)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,Xb)),one_one_real) ).

% one_plus_numeral_commute
tff(fact_2692_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_2693_one__plus__numeral__commute,axiom,
    ! [Xb: num] : plus_p3455044024723400733d_enat(one_on7984719198319812577d_enat,numera1916890842035813515d_enat(Xb)) = plus_p3455044024723400733d_enat(numera1916890842035813515d_enat(Xb),one_on7984719198319812577d_enat) ).

% one_plus_numeral_commute
tff(fact_2694_one__plus__numeral__commute,axiom,
    ! [Xb: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,one_one_Code_integer),aa_num_Code_integer(numera6620942414471956472nteger,Xb)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_num_Code_integer(numera6620942414471956472nteger,Xb)),one_one_Code_integer) ).

% one_plus_numeral_commute
tff(fact_2695_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_2696_subset__Compl__self__eq,axiom,
    ! [A3: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A3),uminus5710092332889474511et_nat(A3))
    <=> ( A3 = bot_bot_set_nat ) ) ).

% subset_Compl_self_eq
tff(fact_2697_subset__Compl__self__eq,axiom,
    ! [A3: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A3),uminus1532241313380277803et_int(A3))
    <=> ( A3 = bot_bot_set_int ) ) ).

% subset_Compl_self_eq
tff(fact_2698_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_2699_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_2700_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_2701_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_2702_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_2703_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_2704_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_2705_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_2706_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_2707_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_2708_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_2709_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_2710_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_2711_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_2712_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_2713_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_2714_lenlex__length,axiom,
    ! [Ms: list_VEBT_VEBT,Ns: list_VEBT_VEBT,R2: set_Pr6192946355708809607T_VEBT] :
      ( member4439316823752958928T_VEBT(produc3897820843166775703T_VEBT(Ms,Ns),lenlex_VEBT_VEBT(R2))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_s6755466524823107622T_VEBT(Ms)),size_s6755466524823107622T_VEBT(Ns)) ) ).

% lenlex_length
tff(fact_2715_lenlex__length,axiom,
    ! [Ms: list_o,Ns: list_o,R2: set_Product_prod_o_o] :
      ( member4159035015898711888list_o(produc8435520187683070743list_o(Ms,Ns),lenlex_o(R2))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_o(Ms)),size_size_list_o(Ns)) ) ).

% lenlex_length
tff(fact_2716_lenlex__length,axiom,
    ! [Ms: list_nat,Ns: list_nat,R2: set_Pr1261947904930325089at_nat] :
      ( member7340969449405702474st_nat(produc2694037385005941721st_nat(Ms,Ns),lenlex_nat(R2))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_nat(Ms)),size_size_list_nat(Ns)) ) ).

% lenlex_length
tff(fact_2717_measures__less,axiom,
    ! [F: fun_Code_integer_nat,Xb: code_integer,Y: code_integer,Fs: list_f8684324736568799034er_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_Code_integer_nat(F,Xb)),aa_Code_integer_nat(F,Y))
     => member157494554546826820nteger(produc1086072967326762835nteger(Xb,Y),measur8870801148506250077nteger(cons_f3120178438656451060er_nat(F,Fs))) ) ).

% measures_less
tff(fact_2718_measures__less,axiom,
    ! [F: fun_Pr2602117396471556196at_nat,Xb: product_prod_nat_nat,Y: product_prod_nat_nat,Fs: list_f8256639991604547178at_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_Pro4243173770044397973at_nat(F,Xb)),aa_Pro4243173770044397973at_nat(F,Y))
     => member8206827879206165904at_nat(produc6161850002892822231at_nat(Xb,Y),measur2679027848233739777at_nat(cons_f462676766793977114at_nat(F,Fs))) ) ).

% measures_less
tff(fact_2719_measures__less,axiom,
    ! [F: fun_nat_nat,Xb: nat,Y: nat,Fs: list_fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,Xb)),aa_nat_nat(F,Y))
     => member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xb),Y),measures_nat(cons_fun_nat_nat(F,Fs))) ) ).

% measures_less
tff(fact_2720_measures__less,axiom,
    ! [F: fun_int_nat,Xb: int,Y: int,Fs: list_fun_int_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(F,Xb)),aa_int_nat(F,Y))
     => member5262025264175285858nt_int(product_Pair_int_int(Xb,Y),measures_int(cons_fun_int_nat(F,Fs))) ) ).

% measures_less
tff(fact_2721_measures__lesseq,axiom,
    ! [F: fun_Code_integer_nat,Xb: code_integer,Y: code_integer,Fs: list_f8684324736568799034er_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_Code_integer_nat(F,Xb)),aa_Code_integer_nat(F,Y))
     => ( member157494554546826820nteger(produc1086072967326762835nteger(Xb,Y),measur8870801148506250077nteger(Fs))
       => member157494554546826820nteger(produc1086072967326762835nteger(Xb,Y),measur8870801148506250077nteger(cons_f3120178438656451060er_nat(F,Fs))) ) ) ).

% measures_lesseq
tff(fact_2722_measures__lesseq,axiom,
    ! [F: fun_Pr2602117396471556196at_nat,Xb: product_prod_nat_nat,Y: product_prod_nat_nat,Fs: list_f8256639991604547178at_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_Pro4243173770044397973at_nat(F,Xb)),aa_Pro4243173770044397973at_nat(F,Y))
     => ( member8206827879206165904at_nat(produc6161850002892822231at_nat(Xb,Y),measur2679027848233739777at_nat(Fs))
       => member8206827879206165904at_nat(produc6161850002892822231at_nat(Xb,Y),measur2679027848233739777at_nat(cons_f462676766793977114at_nat(F,Fs))) ) ) ).

% measures_lesseq
tff(fact_2723_measures__lesseq,axiom,
    ! [F: fun_nat_nat,Xb: nat,Y: nat,Fs: list_fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,Xb)),aa_nat_nat(F,Y))
     => ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xb),Y),measures_nat(Fs))
       => member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xb),Y),measures_nat(cons_fun_nat_nat(F,Fs))) ) ) ).

% measures_lesseq
tff(fact_2724_measures__lesseq,axiom,
    ! [F: fun_int_nat,Xb: int,Y: int,Fs: list_fun_int_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,Xb)),aa_int_nat(F,Y))
     => ( member5262025264175285858nt_int(product_Pair_int_int(Xb,Y),measures_int(Fs))
       => member5262025264175285858nt_int(product_Pair_int_int(Xb,Y),measures_int(cons_fun_int_nat(F,Fs))) ) ) ).

% measures_lesseq
tff(fact_2725_count__le__length,axiom,
    ! [Xs: list_VEBT_VEBT,Xb: vEBT_VEBT] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_VEBT_VEBT_nat(count_list_VEBT_VEBT(Xs),Xb)),size_s6755466524823107622T_VEBT(Xs)) ).

% count_le_length
tff(fact_2726_count__le__length,axiom,
    ! [Xs: list_o,Xb: $o] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_o_nat(count_list_o(Xs),(Xb))),size_size_list_o(Xs)) ).

% count_le_length
tff(fact_2727_count__le__length,axiom,
    ! [Xs: list_nat,Xb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(count_list_nat(Xs),Xb)),size_size_list_nat(Xs)) ).

% count_le_length
tff(fact_2728_Cons__replicate__eq,axiom,
    ! [Xb: int,Xs: list_int,Na: nat,Y: int] :
      ( ( cons_int(Xb,Xs) = replicate_int(Na,Y) )
    <=> ( ( Xb = Y )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
        & ( Xs = replicate_int(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat),Xb) ) ) ) ).

% Cons_replicate_eq
tff(fact_2729_Cons__replicate__eq,axiom,
    ! [Xb: nat,Xs: list_nat,Na: nat,Y: nat] :
      ( ( cons_nat(Xb,Xs) = replicate_nat(Na,Y) )
    <=> ( ( Xb = Y )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
        & ( Xs = replicate_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat),Xb) ) ) ) ).

% Cons_replicate_eq
tff(fact_2730_Cons__replicate__eq,axiom,
    ! [Xb: vEBT_VEBT,Xs: list_VEBT_VEBT,Na: nat,Y: vEBT_VEBT] :
      ( ( cons_VEBT_VEBT(Xb,Xs) = replicate_VEBT_VEBT(Na,Y) )
    <=> ( ( Xb = Y )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
        & ( Xs = replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat),Xb) ) ) ) ).

% Cons_replicate_eq
tff(fact_2731_pred__max,axiom,
    ! [Deg: nat,Ma2: nat,Xb: nat,Mi2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),Xb)
       => ( vEBT_vebt_pred(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Deg,TreeList,Summary),Xb) = some_nat(Ma2) ) ) ) ).

% pred_max
tff(fact_2732_diff__numeral__special_I5_J,axiom,
    ! [Na: num] : aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),numera6690914467698888265omplex(Na)) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(inc,Na))) ).

% diff_numeral_special(5)
tff(fact_2733_diff__numeral__special_I5_J,axiom,
    ! [Na: num] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_num_real(numeral_numeral_real,Na)) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(inc,Na))) ).

% diff_numeral_special(5)
tff(fact_2734_diff__numeral__special_I5_J,axiom,
    ! [Na: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,Na)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(inc,Na))) ).

% diff_numeral_special(5)
tff(fact_2735_diff__numeral__special_I5_J,axiom,
    ! [Na: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_num_Code_integer(numera6620942414471956472nteger,Na)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(inc,Na))) ).

% diff_numeral_special(5)
tff(fact_2736_diff__numeral__special_I5_J,axiom,
    ! [Na: num] : aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_num_rat(numeral_numeral_rat,Na)) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(inc,Na))) ).

% diff_numeral_special(5)
tff(fact_2737_diff__numeral__special_I6_J,axiom,
    ! [M: num] : aa_complex_complex(minus_minus_complex(numera6690914467698888265omplex(M)),aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = numera6690914467698888265omplex(aa_num_num(inc,M)) ).

% diff_numeral_special(6)
tff(fact_2738_diff__numeral__special_I6_J,axiom,
    ! [M: num] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_num_real(numeral_numeral_real,M)),aa_real_real(uminus_uminus_real,one_one_real)) = aa_num_real(numeral_numeral_real,aa_num_num(inc,M)) ).

% diff_numeral_special(6)
tff(fact_2739_diff__numeral__special_I6_J,axiom,
    ! [M: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,one_one_int)) = aa_num_int(numeral_numeral_int,aa_num_num(inc,M)) ).

% diff_numeral_special(6)
tff(fact_2740_diff__numeral__special_I6_J,axiom,
    ! [M: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(inc,M)) ).

% diff_numeral_special(6)
tff(fact_2741_diff__numeral__special_I6_J,axiom,
    ! [M: num] : aa_rat_rat(minus_minus_rat(aa_num_rat(numeral_numeral_rat,M)),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_num_rat(numeral_numeral_rat,aa_num_num(inc,M)) ).

% diff_numeral_special(6)
tff(fact_2742_add__neg__numeral__special_I5_J,axiom,
    ! [Na: num] : aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Na))) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(inc,Na))) ).

% add_neg_numeral_special(5)
tff(fact_2743_add__neg__numeral__special_I5_J,axiom,
    ! [Na: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Na))) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(inc,Na))) ).

% add_neg_numeral_special(5)
tff(fact_2744_add__neg__numeral__special_I5_J,axiom,
    ! [Na: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Na))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(inc,Na))) ).

% add_neg_numeral_special(5)
tff(fact_2745_add__neg__numeral__special_I5_J,axiom,
    ! [Na: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Na))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(inc,Na))) ).

% add_neg_numeral_special(5)
tff(fact_2746_add__neg__numeral__special_I5_J,axiom,
    ! [Na: 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,Na))) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(inc,Na))) ).

% add_neg_numeral_special(5)
tff(fact_2747_add__neg__numeral__special_I6_J,axiom,
    ! [M: num] : aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(M))),aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(inc,M))) ).

% add_neg_numeral_special(6)
tff(fact_2748_add__neg__numeral__special_I6_J,axiom,
    ! [M: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M))),aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(inc,M))) ).

% add_neg_numeral_special(6)
tff(fact_2749_add__neg__numeral__special_I6_J,axiom,
    ! [M: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(inc,M))) ).

% add_neg_numeral_special(6)
tff(fact_2750_add__neg__numeral__special_I6_J,axiom,
    ! [M: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(inc,M))) ).

% add_neg_numeral_special(6)
tff(fact_2751_add__neg__numeral__special_I6_J,axiom,
    ! [M: num] : aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(inc,M))) ).

% add_neg_numeral_special(6)
tff(fact_2752_listrel__iff__nth,axiom,
    ! [Xs: list_Code_integer,Ys2: list_Code_integer,R2: set_Pr4811707699266497531nteger] :
      ( member749217712838834276nteger(produc750622340256944499nteger(Xs,Ys2),listre5734910445319291053nteger(R2))
    <=> ( ( size_s3445333598471063425nteger(Xs) = size_s3445333598471063425nteger(Ys2) )
        & ! [N4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N4),size_s3445333598471063425nteger(Xs))
           => member157494554546826820nteger(produc1086072967326762835nteger(aa_nat_Code_integer(nth_Code_integer(Xs),N4),aa_nat_Code_integer(nth_Code_integer(Ys2),N4)),R2) ) ) ) ).

% listrel_iff_nth
tff(fact_2753_listrel__iff__nth,axiom,
    ! [Xs: list_int,Ys2: list_int,R2: set_Pr958786334691620121nt_int] :
      ( member6698963635872716290st_int(produc364263696895485585st_int(Xs,Ys2),listrel_int_int(R2))
    <=> ( ( size_size_list_int(Xs) = size_size_list_int(Ys2) )
        & ! [N4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N4),size_size_list_int(Xs))
           => member5262025264175285858nt_int(product_Pair_int_int(aa_nat_int(nth_int(Xs),N4),aa_nat_int(nth_int(Ys2),N4)),R2) ) ) ) ).

% listrel_iff_nth
tff(fact_2754_listrel__iff__nth,axiom,
    ! [Xs: list_int,Ys2: list_VEBT_VEBT,R2: set_Pr8044002425091019955T_VEBT] :
      ( member4376149543098372618T_VEBT(produc6743464080745587621T_VEBT(Xs,Ys2),listre8491537028387690453T_VEBT(R2))
    <=> ( ( size_size_list_int(Xs) = size_s6755466524823107622T_VEBT(Ys2) )
        & ! [N4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N4),size_size_list_int(Xs))
           => member2056185340421749780T_VEBT(produc3329399203697025711T_VEBT(aa_nat_int(nth_int(Xs),N4),nth_VEBT_VEBT(Ys2,N4)),R2) ) ) ) ).

% listrel_iff_nth
tff(fact_2755_listrel__iff__nth,axiom,
    ! [Xs: list_int,Ys2: list_o,R2: set_Pr903927857289325719_int_o] :
      ( member9156582987741540206list_o(produc3167582181186427401list_o(Xs,Ys2),listrel_int_o(R2))
    <=> ( ( size_size_list_int(Xs) = size_size_list_o(Ys2) )
        & ! [N4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N4),size_size_list_int(Xs))
           => member4489920277610959864_int_o(product_Pair_int_o(aa_nat_int(nth_int(Xs),N4),aa_nat_o(nth_o(Ys2),N4)),R2) ) ) ) ).

% listrel_iff_nth
tff(fact_2756_listrel__iff__nth,axiom,
    ! [Xs: list_Code_integer,Ys2: list_o,R2: set_Pr448751882837621926eger_o] :
      ( member7510714728986300413list_o(produc2864564883805000344list_o(Xs,Ys2),listre7327554457731897160eger_o(R2))
    <=> ( ( size_s3445333598471063425nteger(Xs) = size_size_list_o(Ys2) )
        & ! [N4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N4),size_s3445333598471063425nteger(Xs))
           => member1379723562493234055eger_o(produc6677183202524767010eger_o(aa_nat_Code_integer(nth_Code_integer(Xs),N4),aa_nat_o(nth_o(Ys2),N4)),R2) ) ) ) ).

% listrel_iff_nth
tff(fact_2757_listrel__iff__nth,axiom,
    ! [Xs: list_int,Ys2: list_nat,R2: set_Pr3448869479623346877nt_nat] :
      ( member9189046780804443046st_nat(produc4542114716404682293st_nat(Xs,Ys2),listrel_int_nat(R2))
    <=> ( ( size_size_list_int(Xs) = size_size_list_nat(Ys2) )
        & ! [N4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N4),size_size_list_int(Xs))
           => member216504246829706758nt_nat(product_Pair_int_nat(aa_nat_int(nth_int(Xs),N4),aa_nat_nat(nth_nat(Ys2),N4)),R2) ) ) ) ).

% listrel_iff_nth
tff(fact_2758_listrel__iff__nth,axiom,
    ! [Xs: list_VEBT_VEBT,Ys2: list_int,R2: set_Pr5066593544530342725BT_int] :
      ( member3703241499402361532st_int(produc1392282695434103839st_int(Xs,Ys2),listre5898179758603845167BT_int(R2))
    <=> ( ( size_s6755466524823107622T_VEBT(Xs) = size_size_list_int(Ys2) )
        & ! [N4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N4),size_s6755466524823107622T_VEBT(Xs))
           => member5419026705395827622BT_int(produc736041933913180425BT_int(nth_VEBT_VEBT(Xs,N4),aa_nat_int(nth_int(Ys2),N4)),R2) ) ) ) ).

% listrel_iff_nth
tff(fact_2759_listrel__iff__nth,axiom,
    ! [Xs: list_VEBT_VEBT,Ys2: list_VEBT_VEBT,R2: set_Pr6192946355708809607T_VEBT] :
      ( member4439316823752958928T_VEBT(produc3897820843166775703T_VEBT(Xs,Ys2),listre1230615542750757617T_VEBT(R2))
    <=> ( ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Ys2) )
        & ! [N4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N4),size_s6755466524823107622T_VEBT(Xs))
           => member568628332442017744T_VEBT(produc537772716801021591T_VEBT(nth_VEBT_VEBT(Xs,N4),nth_VEBT_VEBT(Ys2,N4)),R2) ) ) ) ).

% listrel_iff_nth
tff(fact_2760_listrel__iff__nth,axiom,
    ! [Xs: list_VEBT_VEBT,Ys2: list_o,R2: set_Pr3175402225741728619VEBT_o] :
      ( member3126162362653435956list_o(produc2717590391345394939list_o(Xs,Ys2),listrel_VEBT_VEBT_o(R2))
    <=> ( ( size_s6755466524823107622T_VEBT(Xs) = size_size_list_o(Ys2) )
        & ! [N4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N4),size_s6755466524823107622T_VEBT(Xs))
           => member3307348790968139188VEBT_o(produc8721562602347293563VEBT_o(nth_VEBT_VEBT(Xs,N4),aa_nat_o(nth_o(Ys2),N4)),R2) ) ) ) ).

% listrel_iff_nth
tff(fact_2761_listrel__iff__nth,axiom,
    ! [Xs: list_VEBT_VEBT,Ys2: list_nat,R2: set_Pr7556676689462069481BT_nat] :
      ( member6193324644334088288st_nat(produc5570133714943300547st_nat(Xs,Ys2),listre5900670229112895443BT_nat(R2))
    <=> ( ( size_s6755466524823107622T_VEBT(Xs) = size_size_list_nat(Ys2) )
        & ! [N4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N4),size_s6755466524823107622T_VEBT(Xs))
           => member373505688050248522BT_nat(produc738532404422230701BT_nat(nth_VEBT_VEBT(Xs,N4),aa_nat_nat(nth_nat(Ys2),N4)),R2) ) ) ) ).

% listrel_iff_nth
tff(fact_2762_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_2763_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_2764_dbl__simps_I1_J,axiom,
    ! [K: num] : neg_nu8804712462038260780nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,K))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,neg_nu8804712462038260780nteger(aa_num_Code_integer(numera6620942414471956472nteger,K))) ).

% dbl_simps(1)
tff(fact_2765_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_2766_neg__numeral__le__ceiling,axiom,
    ! [V: num,Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_real_int(archim7802044766580827645g_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))),one_one_real)),Xb) ) ).

% neg_numeral_le_ceiling
tff(fact_2767_neg__numeral__le__ceiling,axiom,
    ! [V: num,Xb: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_rat_int(archim2889992004027027881ng_rat,Xb))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),one_one_rat)),Xb) ) ).

% neg_numeral_le_ceiling
tff(fact_2768_ceiling__less__neg__numeral,axiom,
    ! [Xb: real,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim7802044766580827645g_real,Xb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))),one_one_real)) ) ).

% ceiling_less_neg_numeral
tff(fact_2769_ceiling__less__neg__numeral,axiom,
    ! [Xb: rat,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(archim2889992004027027881ng_rat,Xb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),one_one_rat)) ) ).

% ceiling_less_neg_numeral
tff(fact_2770__092_060open_0622_A_092_060le_062_Ax_092_060close_062,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),xa) ).

% \<open>2 \<le> x\<close>
tff(fact_2771_insert__simp__mima,axiom,
    ! [Xb: nat,Mi2: nat,Ma2: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( Xb = Mi2 )
        | ( Xb = Ma2 ) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)
       => ( vEBT_vebt_insert(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Deg,TreeList,Summary),Xb) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Deg,TreeList,Summary) ) ) ) ).

% insert_simp_mima
tff(fact_2772_zle__add1__eq__le,axiom,
    ! [W2: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,W2),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,W2),Z) ) ).

% zle_add1_eq_le
tff(fact_2773_zle__diff1__eq,axiom,
    ! [W2: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W2),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,W2),Z) ) ).

% zle_diff1_eq
tff(fact_2774_ceiling__zero,axiom,
    aa_rat_int(archim2889992004027027881ng_rat,zero_zero_rat) = zero_zero_int ).

% ceiling_zero
tff(fact_2775_ceiling__zero,axiom,
    aa_real_int(archim7802044766580827645g_real,zero_zero_real) = zero_zero_int ).

% ceiling_zero
tff(fact_2776_ceiling__one,axiom,
    aa_rat_int(archim2889992004027027881ng_rat,one_one_rat) = one_one_int ).

% ceiling_one
tff(fact_2777_ceiling__one,axiom,
    aa_real_int(archim7802044766580827645g_real,one_one_real) = one_one_int ).

% ceiling_one
tff(fact_2778_dbl__simps_I2_J,axiom,
    aa_real_real(neg_numeral_dbl_real,zero_zero_real) = zero_zero_real ).

% dbl_simps(2)
tff(fact_2779_dbl__simps_I2_J,axiom,
    aa_rat_rat(neg_numeral_dbl_rat,zero_zero_rat) = zero_zero_rat ).

% dbl_simps(2)
tff(fact_2780_dbl__simps_I2_J,axiom,
    aa_int_int(neg_numeral_dbl_int,zero_zero_int) = zero_zero_int ).

% dbl_simps(2)
tff(fact_2781_one__eq__numeral__iff,axiom,
    ! [Na: num] :
      ( ( one_one_complex = numera6690914467698888265omplex(Na) )
    <=> ( one = Na ) ) ).

% one_eq_numeral_iff
tff(fact_2782_one__eq__numeral__iff,axiom,
    ! [Na: num] :
      ( ( one_one_rat = aa_num_rat(numeral_numeral_rat,Na) )
    <=> ( one = Na ) ) ).

% one_eq_numeral_iff
tff(fact_2783_one__eq__numeral__iff,axiom,
    ! [Na: num] :
      ( ( one_one_nat = aa_num_nat(numeral_numeral_nat,Na) )
    <=> ( one = Na ) ) ).

% one_eq_numeral_iff
tff(fact_2784_one__eq__numeral__iff,axiom,
    ! [Na: num] :
      ( ( one_one_real = aa_num_real(numeral_numeral_real,Na) )
    <=> ( one = Na ) ) ).

% one_eq_numeral_iff
tff(fact_2785_one__eq__numeral__iff,axiom,
    ! [Na: num] :
      ( ( one_one_int = aa_num_int(numeral_numeral_int,Na) )
    <=> ( one = Na ) ) ).

% one_eq_numeral_iff
tff(fact_2786_one__eq__numeral__iff,axiom,
    ! [Na: num] :
      ( ( one_on7984719198319812577d_enat = numera1916890842035813515d_enat(Na) )
    <=> ( one = Na ) ) ).

% one_eq_numeral_iff
tff(fact_2787_one__eq__numeral__iff,axiom,
    ! [Na: num] :
      ( ( one_one_Code_integer = aa_num_Code_integer(numera6620942414471956472nteger,Na) )
    <=> ( one = Na ) ) ).

% one_eq_numeral_iff
tff(fact_2788_numeral__eq__one__iff,axiom,
    ! [Na: num] :
      ( ( numera6690914467698888265omplex(Na) = one_one_complex )
    <=> ( Na = one ) ) ).

% numeral_eq_one_iff
tff(fact_2789_numeral__eq__one__iff,axiom,
    ! [Na: num] :
      ( ( aa_num_rat(numeral_numeral_rat,Na) = one_one_rat )
    <=> ( Na = one ) ) ).

% numeral_eq_one_iff
tff(fact_2790_numeral__eq__one__iff,axiom,
    ! [Na: num] :
      ( ( aa_num_nat(numeral_numeral_nat,Na) = one_one_nat )
    <=> ( Na = one ) ) ).

% numeral_eq_one_iff
tff(fact_2791_numeral__eq__one__iff,axiom,
    ! [Na: num] :
      ( ( aa_num_real(numeral_numeral_real,Na) = one_one_real )
    <=> ( Na = one ) ) ).

% numeral_eq_one_iff
tff(fact_2792_numeral__eq__one__iff,axiom,
    ! [Na: num] :
      ( ( aa_num_int(numeral_numeral_int,Na) = one_one_int )
    <=> ( Na = one ) ) ).

% numeral_eq_one_iff
tff(fact_2793_numeral__eq__one__iff,axiom,
    ! [Na: num] :
      ( ( numera1916890842035813515d_enat(Na) = one_on7984719198319812577d_enat )
    <=> ( Na = one ) ) ).

% numeral_eq_one_iff
tff(fact_2794_numeral__eq__one__iff,axiom,
    ! [Na: num] :
      ( ( aa_num_Code_integer(numera6620942414471956472nteger,Na) = one_one_Code_integer )
    <=> ( Na = one ) ) ).

% numeral_eq_one_iff
tff(fact_2795_ceiling__add__one,axiom,
    ! [Xb: rat] : aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(plus_plus_rat(Xb),one_one_rat)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim2889992004027027881ng_rat,Xb)),one_one_int) ).

% ceiling_add_one
tff(fact_2796_ceiling__add__one,axiom,
    ! [Xb: real] : aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),one_one_real)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim7802044766580827645g_real,Xb)),one_one_int) ).

% ceiling_add_one
tff(fact_2797_ceiling__diff__one,axiom,
    ! [Xb: rat] : aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(minus_minus_rat(Xb),one_one_rat)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_rat_int(archim2889992004027027881ng_rat,Xb)),one_one_int) ).

% ceiling_diff_one
tff(fact_2798_ceiling__diff__one,axiom,
    ! [Xb: real] : aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),one_one_real)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_real_int(archim7802044766580827645g_real,Xb)),one_one_int) ).

% ceiling_diff_one
tff(fact_2799_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_2800_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_2801_dbl__simps_I5_J,axiom,
    ! [K: num] : neg_nu8804712462038260780nteger(aa_num_Code_integer(numera6620942414471956472nteger,K)) = aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,K)) ).

% dbl_simps(5)
tff(fact_2802_neg__one__eq__numeral__iff,axiom,
    ! [Na: num] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Na)) )
    <=> ( Na = one ) ) ).

% neg_one_eq_numeral_iff
tff(fact_2803_neg__one__eq__numeral__iff,axiom,
    ! [Na: num] :
      ( ( aa_real_real(uminus_uminus_real,one_one_real) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Na)) )
    <=> ( Na = one ) ) ).

% neg_one_eq_numeral_iff
tff(fact_2804_neg__one__eq__numeral__iff,axiom,
    ! [Na: num] :
      ( ( aa_int_int(uminus_uminus_int,one_one_int) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Na)) )
    <=> ( Na = one ) ) ).

% neg_one_eq_numeral_iff
tff(fact_2805_neg__one__eq__numeral__iff,axiom,
    ! [Na: num] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Na)) )
    <=> ( Na = one ) ) ).

% neg_one_eq_numeral_iff
tff(fact_2806_neg__one__eq__numeral__iff,axiom,
    ! [Na: num] :
      ( ( aa_rat_rat(uminus_uminus_rat,one_one_rat) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Na)) )
    <=> ( Na = one ) ) ).

% neg_one_eq_numeral_iff
tff(fact_2807_numeral__eq__neg__one__iff,axiom,
    ! [Na: num] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Na)) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) )
    <=> ( Na = one ) ) ).

% numeral_eq_neg_one_iff
tff(fact_2808_numeral__eq__neg__one__iff,axiom,
    ! [Na: num] :
      ( ( aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Na)) = aa_real_real(uminus_uminus_real,one_one_real) )
    <=> ( Na = one ) ) ).

% numeral_eq_neg_one_iff
tff(fact_2809_numeral__eq__neg__one__iff,axiom,
    ! [Na: num] :
      ( ( aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Na)) = aa_int_int(uminus_uminus_int,one_one_int) )
    <=> ( Na = one ) ) ).

% numeral_eq_neg_one_iff
tff(fact_2810_numeral__eq__neg__one__iff,axiom,
    ! [Na: num] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Na)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) )
    <=> ( Na = one ) ) ).

% numeral_eq_neg_one_iff
tff(fact_2811_numeral__eq__neg__one__iff,axiom,
    ! [Na: num] :
      ( ( aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Na)) = aa_rat_rat(uminus_uminus_rat,one_one_rat) )
    <=> ( Na = one ) ) ).

% numeral_eq_neg_one_iff
tff(fact_2812_ceiling__le__zero,axiom,
    ! [Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,Xb)),zero_zero_int)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real) ) ).

% ceiling_le_zero
tff(fact_2813_ceiling__le__zero,axiom,
    ! [Xb: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,Xb)),zero_zero_int)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),zero_zero_rat) ) ).

% ceiling_le_zero
tff(fact_2814_ceiling__less__one,axiom,
    ! [Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim7802044766580827645g_real,Xb)),one_one_int)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real) ) ).

% ceiling_less_one
tff(fact_2815_ceiling__less__one,axiom,
    ! [Xb: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(archim2889992004027027881ng_rat,Xb)),one_one_int)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),zero_zero_rat) ) ).

% ceiling_less_one
tff(fact_2816_one__le__ceiling,axiom,
    ! [Xb: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_rat_int(archim2889992004027027881ng_rat,Xb))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xb) ) ).

% one_le_ceiling
tff(fact_2817_one__le__ceiling,axiom,
    ! [Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_real_int(archim7802044766580827645g_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb) ) ).

% one_le_ceiling
tff(fact_2818_zero__less__ceiling,axiom,
    ! [Xb: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_rat_int(archim2889992004027027881ng_rat,Xb))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xb) ) ).

% zero_less_ceiling
tff(fact_2819_zero__less__ceiling,axiom,
    ! [Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_real_int(archim7802044766580827645g_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb) ) ).

% zero_less_ceiling
tff(fact_2820_ceiling__le__one,axiom,
    ! [Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,Xb)),one_one_int)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real) ) ).

% ceiling_le_one
tff(fact_2821_ceiling__le__one,axiom,
    ! [Xb: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,Xb)),one_one_int)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),one_one_rat) ) ).

% ceiling_le_one
tff(fact_2822_one__less__ceiling,axiom,
    ! [Xb: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_rat_int(archim2889992004027027881ng_rat,Xb))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),Xb) ) ).

% one_less_ceiling
tff(fact_2823_one__less__ceiling,axiom,
    ! [Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_real_int(archim7802044766580827645g_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb) ) ).

% one_less_ceiling
tff(fact_2824_Suc__numeral,axiom,
    ! [Na: num] : aa_nat_nat(suc,aa_num_nat(numeral_numeral_nat,Na)) = aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,Na),one)) ).

% Suc_numeral
tff(fact_2825_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M: num] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M)))
    <=> ( M != one ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_2826_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M: num] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)))
    <=> ( M != one ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_2827_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M: 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,M)))
    <=> ( M != one ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_2828_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M: num] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)))
    <=> ( M != one ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_2829_neg__numeral__less__neg__one__iff,axiom,
    ! [M: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M))),aa_real_real(uminus_uminus_real,one_one_real))
    <=> ( M != one ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_2830_neg__numeral__less__neg__one__iff,axiom,
    ! [M: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,one_one_int))
    <=> ( M != one ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_2831_neg__numeral__less__neg__one__iff,axiom,
    ! [M: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer))
    <=> ( M != one ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_2832_neg__numeral__less__neg__one__iff,axiom,
    ! [M: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))),aa_rat_rat(uminus_uminus_rat,one_one_rat))
    <=> ( M != one ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_2833_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_2834_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_2835_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_2836_one__add__one,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),one_one_real) = aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)) ).

% one_add_one
tff(fact_2837_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_2838_one__add__one,axiom,
    plus_p3455044024723400733d_enat(one_on7984719198319812577d_enat,one_on7984719198319812577d_enat) = numera1916890842035813515d_enat(aa_num_num(bit0,one)) ).

% one_add_one
tff(fact_2839_one__add__one,axiom,
    aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,one_one_Code_integer),one_one_Code_integer) = aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)) ).

% one_add_one
tff(fact_2840_add__2__eq__Suc,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na) = aa_nat_nat(suc,aa_nat_nat(suc,Na)) ).

% add_2_eq_Suc
tff(fact_2841_add__2__eq__Suc_H,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(suc,aa_nat_nat(suc,Na)) ).

% add_2_eq_Suc'
tff(fact_2842_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_2843_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_2844_fact__2,axiom,
    semiri4449623510593786356d_enat(aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = numera1916890842035813515d_enat(aa_num_num(bit0,one)) ).

% fact_2
tff(fact_2845_fact__2,axiom,
    aa_nat_Code_integer(semiri3624122377584611663nteger,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)) ).

% fact_2
tff(fact_2846_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_2847_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_2848_numeral__plus__one,axiom,
    ! [Na: num] : aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(Na)),one_one_complex) = numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(plus_plus_num,Na),one)) ).

% numeral_plus_one
tff(fact_2849_numeral__plus__one,axiom,
    ! [Na: num] : aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,Na)),one_one_rat) = aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(plus_plus_num,Na),one)) ).

% numeral_plus_one
tff(fact_2850_numeral__plus__one,axiom,
    ! [Na: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,Na)),one_one_nat) = aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,Na),one)) ).

% numeral_plus_one
tff(fact_2851_numeral__plus__one,axiom,
    ! [Na: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,Na)),one_one_real) = aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(plus_plus_num,Na),one)) ).

% numeral_plus_one
tff(fact_2852_numeral__plus__one,axiom,
    ! [Na: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,Na)),one_one_int) = aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,Na),one)) ).

% numeral_plus_one
tff(fact_2853_numeral__plus__one,axiom,
    ! [Na: num] : plus_p3455044024723400733d_enat(numera1916890842035813515d_enat(Na),one_on7984719198319812577d_enat) = numera1916890842035813515d_enat(aa_num_num(aa_num_fun_num_num(plus_plus_num,Na),one)) ).

% numeral_plus_one
tff(fact_2854_numeral__plus__one,axiom,
    ! [Na: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_num_Code_integer(numera6620942414471956472nteger,Na)),one_one_Code_integer) = aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(aa_num_fun_num_num(plus_plus_num,Na),one)) ).

% numeral_plus_one
tff(fact_2855_one__plus__numeral,axiom,
    ! [Na: num] : aa_complex_complex(plus_plus_complex(one_one_complex),numera6690914467698888265omplex(Na)) = numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(plus_plus_num,one),Na)) ).

% one_plus_numeral
tff(fact_2856_one__plus__numeral,axiom,
    ! [Na: num] : aa_rat_rat(plus_plus_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,Na)) = aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(plus_plus_num,one),Na)) ).

% one_plus_numeral
tff(fact_2857_one__plus__numeral,axiom,
    ! [Na: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,Na)) = aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,one),Na)) ).

% one_plus_numeral
tff(fact_2858_one__plus__numeral,axiom,
    ! [Na: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_num_real(numeral_numeral_real,Na)) = aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(plus_plus_num,one),Na)) ).

% one_plus_numeral
tff(fact_2859_one__plus__numeral,axiom,
    ! [Na: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_num_int(numeral_numeral_int,Na)) = aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,one),Na)) ).

% one_plus_numeral
tff(fact_2860_one__plus__numeral,axiom,
    ! [Na: num] : plus_p3455044024723400733d_enat(one_on7984719198319812577d_enat,numera1916890842035813515d_enat(Na)) = numera1916890842035813515d_enat(aa_num_num(aa_num_fun_num_num(plus_plus_num,one),Na)) ).

% one_plus_numeral
tff(fact_2861_one__plus__numeral,axiom,
    ! [Na: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,one_one_Code_integer),aa_num_Code_integer(numera6620942414471956472nteger,Na)) = aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(aa_num_fun_num_num(plus_plus_num,one),Na)) ).

% one_plus_numeral
tff(fact_2862_ceiling__less__zero,axiom,
    ! [Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim7802044766580827645g_real,Xb)),zero_zero_int)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(uminus_uminus_real,one_one_real)) ) ).

% ceiling_less_zero
tff(fact_2863_ceiling__less__zero,axiom,
    ! [Xb: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(archim2889992004027027881ng_rat,Xb)),zero_zero_int)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),aa_rat_rat(uminus_uminus_rat,one_one_rat)) ) ).

% ceiling_less_zero
tff(fact_2864_zero__le__ceiling,axiom,
    ! [Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_real_int(archim7802044766580827645g_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb) ) ).

% zero_le_ceiling
tff(fact_2865_zero__le__ceiling,axiom,
    ! [Xb: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_rat_int(archim2889992004027027881ng_rat,Xb))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),Xb) ) ).

% zero_le_ceiling
tff(fact_2866_ceiling__le__numeral,axiom,
    ! [Xb: real,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,Xb)),aa_num_int(numeral_numeral_int,V))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_num_real(numeral_numeral_real,V)) ) ).

% ceiling_le_numeral
tff(fact_2867_ceiling__le__numeral,axiom,
    ! [Xb: rat,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,Xb)),aa_num_int(numeral_numeral_int,V))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),aa_num_rat(numeral_numeral_rat,V)) ) ).

% ceiling_le_numeral
tff(fact_2868_numeral__less__ceiling,axiom,
    ! [V: num,Xb: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,V)),aa_rat_int(archim2889992004027027881ng_rat,Xb))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,V)),Xb) ) ).

% numeral_less_ceiling
tff(fact_2869_numeral__less__ceiling,axiom,
    ! [V: num,Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,V)),aa_real_int(archim7802044766580827645g_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,V)),Xb) ) ).

% numeral_less_ceiling
tff(fact_2870_numeral__le__one__iff,axiom,
    ! [Na: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,Na)),one_one_real)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Na),one) ) ).

% numeral_le_one_iff
tff(fact_2871_numeral__le__one__iff,axiom,
    ! [Na: num] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(numera1916890842035813515d_enat(Na)),one_on7984719198319812577d_enat)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Na),one) ) ).

% numeral_le_one_iff
tff(fact_2872_numeral__le__one__iff,axiom,
    ! [Na: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_num_Code_integer(numera6620942414471956472nteger,Na)),one_one_Code_integer)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Na),one) ) ).

% numeral_le_one_iff
tff(fact_2873_numeral__le__one__iff,axiom,
    ! [Na: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,Na)),one_one_rat)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Na),one) ) ).

% numeral_le_one_iff
tff(fact_2874_numeral__le__one__iff,axiom,
    ! [Na: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,Na)),one_one_nat)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Na),one) ) ).

% numeral_le_one_iff
tff(fact_2875_numeral__le__one__iff,axiom,
    ! [Na: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Na)),one_one_int)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Na),one) ) ).

% numeral_le_one_iff
tff(fact_2876_one__less__numeral__iff,axiom,
    ! [Na: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_num_rat(numeral_numeral_rat,Na))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,one),Na) ) ).

% one_less_numeral_iff
tff(fact_2877_one__less__numeral__iff,axiom,
    ! [Na: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,Na))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,one),Na) ) ).

% one_less_numeral_iff
tff(fact_2878_one__less__numeral__iff,axiom,
    ! [Na: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_num_real(numeral_numeral_real,Na))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,one),Na) ) ).

% one_less_numeral_iff
tff(fact_2879_one__less__numeral__iff,axiom,
    ! [Na: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_num_int(numeral_numeral_int,Na))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,one),Na) ) ).

% one_less_numeral_iff
tff(fact_2880_one__less__numeral__iff,axiom,
    ! [Na: num] :
      ( aa_Extended_enat_o(ord_le72135733267957522d_enat(one_on7984719198319812577d_enat),numera1916890842035813515d_enat(Na))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,one),Na) ) ).

% one_less_numeral_iff
tff(fact_2881_one__less__numeral__iff,axiom,
    ! [Na: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),aa_num_Code_integer(numera6620942414471956472nteger,Na))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,one),Na) ) ).

% one_less_numeral_iff
tff(fact_2882_dbl__simps_I3_J,axiom,
    aa_complex_complex(neg_nu7009210354673126013omplex,one_one_complex) = numera6690914467698888265omplex(aa_num_num(bit0,one)) ).

% dbl_simps(3)
tff(fact_2883_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_2884_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_2885_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_2886_dbl__simps_I3_J,axiom,
    neg_nu8804712462038260780nteger(one_one_Code_integer) = aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)) ).

% dbl_simps(3)
tff(fact_2887_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_2888_add__neg__numeral__special_I9_J,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% add_neg_numeral_special(9)
tff(fact_2889_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_2890_add__neg__numeral__special_I9_J,axiom,
    aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) ).

% add_neg_numeral_special(9)
tff(fact_2891_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_2892_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_2893_diff__numeral__special_I10_J,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,one_one_real)),one_one_real) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% diff_numeral_special(10)
tff(fact_2894_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_2895_diff__numeral__special_I10_J,axiom,
    aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),one_one_Code_integer) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) ).

% diff_numeral_special(10)
tff(fact_2896_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_2897_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_2898_diff__numeral__special_I11_J,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_real_real(uminus_uminus_real,one_one_real)) = aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)) ).

% diff_numeral_special(11)
tff(fact_2899_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_2900_diff__numeral__special_I11_J,axiom,
    aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)) ).

% diff_numeral_special(11)
tff(fact_2901_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_2902_diff__numeral__special_I3_J,axiom,
    ! [Na: num] : aa_complex_complex(minus_minus_complex(one_one_complex),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Na))) = numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(plus_plus_num,one),Na)) ).

% diff_numeral_special(3)
tff(fact_2903_diff__numeral__special_I3_J,axiom,
    ! [Na: num] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Na))) = aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(plus_plus_num,one),Na)) ).

% diff_numeral_special(3)
tff(fact_2904_diff__numeral__special_I3_J,axiom,
    ! [Na: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Na))) = aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,one),Na)) ).

% diff_numeral_special(3)
tff(fact_2905_diff__numeral__special_I3_J,axiom,
    ! [Na: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Na))) = aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(aa_num_fun_num_num(plus_plus_num,one),Na)) ).

% diff_numeral_special(3)
tff(fact_2906_diff__numeral__special_I3_J,axiom,
    ! [Na: num] : aa_rat_rat(minus_minus_rat(one_one_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Na))) = aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(plus_plus_num,one),Na)) ).

% diff_numeral_special(3)
tff(fact_2907_diff__numeral__special_I4_J,axiom,
    ! [M: num] : aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(M))),one_one_complex) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(aa_num_fun_num_num(plus_plus_num,M),one))) ).

% diff_numeral_special(4)
tff(fact_2908_diff__numeral__special_I4_J,axiom,
    ! [M: num] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M))),one_one_real) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),one))) ).

% diff_numeral_special(4)
tff(fact_2909_diff__numeral__special_I4_J,axiom,
    ! [M: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),one_one_int) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),one))) ).

% diff_numeral_special(4)
tff(fact_2910_diff__numeral__special_I4_J,axiom,
    ! [M: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M))),one_one_Code_integer) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),one))) ).

% diff_numeral_special(4)
tff(fact_2911_diff__numeral__special_I4_J,axiom,
    ! [M: num] : aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))),one_one_rat) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(plus_plus_num,M),one))) ).

% diff_numeral_special(4)
tff(fact_2912_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_2913_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_2914_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_2915_dbl__simps_I4_J,axiom,
    neg_nu8804712462038260780nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) ).

% dbl_simps(4)
tff(fact_2916_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_2917_ceiling__less__numeral,axiom,
    ! [Xb: real,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim7802044766580827645g_real,Xb)),aa_num_int(numeral_numeral_int,V))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_num_real(numeral_numeral_real,V)),one_one_real)) ) ).

% ceiling_less_numeral
tff(fact_2918_ceiling__less__numeral,axiom,
    ! [Xb: rat,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(archim2889992004027027881ng_rat,Xb)),aa_num_int(numeral_numeral_int,V))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),aa_rat_rat(minus_minus_rat(aa_num_rat(numeral_numeral_rat,V)),one_one_rat)) ) ).

% ceiling_less_numeral
tff(fact_2919_numeral__le__ceiling,axiom,
    ! [V: num,Xb: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,V)),aa_rat_int(archim2889992004027027881ng_rat,Xb))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(aa_num_rat(numeral_numeral_rat,V)),one_one_rat)),Xb) ) ).

% numeral_le_ceiling
tff(fact_2920_numeral__le__ceiling,axiom,
    ! [V: num,Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,V)),aa_real_int(archim7802044766580827645g_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_num_real(numeral_numeral_real,V)),one_one_real)),Xb) ) ).

% numeral_le_ceiling
tff(fact_2921_ceiling__le__neg__numeral,axiom,
    ! [Xb: real,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,Xb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))) ) ).

% ceiling_le_neg_numeral
tff(fact_2922_ceiling__le__neg__numeral,axiom,
    ! [Xb: rat,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,Xb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))) ) ).

% ceiling_le_neg_numeral
tff(fact_2923_neg__numeral__less__ceiling,axiom,
    ! [V: num,Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_real_int(archim7802044766580827645g_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))),Xb) ) ).

% neg_numeral_less_ceiling
tff(fact_2924_neg__numeral__less__ceiling,axiom,
    ! [V: num,Xb: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_rat_int(archim2889992004027027881ng_rat,Xb))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),Xb) ) ).

% neg_numeral_less_ceiling
tff(fact_2925_add__One__commute,axiom,
    ! [Na: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,one),Na) = aa_num_num(aa_num_fun_num_num(plus_plus_num,Na),one) ).

% add_One_commute
tff(fact_2926_add__inc,axiom,
    ! [Xb: num,Y: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,Xb),aa_num_num(inc,Y)) = aa_num_num(inc,aa_num_num(aa_num_fun_num_num(plus_plus_num,Xb),Y)) ).

% add_inc
tff(fact_2927_add__One,axiom,
    ! [Xb: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,Xb),one) = aa_num_num(inc,Xb) ).

% add_One
tff(fact_2928_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_2929_zless__add1__eq,axiom,
    ! [W2: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,W2),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,W2),Z)
        | ( W2 = Z ) ) ) ).

% zless_add1_eq
tff(fact_2930_int__gr__induct,axiom,
    ! [K: int,I: int,P: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),I)
     => ( aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,K),one_one_int))
       => ( ! [I2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),I2)
             => ( aa_int_o(P,I2)
               => aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),one_one_int)) ) )
         => aa_int_o(P,I) ) ) ) ).

% int_gr_induct
tff(fact_2931_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_2932_zless__imp__add1__zle,axiom,
    ! [W2: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,W2),Z)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,W2),one_one_int)),Z) ) ).

% zless_imp_add1_zle
tff(fact_2933_add1__zle__eq,axiom,
    ! [W2: 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,W2),one_one_int)),Z)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W2),Z) ) ).

% add1_zle_eq
tff(fact_2934_int__ge__induct,axiom,
    ! [K: int,I: int,P: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),I)
     => ( aa_int_o(P,K)
       => ( ! [I2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),I2)
             => ( aa_int_o(P,I2)
               => aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),one_one_int)) ) )
         => aa_int_o(P,I) ) ) ) ).

% int_ge_induct
tff(fact_2935_imp__le__cong,axiom,
    ! [Xb: int,X6: int,P: $o,P2: $o] :
      ( ( Xb = X6 )
     => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X6)
         => ( (P)
          <=> (P2) ) )
       => ( ( 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),X6)
           => (P2) ) ) ) ) ).

% imp_le_cong
tff(fact_2936_conj__le__cong,axiom,
    ! [Xb: int,X6: int,P: $o,P2: $o] :
      ( ( Xb = X6 )
     => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X6)
         => ( (P)
          <=> (P2) ) )
       => ( ( 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),X6)
            & (P2) ) ) ) ) ).

% conj_le_cong
tff(fact_2937_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_2938_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_2939_verit__la__generic,axiom,
    ! [Aa: int,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),Xb)
      | ( Aa = Xb )
      | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Aa) ) ).

% verit_la_generic
tff(fact_2940_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_2941_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_2942_num__induct,axiom,
    ! [P: fun_num_o,Xb: num] :
      ( aa_num_o(P,one)
     => ( ! [X: num] :
            ( aa_num_o(P,X)
           => aa_num_o(P,aa_num_num(inc,X)) )
       => aa_num_o(P,Xb) ) ) ).

% num_induct
tff(fact_2943_inc_Osimps_I1_J,axiom,
    aa_num_num(inc,one) = aa_num_num(bit0,one) ).

% inc.simps(1)
tff(fact_2944_int__le__induct,axiom,
    ! [I: int,K: int,P: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),K)
     => ( aa_int_o(P,K)
       => ( ! [I2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),K)
             => ( aa_int_o(P,I2)
               => aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,I2),one_one_int)) ) )
         => aa_int_o(P,I) ) ) ) ).

% int_le_induct
tff(fact_2945_int__induct,axiom,
    ! [P: fun_int_o,K: int,I: int] :
      ( aa_int_o(P,K)
     => ( ! [I2: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),I2)
           => ( aa_int_o(P,I2)
             => aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),one_one_int)) ) )
       => ( ! [I2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),K)
             => ( aa_int_o(P,I2)
               => aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,I2),one_one_int)) ) )
         => aa_int_o(P,I) ) ) ) ).

% int_induct
tff(fact_2946_int__less__induct,axiom,
    ! [I: int,K: int,P: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,I),K)
     => ( aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,K),one_one_int))
       => ( ! [I2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,I2),K)
             => ( aa_int_o(P,I2)
               => aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,I2),one_one_int)) ) )
         => aa_int_o(P,I) ) ) ) ).

% int_less_induct
tff(fact_2947_ceiling__mono,axiom,
    ! [Y: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Xb)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,Y)),aa_real_int(archim7802044766580827645g_real,Xb)) ) ).

% ceiling_mono
tff(fact_2948_ceiling__mono,axiom,
    ! [Y: rat,Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Xb)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,Y)),aa_rat_int(archim2889992004027027881ng_rat,Xb)) ) ).

% ceiling_mono
tff(fact_2949_ceiling__less__cancel,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(archim2889992004027027881ng_rat,Xb)),aa_rat_int(archim2889992004027027881ng_rat,Y))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y) ) ).

% ceiling_less_cancel
tff(fact_2950_ceiling__less__cancel,axiom,
    ! [Xb: real,Y: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim7802044766580827645g_real,Xb)),aa_real_int(archim7802044766580827645g_real,Y))
     => aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y) ) ).

% ceiling_less_cancel
tff(fact_2951_ceiling__add__le,axiom,
    ! [Xb: rat,Y: rat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(plus_plus_rat(Xb),Y))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim2889992004027027881ng_rat,Xb)),aa_rat_int(archim2889992004027027881ng_rat,Y))) ).

% ceiling_add_le
tff(fact_2952_ceiling__add__le,axiom,
    ! [Xb: real,Y: real] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim7802044766580827645g_real,Xb)),aa_real_int(archim7802044766580827645g_real,Y))) ).

% ceiling_add_le
tff(fact_2953_num_Osize_I4_J,axiom,
    aa_num_nat(size_size_num,one) = zero_zero_nat ).

% num.size(4)
tff(fact_2954_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_2955_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_2956_less__2__cases__iff,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))
    <=> ( ( Na = zero_zero_nat )
        | ( Na = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% less_2_cases_iff
tff(fact_2957_less__2__cases,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))
     => ( ( Na = zero_zero_nat )
        | ( Na = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% less_2_cases
tff(fact_2958_numeral__Bit0,axiom,
    ! [Na: num] : aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,Na)) = aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,Na)),aa_num_rat(numeral_numeral_rat,Na)) ).

% numeral_Bit0
tff(fact_2959_numeral__Bit0,axiom,
    ! [Na: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Na)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,Na)),aa_num_nat(numeral_numeral_nat,Na)) ).

% numeral_Bit0
tff(fact_2960_numeral__Bit0,axiom,
    ! [Na: num] : aa_num_real(numeral_numeral_real,aa_num_num(bit0,Na)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,Na)),aa_num_real(numeral_numeral_real,Na)) ).

% numeral_Bit0
tff(fact_2961_numeral__Bit0,axiom,
    ! [Na: num] : aa_num_int(numeral_numeral_int,aa_num_num(bit0,Na)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,Na)),aa_num_int(numeral_numeral_int,Na)) ).

% numeral_Bit0
tff(fact_2962_numeral__Bit0,axiom,
    ! [Na: num] : numera1916890842035813515d_enat(aa_num_num(bit0,Na)) = plus_p3455044024723400733d_enat(numera1916890842035813515d_enat(Na),numera1916890842035813515d_enat(Na)) ).

% numeral_Bit0
tff(fact_2963_numeral__Bit0,axiom,
    ! [Na: num] : aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Na)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_num_Code_integer(numera6620942414471956472nteger,Na)),aa_num_Code_integer(numera6620942414471956472nteger,Na)) ).

% numeral_Bit0
tff(fact_2964_dbl__def,axiom,
    ! [Xb: real] : aa_real_real(neg_numeral_dbl_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Xb) ).

% dbl_def
tff(fact_2965_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_2966_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_2967_numeral__One,axiom,
    numera6690914467698888265omplex(one) = one_one_complex ).

% numeral_One
tff(fact_2968_numeral__One,axiom,
    aa_num_rat(numeral_numeral_rat,one) = one_one_rat ).

% numeral_One
tff(fact_2969_numeral__One,axiom,
    aa_num_nat(numeral_numeral_nat,one) = one_one_nat ).

% numeral_One
tff(fact_2970_numeral__One,axiom,
    aa_num_real(numeral_numeral_real,one) = one_one_real ).

% numeral_One
tff(fact_2971_numeral__One,axiom,
    aa_num_int(numeral_numeral_int,one) = one_one_int ).

% numeral_One
tff(fact_2972_numeral__One,axiom,
    numera1916890842035813515d_enat(one) = one_on7984719198319812577d_enat ).

% numeral_One
tff(fact_2973_numeral__One,axiom,
    aa_num_Code_integer(numera6620942414471956472nteger,one) = one_one_Code_integer ).

% numeral_One
tff(fact_2974_numerals_I1_J,axiom,
    aa_num_nat(numeral_numeral_nat,one) = one_one_nat ).

% numerals(1)
tff(fact_2975_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_2976_uminus__numeral__One,axiom,
    aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(one)) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% uminus_numeral_One
tff(fact_2977_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_2978_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_2979_uminus__numeral__One,axiom,
    aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,one)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% uminus_numeral_One
tff(fact_2980_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_2981_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_2982_Suc__nat__number__of__add,axiom,
    ! [V: num,Na: nat] : aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,V)),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(plus_plus_num,V),one))),Na) ).

% Suc_nat_number_of_add
tff(fact_2983_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_2984_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_2985_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_2986_numeral__inc,axiom,
    ! [Xb: num] : aa_num_real(numeral_numeral_real,aa_num_num(inc,Xb)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,Xb)),one_one_real) ).

% numeral_inc
tff(fact_2987_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_2988_numeral__inc,axiom,
    ! [Xb: num] : numera1916890842035813515d_enat(aa_num_num(inc,Xb)) = plus_p3455044024723400733d_enat(numera1916890842035813515d_enat(Xb),one_on7984719198319812577d_enat) ).

% numeral_inc
tff(fact_2989_numeral__inc,axiom,
    ! [Xb: num] : aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(inc,Xb)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_num_Code_integer(numera6620942414471956472nteger,Xb)),one_one_Code_integer) ).

% numeral_inc
tff(fact_2990_semiring__norm_I69_J,axiom,
    ! [M: num] : ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(bit0,M)),one) ).

% semiring_norm(69)
tff(fact_2991_semiring__norm_I76_J,axiom,
    ! [Na: num] : aa_num_o(aa_num_fun_num_o(ord_less_num,one),aa_num_num(bit0,Na)) ).

% semiring_norm(76)
tff(fact_2992_mi__ma__2__deg,axiom,
    ! [Mi2: nat,Ma2: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Na: nat] :
      ( aa_nat_o(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Deg,TreeList,Summary)),Na)
     => ( 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))),Deg)) ) ) ).

% mi_ma_2_deg
tff(fact_2993_nat__induct2,axiom,
    ! [P: fun_nat_o,Na: nat] :
      ( aa_nat_o(P,zero_zero_nat)
     => ( aa_nat_o(P,one_one_nat)
       => ( ! [N: nat] :
              ( aa_nat_o(P,N)
             => aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) )
         => aa_nat_o(P,Na) ) ) ) ).

% nat_induct2
tff(fact_2994_post__member__pre__member,axiom,
    ! [Ta: vEBT_VEBT,Na: nat,Xb: nat,Y: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Na)
     => ( 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))),Na))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))
         => ( aa_nat_o(vEBT_vebt_member(vEBT_vebt_insert(Ta,Xb)),Y)
           => ( aa_nat_o(vEBT_vebt_member(Ta),Y)
              | ( Xb = Y ) ) ) ) ) ) ).

% post_member_pre_member
tff(fact_2995_semiring__norm_I68_J,axiom,
    ! [Na: num] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,one),Na) ).

% semiring_norm(68)
tff(fact_2996_semiring__norm_I75_J,axiom,
    ! [M: num] : ~ aa_num_o(aa_num_fun_num_o(ord_less_num,M),one) ).

% semiring_norm(75)
tff(fact_2997_semiring__norm_I71_J,axiom,
    ! [M: num,Na: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(bit0,M)),aa_num_num(bit0,Na))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,M),Na) ) ).

% semiring_norm(71)
tff(fact_2998_semiring__norm_I78_J,axiom,
    ! [M: num,Na: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(bit0,M)),aa_num_num(bit0,Na))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,M),Na) ) ).

% semiring_norm(78)
tff(fact_2999_power__shift,axiom,
    ! [Xb: nat,Y: nat,Z: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),Y) = Z )
    <=> ( aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_power,some_nat(Xb)),some_nat(Y)) = some_nat(Z) ) ) ).

% power_shift
tff(fact_3000_local_Opower__def,axiom,
    vEBT_VEBT_power = vEBT_V4262088993061758097ft_nat(power_power_nat) ).

% local.power_def
tff(fact_3001_member__bound,axiom,
    ! [Tree: vEBT_VEBT,Xb: nat,Na: nat] :
      ( aa_nat_o(vEBT_vebt_member(Tree),Xb)
     => ( aa_nat_o(vEBT_invar_vebt(Tree),Na)
       => 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))),Na)) ) ) ).

% member_bound
tff(fact_3002_set__n__deg__not__0,axiom,
    ! [TreeList: list_VEBT_VEBT,Na: nat,M: nat] :
      ( ! [X: vEBT_VEBT] :
          ( member_VEBT_VEBT(X,set_VEBT_VEBT2(TreeList))
         => aa_nat_o(vEBT_invar_vebt(X),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) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Na) ) ) ).

% set_n_deg_not_0
tff(fact_3003_valid__insert__both__member__options__add,axiom,
    ! [Ta: vEBT_VEBT,Na: nat,Xb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Na)
     => ( 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))),Na))
       => aa_nat_o(vEBT_V8194947554948674370ptions(vEBT_vebt_insert(Ta,Xb)),Xb) ) ) ).

% valid_insert_both_member_options_add
tff(fact_3004_valid__insert__both__member__options__pres,axiom,
    ! [Ta: vEBT_VEBT,Na: nat,Xb: nat,Y: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Na)
     => ( 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))),Na))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))
         => ( aa_nat_o(vEBT_V8194947554948674370ptions(Ta),Xb)
           => aa_nat_o(vEBT_V8194947554948674370ptions(vEBT_vebt_insert(Ta,Y)),Xb) ) ) ) ) ).

% valid_insert_both_member_options_pres
tff(fact_3005_misiz,axiom,
    ! [Ta: vEBT_VEBT,Na: nat,M: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Na)
     => ( ( some_nat(M) = vEBT_vebt_mint(Ta) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) ) ) ).

% misiz
tff(fact_3006_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => ( aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),K)) = aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_3007_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => ( aa_nat_real(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,Na),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,Na),K)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_3008_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),K)) = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_3009_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),K)) = aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_3010_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => ( aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),K)) = aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_3011_invar__vebt_Ointros_I2_J,axiom,
    ! [TreeList: list_VEBT_VEBT,Na: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
      ( ! [X: vEBT_VEBT] :
          ( member_VEBT_VEBT(X,set_VEBT_VEBT2(TreeList))
         => aa_nat_o(vEBT_invar_vebt(X),Na) )
     => ( aa_nat_o(vEBT_invar_vebt(Summary),M)
       => ( ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M) )
         => ( ( M = Na )
           => ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),M) )
             => ( ~ ? [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) )
                 => aa_nat_o(vEBT_invar_vebt(vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList,Summary)),Deg) ) ) ) ) ) ) ) ).

% invar_vebt.intros(2)
tff(fact_3012_invar__vebt_Ointros_I3_J,axiom,
    ! [TreeList: list_VEBT_VEBT,Na: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
      ( ! [X: vEBT_VEBT] :
          ( member_VEBT_VEBT(X,set_VEBT_VEBT2(TreeList))
         => aa_nat_o(vEBT_invar_vebt(X),Na) )
     => ( aa_nat_o(vEBT_invar_vebt(Summary),M)
       => ( ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M) )
         => ( ( M = aa_nat_nat(suc,Na) )
           => ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),M) )
             => ( ~ ? [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) )
                 => aa_nat_o(vEBT_invar_vebt(vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList,Summary)),Deg) ) ) ) ) ) ) ) ).

% invar_vebt.intros(3)
tff(fact_3013_sum__power2__eq__zero__iff,axiom,
    ! [Xb: rat,Y: 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(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = zero_zero_rat )
    <=> ( ( Xb = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_power2_eq_zero_iff
tff(fact_3014_sum__power2__eq__zero__iff,axiom,
    ! [Xb: real,Y: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_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,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = zero_zero_real )
    <=> ( ( Xb = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_power2_eq_zero_iff
tff(fact_3015_sum__power2__eq__zero__iff,axiom,
    ! [Xb: int,Y: 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(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = zero_zero_int )
    <=> ( ( Xb = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_power2_eq_zero_iff
tff(fact_3016_zero__less__power2,axiom,
    ! [Aa: 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,Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
    <=> ( Aa != zero_zero_real ) ) ).

% zero_less_power2
tff(fact_3017_zero__less__power2,axiom,
    ! [Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
    <=> ( Aa != zero_zero_rat ) ) ).

% zero_less_power2
tff(fact_3018_zero__less__power2,axiom,
    ! [Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
    <=> ( Aa != zero_zero_int ) ) ).

% zero_less_power2
tff(fact_3019_power2__less__eq__zero__iff,axiom,
    ! [Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_real)
    <=> ( Aa = zero_zero_real ) ) ).

% power2_less_eq_zero_iff
tff(fact_3020_power2__less__eq__zero__iff,axiom,
    ! [Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_rat)
    <=> ( Aa = zero_zero_rat ) ) ).

% power2_less_eq_zero_iff
tff(fact_3021_power2__less__eq__zero__iff,axiom,
    ! [Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_int)
    <=> ( Aa = zero_zero_int ) ) ).

% power2_less_eq_zero_iff
tff(fact_3022_power2__eq__iff__nonneg,axiom,
    ! [Xb: real,Y: 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),Y)
       => ( ( 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,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
        <=> ( Xb = Y ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_3023_power2__eq__iff__nonneg,axiom,
    ! [Xb: rat,Y: 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),Y)
       => ( ( aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
        <=> ( Xb = Y ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_3024_power2__eq__iff__nonneg,axiom,
    ! [Xb: nat,Y: 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),Y)
       => ( ( 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,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
        <=> ( Xb = Y ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_3025_power2__eq__iff__nonneg,axiom,
    ! [Xb: int,Y: 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),Y)
       => ( ( aa_nat_int(power_power_int(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
        <=> ( Xb = Y ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_3026_power__decreasing__iff,axiom,
    ! [Ba: real,M: nat,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Ba)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Ba),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,Ba),M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ba),Na))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M) ) ) ) ).

% power_decreasing_iff
tff(fact_3027_power__decreasing__iff,axiom,
    ! [Ba: rat,M: nat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Ba)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ba),one_one_rat)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(Ba),M)),aa_nat_rat(power_power_rat(Ba),Na))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M) ) ) ) ).

% power_decreasing_iff
tff(fact_3028_power__decreasing__iff,axiom,
    ! [Ba: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Ba)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ba),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,Ba),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Ba),Na))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M) ) ) ) ).

% power_decreasing_iff
tff(fact_3029_power__decreasing__iff,axiom,
    ! [Ba: int,M: nat,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Ba)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Ba),one_one_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(Ba),M)),aa_nat_int(power_power_int(Ba),Na))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M) ) ) ) ).

% power_decreasing_iff
tff(fact_3030_zero__eq__power2,axiom,
    ! [Aa: rat] :
      ( ( aa_nat_rat(power_power_rat(Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_rat )
    <=> ( Aa = zero_zero_rat ) ) ).

% zero_eq_power2
tff(fact_3031_zero__eq__power2,axiom,
    ! [Aa: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat )
    <=> ( Aa = zero_zero_nat ) ) ).

% zero_eq_power2
tff(fact_3032_zero__eq__power2,axiom,
    ! [Aa: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_real )
    <=> ( Aa = zero_zero_real ) ) ).

% zero_eq_power2
tff(fact_3033_zero__eq__power2,axiom,
    ! [Aa: int] :
      ( ( aa_nat_int(power_power_int(Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_int )
    <=> ( Aa = zero_zero_int ) ) ).

% zero_eq_power2
tff(fact_3034_zero__eq__power2,axiom,
    ! [Aa: complex] :
      ( ( aa_nat_complex(power_power_complex(Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_complex )
    <=> ( Aa = zero_zero_complex ) ) ).

% zero_eq_power2
tff(fact_3035_power__mono__iff,axiom,
    ! [Aa: real,Ba: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Aa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Ba)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),Na)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ba),Na))
          <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),Ba) ) ) ) ) ).

% power_mono_iff
tff(fact_3036_power__mono__iff,axiom,
    ! [Aa: rat,Ba: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Aa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Ba)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(Aa),Na)),aa_nat_rat(power_power_rat(Ba),Na))
          <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),Ba) ) ) ) ) ).

% power_mono_iff
tff(fact_3037_power__mono__iff,axiom,
    ! [Aa: nat,Ba: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Aa)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Ba)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Aa),Na)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Ba),Na))
          <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),Ba) ) ) ) ) ).

% power_mono_iff
tff(fact_3038_power__mono__iff,axiom,
    ! [Aa: int,Ba: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Aa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Ba)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(Aa),Na)),aa_nat_int(power_power_int(Ba),Na))
          <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),Ba) ) ) ) ) ).

% power_mono_iff
tff(fact_3039_power__increasing__iff,axiom,
    ! [Ba: real,Xb: nat,Y: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Ba)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Ba),Xb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ba),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y) ) ) ).

% power_increasing_iff
tff(fact_3040_power__increasing__iff,axiom,
    ! [Ba: rat,Xb: nat,Y: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),Ba)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(Ba),Xb)),aa_nat_rat(power_power_rat(Ba),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y) ) ) ).

% power_increasing_iff
tff(fact_3041_power__increasing__iff,axiom,
    ! [Ba: nat,Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),Ba)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Ba),Xb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Ba),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y) ) ) ).

% power_increasing_iff
tff(fact_3042_power__increasing__iff,axiom,
    ! [Ba: int,Xb: nat,Y: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),Ba)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(Ba),Xb)),aa_nat_int(power_power_int(Ba),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y) ) ) ).

% power_increasing_iff
tff(fact_3043_power__strict__decreasing__iff,axiom,
    ! [Ba: real,M: nat,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Ba)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Ba),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,Ba),M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ba),Na))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_3044_power__strict__decreasing__iff,axiom,
    ! [Ba: rat,M: nat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Ba)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ba),one_one_rat)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(Ba),M)),aa_nat_rat(power_power_rat(Ba),Na))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_3045_power__strict__decreasing__iff,axiom,
    ! [Ba: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Ba)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ba),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,Ba),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Ba),Na))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_3046_power__strict__decreasing__iff,axiom,
    ! [Ba: int,M: nat,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Ba)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Ba),one_one_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(Ba),M)),aa_nat_int(power_power_int(Ba),Na))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_3047_power__one,axiom,
    ! [Na: nat] : aa_nat_rat(power_power_rat(one_one_rat),Na) = one_one_rat ).

% power_one
tff(fact_3048_power__one,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,one_one_nat),Na) = one_one_nat ).

% power_one
tff(fact_3049_power__one,axiom,
    ! [Na: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,one_one_real),Na) = one_one_real ).

% power_one
tff(fact_3050_power__one,axiom,
    ! [Na: nat] : aa_nat_int(power_power_int(one_one_int),Na) = one_one_int ).

% power_one
tff(fact_3051_power__one,axiom,
    ! [Na: nat] : aa_nat_complex(power_power_complex(one_one_complex),Na) = one_one_complex ).

% power_one
tff(fact_3052_power__one__right,axiom,
    ! [Aa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Aa),one_one_nat) = Aa ).

% power_one_right
tff(fact_3053_power__one__right,axiom,
    ! [Aa: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),one_one_nat) = Aa ).

% power_one_right
tff(fact_3054_power__one__right,axiom,
    ! [Aa: int] : aa_nat_int(power_power_int(Aa),one_one_nat) = Aa ).

% power_one_right
tff(fact_3055_power__one__right,axiom,
    ! [Aa: complex] : aa_nat_complex(power_power_complex(Aa),one_one_nat) = Aa ).

% power_one_right
tff(fact_3056_power__inject__exp,axiom,
    ! [Aa: real,M: nat,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Aa)
     => ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),M) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),Na) )
      <=> ( M = Na ) ) ) ).

% power_inject_exp
tff(fact_3057_power__inject__exp,axiom,
    ! [Aa: rat,M: nat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),Aa)
     => ( ( aa_nat_rat(power_power_rat(Aa),M) = aa_nat_rat(power_power_rat(Aa),Na) )
      <=> ( M = Na ) ) ) ).

% power_inject_exp
tff(fact_3058_power__inject__exp,axiom,
    ! [Aa: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),Aa)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Aa),M) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Aa),Na) )
      <=> ( M = Na ) ) ) ).

% power_inject_exp
tff(fact_3059_power__inject__exp,axiom,
    ! [Aa: int,M: nat,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),Aa)
     => ( ( aa_nat_int(power_power_int(Aa),M) = aa_nat_int(power_power_int(Aa),Na) )
      <=> ( M = Na ) ) ) ).

% power_inject_exp
tff(fact_3060_power__0__Suc,axiom,
    ! [Na: nat] : aa_nat_rat(power_power_rat(zero_zero_rat),aa_nat_nat(suc,Na)) = zero_zero_rat ).

% power_0_Suc
tff(fact_3061_power__0__Suc,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),aa_nat_nat(suc,Na)) = zero_zero_nat ).

% power_0_Suc
tff(fact_3062_power__0__Suc,axiom,
    ! [Na: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),aa_nat_nat(suc,Na)) = zero_zero_real ).

% power_0_Suc
tff(fact_3063_power__0__Suc,axiom,
    ! [Na: nat] : aa_nat_int(power_power_int(zero_zero_int),aa_nat_nat(suc,Na)) = zero_zero_int ).

% power_0_Suc
tff(fact_3064_power__0__Suc,axiom,
    ! [Na: nat] : aa_nat_complex(power_power_complex(zero_zero_complex),aa_nat_nat(suc,Na)) = zero_zero_complex ).

% power_0_Suc
tff(fact_3065_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_3066_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_3067_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_3068_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_3069_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_3070_power__Suc0__right,axiom,
    ! [Aa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Aa),aa_nat_nat(suc,zero_zero_nat)) = Aa ).

% power_Suc0_right
tff(fact_3071_power__Suc0__right,axiom,
    ! [Aa: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),aa_nat_nat(suc,zero_zero_nat)) = Aa ).

% power_Suc0_right
tff(fact_3072_power__Suc0__right,axiom,
    ! [Aa: int] : aa_nat_int(power_power_int(Aa),aa_nat_nat(suc,zero_zero_nat)) = Aa ).

% power_Suc0_right
tff(fact_3073_power__Suc0__right,axiom,
    ! [Aa: complex] : aa_nat_complex(power_power_complex(Aa),aa_nat_nat(suc,zero_zero_nat)) = Aa ).

% power_Suc0_right
tff(fact_3074_nat__power__eq__Suc__0__iff,axiom,
    ! [Xb: nat,M: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),M) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ( ( M = zero_zero_nat )
        | ( Xb = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% nat_power_eq_Suc_0_iff
tff(fact_3075_power__Suc__0,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(suc,zero_zero_nat)),Na) = aa_nat_nat(suc,zero_zero_nat) ).

% power_Suc_0
tff(fact_3076_nat__zero__less__power__iff,axiom,
    ! [Xb: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),Na))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Xb)
        | ( Na = zero_zero_nat ) ) ) ).

% nat_zero_less_power_iff
tff(fact_3077_power__strict__increasing__iff,axiom,
    ! [Ba: real,Xb: nat,Y: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Ba)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Ba),Xb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ba),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y) ) ) ).

% power_strict_increasing_iff
tff(fact_3078_power__strict__increasing__iff,axiom,
    ! [Ba: rat,Xb: nat,Y: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),Ba)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(Ba),Xb)),aa_nat_rat(power_power_rat(Ba),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y) ) ) ).

% power_strict_increasing_iff
tff(fact_3079_power__strict__increasing__iff,axiom,
    ! [Ba: nat,Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),Ba)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Ba),Xb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Ba),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y) ) ) ).

% power_strict_increasing_iff
tff(fact_3080_power__strict__increasing__iff,axiom,
    ! [Ba: int,Xb: nat,Y: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),Ba)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(Ba),Xb)),aa_nat_int(power_power_int(Ba),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y) ) ) ).

% power_strict_increasing_iff
tff(fact_3081_power__eq__0__iff,axiom,
    ! [Aa: rat,Na: nat] :
      ( ( aa_nat_rat(power_power_rat(Aa),Na) = zero_zero_rat )
    <=> ( ( Aa = zero_zero_rat )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ) ).

% power_eq_0_iff
tff(fact_3082_power__eq__0__iff,axiom,
    ! [Aa: nat,Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Aa),Na) = zero_zero_nat )
    <=> ( ( Aa = zero_zero_nat )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ) ).

% power_eq_0_iff
tff(fact_3083_power__eq__0__iff,axiom,
    ! [Aa: real,Na: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),Na) = zero_zero_real )
    <=> ( ( Aa = zero_zero_real )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ) ).

% power_eq_0_iff
tff(fact_3084_power__eq__0__iff,axiom,
    ! [Aa: int,Na: nat] :
      ( ( aa_nat_int(power_power_int(Aa),Na) = zero_zero_int )
    <=> ( ( Aa = zero_zero_int )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ) ).

% power_eq_0_iff
tff(fact_3085_power__eq__0__iff,axiom,
    ! [Aa: complex,Na: nat] :
      ( ( aa_nat_complex(power_power_complex(Aa),Na) = zero_zero_complex )
    <=> ( ( Aa = zero_zero_complex )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ) ).

% power_eq_0_iff
tff(fact_3086_two__realpow__ge__one,axiom,
    ! [Na: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Na)) ).

% two_realpow_ge_one
tff(fact_3087_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_3088_power__not__zero,axiom,
    ! [Aa: rat,Na: nat] :
      ( ( Aa != zero_zero_rat )
     => ( aa_nat_rat(power_power_rat(Aa),Na) != zero_zero_rat ) ) ).

% power_not_zero
tff(fact_3089_power__not__zero,axiom,
    ! [Aa: nat,Na: nat] :
      ( ( Aa != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Aa),Na) != zero_zero_nat ) ) ).

% power_not_zero
tff(fact_3090_power__not__zero,axiom,
    ! [Aa: real,Na: nat] :
      ( ( Aa != zero_zero_real )
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),Na) != zero_zero_real ) ) ).

% power_not_zero
tff(fact_3091_power__not__zero,axiom,
    ! [Aa: int,Na: nat] :
      ( ( Aa != zero_zero_int )
     => ( aa_nat_int(power_power_int(Aa),Na) != zero_zero_int ) ) ).

% power_not_zero
tff(fact_3092_power__not__zero,axiom,
    ! [Aa: complex,Na: nat] :
      ( ( Aa != zero_zero_complex )
     => ( aa_nat_complex(power_power_complex(Aa),Na) != zero_zero_complex ) ) ).

% power_not_zero
tff(fact_3093_zero__le__power,axiom,
    ! [Aa: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Aa)
     => 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),Na)) ) ).

% zero_le_power
tff(fact_3094_zero__le__power,axiom,
    ! [Aa: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Aa)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(Aa),Na)) ) ).

% zero_le_power
tff(fact_3095_zero__le__power,axiom,
    ! [Aa: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Aa)
     => 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,Aa),Na)) ) ).

% zero_le_power
tff(fact_3096_zero__le__power,axiom,
    ! [Aa: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Aa)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(Aa),Na)) ) ).

% zero_le_power
tff(fact_3097_power__mono,axiom,
    ! [Aa: real,Ba: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),Ba)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Aa)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),Na)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ba),Na)) ) ) ).

% power_mono
tff(fact_3098_power__mono,axiom,
    ! [Aa: rat,Ba: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),Ba)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Aa)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(Aa),Na)),aa_nat_rat(power_power_rat(Ba),Na)) ) ) ).

% power_mono
tff(fact_3099_power__mono,axiom,
    ! [Aa: nat,Ba: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),Ba)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Aa)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Aa),Na)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Ba),Na)) ) ) ).

% power_mono
tff(fact_3100_power__mono,axiom,
    ! [Aa: int,Ba: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),Ba)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Aa)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(Aa),Na)),aa_nat_int(power_power_int(Ba),Na)) ) ) ).

% power_mono
tff(fact_3101_zero__less__power,axiom,
    ! [Aa: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Aa)
     => 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),Na)) ) ).

% zero_less_power
tff(fact_3102_zero__less__power,axiom,
    ! [Aa: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Aa)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(Aa),Na)) ) ).

% zero_less_power
tff(fact_3103_zero__less__power,axiom,
    ! [Aa: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Aa)
     => 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),Na)) ) ).

% zero_less_power
tff(fact_3104_zero__less__power,axiom,
    ! [Aa: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Aa)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(Aa),Na)) ) ).

% zero_less_power
tff(fact_3105_one__le__power,axiom,
    ! [Aa: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Aa)
     => 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),Na)) ) ).

% one_le_power
tff(fact_3106_one__le__power,axiom,
    ! [Aa: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),Aa)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_nat_rat(power_power_rat(Aa),Na)) ) ).

% one_le_power
tff(fact_3107_one__le__power,axiom,
    ! [Aa: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Aa)
     => 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,Aa),Na)) ) ).

% one_le_power
tff(fact_3108_one__le__power,axiom,
    ! [Aa: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),Aa)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_nat_int(power_power_int(Aa),Na)) ) ).

% one_le_power
tff(fact_3109_power__0,axiom,
    ! [Aa: rat] : aa_nat_rat(power_power_rat(Aa),zero_zero_nat) = one_one_rat ).

% power_0
tff(fact_3110_power__0,axiom,
    ! [Aa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Aa),zero_zero_nat) = one_one_nat ).

% power_0
tff(fact_3111_power__0,axiom,
    ! [Aa: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),zero_zero_nat) = one_one_real ).

% power_0
tff(fact_3112_power__0,axiom,
    ! [Aa: int] : aa_nat_int(power_power_int(Aa),zero_zero_nat) = one_one_int ).

% power_0
tff(fact_3113_power__0,axiom,
    ! [Aa: complex] : aa_nat_complex(power_power_complex(Aa),zero_zero_nat) = one_one_complex ).

% power_0
tff(fact_3114_nat__power__less__imp__less,axiom,
    ! [I: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),I)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),Na))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ) ).

% nat_power_less_imp_less
tff(fact_3115_power__less__imp__less__base,axiom,
    ! [Aa: real,Na: nat,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),Na)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ba),Na))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Ba)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),Ba) ) ) ).

% power_less_imp_less_base
tff(fact_3116_power__less__imp__less__base,axiom,
    ! [Aa: rat,Na: nat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(Aa),Na)),aa_nat_rat(power_power_rat(Ba),Na))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Ba)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),Ba) ) ) ).

% power_less_imp_less_base
tff(fact_3117_power__less__imp__less__base,axiom,
    ! [Aa: nat,Na: nat,Ba: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Aa),Na)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Ba),Na))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Ba)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),Ba) ) ) ).

% power_less_imp_less_base
tff(fact_3118_power__less__imp__less__base,axiom,
    ! [Aa: int,Na: nat,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(Aa),Na)),aa_nat_int(power_power_int(Ba),Na))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Ba)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),Ba) ) ) ).

% power_less_imp_less_base
tff(fact_3119_power__le__one,axiom,
    ! [Aa: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Aa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),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,Aa),Na)),one_one_real) ) ) ).

% power_le_one
tff(fact_3120_power__le__one,axiom,
    ! [Aa: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Aa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),one_one_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(Aa),Na)),one_one_rat) ) ) ).

% power_le_one
tff(fact_3121_power__le__one,axiom,
    ! [Aa: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Aa)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),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,Aa),Na)),one_one_nat) ) ) ).

% power_le_one
tff(fact_3122_power__le__one,axiom,
    ! [Aa: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Aa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),one_one_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(Aa),Na)),one_one_int) ) ) ).

% power_le_one
tff(fact_3123_power__le__imp__le__base,axiom,
    ! [Aa: real,Na: nat,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),aa_nat_nat(suc,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ba),aa_nat_nat(suc,Na)))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Ba)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),Ba) ) ) ).

% power_le_imp_le_base
tff(fact_3124_power__le__imp__le__base,axiom,
    ! [Aa: rat,Na: nat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(Aa),aa_nat_nat(suc,Na))),aa_nat_rat(power_power_rat(Ba),aa_nat_nat(suc,Na)))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Ba)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),Ba) ) ) ).

% power_le_imp_le_base
tff(fact_3125_power__le__imp__le__base,axiom,
    ! [Aa: nat,Na: nat,Ba: 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),aa_nat_nat(suc,Na))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Ba),aa_nat_nat(suc,Na)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Ba)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),Ba) ) ) ).

% power_le_imp_le_base
tff(fact_3126_power__le__imp__le__base,axiom,
    ! [Aa: int,Na: nat,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(Aa),aa_nat_nat(suc,Na))),aa_nat_int(power_power_int(Ba),aa_nat_nat(suc,Na)))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Ba)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),Ba) ) ) ).

% power_le_imp_le_base
tff(fact_3127_power__inject__base,axiom,
    ! [Aa: real,Na: nat,Ba: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),aa_nat_nat(suc,Na)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Ba),aa_nat_nat(suc,Na)) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Aa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Ba)
         => ( Aa = Ba ) ) ) ) ).

% power_inject_base
tff(fact_3128_power__inject__base,axiom,
    ! [Aa: rat,Na: nat,Ba: rat] :
      ( ( aa_nat_rat(power_power_rat(Aa),aa_nat_nat(suc,Na)) = aa_nat_rat(power_power_rat(Ba),aa_nat_nat(suc,Na)) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Aa)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Ba)
         => ( Aa = Ba ) ) ) ) ).

% power_inject_base
tff(fact_3129_power__inject__base,axiom,
    ! [Aa: nat,Na: nat,Ba: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Aa),aa_nat_nat(suc,Na)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Ba),aa_nat_nat(suc,Na)) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Aa)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Ba)
         => ( Aa = Ba ) ) ) ) ).

% power_inject_base
tff(fact_3130_power__inject__base,axiom,
    ! [Aa: int,Na: nat,Ba: int] :
      ( ( aa_nat_int(power_power_int(Aa),aa_nat_nat(suc,Na)) = aa_nat_int(power_power_int(Ba),aa_nat_nat(suc,Na)) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Aa)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Ba)
         => ( Aa = Ba ) ) ) ) ).

% power_inject_base
tff(fact_3131_power__gt1,axiom,
    ! [Aa: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Aa)
     => 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,Aa),aa_nat_nat(suc,Na))) ) ).

% power_gt1
tff(fact_3132_power__gt1,axiom,
    ! [Aa: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),Aa)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_nat_rat(power_power_rat(Aa),aa_nat_nat(suc,Na))) ) ).

% power_gt1
tff(fact_3133_power__gt1,axiom,
    ! [Aa: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),Aa)
     => 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,Aa),aa_nat_nat(suc,Na))) ) ).

% power_gt1
tff(fact_3134_power__gt1,axiom,
    ! [Aa: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),Aa)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_nat_int(power_power_int(Aa),aa_nat_nat(suc,Na))) ) ).

% power_gt1
tff(fact_3135_power__0__left,axiom,
    ! [Na: nat] :
      aa_nat_rat(power_power_rat(zero_zero_rat),Na) = $ite(Na = zero_zero_nat,one_one_rat,zero_zero_rat) ).

% power_0_left
tff(fact_3136_power__0__left,axiom,
    ! [Na: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),Na) = $ite(Na = zero_zero_nat,one_one_nat,zero_zero_nat) ).

% power_0_left
tff(fact_3137_power__0__left,axiom,
    ! [Na: nat] :
      aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),Na) = $ite(Na = zero_zero_nat,one_one_real,zero_zero_real) ).

% power_0_left
tff(fact_3138_power__0__left,axiom,
    ! [Na: nat] :
      aa_nat_int(power_power_int(zero_zero_int),Na) = $ite(Na = zero_zero_nat,one_one_int,zero_zero_int) ).

% power_0_left
tff(fact_3139_power__0__left,axiom,
    ! [Na: nat] :
      aa_nat_complex(power_power_complex(zero_zero_complex),Na) = $ite(Na = zero_zero_nat,one_one_complex,zero_zero_complex) ).

% power_0_left
tff(fact_3140_power__strict__increasing,axiom,
    ! [Na: nat,N3: nat,Aa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),N3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Aa)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),Na)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),N3)) ) ) ).

% power_strict_increasing
tff(fact_3141_power__strict__increasing,axiom,
    ! [Na: nat,N3: nat,Aa: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),N3)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),Aa)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(Aa),Na)),aa_nat_rat(power_power_rat(Aa),N3)) ) ) ).

% power_strict_increasing
tff(fact_3142_power__strict__increasing,axiom,
    ! [Na: nat,N3: nat,Aa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),N3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),Aa)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Aa),Na)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Aa),N3)) ) ) ).

% power_strict_increasing
tff(fact_3143_power__strict__increasing,axiom,
    ! [Na: nat,N3: nat,Aa: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),N3)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),Aa)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(Aa),Na)),aa_nat_int(power_power_int(Aa),N3)) ) ) ).

% power_strict_increasing
tff(fact_3144_power__less__imp__less__exp,axiom,
    ! [Aa: real,M: nat,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Aa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),Na))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ) ).

% power_less_imp_less_exp
tff(fact_3145_power__less__imp__less__exp,axiom,
    ! [Aa: rat,M: nat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),Aa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(Aa),M)),aa_nat_rat(power_power_rat(Aa),Na))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ) ).

% power_less_imp_less_exp
tff(fact_3146_power__less__imp__less__exp,axiom,
    ! [Aa: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),Aa)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Aa),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Aa),Na))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ) ).

% power_less_imp_less_exp
tff(fact_3147_power__less__imp__less__exp,axiom,
    ! [Aa: int,M: nat,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),Aa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(Aa),M)),aa_nat_int(power_power_int(Aa),Na))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ) ).

% power_less_imp_less_exp
tff(fact_3148_power__increasing,axiom,
    ! [Na: nat,N3: nat,Aa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),N3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Aa)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),Na)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),N3)) ) ) ).

% power_increasing
tff(fact_3149_power__increasing,axiom,
    ! [Na: nat,N3: nat,Aa: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),N3)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),Aa)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(Aa),Na)),aa_nat_rat(power_power_rat(Aa),N3)) ) ) ).

% power_increasing
tff(fact_3150_power__increasing,axiom,
    ! [Na: nat,N3: nat,Aa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),N3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Aa)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Aa),Na)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Aa),N3)) ) ) ).

% power_increasing
tff(fact_3151_power__increasing,axiom,
    ! [Na: nat,N3: nat,Aa: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),N3)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),Aa)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(Aa),Na)),aa_nat_int(power_power_int(Aa),N3)) ) ) ).

% power_increasing
tff(fact_3152_zero__power,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_rat(power_power_rat(zero_zero_rat),Na) = zero_zero_rat ) ) ).

% zero_power
tff(fact_3153_zero__power,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),Na) = zero_zero_nat ) ) ).

% zero_power
tff(fact_3154_zero__power,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),Na) = zero_zero_real ) ) ).

% zero_power
tff(fact_3155_zero__power,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_int(power_power_int(zero_zero_int),Na) = zero_zero_int ) ) ).

% zero_power
tff(fact_3156_zero__power,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_complex(power_power_complex(zero_zero_complex),Na) = zero_zero_complex ) ) ).

% zero_power
tff(fact_3157_power__gt__expt,axiom,
    ! [Na: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Na),K)) ) ).

% power_gt_expt
tff(fact_3158_nat__one__le__power,axiom,
    ! [I: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),I)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),Na)) ) ).

% nat_one_le_power
tff(fact_3159_power__Suc__le__self,axiom,
    ! [Aa: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Aa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),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,Aa),aa_nat_nat(suc,Na))),Aa) ) ) ).

% power_Suc_le_self
tff(fact_3160_power__Suc__le__self,axiom,
    ! [Aa: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Aa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),one_one_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(Aa),aa_nat_nat(suc,Na))),Aa) ) ) ).

% power_Suc_le_self
tff(fact_3161_power__Suc__le__self,axiom,
    ! [Aa: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Aa)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),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,Aa),aa_nat_nat(suc,Na))),Aa) ) ) ).

% power_Suc_le_self
tff(fact_3162_power__Suc__le__self,axiom,
    ! [Aa: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Aa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),one_one_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(Aa),aa_nat_nat(suc,Na))),Aa) ) ) ).

% power_Suc_le_self
tff(fact_3163_power__Suc__less__one,axiom,
    ! [Aa: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Aa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),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,Aa),aa_nat_nat(suc,Na))),one_one_real) ) ) ).

% power_Suc_less_one
tff(fact_3164_power__Suc__less__one,axiom,
    ! [Aa: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Aa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),one_one_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(Aa),aa_nat_nat(suc,Na))),one_one_rat) ) ) ).

% power_Suc_less_one
tff(fact_3165_power__Suc__less__one,axiom,
    ! [Aa: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Aa)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),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,Aa),aa_nat_nat(suc,Na))),one_one_nat) ) ) ).

% power_Suc_less_one
tff(fact_3166_power__Suc__less__one,axiom,
    ! [Aa: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Aa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),one_one_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(Aa),aa_nat_nat(suc,Na))),one_one_int) ) ) ).

% power_Suc_less_one
tff(fact_3167_power__strict__decreasing,axiom,
    ! [Na: nat,N3: nat,Aa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),N3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Aa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),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,Aa),N3)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),Na)) ) ) ) ).

% power_strict_decreasing
tff(fact_3168_power__strict__decreasing,axiom,
    ! [Na: nat,N3: nat,Aa: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),N3)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Aa)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),one_one_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(Aa),N3)),aa_nat_rat(power_power_rat(Aa),Na)) ) ) ) ).

% power_strict_decreasing
tff(fact_3169_power__strict__decreasing,axiom,
    ! [Na: nat,N3: nat,Aa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),N3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Aa)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),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,Aa),N3)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Aa),Na)) ) ) ) ).

% power_strict_decreasing
tff(fact_3170_power__strict__decreasing,axiom,
    ! [Na: nat,N3: nat,Aa: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),N3)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Aa)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),one_one_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(Aa),N3)),aa_nat_int(power_power_int(Aa),Na)) ) ) ) ).

% power_strict_decreasing
tff(fact_3171_power__decreasing,axiom,
    ! [Na: nat,N3: nat,Aa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),N3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Aa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),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,Aa),N3)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),Na)) ) ) ) ).

% power_decreasing
tff(fact_3172_power__decreasing,axiom,
    ! [Na: nat,N3: nat,Aa: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),N3)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Aa)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),one_one_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(Aa),N3)),aa_nat_rat(power_power_rat(Aa),Na)) ) ) ) ).

% power_decreasing
tff(fact_3173_power__decreasing,axiom,
    ! [Na: nat,N3: nat,Aa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),N3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Aa)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),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,Aa),N3)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Aa),Na)) ) ) ) ).

% power_decreasing
tff(fact_3174_power__decreasing,axiom,
    ! [Na: nat,N3: nat,Aa: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),N3)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Aa)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),one_one_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(Aa),N3)),aa_nat_int(power_power_int(Aa),Na)) ) ) ) ).

% power_decreasing
tff(fact_3175_power__le__imp__le__exp,axiom,
    ! [Aa: real,M: nat,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Aa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),Na))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ) ).

% power_le_imp_le_exp
tff(fact_3176_power__le__imp__le__exp,axiom,
    ! [Aa: rat,M: nat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),Aa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(Aa),M)),aa_nat_rat(power_power_rat(Aa),Na))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ) ).

% power_le_imp_le_exp
tff(fact_3177_power__le__imp__le__exp,axiom,
    ! [Aa: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),Aa)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Aa),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Aa),Na))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ) ).

% power_le_imp_le_exp
tff(fact_3178_power__le__imp__le__exp,axiom,
    ! [Aa: int,M: nat,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),Aa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(Aa),M)),aa_nat_int(power_power_int(Aa),Na))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ) ).

% power_le_imp_le_exp
tff(fact_3179_power__eq__iff__eq__base,axiom,
    ! [Na: nat,Aa: real,Ba: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Aa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Ba)
         => ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),Na) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Ba),Na) )
          <=> ( Aa = Ba ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_3180_power__eq__iff__eq__base,axiom,
    ! [Na: nat,Aa: rat,Ba: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Aa)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Ba)
         => ( ( aa_nat_rat(power_power_rat(Aa),Na) = aa_nat_rat(power_power_rat(Ba),Na) )
          <=> ( Aa = Ba ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_3181_power__eq__iff__eq__base,axiom,
    ! [Na: nat,Aa: nat,Ba: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Aa)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Ba)
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Aa),Na) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Ba),Na) )
          <=> ( Aa = Ba ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_3182_power__eq__iff__eq__base,axiom,
    ! [Na: nat,Aa: int,Ba: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Aa)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Ba)
         => ( ( aa_nat_int(power_power_int(Aa),Na) = aa_nat_int(power_power_int(Ba),Na) )
          <=> ( Aa = Ba ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_3183_power__eq__imp__eq__base,axiom,
    ! [Aa: real,Na: nat,Ba: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),Na) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Ba),Na) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Aa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Ba)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
           => ( Aa = Ba ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_3184_power__eq__imp__eq__base,axiom,
    ! [Aa: rat,Na: nat,Ba: rat] :
      ( ( aa_nat_rat(power_power_rat(Aa),Na) = aa_nat_rat(power_power_rat(Ba),Na) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Aa)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Ba)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
           => ( Aa = Ba ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_3185_power__eq__imp__eq__base,axiom,
    ! [Aa: nat,Na: nat,Ba: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Aa),Na) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Ba),Na) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Aa)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Ba)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
           => ( Aa = Ba ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_3186_power__eq__imp__eq__base,axiom,
    ! [Aa: int,Na: nat,Ba: int] :
      ( ( aa_nat_int(power_power_int(Aa),Na) = aa_nat_int(power_power_int(Ba),Na) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Aa)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Ba)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
           => ( Aa = Ba ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_3187_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_3188_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_3189_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_3190_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_3191_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_3192_self__le__power,axiom,
    ! [Aa: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Aa)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),Na)) ) ) ).

% self_le_power
tff(fact_3193_self__le__power,axiom,
    ! [Aa: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),Aa)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),aa_nat_rat(power_power_rat(Aa),Na)) ) ) ).

% self_le_power
tff(fact_3194_self__le__power,axiom,
    ! [Aa: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Aa)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Aa),Na)) ) ) ).

% self_le_power
tff(fact_3195_self__le__power,axiom,
    ! [Aa: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),Aa)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),aa_nat_int(power_power_int(Aa),Na)) ) ) ).

% self_le_power
tff(fact_3196_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_3197_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_3198_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_3199_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_3200_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_3201_one__less__power,axiom,
    ! [Aa: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Aa)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),Na)) ) ) ).

% one_less_power
tff(fact_3202_one__less__power,axiom,
    ! [Aa: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),Aa)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_nat_rat(power_power_rat(Aa),Na)) ) ) ).

% one_less_power
tff(fact_3203_one__less__power,axiom,
    ! [Aa: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),Aa)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Aa),Na)) ) ) ).

% one_less_power
tff(fact_3204_one__less__power,axiom,
    ! [Aa: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),Aa)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_nat_int(power_power_int(Aa),Na)) ) ) ).

% one_less_power
tff(fact_3205_less__exp,axiom,
    ! [Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) ).

% less_exp
tff(fact_3206_power2__nat__le__imp__le,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ).

% power2_nat_le_imp_le
tff(fact_3207_power2__nat__le__eq__le,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ).

% power2_nat_le_eq_le
tff(fact_3208_self__le__ge2__pow,axiom,
    ! [K: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K),M)) ) ).

% self_le_ge2_pow
tff(fact_3209_power2__le__imp__le,axiom,
    ! [Xb: real,Y: 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,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y) ) ) ).

% power2_le_imp_le
tff(fact_3210_power2__le__imp__le,axiom,
    ! [Xb: rat,Y: 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(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y) ) ) ).

% power2_le_imp_le
tff(fact_3211_power2__le__imp__le,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Y)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y) ) ) ).

% power2_le_imp_le
tff(fact_3212_power2__le__imp__le,axiom,
    ! [Xb: int,Y: 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(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y) ) ) ).

% power2_le_imp_le
tff(fact_3213_power2__eq__imp__eq,axiom,
    ! [Xb: real,Y: 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,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
         => ( Xb = Y ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_3214_power2__eq__imp__eq,axiom,
    ! [Xb: rat,Y: 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(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xb)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
         => ( Xb = Y ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_3215_power2__eq__imp__eq,axiom,
    ! [Xb: nat,Y: 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,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Xb)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Y)
         => ( Xb = Y ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_3216_power2__eq__imp__eq,axiom,
    ! [Xb: int,Y: 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(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
         => ( Xb = Y ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_3217_zero__le__power2,axiom,
    ! [Aa: 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,Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% zero_le_power2
tff(fact_3218_zero__le__power2,axiom,
    ! [Aa: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% zero_le_power2
tff(fact_3219_zero__le__power2,axiom,
    ! [Aa: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% zero_le_power2
tff(fact_3220_power__strict__mono,axiom,
    ! [Aa: real,Ba: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),Ba)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Aa)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),Na)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ba),Na)) ) ) ) ).

% power_strict_mono
tff(fact_3221_power__strict__mono,axiom,
    ! [Aa: rat,Ba: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),Ba)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Aa)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(Aa),Na)),aa_nat_rat(power_power_rat(Ba),Na)) ) ) ) ).

% power_strict_mono
tff(fact_3222_power__strict__mono,axiom,
    ! [Aa: nat,Ba: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),Ba)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Aa)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Aa),Na)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Ba),Na)) ) ) ) ).

% power_strict_mono
tff(fact_3223_power__strict__mono,axiom,
    ! [Aa: int,Ba: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),Ba)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Aa)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(Aa),Na)),aa_nat_int(power_power_int(Ba),Na)) ) ) ) ).

% power_strict_mono
tff(fact_3224_power2__less__0,axiom,
    ! [Aa: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_real) ).

% power2_less_0
tff(fact_3225_power2__less__0,axiom,
    ! [Aa: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_rat) ).

% power2_less_0
tff(fact_3226_power2__less__0,axiom,
    ! [Aa: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_int) ).

% power2_less_0
tff(fact_3227_power2__eq__1__iff,axiom,
    ! [Aa: complex] :
      ( ( aa_nat_complex(power_power_complex(Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_complex )
    <=> ( ( Aa = one_one_complex )
        | ( Aa = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ) ) ) ).

% power2_eq_1_iff
tff(fact_3228_power2__eq__1__iff,axiom,
    ! [Aa: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_real )
    <=> ( ( Aa = one_one_real )
        | ( Aa = aa_real_real(uminus_uminus_real,one_one_real) ) ) ) ).

% power2_eq_1_iff
tff(fact_3229_power2__eq__1__iff,axiom,
    ! [Aa: int] :
      ( ( aa_nat_int(power_power_int(Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_int )
    <=> ( ( Aa = one_one_int )
        | ( Aa = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% power2_eq_1_iff
tff(fact_3230_power2__eq__1__iff,axiom,
    ! [Aa: code_integer] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_Code_integer )
    <=> ( ( Aa = one_one_Code_integer )
        | ( Aa = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ) ) ) ).

% power2_eq_1_iff
tff(fact_3231_power2__eq__1__iff,axiom,
    ! [Aa: rat] :
      ( ( aa_nat_rat(power_power_rat(Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_rat )
    <=> ( ( Aa = one_one_rat )
        | ( Aa = aa_rat_rat(uminus_uminus_rat,one_one_rat) ) ) ) ).

% power2_eq_1_iff
tff(fact_3232_diff__le__diff__pow,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K),Na))) ) ).

% diff_le_diff_pow
tff(fact_3233_power2__less__imp__less,axiom,
    ! [Xb: real,Y: 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,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y) ) ) ).

% power2_less_imp_less
tff(fact_3234_power2__less__imp__less,axiom,
    ! [Xb: rat,Y: 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(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y) ) ) ).

% power2_less_imp_less
tff(fact_3235_power2__less__imp__less,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Y)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y) ) ) ).

% power2_less_imp_less
tff(fact_3236_power2__less__imp__less,axiom,
    ! [Xb: int,Y: 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(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y) ) ) ).

% power2_less_imp_less
tff(fact_3237_sum__power2__le__zero__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(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,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_real)
    <=> ( ( Xb = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_power2_le_zero_iff
tff(fact_3238_sum__power2__le__zero__iff,axiom,
    ! [Xb: rat,Y: 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(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_rat)
    <=> ( ( Xb = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_power2_le_zero_iff
tff(fact_3239_sum__power2__le__zero__iff,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(power_power_int(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_int)
    <=> ( ( Xb = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_power2_le_zero_iff
tff(fact_3240_sum__power2__ge__zero,axiom,
    ! [Xb: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(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,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% sum_power2_ge_zero
tff(fact_3241_sum__power2__ge__zero,axiom,
    ! [Xb: rat,Y: 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(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% sum_power2_ge_zero
tff(fact_3242_sum__power2__ge__zero,axiom,
    ! [Xb: int,Y: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(power_power_int(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% sum_power2_ge_zero
tff(fact_3243_not__sum__power2__lt__zero,axiom,
    ! [Xb: real,Y: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(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,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_real) ).

% not_sum_power2_lt_zero
tff(fact_3244_not__sum__power2__lt__zero,axiom,
    ! [Xb: rat,Y: 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(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_rat) ).

% not_sum_power2_lt_zero
tff(fact_3245_not__sum__power2__lt__zero,axiom,
    ! [Xb: int,Y: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(power_power_int(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_int) ).

% not_sum_power2_lt_zero
tff(fact_3246_sum__power2__gt__zero__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(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,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
    <=> ( ( Xb != zero_zero_real )
        | ( Y != zero_zero_real ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_3247_sum__power2__gt__zero__iff,axiom,
    ! [Xb: rat,Y: 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(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
    <=> ( ( Xb != zero_zero_rat )
        | ( Y != zero_zero_rat ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_3248_sum__power2__gt__zero__iff,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(power_power_int(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
    <=> ( ( Xb != zero_zero_int )
        | ( Y != zero_zero_int ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_3249_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_3250_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_3251_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_3252_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_3253_ex__power__ivl2,axiom,
    ! [Ba: 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))),Ba)
     => ( 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,Ba),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,Ba),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat))) ) ) ) ).

% ex_power_ivl2
tff(fact_3254_ex__power__ivl1,axiom,
    ! [Ba: 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))),Ba)
     => ( 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,Ba),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,Ba),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat))) ) ) ) ).

% ex_power_ivl1
tff(fact_3255_mintlistlength,axiom,
    ! [Mi2: nat,Ma2: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Na: nat] :
      ( aa_nat_o(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Deg,TreeList,Summary)),Na)
     => ( ( Mi2 != Ma2 )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi2),Ma2)
          & ? [M3: nat] :
              ( ( some_nat(M3) = vEBT_vebt_mint(Summary) )
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ) ) ).

% mintlistlength
tff(fact_3256_inrange,axiom,
    ! [Ta: vEBT_VEBT,Na: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Na)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,vEBT_VEBT_set_vebt(Ta)),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))),Na)),one_one_nat))) ) ).

% inrange
tff(fact_3257_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [Na: nat,M: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na) != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),M)) != zero_zero_nat ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
tff(fact_3258_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [Na: nat,M: nat] :
      ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na) != zero_zero_int )
     => ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),M)) != zero_zero_int ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
tff(fact_3259_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [Na: nat,M: nat] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Na) != zero_z3403309356797280102nteger )
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),M)) != zero_z3403309356797280102nteger ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
tff(fact_3260_exp__add__not__zero__imp__left,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na)) != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M) != zero_zero_nat ) ) ).

% exp_add_not_zero_imp_left
tff(fact_3261_exp__add__not__zero__imp__left,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na)) != zero_zero_int )
     => ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M) != zero_zero_int ) ) ).

% exp_add_not_zero_imp_left
tff(fact_3262_exp__add__not__zero__imp__left,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na)) != zero_z3403309356797280102nteger )
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),M) != zero_z3403309356797280102nteger ) ) ).

% exp_add_not_zero_imp_left
tff(fact_3263_exp__add__not__zero__imp__right,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na)) != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na) != zero_zero_nat ) ) ).

% exp_add_not_zero_imp_right
tff(fact_3264_exp__add__not__zero__imp__right,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na)) != zero_zero_int )
     => ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na) != zero_zero_int ) ) ).

% exp_add_not_zero_imp_right
tff(fact_3265_exp__add__not__zero__imp__right,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na)) != zero_z3403309356797280102nteger )
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Na) != zero_z3403309356797280102nteger ) ) ).

% exp_add_not_zero_imp_right
tff(fact_3266_not__exp__less__eq__0__int,axiom,
    ! [Na: nat] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)),zero_zero_int) ).

% not_exp_less_eq_0_int
tff(fact_3267_enat__ord__number_I1_J,axiom,
    ! [M: num,Na: num] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(numera1916890842035813515d_enat(M)),numera1916890842035813515d_enat(Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,Na)) ) ).

% enat_ord_number(1)
tff(fact_3268_enat__ord__number_I2_J,axiom,
    ! [M: num,Na: num] :
      ( aa_Extended_enat_o(ord_le72135733267957522d_enat(numera1916890842035813515d_enat(M)),numera1916890842035813515d_enat(Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,Na)) ) ).

% enat_ord_number(2)
tff(fact_3269_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_3270_i0__less,axiom,
    ! [Na: extended_enat] :
      ( aa_Extended_enat_o(ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat),Na)
    <=> ( Na != zero_z5237406670263579293d_enat ) ) ).

% i0_less
tff(fact_3271_pow__sum,axiom,
    ! [Aa: nat,Ba: 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,Aa),Ba))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Aa)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Ba) ).

% pow_sum
tff(fact_3272_power__minus__is__div,axiom,
    ! [Ba: nat,Aa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ba),Aa)
     => ( 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,Aa),Ba)) = 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)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Ba)) ) ) ).

% power_minus_is_div
tff(fact_3273_bits__div__by__0,axiom,
    ! [Aa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Aa),zero_zero_nat) = zero_zero_nat ).

% bits_div_by_0
tff(fact_3274_bits__div__by__0,axiom,
    ! [Aa: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),zero_zero_int) = zero_zero_int ).

% bits_div_by_0
tff(fact_3275_bits__div__by__0,axiom,
    ! [Aa: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Aa),zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% bits_div_by_0
tff(fact_3276_bits__div__0,axiom,
    ! [Aa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,zero_zero_nat),Aa) = zero_zero_nat ).

% bits_div_0
tff(fact_3277_bits__div__0,axiom,
    ! [Aa: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,zero_zero_int),Aa) = zero_zero_int ).

% bits_div_0
tff(fact_3278_bits__div__0,axiom,
    ! [Aa: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,zero_z3403309356797280102nteger),Aa) = zero_z3403309356797280102nteger ).

% bits_div_0
tff(fact_3279_division__ring__divide__zero,axiom,
    ! [Aa: rat] : aa_rat_rat(divide_divide_rat(Aa),zero_zero_rat) = zero_zero_rat ).

% division_ring_divide_zero
tff(fact_3280_division__ring__divide__zero,axiom,
    ! [Aa: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Aa),zero_zero_real) = zero_zero_real ).

% division_ring_divide_zero
tff(fact_3281_divide__cancel__right,axiom,
    ! [Aa: rat,C: rat,Ba: rat] :
      ( ( aa_rat_rat(divide_divide_rat(Aa),C) = aa_rat_rat(divide_divide_rat(Ba),C) )
    <=> ( ( C = zero_zero_rat )
        | ( Aa = Ba ) ) ) ).

% divide_cancel_right
tff(fact_3282_divide__cancel__right,axiom,
    ! [Aa: real,C: real,Ba: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Aa),C) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Ba),C) )
    <=> ( ( C = zero_zero_real )
        | ( Aa = Ba ) ) ) ).

% divide_cancel_right
tff(fact_3283_divide__cancel__left,axiom,
    ! [C: rat,Aa: rat,Ba: rat] :
      ( ( aa_rat_rat(divide_divide_rat(C),Aa) = aa_rat_rat(divide_divide_rat(C),Ba) )
    <=> ( ( C = zero_zero_rat )
        | ( Aa = Ba ) ) ) ).

% divide_cancel_left
tff(fact_3284_divide__cancel__left,axiom,
    ! [C: real,Aa: real,Ba: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),Aa) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),Ba) )
    <=> ( ( C = zero_zero_real )
        | ( Aa = Ba ) ) ) ).

% divide_cancel_left
tff(fact_3285_divide__eq__0__iff,axiom,
    ! [Aa: rat,Ba: rat] :
      ( ( aa_rat_rat(divide_divide_rat(Aa),Ba) = zero_zero_rat )
    <=> ( ( Aa = zero_zero_rat )
        | ( Ba = zero_zero_rat ) ) ) ).

% divide_eq_0_iff
tff(fact_3286_divide__eq__0__iff,axiom,
    ! [Aa: real,Ba: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Aa),Ba) = zero_zero_real )
    <=> ( ( Aa = zero_zero_real )
        | ( Ba = zero_zero_real ) ) ) ).

% divide_eq_0_iff
tff(fact_3287_div__by__0,axiom,
    ! [Aa: rat] : aa_rat_rat(divide_divide_rat(Aa),zero_zero_rat) = zero_zero_rat ).

% div_by_0
tff(fact_3288_div__by__0,axiom,
    ! [Aa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Aa),zero_zero_nat) = zero_zero_nat ).

% div_by_0
tff(fact_3289_div__by__0,axiom,
    ! [Aa: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),zero_zero_int) = zero_zero_int ).

% div_by_0
tff(fact_3290_div__by__0,axiom,
    ! [Aa: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Aa),zero_zero_real) = zero_zero_real ).

% div_by_0
tff(fact_3291_div__by__0,axiom,
    ! [Aa: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Aa),zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% div_by_0
tff(fact_3292_div__0,axiom,
    ! [Aa: rat] : aa_rat_rat(divide_divide_rat(zero_zero_rat),Aa) = zero_zero_rat ).

% div_0
tff(fact_3293_div__0,axiom,
    ! [Aa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,zero_zero_nat),Aa) = zero_zero_nat ).

% div_0
tff(fact_3294_div__0,axiom,
    ! [Aa: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,zero_zero_int),Aa) = zero_zero_int ).

% div_0
tff(fact_3295_div__0,axiom,
    ! [Aa: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,zero_zero_real),Aa) = zero_zero_real ).

% div_0
tff(fact_3296_div__0,axiom,
    ! [Aa: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,zero_z3403309356797280102nteger),Aa) = zero_z3403309356797280102nteger ).

% div_0
tff(fact_3297_bits__div__by__1,axiom,
    ! [Aa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Aa),one_one_nat) = Aa ).

% bits_div_by_1
tff(fact_3298_bits__div__by__1,axiom,
    ! [Aa: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),one_one_int) = Aa ).

% bits_div_by_1
tff(fact_3299_bits__div__by__1,axiom,
    ! [Aa: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Aa),one_one_Code_integer) = Aa ).

% bits_div_by_1
tff(fact_3300_div__by__1,axiom,
    ! [Aa: complex] : aa_complex_complex(divide1717551699836669952omplex(Aa),one_one_complex) = Aa ).

% div_by_1
tff(fact_3301_div__by__1,axiom,
    ! [Aa: rat] : aa_rat_rat(divide_divide_rat(Aa),one_one_rat) = Aa ).

% div_by_1
tff(fact_3302_div__by__1,axiom,
    ! [Aa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Aa),one_one_nat) = Aa ).

% div_by_1
tff(fact_3303_div__by__1,axiom,
    ! [Aa: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),one_one_int) = Aa ).

% div_by_1
tff(fact_3304_div__by__1,axiom,
    ! [Aa: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Aa),one_one_real) = Aa ).

% div_by_1
tff(fact_3305_div__by__1,axiom,
    ! [Aa: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Aa),one_one_Code_integer) = Aa ).

% div_by_1
tff(fact_3306_div__minus__minus,axiom,
    ! [Aa: int,Ba: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,Aa)),aa_int_int(uminus_uminus_int,Ba)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),Ba) ).

% div_minus_minus
tff(fact_3307_div__minus__minus,axiom,
    ! [Aa: code_integer,Ba: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Ba)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Aa),Ba) ).

% div_minus_minus
tff(fact_3308_atLeastAtMost__iff,axiom,
    ! [I: set_nat,L: set_nat,U: set_nat] :
      ( member_set_nat(I,set_or4548717258645045905et_nat(L,U))
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,L),I)
        & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,I),U) ) ) ).

% atLeastAtMost_iff
tff(fact_3309_atLeastAtMost__iff,axiom,
    ! [I: set_int,L: set_int,U: set_int] :
      ( member_set_int(I,set_or370866239135849197et_int(L,U))
    <=> ( aa_set_int_o(ord_less_eq_set_int(L),I)
        & aa_set_int_o(ord_less_eq_set_int(I),U) ) ) ).

% atLeastAtMost_iff
tff(fact_3310_atLeastAtMost__iff,axiom,
    ! [I: rat,L: rat,U: rat] :
      ( member_rat(I,set_or633870826150836451st_rat(L,U))
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,L),I)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,I),U) ) ) ).

% atLeastAtMost_iff
tff(fact_3311_atLeastAtMost__iff,axiom,
    ! [I: num,L: num,U: num] :
      ( member_num(I,set_or7049704709247886629st_num(L,U))
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,L),I)
        & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,I),U) ) ) ).

% atLeastAtMost_iff
tff(fact_3312_atLeastAtMost__iff,axiom,
    ! [I: nat,L: nat,U: nat] :
      ( aa_set_nat_o(member_nat(I),set_or1269000886237332187st_nat(L,U))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,L),I)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),U) ) ) ).

% atLeastAtMost_iff
tff(fact_3313_atLeastAtMost__iff,axiom,
    ! [I: int,L: int,U: int] :
      ( aa_set_int_o(member_int(I),set_or1266510415728281911st_int(L,U))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L),I)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),U) ) ) ).

% atLeastAtMost_iff
tff(fact_3314_atLeastAtMost__iff,axiom,
    ! [I: real,L: real,U: real] :
      ( member_real(I,set_or1222579329274155063t_real(L,U))
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L),I)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,I),U) ) ) ).

% atLeastAtMost_iff
tff(fact_3315_Icc__eq__Icc,axiom,
    ! [L: set_int,H: set_int,L2: set_int,H2: set_int] :
      ( ( set_or370866239135849197et_int(L,H) = set_or370866239135849197et_int(L2,H2) )
    <=> ( ( ( L = L2 )
          & ( H = H2 ) )
        | ( ~ aa_set_int_o(ord_less_eq_set_int(L),H)
          & ~ aa_set_int_o(ord_less_eq_set_int(L2),H2) ) ) ) ).

% Icc_eq_Icc
tff(fact_3316_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_3317_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_3318_Icc__eq__Icc,axiom,
    ! [L: nat,H: nat,L2: nat,H2: nat] :
      ( ( set_or1269000886237332187st_nat(L,H) = 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_3319_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_3320_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_3321_finite__atLeastAtMost,axiom,
    ! [L: nat,U: nat] : aa_set_nat_o(finite_finite_nat,set_or1269000886237332187st_nat(L,U)) ).

% finite_atLeastAtMost
tff(fact_3322_divide__eq__1__iff,axiom,
    ! [Aa: complex,Ba: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(Aa),Ba) = one_one_complex )
    <=> ( ( Ba != zero_zero_complex )
        & ( Aa = Ba ) ) ) ).

% divide_eq_1_iff
tff(fact_3323_divide__eq__1__iff,axiom,
    ! [Aa: rat,Ba: rat] :
      ( ( aa_rat_rat(divide_divide_rat(Aa),Ba) = one_one_rat )
    <=> ( ( Ba != zero_zero_rat )
        & ( Aa = Ba ) ) ) ).

% divide_eq_1_iff
tff(fact_3324_divide__eq__1__iff,axiom,
    ! [Aa: real,Ba: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Aa),Ba) = one_one_real )
    <=> ( ( Ba != zero_zero_real )
        & ( Aa = Ba ) ) ) ).

% divide_eq_1_iff
tff(fact_3325_one__eq__divide__iff,axiom,
    ! [Aa: complex,Ba: complex] :
      ( ( one_one_complex = aa_complex_complex(divide1717551699836669952omplex(Aa),Ba) )
    <=> ( ( Ba != zero_zero_complex )
        & ( Aa = Ba ) ) ) ).

% one_eq_divide_iff
tff(fact_3326_one__eq__divide__iff,axiom,
    ! [Aa: rat,Ba: rat] :
      ( ( one_one_rat = aa_rat_rat(divide_divide_rat(Aa),Ba) )
    <=> ( ( Ba != zero_zero_rat )
        & ( Aa = Ba ) ) ) ).

% one_eq_divide_iff
tff(fact_3327_one__eq__divide__iff,axiom,
    ! [Aa: real,Ba: real] :
      ( ( one_one_real = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Aa),Ba) )
    <=> ( ( Ba != zero_zero_real )
        & ( Aa = Ba ) ) ) ).

% one_eq_divide_iff
tff(fact_3328_divide__self,axiom,
    ! [Aa: complex] :
      ( ( Aa != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(Aa),Aa) = one_one_complex ) ) ).

% divide_self
tff(fact_3329_divide__self,axiom,
    ! [Aa: rat] :
      ( ( Aa != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(Aa),Aa) = one_one_rat ) ) ).

% divide_self
tff(fact_3330_divide__self,axiom,
    ! [Aa: real] :
      ( ( Aa != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Aa),Aa) = one_one_real ) ) ).

% divide_self
tff(fact_3331_divide__self__if,axiom,
    ! [Aa: complex] :
      aa_complex_complex(divide1717551699836669952omplex(Aa),Aa) = $ite(Aa = zero_zero_complex,zero_zero_complex,one_one_complex) ).

% divide_self_if
tff(fact_3332_divide__self__if,axiom,
    ! [Aa: rat] :
      aa_rat_rat(divide_divide_rat(Aa),Aa) = $ite(Aa = zero_zero_rat,zero_zero_rat,one_one_rat) ).

% divide_self_if
tff(fact_3333_divide__self__if,axiom,
    ! [Aa: real] :
      aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Aa),Aa) = $ite(Aa = zero_zero_real,zero_zero_real,one_one_real) ).

% divide_self_if
tff(fact_3334_divide__eq__eq__1,axiom,
    ! [Ba: rat,Aa: rat] :
      ( ( aa_rat_rat(divide_divide_rat(Ba),Aa) = one_one_rat )
    <=> ( ( Aa != zero_zero_rat )
        & ( Aa = Ba ) ) ) ).

% divide_eq_eq_1
tff(fact_3335_divide__eq__eq__1,axiom,
    ! [Ba: real,Aa: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Ba),Aa) = one_one_real )
    <=> ( ( Aa != zero_zero_real )
        & ( Aa = Ba ) ) ) ).

% divide_eq_eq_1
tff(fact_3336_eq__divide__eq__1,axiom,
    ! [Ba: rat,Aa: rat] :
      ( ( one_one_rat = aa_rat_rat(divide_divide_rat(Ba),Aa) )
    <=> ( ( Aa != zero_zero_rat )
        & ( Aa = Ba ) ) ) ).

% eq_divide_eq_1
tff(fact_3337_eq__divide__eq__1,axiom,
    ! [Ba: real,Aa: real] :
      ( ( one_one_real = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Ba),Aa) )
    <=> ( ( Aa != zero_zero_real )
        & ( Aa = Ba ) ) ) ).

% eq_divide_eq_1
tff(fact_3338_one__divide__eq__0__iff,axiom,
    ! [Aa: rat] :
      ( ( aa_rat_rat(divide_divide_rat(one_one_rat),Aa) = zero_zero_rat )
    <=> ( Aa = zero_zero_rat ) ) ).

% one_divide_eq_0_iff
tff(fact_3339_one__divide__eq__0__iff,axiom,
    ! [Aa: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),Aa) = zero_zero_real )
    <=> ( Aa = zero_zero_real ) ) ).

% one_divide_eq_0_iff
tff(fact_3340_zero__eq__1__divide__iff,axiom,
    ! [Aa: rat] :
      ( ( zero_zero_rat = aa_rat_rat(divide_divide_rat(one_one_rat),Aa) )
    <=> ( Aa = zero_zero_rat ) ) ).

% zero_eq_1_divide_iff
tff(fact_3341_zero__eq__1__divide__iff,axiom,
    ! [Aa: real] :
      ( ( zero_zero_real = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),Aa) )
    <=> ( Aa = zero_zero_real ) ) ).

% zero_eq_1_divide_iff
tff(fact_3342_div__self,axiom,
    ! [Aa: complex] :
      ( ( Aa != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(Aa),Aa) = one_one_complex ) ) ).

% div_self
tff(fact_3343_div__self,axiom,
    ! [Aa: rat] :
      ( ( Aa != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(Aa),Aa) = one_one_rat ) ) ).

% div_self
tff(fact_3344_div__self,axiom,
    ! [Aa: nat] :
      ( ( Aa != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Aa),Aa) = one_one_nat ) ) ).

% div_self
tff(fact_3345_div__self,axiom,
    ! [Aa: int] :
      ( ( Aa != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),Aa) = one_one_int ) ) ).

% div_self
tff(fact_3346_div__self,axiom,
    ! [Aa: real] :
      ( ( Aa != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Aa),Aa) = one_one_real ) ) ).

% div_self
tff(fact_3347_div__self,axiom,
    ! [Aa: code_integer] :
      ( ( Aa != zero_z3403309356797280102nteger )
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Aa),Aa) = one_one_Code_integer ) ) ).

% div_self
tff(fact_3348_div__minus1__right,axiom,
    ! [Aa: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,Aa) ).

% div_minus1_right
tff(fact_3349_div__minus1__right,axiom,
    ! [Aa: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Aa),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa) ).

% div_minus1_right
tff(fact_3350_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_3351_divide__minus1,axiom,
    ! [Xb: real] : aa_real_real(aa_rea7340605660300375871l_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_3352_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_3353_atLeastatMost__empty__iff2,axiom,
    ! [Aa: set_int,Ba: set_int] :
      ( ( bot_bot_set_set_int = set_or370866239135849197et_int(Aa,Ba) )
    <=> ~ aa_set_int_o(ord_less_eq_set_int(Aa),Ba) ) ).

% atLeastatMost_empty_iff2
tff(fact_3354_atLeastatMost__empty__iff2,axiom,
    ! [Aa: rat,Ba: rat] :
      ( ( bot_bot_set_rat = set_or633870826150836451st_rat(Aa,Ba) )
    <=> ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),Ba) ) ).

% atLeastatMost_empty_iff2
tff(fact_3355_atLeastatMost__empty__iff2,axiom,
    ! [Aa: num,Ba: num] :
      ( ( bot_bot_set_num = set_or7049704709247886629st_num(Aa,Ba) )
    <=> ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Aa),Ba) ) ).

% atLeastatMost_empty_iff2
tff(fact_3356_atLeastatMost__empty__iff2,axiom,
    ! [Aa: nat,Ba: nat] :
      ( ( bot_bot_set_nat = set_or1269000886237332187st_nat(Aa,Ba) )
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),Ba) ) ).

% atLeastatMost_empty_iff2
tff(fact_3357_atLeastatMost__empty__iff2,axiom,
    ! [Aa: int,Ba: int] :
      ( ( bot_bot_set_int = set_or1266510415728281911st_int(Aa,Ba) )
    <=> ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),Ba) ) ).

% atLeastatMost_empty_iff2
tff(fact_3358_atLeastatMost__empty__iff2,axiom,
    ! [Aa: real,Ba: real] :
      ( ( bot_bot_set_real = set_or1222579329274155063t_real(Aa,Ba) )
    <=> ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),Ba) ) ).

% atLeastatMost_empty_iff2
tff(fact_3359_atLeastatMost__empty__iff,axiom,
    ! [Aa: set_int,Ba: set_int] :
      ( ( set_or370866239135849197et_int(Aa,Ba) = bot_bot_set_set_int )
    <=> ~ aa_set_int_o(ord_less_eq_set_int(Aa),Ba) ) ).

% atLeastatMost_empty_iff
tff(fact_3360_atLeastatMost__empty__iff,axiom,
    ! [Aa: rat,Ba: rat] :
      ( ( set_or633870826150836451st_rat(Aa,Ba) = bot_bot_set_rat )
    <=> ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),Ba) ) ).

% atLeastatMost_empty_iff
tff(fact_3361_atLeastatMost__empty__iff,axiom,
    ! [Aa: num,Ba: num] :
      ( ( set_or7049704709247886629st_num(Aa,Ba) = bot_bot_set_num )
    <=> ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Aa),Ba) ) ).

% atLeastatMost_empty_iff
tff(fact_3362_atLeastatMost__empty__iff,axiom,
    ! [Aa: nat,Ba: nat] :
      ( ( set_or1269000886237332187st_nat(Aa,Ba) = bot_bot_set_nat )
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),Ba) ) ).

% atLeastatMost_empty_iff
tff(fact_3363_atLeastatMost__empty__iff,axiom,
    ! [Aa: int,Ba: int] :
      ( ( set_or1266510415728281911st_int(Aa,Ba) = bot_bot_set_int )
    <=> ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),Ba) ) ).

% atLeastatMost_empty_iff
tff(fact_3364_atLeastatMost__empty__iff,axiom,
    ! [Aa: real,Ba: real] :
      ( ( set_or1222579329274155063t_real(Aa,Ba) = bot_bot_set_real )
    <=> ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),Ba) ) ).

% atLeastatMost_empty_iff
tff(fact_3365_atLeastatMost__subset__iff,axiom,
    ! [Aa: set_int,Ba: set_int,C: set_int,D: set_int] :
      ( ord_le4403425263959731960et_int(set_or370866239135849197et_int(Aa,Ba),set_or370866239135849197et_int(C,D))
    <=> ( ~ aa_set_int_o(ord_less_eq_set_int(Aa),Ba)
        | ( aa_set_int_o(ord_less_eq_set_int(C),Aa)
          & aa_set_int_o(ord_less_eq_set_int(Ba),D) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_3366_atLeastatMost__subset__iff,axiom,
    ! [Aa: rat,Ba: rat,C: rat,D: rat] :
      ( ord_less_eq_set_rat(set_or633870826150836451st_rat(Aa,Ba),set_or633870826150836451st_rat(C,D))
    <=> ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),Ba)
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),Aa)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ba),D) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_3367_atLeastatMost__subset__iff,axiom,
    ! [Aa: num,Ba: num,C: num,D: num] :
      ( ord_less_eq_set_num(set_or7049704709247886629st_num(Aa,Ba),set_or7049704709247886629st_num(C,D))
    <=> ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Aa),Ba)
        | ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),Aa)
          & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ba),D) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_3368_atLeastatMost__subset__iff,axiom,
    ! [Aa: nat,Ba: nat,C: nat,D: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,set_or1269000886237332187st_nat(Aa,Ba)),set_or1269000886237332187st_nat(C,D))
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),Ba)
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),Aa)
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ba),D) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_3369_atLeastatMost__subset__iff,axiom,
    ! [Aa: int,Ba: int,C: int,D: int] :
      ( aa_set_int_o(ord_less_eq_set_int(set_or1266510415728281911st_int(Aa,Ba)),set_or1266510415728281911st_int(C,D))
    <=> ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),Ba)
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),Aa)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ba),D) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_3370_atLeastatMost__subset__iff,axiom,
    ! [Aa: real,Ba: real,C: real,D: real] :
      ( ord_less_eq_set_real(set_or1222579329274155063t_real(Aa,Ba),set_or1222579329274155063t_real(C,D))
    <=> ( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),Ba)
        | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),Aa)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ba),D) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_3371_atLeastatMost__empty,axiom,
    ! [Ba: rat,Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ba),Aa)
     => ( set_or633870826150836451st_rat(Aa,Ba) = bot_bot_set_rat ) ) ).

% atLeastatMost_empty
tff(fact_3372_atLeastatMost__empty,axiom,
    ! [Ba: num,Aa: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Ba),Aa)
     => ( set_or7049704709247886629st_num(Aa,Ba) = bot_bot_set_num ) ) ).

% atLeastatMost_empty
tff(fact_3373_atLeastatMost__empty,axiom,
    ! [Ba: nat,Aa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ba),Aa)
     => ( set_or1269000886237332187st_nat(Aa,Ba) = bot_bot_set_nat ) ) ).

% atLeastatMost_empty
tff(fact_3374_atLeastatMost__empty,axiom,
    ! [Ba: int,Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Ba),Aa)
     => ( set_or1266510415728281911st_int(Aa,Ba) = bot_bot_set_int ) ) ).

% atLeastatMost_empty
tff(fact_3375_atLeastatMost__empty,axiom,
    ! [Ba: real,Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Ba),Aa)
     => ( set_or1222579329274155063t_real(Aa,Ba) = bot_bot_set_real ) ) ).

% atLeastatMost_empty
tff(fact_3376_infinite__Icc__iff,axiom,
    ! [Aa: rat,Ba: rat] :
      ( ~ finite_finite_rat(set_or633870826150836451st_rat(Aa,Ba))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),Ba) ) ).

% infinite_Icc_iff
tff(fact_3377_infinite__Icc__iff,axiom,
    ! [Aa: real,Ba: real] :
      ( ~ finite_finite_real(set_or1222579329274155063t_real(Aa,Ba))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),Ba) ) ).

% infinite_Icc_iff
tff(fact_3378_div__by__Suc__0,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_nat_nat(suc,zero_zero_nat)) = M ).

% div_by_Suc_0
tff(fact_3379_div__less,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na) = zero_zero_nat ) ) ).

% div_less
tff(fact_3380_divide__le__0__1__iff,axiom,
    ! [Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),Aa)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),zero_zero_real) ) ).

% divide_le_0_1_iff
tff(fact_3381_divide__le__0__1__iff,axiom,
    ! [Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(one_one_rat),Aa)),zero_zero_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),zero_zero_rat) ) ).

% divide_le_0_1_iff
tff(fact_3382_zero__le__divide__1__iff,axiom,
    ! [Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),Aa))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Aa) ) ).

% zero_le_divide_1_iff
tff(fact_3383_zero__le__divide__1__iff,axiom,
    ! [Aa: 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),Aa))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Aa) ) ).

% zero_le_divide_1_iff
tff(fact_3384_divide__less__0__1__iff,axiom,
    ! [Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(one_one_rat),Aa)),zero_zero_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),zero_zero_rat) ) ).

% divide_less_0_1_iff
tff(fact_3385_divide__less__0__1__iff,axiom,
    ! [Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),Aa)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),zero_zero_real) ) ).

% divide_less_0_1_iff
tff(fact_3386_divide__less__eq__1__neg,axiom,
    ! [Aa: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(Ba),Aa)),one_one_rat)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),Ba) ) ) ).

% divide_less_eq_1_neg
tff(fact_3387_divide__less__eq__1__neg,axiom,
    ! [Aa: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Ba),Aa)),one_one_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),Ba) ) ) ).

% divide_less_eq_1_neg
tff(fact_3388_divide__less__eq__1__pos,axiom,
    ! [Aa: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Aa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(Ba),Aa)),one_one_rat)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ba),Aa) ) ) ).

% divide_less_eq_1_pos
tff(fact_3389_divide__less__eq__1__pos,axiom,
    ! [Aa: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Aa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Ba),Aa)),one_one_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Ba),Aa) ) ) ).

% divide_less_eq_1_pos
tff(fact_3390_less__divide__eq__1__neg,axiom,
    ! [Aa: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_rat_rat(divide_divide_rat(Ba),Aa))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ba),Aa) ) ) ).

% less_divide_eq_1_neg
tff(fact_3391_less__divide__eq__1__neg,axiom,
    ! [Aa: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Ba),Aa))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Ba),Aa) ) ) ).

% less_divide_eq_1_neg
tff(fact_3392_less__divide__eq__1__pos,axiom,
    ! [Aa: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Aa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_rat_rat(divide_divide_rat(Ba),Aa))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),Ba) ) ) ).

% less_divide_eq_1_pos
tff(fact_3393_less__divide__eq__1__pos,axiom,
    ! [Aa: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Aa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Ba),Aa))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),Ba) ) ) ).

% less_divide_eq_1_pos
tff(fact_3394_zero__less__divide__1__iff,axiom,
    ! [Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(one_one_rat),Aa))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Aa) ) ).

% zero_less_divide_1_iff
tff(fact_3395_zero__less__divide__1__iff,axiom,
    ! [Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),Aa))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Aa) ) ).

% zero_less_divide_1_iff
tff(fact_3396_divide__le__eq__1__neg,axiom,
    ! [Aa: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Ba),Aa)),one_one_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),Ba) ) ) ).

% divide_le_eq_1_neg
tff(fact_3397_divide__le__eq__1__neg,axiom,
    ! [Aa: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(Ba),Aa)),one_one_rat)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),Ba) ) ) ).

% divide_le_eq_1_neg
tff(fact_3398_divide__le__eq__1__pos,axiom,
    ! [Aa: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Aa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Ba),Aa)),one_one_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ba),Aa) ) ) ).

% divide_le_eq_1_pos
tff(fact_3399_divide__le__eq__1__pos,axiom,
    ! [Aa: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Aa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(Ba),Aa)),one_one_rat)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ba),Aa) ) ) ).

% divide_le_eq_1_pos
tff(fact_3400_le__divide__eq__1__neg,axiom,
    ! [Aa: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Ba),Aa))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ba),Aa) ) ) ).

% le_divide_eq_1_neg
tff(fact_3401_le__divide__eq__1__neg,axiom,
    ! [Aa: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_rat_rat(divide_divide_rat(Ba),Aa))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ba),Aa) ) ) ).

% le_divide_eq_1_neg
tff(fact_3402_le__divide__eq__1__pos,axiom,
    ! [Aa: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Aa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Ba),Aa))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),Ba) ) ) ).

% le_divide_eq_1_pos
tff(fact_3403_le__divide__eq__1__pos,axiom,
    ! [Aa: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Aa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_rat_rat(divide_divide_rat(Ba),Aa))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),Ba) ) ) ).

% le_divide_eq_1_pos
tff(fact_3404_div2__Suc__Suc,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,M))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% div2_Suc_Suc
tff(fact_3405_add__self__div__2,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),M)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = M ).

% add_self_div_2
tff(fact_3406_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_3407_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_3408_bits__1__div__2,axiom,
    aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,one_one_Code_integer),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = zero_z3403309356797280102nteger ).

% bits_1_div_2
tff(fact_3409_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_3410_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_3411_one__div__two__eq__zero,axiom,
    aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,one_one_Code_integer),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = zero_z3403309356797280102nteger ).

% one_div_two_eq_zero
tff(fact_3412_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_3413_minus__1__div__2__eq,axiom,
    aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% minus_1_div_2_eq
tff(fact_3414_set__decode__Suc,axiom,
    ! [Na: nat,Xb: nat] :
      ( aa_set_nat_o(member_nat(aa_nat_nat(suc,Na)),nat_set_decode(Xb))
    <=> aa_set_nat_o(member_nat(Na),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_3415_real__arch__pow,axiom,
    ! [Xb: real,Y: 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,Y),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),N)) ) ).

% real_arch_pow
tff(fact_3416_real__arch__pow__inv,axiom,
    ! [Y: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,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)),Y) ) ) ).

% real_arch_pow_inv
tff(fact_3417_enat__less__induct,axiom,
    ! [P: fun_Extended_enat_o,Na: extended_enat] :
      ( ! [N: extended_enat] :
          ( ! [M2: extended_enat] :
              ( aa_Extended_enat_o(ord_le72135733267957522d_enat(M2),N)
             => aa_Extended_enat_o(P,M2) )
         => aa_Extended_enat_o(P,N) )
     => aa_Extended_enat_o(P,Na) ) ).

% enat_less_induct
tff(fact_3418_add__diff__assoc__enat,axiom,
    ! [Z: extended_enat,Y: extended_enat,Xb: extended_enat] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(Z),Y)
     => ( plus_p3455044024723400733d_enat(Xb,minus_3235023915231533773d_enat(Y,Z)) = minus_3235023915231533773d_enat(plus_p3455044024723400733d_enat(Xb,Y),Z) ) ) ).

% add_diff_assoc_enat
tff(fact_3419_not__iless0,axiom,
    ! [Na: extended_enat] : ~ aa_Extended_enat_o(ord_le72135733267957522d_enat(Na),zero_z5237406670263579293d_enat) ).

% not_iless0
tff(fact_3420_i0__lb,axiom,
    ! [Na: extended_enat] : aa_Extended_enat_o(ord_le2932123472753598470d_enat(zero_z5237406670263579293d_enat),Na) ).

% i0_lb
tff(fact_3421_ile0__eq,axiom,
    ! [Na: extended_enat] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(Na),zero_z5237406670263579293d_enat)
    <=> ( Na = zero_z5237406670263579293d_enat ) ) ).

% ile0_eq
tff(fact_3422_complete__real,axiom,
    ! [S2: set_real] :
      ( ? [X3: real] : member_real(X3,S2)
     => ( ? [Z4: real] :
          ! [X: real] :
            ( member_real(X,S2)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Z4) )
       => ? [Y3: real] :
            ( ! [X3: real] :
                ( member_real(X3,S2)
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),Y3) )
            & ! [Z4: real] :
                ( ! [X: real] :
                    ( member_real(X,S2)
                   => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Z4) )
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y3),Z4) ) ) ) ) ).

% complete_real
tff(fact_3423_less__eq__real__def,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
        | ( Xb = Y ) ) ) ).

% less_eq_real_def
tff(fact_3424_real__0__less__add__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,Xb)),Y) ) ).

% real_0_less_add_iff
tff(fact_3425_real__add__less__0__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(uminus_uminus_real,Xb)) ) ).

% real_add_less_0_iff
tff(fact_3426_real__add__le__0__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(uminus_uminus_real,Xb)) ) ).

% real_add_le_0_iff
tff(fact_3427_real__0__le__add__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,Xb)),Y) ) ).

% real_0_le_add_iff
tff(fact_3428_div__minus__right,axiom,
    ! [Aa: int,Ba: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),aa_int_int(uminus_uminus_int,Ba)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,Aa)),Ba) ).

% div_minus_right
tff(fact_3429_div__minus__right,axiom,
    ! [Aa: code_integer,Ba: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Aa),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Ba)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa)),Ba) ).

% div_minus_right
tff(fact_3430_div__le__mono,axiom,
    ! [M: nat,Na: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),K)) ) ).

% div_le_mono
tff(fact_3431_div__le__dividend,axiom,
    ! [M: nat,Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na)),M) ).

% div_le_dividend
tff(fact_3432_infinite__Icc,axiom,
    ! [Aa: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),Ba)
     => ~ finite_finite_rat(set_or633870826150836451st_rat(Aa,Ba)) ) ).

% infinite_Icc
tff(fact_3433_infinite__Icc,axiom,
    ! [Aa: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),Ba)
     => ~ finite_finite_real(set_or1222579329274155063t_real(Aa,Ba)) ) ).

% infinite_Icc
tff(fact_3434_divide__le__0__iff,axiom,
    ! [Aa: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Aa),Ba)),zero_zero_real)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Aa)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ba),zero_zero_real) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Ba) ) ) ) ).

% divide_le_0_iff
tff(fact_3435_divide__le__0__iff,axiom,
    ! [Aa: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(Aa),Ba)),zero_zero_rat)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Aa)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ba),zero_zero_rat) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Ba) ) ) ) ).

% divide_le_0_iff
tff(fact_3436_divide__right__mono,axiom,
    ! [Aa: real,Ba: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),Ba)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Aa),C)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Ba),C)) ) ) ).

% divide_right_mono
tff(fact_3437_divide__right__mono,axiom,
    ! [Aa: rat,Ba: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),Ba)
     => ( 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(Aa),C)),aa_rat_rat(divide_divide_rat(Ba),C)) ) ) ).

% divide_right_mono
tff(fact_3438_zero__le__divide__iff,axiom,
    ! [Aa: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Aa),Ba))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Aa)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Ba) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ba),zero_zero_real) ) ) ) ).

% zero_le_divide_iff
tff(fact_3439_zero__le__divide__iff,axiom,
    ! [Aa: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(Aa),Ba))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Aa)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Ba) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ba),zero_zero_rat) ) ) ) ).

% zero_le_divide_iff
tff(fact_3440_divide__nonneg__nonneg,axiom,
    ! [Xb: real,Y: 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),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)) ) ) ).

% divide_nonneg_nonneg
tff(fact_3441_divide__nonneg__nonneg,axiom,
    ! [Xb: rat,Y: 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),Y)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(Xb),Y)) ) ) ).

% divide_nonneg_nonneg
tff(fact_3442_divide__nonneg__nonpos,axiom,
    ! [Xb: real,Y: 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,Y),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)),zero_zero_real) ) ) ).

% divide_nonneg_nonpos
tff(fact_3443_divide__nonneg__nonpos,axiom,
    ! [Xb: rat,Y: 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,Y),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(Xb),Y)),zero_zero_rat) ) ) ).

% divide_nonneg_nonpos
tff(fact_3444_divide__nonpos__nonneg,axiom,
    ! [Xb: real,Y: 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),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)),zero_zero_real) ) ) ).

% divide_nonpos_nonneg
tff(fact_3445_divide__nonpos__nonneg,axiom,
    ! [Xb: rat,Y: 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),Y)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(Xb),Y)),zero_zero_rat) ) ) ).

% divide_nonpos_nonneg
tff(fact_3446_divide__nonpos__nonpos,axiom,
    ! [Xb: real,Y: 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,Y),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)) ) ) ).

% divide_nonpos_nonpos
tff(fact_3447_divide__nonpos__nonpos,axiom,
    ! [Xb: rat,Y: 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,Y),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),Y)) ) ) ).

% divide_nonpos_nonpos
tff(fact_3448_divide__right__mono__neg,axiom,
    ! [Aa: real,Ba: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),Ba)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Ba),C)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Aa),C)) ) ) ).

% divide_right_mono_neg
tff(fact_3449_divide__right__mono__neg,axiom,
    ! [Aa: rat,Ba: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),Ba)
     => ( 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(Ba),C)),aa_rat_rat(divide_divide_rat(Aa),C)) ) ) ).

% divide_right_mono_neg
tff(fact_3450_divide__neg__neg,axiom,
    ! [Xb: rat,Y: 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,Y),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(Xb),Y)) ) ) ).

% divide_neg_neg
tff(fact_3451_divide__neg__neg,axiom,
    ! [Xb: real,Y: 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,Y),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)) ) ) ).

% divide_neg_neg
tff(fact_3452_divide__neg__pos,axiom,
    ! [Xb: rat,Y: 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),Y)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(Xb),Y)),zero_zero_rat) ) ) ).

% divide_neg_pos
tff(fact_3453_divide__neg__pos,axiom,
    ! [Xb: real,Y: 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),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)),zero_zero_real) ) ) ).

% divide_neg_pos
tff(fact_3454_divide__pos__neg,axiom,
    ! [Xb: rat,Y: 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,Y),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(Xb),Y)),zero_zero_rat) ) ) ).

% divide_pos_neg
tff(fact_3455_divide__pos__neg,axiom,
    ! [Xb: real,Y: 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,Y),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)),zero_zero_real) ) ) ).

% divide_pos_neg
tff(fact_3456_divide__pos__pos,axiom,
    ! [Xb: rat,Y: 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),Y)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(Xb),Y)) ) ) ).

% divide_pos_pos
tff(fact_3457_divide__pos__pos,axiom,
    ! [Xb: real,Y: 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),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)) ) ) ).

% divide_pos_pos
tff(fact_3458_divide__less__0__iff,axiom,
    ! [Aa: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(Aa),Ba)),zero_zero_rat)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Aa)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ba),zero_zero_rat) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Ba) ) ) ) ).

% divide_less_0_iff
tff(fact_3459_divide__less__0__iff,axiom,
    ! [Aa: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Aa),Ba)),zero_zero_real)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Aa)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,Ba),zero_zero_real) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Ba) ) ) ) ).

% divide_less_0_iff
tff(fact_3460_divide__less__cancel,axiom,
    ! [Aa: rat,C: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(Aa),C)),aa_rat_rat(divide_divide_rat(Ba),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,Aa),Ba) )
        & ( 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,Ba),Aa) )
        & ( C != zero_zero_rat ) ) ) ).

% divide_less_cancel
tff(fact_3461_divide__less__cancel,axiom,
    ! [Aa: real,C: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Aa),C)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Ba),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,Aa),Ba) )
        & ( 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,Ba),Aa) )
        & ( C != zero_zero_real ) ) ) ).

% divide_less_cancel
tff(fact_3462_zero__less__divide__iff,axiom,
    ! [Aa: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(Aa),Ba))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Aa)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Ba) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ba),zero_zero_rat) ) ) ) ).

% zero_less_divide_iff
tff(fact_3463_zero__less__divide__iff,axiom,
    ! [Aa: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Aa),Ba))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Aa)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Ba) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,Ba),zero_zero_real) ) ) ) ).

% zero_less_divide_iff
tff(fact_3464_divide__strict__right__mono,axiom,
    ! [Aa: rat,Ba: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),Ba)
     => ( 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(Aa),C)),aa_rat_rat(divide_divide_rat(Ba),C)) ) ) ).

% divide_strict_right_mono
tff(fact_3465_divide__strict__right__mono,axiom,
    ! [Aa: real,Ba: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),Ba)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Aa),C)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Ba),C)) ) ) ).

% divide_strict_right_mono
tff(fact_3466_divide__strict__right__mono__neg,axiom,
    ! [Ba: rat,Aa: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ba),Aa)
     => ( 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(Aa),C)),aa_rat_rat(divide_divide_rat(Ba),C)) ) ) ).

% divide_strict_right_mono_neg
tff(fact_3467_divide__strict__right__mono__neg,axiom,
    ! [Ba: real,Aa: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Ba),Aa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Aa),C)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Ba),C)) ) ) ).

% divide_strict_right_mono_neg
tff(fact_3468_ex__nat__less,axiom,
    ! [Na: nat,P: fun_nat_o] :
      ( ? [M6: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M6),Na)
          & aa_nat_o(P,M6) )
    <=> ? [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),set_or1269000886237332187st_nat(zero_zero_nat,Na))
          & aa_nat_o(P,X4) ) ) ).

% ex_nat_less
tff(fact_3469_all__nat__less,axiom,
    ! [Na: nat,P: fun_nat_o] :
      ( ! [M6: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M6),Na)
         => aa_nat_o(P,M6) )
    <=> ! [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),set_or1269000886237332187st_nat(zero_zero_nat,Na))
         => aa_nat_o(P,X4) ) ) ).

% all_nat_less
tff(fact_3470_right__inverse__eq,axiom,
    ! [Ba: complex,Aa: complex] :
      ( ( Ba != zero_zero_complex )
     => ( ( aa_complex_complex(divide1717551699836669952omplex(Aa),Ba) = one_one_complex )
      <=> ( Aa = Ba ) ) ) ).

% right_inverse_eq
tff(fact_3471_right__inverse__eq,axiom,
    ! [Ba: rat,Aa: rat] :
      ( ( Ba != zero_zero_rat )
     => ( ( aa_rat_rat(divide_divide_rat(Aa),Ba) = one_one_rat )
      <=> ( Aa = Ba ) ) ) ).

% right_inverse_eq
tff(fact_3472_right__inverse__eq,axiom,
    ! [Ba: real,Aa: real] :
      ( ( Ba != zero_zero_real )
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Aa),Ba) = one_one_real )
      <=> ( Aa = Ba ) ) ) ).

% right_inverse_eq
tff(fact_3473_nonzero__minus__divide__right,axiom,
    ! [Ba: real,Aa: real] :
      ( ( Ba != zero_zero_real )
     => ( aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Aa),Ba)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Aa),aa_real_real(uminus_uminus_real,Ba)) ) ) ).

% nonzero_minus_divide_right
tff(fact_3474_nonzero__minus__divide__right,axiom,
    ! [Ba: rat,Aa: rat] :
      ( ( Ba != zero_zero_rat )
     => ( aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(Aa),Ba)) = aa_rat_rat(divide_divide_rat(Aa),aa_rat_rat(uminus_uminus_rat,Ba)) ) ) ).

% nonzero_minus_divide_right
tff(fact_3475_nonzero__minus__divide__divide,axiom,
    ! [Ba: real,Aa: real] :
      ( ( Ba != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,Aa)),aa_real_real(uminus_uminus_real,Ba)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Aa),Ba) ) ) ).

% nonzero_minus_divide_divide
tff(fact_3476_nonzero__minus__divide__divide,axiom,
    ! [Ba: rat,Aa: rat] :
      ( ( Ba != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(uminus_uminus_rat,Aa)),aa_rat_rat(uminus_uminus_rat,Ba)) = aa_rat_rat(divide_divide_rat(Aa),Ba) ) ) ).

% nonzero_minus_divide_divide
tff(fact_3477_divide__numeral__1,axiom,
    ! [Aa: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Aa),aa_num_real(numeral_numeral_real,one)) = Aa ).

% divide_numeral_1
tff(fact_3478_power__one__over,axiom,
    ! [Aa: rat,Na: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(divide_divide_rat(one_one_rat),Aa)),Na) = aa_rat_rat(divide_divide_rat(one_one_rat),aa_nat_rat(power_power_rat(Aa),Na)) ).

% power_one_over
tff(fact_3479_power__one__over,axiom,
    ! [Aa: complex,Na: nat] : aa_nat_complex(power_power_complex(aa_complex_complex(divide1717551699836669952omplex(one_one_complex),Aa)),Na) = aa_complex_complex(divide1717551699836669952omplex(one_one_complex),aa_nat_complex(power_power_complex(Aa),Na)) ).

% power_one_over
tff(fact_3480_power__one__over,axiom,
    ! [Aa: real,Na: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),Aa)),Na) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),Na)) ).

% power_one_over
tff(fact_3481_Euclidean__Division_Odiv__eq__0__iff,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na) = zero_zero_nat )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
        | ( Na = zero_zero_nat ) ) ) ).

% Euclidean_Division.div_eq_0_iff
tff(fact_3482_Suc__div__le__mono,axiom,
    ! [M: nat,Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,M)),Na)) ).

% Suc_div_le_mono
tff(fact_3483_atLeastatMost__psubset__iff,axiom,
    ! [Aa: set_int,Ba: set_int,C: set_int,D: set_int] :
      ( ord_less_set_set_int(set_or370866239135849197et_int(Aa,Ba),set_or370866239135849197et_int(C,D))
    <=> ( ( ~ aa_set_int_o(ord_less_eq_set_int(Aa),Ba)
          | ( aa_set_int_o(ord_less_eq_set_int(C),Aa)
            & aa_set_int_o(ord_less_eq_set_int(Ba),D)
            & ( aa_set_int_o(ord_less_set_int(C),Aa)
              | aa_set_int_o(ord_less_set_int(Ba),D) ) ) )
        & aa_set_int_o(ord_less_eq_set_int(C),D) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_3484_atLeastatMost__psubset__iff,axiom,
    ! [Aa: rat,Ba: rat,C: rat,D: rat] :
      ( ord_less_set_rat(set_or633870826150836451st_rat(Aa,Ba),set_or633870826150836451st_rat(C,D))
    <=> ( ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),Ba)
          | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),Aa)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ba),D)
            & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),Aa)
              | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ba),D) ) ) )
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),D) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_3485_atLeastatMost__psubset__iff,axiom,
    ! [Aa: num,Ba: num,C: num,D: num] :
      ( ord_less_set_num(set_or7049704709247886629st_num(Aa,Ba),set_or7049704709247886629st_num(C,D))
    <=> ( ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Aa),Ba)
          | ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),Aa)
            & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ba),D)
            & ( aa_num_o(aa_num_fun_num_o(ord_less_num,C),Aa)
              | aa_num_o(aa_num_fun_num_o(ord_less_num,Ba),D) ) ) )
        & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),D) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_3486_atLeastatMost__psubset__iff,axiom,
    ! [Aa: nat,Ba: nat,C: nat,D: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,set_or1269000886237332187st_nat(Aa,Ba)),set_or1269000886237332187st_nat(C,D))
    <=> ( ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),Ba)
          | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),Aa)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ba),D)
            & ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),Aa)
              | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ba),D) ) ) )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),D) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_3487_atLeastatMost__psubset__iff,axiom,
    ! [Aa: int,Ba: int,C: int,D: int] :
      ( aa_set_int_o(ord_less_set_int(set_or1266510415728281911st_int(Aa,Ba)),set_or1266510415728281911st_int(C,D))
    <=> ( ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),Ba)
          | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),Aa)
            & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ba),D)
            & ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),Aa)
              | aa_int_o(aa_int_fun_int_o(ord_less_int,Ba),D) ) ) )
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),D) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_3488_atLeastatMost__psubset__iff,axiom,
    ! [Aa: real,Ba: real,C: real,D: real] :
      ( ord_less_set_real(set_or1222579329274155063t_real(Aa,Ba),set_or1222579329274155063t_real(C,D))
    <=> ( ( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),Ba)
          | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),Aa)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ba),D)
            & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),Aa)
              | aa_real_o(aa_real_fun_real_o(ord_less_real,Ba),D) ) ) )
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),D) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_3489_frac__le,axiom,
    ! [Y: real,Xb: real,W2: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),W2)
         => ( 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,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Z)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Y),W2)) ) ) ) ) ).

% frac_le
tff(fact_3490_frac__le,axiom,
    ! [Y: rat,Xb: rat,W2: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),W2)
         => ( 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,aa_rat_rat(divide_divide_rat(Xb),Z)),aa_rat_rat(divide_divide_rat(Y),W2)) ) ) ) ) ).

% frac_le
tff(fact_3491_frac__less,axiom,
    ! [Xb: real,Y: real,W2: 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),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),W2)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,W2),Z)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Z)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Y),W2)) ) ) ) ) ).

% frac_less
tff(fact_3492_frac__less,axiom,
    ! [Xb: rat,Y: rat,W2: 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),Y)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),W2)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,W2),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(Y),W2)) ) ) ) ) ).

% frac_less
tff(fact_3493_frac__less2,axiom,
    ! [Xb: real,Y: real,W2: 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),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),W2)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,W2),Z)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Z)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Y),W2)) ) ) ) ) ).

% frac_less2
tff(fact_3494_frac__less2,axiom,
    ! [Xb: rat,Y: rat,W2: 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),Y)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),W2)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,W2),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(Y),W2)) ) ) ) ) ).

% frac_less2
tff(fact_3495_divide__le__cancel,axiom,
    ! [Aa: real,C: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Aa),C)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Ba),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,Aa),Ba) )
        & ( 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,Ba),Aa) ) ) ) ).

% divide_le_cancel
tff(fact_3496_divide__le__cancel,axiom,
    ! [Aa: rat,C: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(Aa),C)),aa_rat_rat(divide_divide_rat(Ba),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,Aa),Ba) )
        & ( 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,Ba),Aa) ) ) ) ).

% divide_le_cancel
tff(fact_3497_divide__nonneg__neg,axiom,
    ! [Xb: real,Y: 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,Y),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)),zero_zero_real) ) ) ).

% divide_nonneg_neg
tff(fact_3498_divide__nonneg__neg,axiom,
    ! [Xb: rat,Y: 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,Y),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(Xb),Y)),zero_zero_rat) ) ) ).

% divide_nonneg_neg
tff(fact_3499_divide__nonneg__pos,axiom,
    ! [Xb: real,Y: 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),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)) ) ) ).

% divide_nonneg_pos
tff(fact_3500_divide__nonneg__pos,axiom,
    ! [Xb: rat,Y: 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),Y)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(Xb),Y)) ) ) ).

% divide_nonneg_pos
tff(fact_3501_divide__nonpos__neg,axiom,
    ! [Xb: real,Y: 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,Y),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)) ) ) ).

% divide_nonpos_neg
tff(fact_3502_divide__nonpos__neg,axiom,
    ! [Xb: rat,Y: 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,Y),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),Y)) ) ) ).

% divide_nonpos_neg
tff(fact_3503_divide__nonpos__pos,axiom,
    ! [Xb: real,Y: 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),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)),zero_zero_real) ) ) ).

% divide_nonpos_pos
tff(fact_3504_divide__nonpos__pos,axiom,
    ! [Xb: rat,Y: 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),Y)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(Xb),Y)),zero_zero_rat) ) ) ).

% divide_nonpos_pos
tff(fact_3505_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
    ! [Aa: code_integer,Ba: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Aa)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Aa),Ba)
       => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Aa),Ba) = zero_z3403309356797280102nteger ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_less
tff(fact_3506_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
    ! [Aa: nat,Ba: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Aa)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),Ba)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Aa),Ba) = zero_zero_nat ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_less
tff(fact_3507_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
    ! [Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Aa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),Ba)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),Ba) = zero_zero_int ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_less
tff(fact_3508_div__positive,axiom,
    ! [Ba: code_integer,Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),Ba)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Ba),Aa)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Aa),Ba)) ) ) ).

% div_positive
tff(fact_3509_div__positive,axiom,
    ! [Ba: nat,Aa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Ba)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ba),Aa)
       => 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),Ba)) ) ) ).

% div_positive
tff(fact_3510_div__positive,axiom,
    ! [Ba: int,Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Ba)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ba),Aa)
       => 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,Aa),Ba)) ) ) ).

% div_positive
tff(fact_3511_divide__less__eq__1,axiom,
    ! [Ba: rat,Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(Ba),Aa)),one_one_rat)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Aa)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ba),Aa) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),Ba) )
        | ( Aa = zero_zero_rat ) ) ) ).

% divide_less_eq_1
tff(fact_3512_divide__less__eq__1,axiom,
    ! [Ba: real,Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Ba),Aa)),one_one_real)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Aa)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,Ba),Aa) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),Ba) )
        | ( Aa = zero_zero_real ) ) ) ).

% divide_less_eq_1
tff(fact_3513_less__divide__eq__1,axiom,
    ! [Ba: rat,Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_rat_rat(divide_divide_rat(Ba),Aa))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Aa)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),Ba) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ba),Aa) ) ) ) ).

% less_divide_eq_1
tff(fact_3514_less__divide__eq__1,axiom,
    ! [Ba: real,Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Ba),Aa))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Aa)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),Ba) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,Ba),Aa) ) ) ) ).

% less_divide_eq_1
tff(fact_3515_div__add__self1,axiom,
    ! [Ba: nat,Aa: nat] :
      ( ( Ba != 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,Ba),Aa)),Ba) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Aa),Ba)),one_one_nat) ) ) ).

% div_add_self1
tff(fact_3516_div__add__self1,axiom,
    ! [Ba: int,Aa: int] :
      ( ( Ba != 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,Ba),Aa)),Ba) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),Ba)),one_one_int) ) ) ).

% div_add_self1
tff(fact_3517_div__add__self1,axiom,
    ! [Ba: code_integer,Aa: code_integer] :
      ( ( Ba != zero_z3403309356797280102nteger )
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Ba),Aa)),Ba) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Aa),Ba)),one_one_Code_integer) ) ) ).

% div_add_self1
tff(fact_3518_div__add__self2,axiom,
    ! [Ba: nat,Aa: nat] :
      ( ( Ba != 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),Ba)),Ba) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Aa),Ba)),one_one_nat) ) ) ).

% div_add_self2
tff(fact_3519_div__add__self2,axiom,
    ! [Ba: int,Aa: int] :
      ( ( Ba != 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),Ba)),Ba) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),Ba)),one_one_int) ) ) ).

% div_add_self2
tff(fact_3520_div__add__self2,axiom,
    ! [Ba: code_integer,Aa: code_integer] :
      ( ( Ba != zero_z3403309356797280102nteger )
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Aa),Ba)),Ba) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Aa),Ba)),one_one_Code_integer) ) ) ).

% div_add_self2
tff(fact_3521_gt__half__sum,axiom,
    ! [Aa: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),Ba)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(Aa),Ba)),aa_rat_rat(plus_plus_rat(one_one_rat),one_one_rat))),Ba) ) ).

% gt_half_sum
tff(fact_3522_gt__half__sum,axiom,
    ! [Aa: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),Ba)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),Ba)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),one_one_real))),Ba) ) ).

% gt_half_sum
tff(fact_3523_less__half__sum,axiom,
    ! [Aa: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),Ba)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(Aa),Ba)),aa_rat_rat(plus_plus_rat(one_one_rat),one_one_rat))) ) ).

% less_half_sum
tff(fact_3524_less__half__sum,axiom,
    ! [Aa: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),Ba)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),Ba)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),one_one_real))) ) ).

% less_half_sum
tff(fact_3525_divide__eq__minus__1__iff,axiom,
    ! [Aa: complex,Ba: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(Aa),Ba) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) )
    <=> ( ( Ba != zero_zero_complex )
        & ( Aa = aa_complex_complex(uminus1482373934393186551omplex,Ba) ) ) ) ).

% divide_eq_minus_1_iff
tff(fact_3526_divide__eq__minus__1__iff,axiom,
    ! [Aa: real,Ba: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Aa),Ba) = aa_real_real(uminus_uminus_real,one_one_real) )
    <=> ( ( Ba != zero_zero_real )
        & ( Aa = aa_real_real(uminus_uminus_real,Ba) ) ) ) ).

% divide_eq_minus_1_iff
tff(fact_3527_divide__eq__minus__1__iff,axiom,
    ! [Aa: rat,Ba: rat] :
      ( ( aa_rat_rat(divide_divide_rat(Aa),Ba) = aa_rat_rat(uminus_uminus_rat,one_one_rat) )
    <=> ( ( Ba != zero_zero_rat )
        & ( Aa = aa_rat_rat(uminus_uminus_rat,Ba) ) ) ) ).

% divide_eq_minus_1_iff
tff(fact_3528_numeral__Bit0__div__2,axiom,
    ! [Na: num] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Na))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_num_nat(numeral_numeral_nat,Na) ).

% numeral_Bit0_div_2
tff(fact_3529_numeral__Bit0__div__2,axiom,
    ! [Na: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Na))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_num_int(numeral_numeral_int,Na) ).

% numeral_Bit0_div_2
tff(fact_3530_numeral__Bit0__div__2,axiom,
    ! [Na: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Na))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = aa_num_Code_integer(numera6620942414471956472nteger,Na) ).

% numeral_Bit0_div_2
tff(fact_3531_div__greater__zero__iff,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ) ).

% div_greater_zero_iff
tff(fact_3532_div__le__mono2,axiom,
    ! [M: nat,Na: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,K),Na)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,K),M)) ) ) ).

% div_le_mono2
tff(fact_3533_div__less__dividend,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na)),M) ) ) ).

% div_less_dividend
tff(fact_3534_div__eq__dividend__iff,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na) = M )
      <=> ( Na = one_one_nat ) ) ) ).

% div_eq_dividend_iff
tff(fact_3535_realpow__pos__nth2,axiom,
    ! [Aa: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Aa)
     => ? [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,Na)) = Aa ) ) ) ).

% realpow_pos_nth2
tff(fact_3536_subset__eq__atLeast0__atMost__finite,axiom,
    ! [N3: set_nat,Na: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,N3),set_or1269000886237332187st_nat(zero_zero_nat,Na))
     => aa_set_nat_o(finite_finite_nat,N3) ) ).

% subset_eq_atLeast0_atMost_finite
tff(fact_3537_divide__le__eq__1,axiom,
    ! [Ba: real,Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Ba),Aa)),one_one_real)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Aa)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ba),Aa) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),Ba) )
        | ( Aa = zero_zero_real ) ) ) ).

% divide_le_eq_1
tff(fact_3538_divide__le__eq__1,axiom,
    ! [Ba: rat,Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(Ba),Aa)),one_one_rat)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Aa)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ba),Aa) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),Ba) )
        | ( Aa = zero_zero_rat ) ) ) ).

% divide_le_eq_1
tff(fact_3539_le__divide__eq__1,axiom,
    ! [Ba: real,Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Ba),Aa))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Aa)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),Ba) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ba),Aa) ) ) ) ).

% le_divide_eq_1
tff(fact_3540_le__divide__eq__1,axiom,
    ! [Ba: rat,Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_rat_rat(divide_divide_rat(Ba),Aa))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Aa)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),Ba) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ba),Aa) ) ) ) ).

% le_divide_eq_1
tff(fact_3541_power__diff,axiom,
    ! [Aa: rat,Na: nat,M: nat] :
      ( ( Aa != zero_zero_rat )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
       => ( aa_nat_rat(power_power_rat(Aa),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na)) = aa_rat_rat(divide_divide_rat(aa_nat_rat(power_power_rat(Aa),M)),aa_nat_rat(power_power_rat(Aa),Na)) ) ) ) ).

% power_diff
tff(fact_3542_power__diff,axiom,
    ! [Aa: complex,Na: nat,M: nat] :
      ( ( Aa != zero_zero_complex )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
       => ( aa_nat_complex(power_power_complex(Aa),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na)) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(power_power_complex(Aa),M)),aa_nat_complex(power_power_complex(Aa),Na)) ) ) ) ).

% power_diff
tff(fact_3543_power__diff,axiom,
    ! [Aa: nat,Na: nat,M: nat] :
      ( ( Aa != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Aa),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Aa),Na)) ) ) ) ).

% power_diff
tff(fact_3544_power__diff,axiom,
    ! [Aa: int,Na: nat,M: nat] :
      ( ( Aa != zero_zero_int )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
       => ( aa_nat_int(power_power_int(Aa),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(power_power_int(Aa),M)),aa_nat_int(power_power_int(Aa),Na)) ) ) ) ).

% power_diff
tff(fact_3545_power__diff,axiom,
    ! [Aa: real,Na: nat,M: nat] :
      ( ( Aa != zero_zero_real )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),Na)) ) ) ) ).

% power_diff
tff(fact_3546_power__diff,axiom,
    ! [Aa: code_integer,Na: nat,M: nat] :
      ( ( Aa != zero_z3403309356797280102nteger )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
       => ( aa_nat_Code_integer(power_8256067586552552935nteger(Aa),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_nat_Code_integer(power_8256067586552552935nteger(Aa),M)),aa_nat_Code_integer(power_8256067586552552935nteger(Aa),Na)) ) ) ) ).

% power_diff
tff(fact_3547_div__geq,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na)),Na)) ) ) ) ).

% div_geq
tff(fact_3548_div__if,axiom,
    ! [M: nat,Na: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na) = $ite(
        ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
        | ( Na = zero_zero_nat ) ),
        zero_zero_nat,
        aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na)),Na)) ) ).

% div_if
tff(fact_3549_fact__div__fact__le__pow,axiom,
    ! [R2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,R2),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Na)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),R2)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Na),R2)) ) ).

% fact_div_fact_le_pow
tff(fact_3550_half__gt__zero,axiom,
    ! [Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Aa)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(Aa),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))) ) ).

% half_gt_zero
tff(fact_3551_half__gt__zero,axiom,
    ! [Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Aa)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Aa),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ).

% half_gt_zero
tff(fact_3552_half__gt__zero__iff,axiom,
    ! [Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(Aa),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),Aa) ) ).

% half_gt_zero_iff
tff(fact_3553_half__gt__zero__iff,axiom,
    ! [Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Aa),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) ) ).

% half_gt_zero_iff
tff(fact_3554_field__less__half__sum,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
     => 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),Y)),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))) ) ).

% field_less_half_sum
tff(fact_3555_field__less__half__sum,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ).

% field_less_half_sum
tff(fact_3556_power__diff__power__eq,axiom,
    ! [Aa: nat,M: nat,Na: nat] :
      ( ( Aa != 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,Aa),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Aa),Na)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),M)))) ) ) ).

% power_diff_power_eq
tff(fact_3557_power__diff__power__eq,axiom,
    ! [Aa: int,M: nat,Na: nat] :
      ( ( Aa != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(power_power_int(Aa),M)),aa_nat_int(power_power_int(Aa),Na)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M),aa_nat_int(power_power_int(Aa),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na)),aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_nat_int(power_power_int(Aa),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),M)))) ) ) ).

% power_diff_power_eq
tff(fact_3558_power__diff__power__eq,axiom,
    ! [Aa: code_integer,M: nat,Na: nat] :
      ( ( Aa != zero_z3403309356797280102nteger )
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_nat_Code_integer(power_8256067586552552935nteger(Aa),M)),aa_nat_Code_integer(power_8256067586552552935nteger(Aa),Na)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M),aa_nat_Code_integer(power_8256067586552552935nteger(Aa),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,one_one_Code_integer),aa_nat_Code_integer(power_8256067586552552935nteger(Aa),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),M)))) ) ) ).

% power_diff_power_eq
tff(fact_3559_le__div__geq,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na)),Na)) ) ) ) ).

% le_div_geq
tff(fact_3560_Suc__n__div__2__gt__zero,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,Na)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% Suc_n_div_2_gt_zero
tff(fact_3561_div__2__gt__zero,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% div_2_gt_zero
tff(fact_3562_realpow__pos__nth,axiom,
    ! [Na: nat,Aa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Aa)
       => ? [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),Na) = Aa ) ) ) ) ).

% realpow_pos_nth
tff(fact_3563_realpow__pos__nth__unique,axiom,
    ! [Na: nat,Aa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Aa)
       => ? [X: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
            & ( aa_nat_real(aa_real_fun_nat_real(power_power_real,X),Na) = Aa )
            & ! [Y4: real] :
                ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y4)
                  & ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Y4),Na) = Aa ) )
               => ( Y4 = X ) ) ) ) ) ).

% realpow_pos_nth_unique
tff(fact_3564_pred__list__to__short,axiom,
    ! [Deg: nat,Xb: nat,Ma2: nat,TreeList: list_VEBT_VEBT,Mi2: nat,Summary: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Ma2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_s6755466524823107622T_VEBT(TreeList)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
         => ( vEBT_vebt_pred(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Deg,TreeList,Summary),Xb) = none_nat ) ) ) ) ).

% pred_list_to_short
tff(fact_3565_high__bound__aux,axiom,
    ! [Ma2: nat,Na: nat,M: nat] :
      ( 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))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),M)))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma2),Na)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) ) ).

% high_bound_aux
tff(fact_3566_high__def,axiom,
    ! [Xb: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),Na) = 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))),Na)) ).

% high_def
tff(fact_3567_even__succ__div__exp,axiom,
    ! [Aa: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Aa)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),Aa)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) ) ) ) ).

% even_succ_div_exp
tff(fact_3568_even__succ__div__exp,axiom,
    ! [Aa: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Aa)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),Aa)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)) ) ) ) ).

% even_succ_div_exp
tff(fact_3569_even__succ__div__exp,axiom,
    ! [Aa: code_integer,Na: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Aa)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,one_one_Code_integer),Aa)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Na)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Aa),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Na)) ) ) ) ).

% even_succ_div_exp
tff(fact_3570_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_3571_nat__dvd__1__iff__1,axiom,
    ! [M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),one_one_nat)
    <=> ( M = one_one_nat ) ) ).

% nat_dvd_1_iff_1
tff(fact_3572_dvd__0__right,axiom,
    ! [Aa: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),zero_z3403309356797280102nteger) ).

% dvd_0_right
tff(fact_3573_dvd__0__right,axiom,
    ! [Aa: real] : aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Aa),zero_zero_real) ).

% dvd_0_right
tff(fact_3574_dvd__0__right,axiom,
    ! [Aa: rat] : aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Aa),zero_zero_rat) ).

% dvd_0_right
tff(fact_3575_dvd__0__right,axiom,
    ! [Aa: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),zero_zero_nat) ).

% dvd_0_right
tff(fact_3576_dvd__0__right,axiom,
    ! [Aa: int] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),zero_zero_int) ).

% dvd_0_right
tff(fact_3577_dvd__0__left__iff,axiom,
    ! [Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,zero_z3403309356797280102nteger),Aa)
    <=> ( Aa = zero_z3403309356797280102nteger ) ) ).

% dvd_0_left_iff
tff(fact_3578_dvd__0__left__iff,axiom,
    ! [Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,zero_zero_real),Aa)
    <=> ( Aa = zero_zero_real ) ) ).

% dvd_0_left_iff
tff(fact_3579_dvd__0__left__iff,axiom,
    ! [Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,zero_zero_rat),Aa)
    <=> ( Aa = zero_zero_rat ) ) ).

% dvd_0_left_iff
tff(fact_3580_dvd__0__left__iff,axiom,
    ! [Aa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,zero_zero_nat),Aa)
    <=> ( Aa = zero_zero_nat ) ) ).

% dvd_0_left_iff
tff(fact_3581_dvd__0__left__iff,axiom,
    ! [Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,zero_zero_int),Aa)
    <=> ( Aa = zero_zero_int ) ) ).

% dvd_0_left_iff
tff(fact_3582_dvd__add__triv__right__iff,axiom,
    ! [Aa: code_integer,Ba: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Ba),Aa))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),Ba) ) ).

% dvd_add_triv_right_iff
tff(fact_3583_dvd__add__triv__right__iff,axiom,
    ! [Aa: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Aa),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Ba),Aa))
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Aa),Ba) ) ).

% dvd_add_triv_right_iff
tff(fact_3584_dvd__add__triv__right__iff,axiom,
    ! [Aa: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Aa),aa_rat_rat(plus_plus_rat(Ba),Aa))
    <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Aa),Ba) ) ).

% dvd_add_triv_right_iff
tff(fact_3585_dvd__add__triv__right__iff,axiom,
    ! [Aa: nat,Ba: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ba),Aa))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),Ba) ) ).

% dvd_add_triv_right_iff
tff(fact_3586_dvd__add__triv__right__iff,axiom,
    ! [Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),aa_int_int(aa_int_fun_int_int(plus_plus_int,Ba),Aa))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),Ba) ) ).

% dvd_add_triv_right_iff
tff(fact_3587_dvd__add__triv__left__iff,axiom,
    ! [Aa: code_integer,Ba: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Aa),Ba))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),Ba) ) ).

% dvd_add_triv_left_iff
tff(fact_3588_dvd__add__triv__left__iff,axiom,
    ! [Aa: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Aa),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),Ba))
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Aa),Ba) ) ).

% dvd_add_triv_left_iff
tff(fact_3589_dvd__add__triv__left__iff,axiom,
    ! [Aa: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Aa),aa_rat_rat(plus_plus_rat(Aa),Ba))
    <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Aa),Ba) ) ).

% dvd_add_triv_left_iff
tff(fact_3590_dvd__add__triv__left__iff,axiom,
    ! [Aa: nat,Ba: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),Ba))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),Ba) ) ).

% dvd_add_triv_left_iff
tff(fact_3591_dvd__add__triv__left__iff,axiom,
    ! [Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),Ba))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),Ba) ) ).

% dvd_add_triv_left_iff
tff(fact_3592_dvd__1__iff__1,axiom,
    ! [M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),aa_nat_nat(suc,zero_zero_nat))
    <=> ( M = aa_nat_nat(suc,zero_zero_nat) ) ) ).

% dvd_1_iff_1
tff(fact_3593_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_3594_div__dvd__div,axiom,
    ! [Aa: nat,Ba: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),Ba)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),C)
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Ba),Aa)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),Aa))
        <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Ba),C) ) ) ) ).

% div_dvd_div
tff(fact_3595_div__dvd__div,axiom,
    ! [Aa: int,Ba: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),Ba)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),C)
       => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,Ba),Aa)),aa_int_int(aa_int_fun_int_int(divide_divide_int,C),Aa))
        <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Ba),C) ) ) ) ).

% div_dvd_div
tff(fact_3596_div__dvd__div,axiom,
    ! [Aa: code_integer,Ba: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),Ba)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),C)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Ba),Aa)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,C),Aa))
        <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Ba),C) ) ) ) ).

% div_dvd_div
tff(fact_3597_minus__dvd__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_real_real(uminus_uminus_real,Xb)),Y)
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Xb),Y) ) ).

% minus_dvd_iff
tff(fact_3598_minus__dvd__iff,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(uminus_uminus_int,Xb)),Y)
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xb),Y) ) ).

% minus_dvd_iff
tff(fact_3599_minus__dvd__iff,axiom,
    ! [Xb: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Xb)),Y)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Xb),Y) ) ).

% minus_dvd_iff
tff(fact_3600_minus__dvd__iff,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(uminus_uminus_rat,Xb)),Y)
    <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Xb),Y) ) ).

% minus_dvd_iff
tff(fact_3601_dvd__minus__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Xb),aa_real_real(uminus_uminus_real,Y))
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Xb),Y) ) ).

% dvd_minus_iff
tff(fact_3602_dvd__minus__iff,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xb),aa_int_int(uminus_uminus_int,Y))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xb),Y) ) ).

% dvd_minus_iff
tff(fact_3603_dvd__minus__iff,axiom,
    ! [Xb: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Xb),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Y))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Xb),Y) ) ).

% dvd_minus_iff
tff(fact_3604_dvd__minus__iff,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Xb),aa_rat_rat(uminus_uminus_rat,Y))
    <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Xb),Y) ) ).

% dvd_minus_iff
tff(fact_3605_unit__div__1__div__1,axiom,
    ! [Aa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),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),Aa)) = Aa ) ) ).

% unit_div_1_div_1
tff(fact_3606_unit__div__1__div__1,axiom,
    ! [Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),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),Aa)) = Aa ) ) ).

% unit_div_1_div_1
tff(fact_3607_unit__div__1__div__1,axiom,
    ! [Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),one_one_Code_integer)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,one_one_Code_integer),Aa)) = Aa ) ) ).

% unit_div_1_div_1
tff(fact_3608_unit__div__1__unit,axiom,
    ! [Aa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),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),Aa)),one_one_nat) ) ).

% unit_div_1_unit
tff(fact_3609_unit__div__1__unit,axiom,
    ! [Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),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),Aa)),one_one_int) ) ).

% unit_div_1_unit
tff(fact_3610_unit__div__1__unit,axiom,
    ! [Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),one_one_Code_integer)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,one_one_Code_integer),Aa)),one_one_Code_integer) ) ).

% unit_div_1_unit
tff(fact_3611_unit__div,axiom,
    ! [Aa: nat,Ba: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Ba),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,Aa),Ba)),one_one_nat) ) ) ).

% unit_div
tff(fact_3612_unit__div,axiom,
    ! [Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Ba),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,Aa),Ba)),one_one_int) ) ) ).

% unit_div
tff(fact_3613_unit__div,axiom,
    ! [Aa: code_integer,Ba: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Ba),one_one_Code_integer)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Aa),Ba)),one_one_Code_integer) ) ) ).

% unit_div
tff(fact_3614_div__add,axiom,
    ! [C: nat,Aa: nat,Ba: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),Aa)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),Ba)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),Ba)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Aa),C)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Ba),C)) ) ) ) ).

% div_add
tff(fact_3615_div__add,axiom,
    ! [C: int,Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),Aa)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),Ba)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),Ba)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),C)),aa_int_int(aa_int_fun_int_int(divide_divide_int,Ba),C)) ) ) ) ).

% div_add
tff(fact_3616_div__add,axiom,
    ! [C: code_integer,Aa: code_integer,Ba: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),Aa)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),Ba)
       => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Aa),Ba)),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Aa),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Ba),C)) ) ) ) ).

% div_add
tff(fact_3617_div__diff,axiom,
    ! [C: int,Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),Aa)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),Ba)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Aa),Ba)),C) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),C)),aa_int_int(aa_int_fun_int_int(divide_divide_int,Ba),C)) ) ) ) ).

% div_diff
tff(fact_3618_div__diff,axiom,
    ! [C: code_integer,Aa: code_integer,Ba: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),Aa)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),Ba)
       => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,Aa),Ba)),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Aa),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Ba),C)) ) ) ) ).

% div_diff
tff(fact_3619_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_3620_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_3621_even__Suc,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,Na))
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na) ) ).

% even_Suc
tff(fact_3622_even__Suc__Suc__iff,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Na)))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na) ) ).

% even_Suc_Suc_iff
tff(fact_3623_pow__divides__pow__iff,axiom,
    ! [Na: nat,Aa: nat,Ba: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Aa),Na)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Ba),Na))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),Ba) ) ) ).

% pow_divides_pow_iff
tff(fact_3624_pow__divides__pow__iff,axiom,
    ! [Na: nat,Aa: int,Ba: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(Aa),Na)),aa_nat_int(power_power_int(Ba),Na))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),Ba) ) ) ).

% pow_divides_pow_iff
tff(fact_3625_odd__Suc__div__two,axiom,
    ! [Na: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,Na)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ).

% odd_Suc_div_two
tff(fact_3626_even__Suc__div__two,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,Na)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ) ) ).

% even_Suc_div_two
tff(fact_3627_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_3628_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_3629_set__decode__0,axiom,
    ! [Xb: nat] :
      ( aa_set_nat_o(member_nat(zero_zero_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_3630_zero__le__power__eq__numeral,axiom,
    ! [Aa: real,W2: 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,Aa),aa_num_nat(numeral_numeral_nat,W2)))
    <=> ( 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,W2))
        | ( ~ 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,W2))
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Aa) ) ) ) ).

% zero_le_power_eq_numeral
tff(fact_3631_zero__le__power__eq__numeral,axiom,
    ! [Aa: rat,W2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(Aa),aa_num_nat(numeral_numeral_nat,W2)))
    <=> ( 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,W2))
        | ( ~ 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,W2))
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Aa) ) ) ) ).

% zero_le_power_eq_numeral
tff(fact_3632_zero__le__power__eq__numeral,axiom,
    ! [Aa: int,W2: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(Aa),aa_num_nat(numeral_numeral_nat,W2)))
    <=> ( 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,W2))
        | ( ~ 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,W2))
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Aa) ) ) ) ).

% zero_le_power_eq_numeral
tff(fact_3633_power__less__zero__eq,axiom,
    ! [Aa: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),Na)),zero_zero_real)
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),zero_zero_real) ) ) ).

% power_less_zero_eq
tff(fact_3634_power__less__zero__eq,axiom,
    ! [Aa: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(Aa),Na)),zero_zero_rat)
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),zero_zero_rat) ) ) ).

% power_less_zero_eq
tff(fact_3635_power__less__zero__eq,axiom,
    ! [Aa: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(Aa),Na)),zero_zero_int)
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),zero_zero_int) ) ) ).

% power_less_zero_eq
tff(fact_3636_power__less__zero__eq__numeral,axiom,
    ! [Aa: real,W2: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),aa_num_nat(numeral_numeral_nat,W2))),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,W2))
        & aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),zero_zero_real) ) ) ).

% power_less_zero_eq_numeral
tff(fact_3637_power__less__zero__eq__numeral,axiom,
    ! [Aa: rat,W2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(Aa),aa_num_nat(numeral_numeral_nat,W2))),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,W2))
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),zero_zero_rat) ) ) ).

% power_less_zero_eq_numeral
tff(fact_3638_power__less__zero__eq__numeral,axiom,
    ! [Aa: int,W2: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(Aa),aa_num_nat(numeral_numeral_nat,W2))),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,W2))
        & aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),zero_zero_int) ) ) ).

% power_less_zero_eq_numeral
tff(fact_3639_even__plus__one__iff,axiom,
    ! [Aa: 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,Aa),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))),Aa) ) ).

% even_plus_one_iff
tff(fact_3640_even__plus__one__iff,axiom,
    ! [Aa: 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),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))),Aa) ) ).

% even_plus_one_iff
tff(fact_3641_even__plus__one__iff,axiom,
    ! [Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Aa),one_one_Code_integer))
    <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Aa) ) ).

% even_plus_one_iff
tff(fact_3642_neg__one__odd__power,axiom,
    ! [Na: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Na) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ) ) ).

% neg_one_odd_power
tff(fact_3643_neg__one__odd__power,axiom,
    ! [Na: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Na) = aa_real_real(uminus_uminus_real,one_one_real) ) ) ).

% neg_one_odd_power
tff(fact_3644_neg__one__odd__power,axiom,
    ! [Na: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),Na) = aa_int_int(uminus_uminus_int,one_one_int) ) ) ).

% neg_one_odd_power
tff(fact_3645_neg__one__odd__power,axiom,
    ! [Na: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Na) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ) ) ).

% neg_one_odd_power
tff(fact_3646_neg__one__odd__power,axiom,
    ! [Na: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Na) = aa_rat_rat(uminus_uminus_rat,one_one_rat) ) ) ).

% neg_one_odd_power
tff(fact_3647_neg__one__even__power,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Na) = one_one_complex ) ) ).

% neg_one_even_power
tff(fact_3648_neg__one__even__power,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Na) = one_one_real ) ) ).

% neg_one_even_power
tff(fact_3649_neg__one__even__power,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),Na) = one_one_int ) ) ).

% neg_one_even_power
tff(fact_3650_neg__one__even__power,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Na) = one_one_Code_integer ) ) ).

% neg_one_even_power
tff(fact_3651_neg__one__even__power,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Na) = one_one_rat ) ) ).

% neg_one_even_power
tff(fact_3652_odd__Suc__minus__one,axiom,
    ! [Na: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(suc,zero_zero_nat))) = Na ) ) ).

% odd_Suc_minus_one
tff(fact_3653_even__diff__nat,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
        | aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na)) ) ) ).

% even_diff_nat
tff(fact_3654_zero__less__power__eq__numeral,axiom,
    ! [Aa: real,W2: 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,Aa),aa_num_nat(numeral_numeral_nat,W2)))
    <=> ( ( aa_num_nat(numeral_numeral_nat,W2) = 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,W2))
          & ( Aa != 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,W2))
          & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Aa) ) ) ) ).

% zero_less_power_eq_numeral
tff(fact_3655_zero__less__power__eq__numeral,axiom,
    ! [Aa: rat,W2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(Aa),aa_num_nat(numeral_numeral_nat,W2)))
    <=> ( ( aa_num_nat(numeral_numeral_nat,W2) = 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,W2))
          & ( Aa != 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,W2))
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Aa) ) ) ) ).

% zero_less_power_eq_numeral
tff(fact_3656_zero__less__power__eq__numeral,axiom,
    ! [Aa: int,W2: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(Aa),aa_num_nat(numeral_numeral_nat,W2)))
    <=> ( ( aa_num_nat(numeral_numeral_nat,W2) = 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,W2))
          & ( Aa != 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,W2))
          & aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Aa) ) ) ) ).

% zero_less_power_eq_numeral
tff(fact_3657_odd__succ__div__two,axiom,
    ! [Aa: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Aa)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),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,Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat) ) ) ).

% odd_succ_div_two
tff(fact_3658_odd__succ__div__two,axiom,
    ! [Aa: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Aa)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),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,Aa),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ) ) ).

% odd_succ_div_two
tff(fact_3659_odd__succ__div__two,axiom,
    ! [Aa: code_integer] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Aa)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Aa),one_one_Code_integer)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Aa),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))),one_one_Code_integer) ) ) ).

% odd_succ_div_two
tff(fact_3660_even__succ__div__two,axiom,
    ! [Aa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Aa)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),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,Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ) ) ).

% even_succ_div_two
tff(fact_3661_even__succ__div__two,axiom,
    ! [Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Aa)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),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,Aa),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ) ) ).

% even_succ_div_two
tff(fact_3662_even__succ__div__two,axiom,
    ! [Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Aa)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Aa),one_one_Code_integer)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Aa),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) ) ) ).

% even_succ_div_two
tff(fact_3663_even__succ__div__2,axiom,
    ! [Aa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Aa)
     => ( 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),Aa)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ) ) ).

% even_succ_div_2
tff(fact_3664_even__succ__div__2,axiom,
    ! [Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Aa)
     => ( 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)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ) ) ).

% even_succ_div_2
tff(fact_3665_even__succ__div__2,axiom,
    ! [Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Aa)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,one_one_Code_integer),Aa)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Aa),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) ) ) ).

% even_succ_div_2
tff(fact_3666_even__power,axiom,
    ! [Aa: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Aa),Na))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Aa)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ) ).

% even_power
tff(fact_3667_even__power,axiom,
    ! [Aa: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_int(power_power_int(Aa),Na))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Aa)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ) ).

% even_power
tff(fact_3668_even__power,axiom,
    ! [Aa: code_integer,Na: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_nat_Code_integer(power_8256067586552552935nteger(Aa),Na))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Aa)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ) ).

% even_power
tff(fact_3669_power__le__zero__eq__numeral,axiom,
    ! [Aa: real,W2: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),aa_num_nat(numeral_numeral_nat,W2))),zero_zero_real)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,W2))
        & ( ( ~ 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,W2))
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),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,W2))
            & ( Aa = zero_zero_real ) ) ) ) ) ).

% power_le_zero_eq_numeral
tff(fact_3670_power__le__zero__eq__numeral,axiom,
    ! [Aa: rat,W2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(Aa),aa_num_nat(numeral_numeral_nat,W2))),zero_zero_rat)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,W2))
        & ( ( ~ 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,W2))
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),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,W2))
            & ( Aa = zero_zero_rat ) ) ) ) ) ).

% power_le_zero_eq_numeral
tff(fact_3671_power__le__zero__eq__numeral,axiom,
    ! [Aa: int,W2: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(Aa),aa_num_nat(numeral_numeral_nat,W2))),zero_zero_int)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,W2))
        & ( ( ~ 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,W2))
            & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),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,W2))
            & ( Aa = zero_zero_int ) ) ) ) ) ).

% power_le_zero_eq_numeral
tff(fact_3672_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)),one_one_nat))
    <=> ( Na = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
tff(fact_3673_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)),one_one_int))
    <=> ( Na = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
tff(fact_3674_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [Na: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Na)),one_one_Code_integer))
    <=> ( Na = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
tff(fact_3675_bot__enat__def,axiom,
    bot_bo4199563552545308370d_enat = zero_z5237406670263579293d_enat ).

% bot_enat_def
tff(fact_3676_dvd__trans,axiom,
    ! [Aa: nat,Ba: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),Ba)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Ba),C)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),C) ) ) ).

% dvd_trans
tff(fact_3677_dvd__trans,axiom,
    ! [Aa: int,Ba: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),Ba)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Ba),C)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),C) ) ) ).

% dvd_trans
tff(fact_3678_dvd__trans,axiom,
    ! [Aa: code_integer,Ba: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),Ba)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Ba),C)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),C) ) ) ).

% dvd_trans
tff(fact_3679_dvd__refl,axiom,
    ! [Aa: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),Aa) ).

% dvd_refl
tff(fact_3680_dvd__refl,axiom,
    ! [Aa: int] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),Aa) ).

% dvd_refl
tff(fact_3681_dvd__refl,axiom,
    ! [Aa: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),Aa) ).

% dvd_refl
tff(fact_3682_dvd__0__left,axiom,
    ! [Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,zero_z3403309356797280102nteger),Aa)
     => ( Aa = zero_z3403309356797280102nteger ) ) ).

% dvd_0_left
tff(fact_3683_dvd__0__left,axiom,
    ! [Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,zero_zero_real),Aa)
     => ( Aa = zero_zero_real ) ) ).

% dvd_0_left
tff(fact_3684_dvd__0__left,axiom,
    ! [Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,zero_zero_rat),Aa)
     => ( Aa = zero_zero_rat ) ) ).

% dvd_0_left
tff(fact_3685_dvd__0__left,axiom,
    ! [Aa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,zero_zero_nat),Aa)
     => ( Aa = zero_zero_nat ) ) ).

% dvd_0_left
tff(fact_3686_dvd__0__left,axiom,
    ! [Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,zero_zero_int),Aa)
     => ( Aa = zero_zero_int ) ) ).

% dvd_0_left
tff(fact_3687_dvd__field__iff,axiom,
    ! [Aa: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Aa),Ba)
    <=> ( ( Aa = zero_zero_real )
       => ( Ba = zero_zero_real ) ) ) ).

% dvd_field_iff
tff(fact_3688_dvd__field__iff,axiom,
    ! [Aa: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Aa),Ba)
    <=> ( ( Aa = zero_zero_rat )
       => ( Ba = zero_zero_rat ) ) ) ).

% dvd_field_iff
tff(fact_3689_dvd__unit__imp__unit,axiom,
    ! [Aa: code_integer,Ba: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),Ba)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Ba),one_one_Code_integer)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),one_one_Code_integer) ) ) ).

% dvd_unit_imp_unit
tff(fact_3690_dvd__unit__imp__unit,axiom,
    ! [Aa: nat,Ba: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),Ba)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Ba),one_one_nat)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),one_one_nat) ) ) ).

% dvd_unit_imp_unit
tff(fact_3691_dvd__unit__imp__unit,axiom,
    ! [Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),Ba)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Ba),one_one_int)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),one_one_int) ) ) ).

% dvd_unit_imp_unit
tff(fact_3692_unit__imp__dvd,axiom,
    ! [Ba: code_integer,Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Ba),one_one_Code_integer)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Ba),Aa) ) ).

% unit_imp_dvd
tff(fact_3693_unit__imp__dvd,axiom,
    ! [Ba: nat,Aa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Ba),one_one_nat)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Ba),Aa) ) ).

% unit_imp_dvd
tff(fact_3694_unit__imp__dvd,axiom,
    ! [Ba: int,Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Ba),one_one_int)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Ba),Aa) ) ).

% unit_imp_dvd
tff(fact_3695_one__dvd,axiom,
    ! [Aa: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,one_one_Code_integer),Aa) ).

% one_dvd
tff(fact_3696_one__dvd,axiom,
    ! [Aa: complex] : aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,one_one_complex),Aa) ).

% one_dvd
tff(fact_3697_one__dvd,axiom,
    ! [Aa: real] : aa_real_o(aa_real_fun_real_o(dvd_dvd_real,one_one_real),Aa) ).

% one_dvd
tff(fact_3698_one__dvd,axiom,
    ! [Aa: rat] : aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,one_one_rat),Aa) ).

% one_dvd
tff(fact_3699_one__dvd,axiom,
    ! [Aa: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,one_one_nat),Aa) ).

% one_dvd
tff(fact_3700_one__dvd,axiom,
    ! [Aa: int] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,one_one_int),Aa) ).

% one_dvd
tff(fact_3701_dvd__add__right__iff,axiom,
    ! [Aa: code_integer,Ba: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),Ba)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Ba),C))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),C) ) ) ).

% dvd_add_right_iff
tff(fact_3702_dvd__add__right__iff,axiom,
    ! [Aa: real,Ba: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Aa),Ba)
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Aa),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Ba),C))
      <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Aa),C) ) ) ).

% dvd_add_right_iff
tff(fact_3703_dvd__add__right__iff,axiom,
    ! [Aa: rat,Ba: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Aa),Ba)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Aa),aa_rat_rat(plus_plus_rat(Ba),C))
      <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Aa),C) ) ) ).

% dvd_add_right_iff
tff(fact_3704_dvd__add__right__iff,axiom,
    ! [Aa: nat,Ba: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),Ba)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ba),C))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),C) ) ) ).

% dvd_add_right_iff
tff(fact_3705_dvd__add__right__iff,axiom,
    ! [Aa: int,Ba: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),Ba)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),aa_int_int(aa_int_fun_int_int(plus_plus_int,Ba),C))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),C) ) ) ).

% dvd_add_right_iff
tff(fact_3706_dvd__add__left__iff,axiom,
    ! [Aa: code_integer,C: code_integer,Ba: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),C)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Ba),C))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),Ba) ) ) ).

% dvd_add_left_iff
tff(fact_3707_dvd__add__left__iff,axiom,
    ! [Aa: real,C: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Aa),C)
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Aa),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Ba),C))
      <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Aa),Ba) ) ) ).

% dvd_add_left_iff
tff(fact_3708_dvd__add__left__iff,axiom,
    ! [Aa: rat,C: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Aa),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Aa),aa_rat_rat(plus_plus_rat(Ba),C))
      <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Aa),Ba) ) ) ).

% dvd_add_left_iff
tff(fact_3709_dvd__add__left__iff,axiom,
    ! [Aa: nat,C: nat,Ba: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),C)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ba),C))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),Ba) ) ) ).

% dvd_add_left_iff
tff(fact_3710_dvd__add__left__iff,axiom,
    ! [Aa: int,C: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),C)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),aa_int_int(aa_int_fun_int_int(plus_plus_int,Ba),C))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),Ba) ) ) ).

% dvd_add_left_iff
tff(fact_3711_dvd__add,axiom,
    ! [Aa: code_integer,Ba: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),Ba)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),C)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Ba),C)) ) ) ).

% dvd_add
tff(fact_3712_dvd__add,axiom,
    ! [Aa: real,Ba: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Aa),Ba)
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Aa),C)
       => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Aa),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Ba),C)) ) ) ).

% dvd_add
tff(fact_3713_dvd__add,axiom,
    ! [Aa: rat,Ba: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Aa),Ba)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Aa),C)
       => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Aa),aa_rat_rat(plus_plus_rat(Ba),C)) ) ) ).

% dvd_add
tff(fact_3714_dvd__add,axiom,
    ! [Aa: nat,Ba: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),Ba)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),C)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ba),C)) ) ) ).

% dvd_add
tff(fact_3715_dvd__add,axiom,
    ! [Aa: int,Ba: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),Ba)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),C)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),aa_int_int(aa_int_fun_int_int(plus_plus_int,Ba),C)) ) ) ).

% dvd_add
tff(fact_3716_dvd__diff__commute,axiom,
    ! [Aa: code_integer,C: code_integer,Ba: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,C),Ba))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,Ba),C)) ) ).

% dvd_diff_commute
tff(fact_3717_dvd__diff__commute,axiom,
    ! [Aa: int,C: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),Ba))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),aa_int_int(aa_int_fun_int_int(minus_minus_int,Ba),C)) ) ).

% dvd_diff_commute
tff(fact_3718_dvd__diff,axiom,
    ! [Xb: code_integer,Y: code_integer,Z: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Xb),Y)
     => ( 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),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,Y),Z)) ) ) ).

% dvd_diff
tff(fact_3719_dvd__diff,axiom,
    ! [Xb: real,Y: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Xb),Y)
     => ( 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(aa_rea7340605660300375871l_real(minus_minus_real,Y),Z)) ) ) ).

% dvd_diff
tff(fact_3720_dvd__diff,axiom,
    ! [Xb: rat,Y: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Xb),Y)
     => ( 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(Y),Z)) ) ) ).

% dvd_diff
tff(fact_3721_dvd__diff,axiom,
    ! [Xb: int,Y: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xb),Y)
     => ( 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,Y),Z)) ) ) ).

% dvd_diff
tff(fact_3722_div__div__div__same,axiom,
    ! [D: nat,Ba: nat,Aa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),Ba)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Ba),Aa)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Aa),D)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Ba),D)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Aa),Ba) ) ) ) ).

% div_div_div_same
tff(fact_3723_div__div__div__same,axiom,
    ! [D: int,Ba: int,Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),Ba)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Ba),Aa)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),D)),aa_int_int(aa_int_fun_int_int(divide_divide_int,Ba),D)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),Ba) ) ) ) ).

% div_div_div_same
tff(fact_3724_div__div__div__same,axiom,
    ! [D: code_integer,Ba: code_integer,Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),Ba)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Ba),Aa)
       => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Aa),D)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Ba),D)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Aa),Ba) ) ) ) ).

% div_div_div_same
tff(fact_3725_dvd__div__eq__cancel,axiom,
    ! [Aa: nat,C: nat,Ba: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Aa),C) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Ba),C) )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),Aa)
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),Ba)
         => ( Aa = Ba ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_3726_dvd__div__eq__cancel,axiom,
    ! [Aa: int,C: int,Ba: int] :
      ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),C) = aa_int_int(aa_int_fun_int_int(divide_divide_int,Ba),C) )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),Aa)
       => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),Ba)
         => ( Aa = Ba ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_3727_dvd__div__eq__cancel,axiom,
    ! [Aa: real,C: real,Ba: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Aa),C) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Ba),C) )
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,C),Aa)
       => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,C),Ba)
         => ( Aa = Ba ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_3728_dvd__div__eq__cancel,axiom,
    ! [Aa: code_integer,C: code_integer,Ba: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Aa),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Ba),C) )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),Aa)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),Ba)
         => ( Aa = Ba ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_3729_dvd__div__eq__iff,axiom,
    ! [C: nat,Aa: nat,Ba: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),Aa)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),Ba)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Aa),C) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Ba),C) )
        <=> ( Aa = Ba ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_3730_dvd__div__eq__iff,axiom,
    ! [C: int,Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),Aa)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),Ba)
       => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),C) = aa_int_int(aa_int_fun_int_int(divide_divide_int,Ba),C) )
        <=> ( Aa = Ba ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_3731_dvd__div__eq__iff,axiom,
    ! [C: real,Aa: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,C),Aa)
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,C),Ba)
       => ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Aa),C) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Ba),C) )
        <=> ( Aa = Ba ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_3732_dvd__div__eq__iff,axiom,
    ! [C: code_integer,Aa: code_integer,Ba: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),Aa)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),Ba)
       => ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Aa),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Ba),C) )
        <=> ( Aa = Ba ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_3733_gcd__nat_Oextremum,axiom,
    ! [Aa: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),zero_zero_nat) ).

% gcd_nat.extremum
tff(fact_3734_gcd__nat_Oextremum__strict,axiom,
    ! [Aa: nat] :
      ~ ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,zero_zero_nat),Aa)
        & ( zero_zero_nat != Aa ) ) ).

% gcd_nat.extremum_strict
tff(fact_3735_gcd__nat_Oextremum__unique,axiom,
    ! [Aa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,zero_zero_nat),Aa)
    <=> ( Aa = zero_zero_nat ) ) ).

% gcd_nat.extremum_unique
tff(fact_3736_gcd__nat_Onot__eq__extremum,axiom,
    ! [Aa: nat] :
      ( ( Aa != zero_zero_nat )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),zero_zero_nat)
        & ( Aa != zero_zero_nat ) ) ) ).

% gcd_nat.not_eq_extremum
tff(fact_3737_gcd__nat_Oextremum__uniqueI,axiom,
    ! [Aa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,zero_zero_nat),Aa)
     => ( Aa = zero_zero_nat ) ) ).

% gcd_nat.extremum_uniqueI
tff(fact_3738_dvd__diff__nat,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),Na)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na)) ) ) ).

% dvd_diff_nat
tff(fact_3739_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_3740_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_3741_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_3742_pinf_I9_J,axiom,
    ! [D: code_integer,S3: code_integer] :
    ? [Z3: code_integer] :
    ! [X3: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Z3),X3)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,X3),S3))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,X3),S3)) ) ) ).

% pinf(9)
tff(fact_3743_pinf_I9_J,axiom,
    ! [D: real,S3: real] :
    ? [Z3: real] :
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),X3)
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X3),S3))
      <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X3),S3)) ) ) ).

% pinf(9)
tff(fact_3744_pinf_I9_J,axiom,
    ! [D: rat,S3: rat] :
    ? [Z3: rat] :
    ! [X3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z3),X3)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X3),S3))
      <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X3),S3)) ) ) ).

% pinf(9)
tff(fact_3745_pinf_I9_J,axiom,
    ! [D: nat,S3: nat] :
    ? [Z3: nat] :
    ! [X3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z3),X3)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X3),S3))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X3),S3)) ) ) ).

% pinf(9)
tff(fact_3746_pinf_I9_J,axiom,
    ! [D: int,S3: int] :
    ? [Z3: int] :
    ! [X3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),X3)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),S3))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),S3)) ) ) ).

% pinf(9)
tff(fact_3747_pinf_I10_J,axiom,
    ! [D: code_integer,S3: code_integer] :
    ? [Z3: code_integer] :
    ! [X3: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Z3),X3)
     => ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,X3),S3))
      <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,X3),S3)) ) ) ).

% pinf(10)
tff(fact_3748_pinf_I10_J,axiom,
    ! [D: real,S3: real] :
    ? [Z3: real] :
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),X3)
     => ( ~ aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X3),S3))
      <=> ~ aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X3),S3)) ) ) ).

% pinf(10)
tff(fact_3749_pinf_I10_J,axiom,
    ! [D: rat,S3: rat] :
    ? [Z3: rat] :
    ! [X3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z3),X3)
     => ( ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X3),S3))
      <=> ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X3),S3)) ) ) ).

% pinf(10)
tff(fact_3750_pinf_I10_J,axiom,
    ! [D: nat,S3: nat] :
    ? [Z3: nat] :
    ! [X3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z3),X3)
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X3),S3))
      <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X3),S3)) ) ) ).

% pinf(10)
tff(fact_3751_pinf_I10_J,axiom,
    ! [D: int,S3: int] :
    ? [Z3: int] :
    ! [X3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),X3)
     => ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),S3))
      <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),S3)) ) ) ).

% pinf(10)
tff(fact_3752_minf_I9_J,axiom,
    ! [D: code_integer,S3: code_integer] :
    ? [Z3: code_integer] :
    ! [X3: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X3),Z3)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,X3),S3))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,X3),S3)) ) ) ).

% minf(9)
tff(fact_3753_minf_I9_J,axiom,
    ! [D: real,S3: real] :
    ? [Z3: real] :
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Z3)
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X3),S3))
      <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X3),S3)) ) ) ).

% minf(9)
tff(fact_3754_minf_I9_J,axiom,
    ! [D: rat,S3: rat] :
    ? [Z3: rat] :
    ! [X3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Z3)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X3),S3))
      <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X3),S3)) ) ) ).

% minf(9)
tff(fact_3755_minf_I9_J,axiom,
    ! [D: nat,S3: nat] :
    ? [Z3: nat] :
    ! [X3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Z3)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X3),S3))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X3),S3)) ) ) ).

% minf(9)
tff(fact_3756_minf_I9_J,axiom,
    ! [D: int,S3: int] :
    ? [Z3: int] :
    ! [X3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Z3)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),S3))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),S3)) ) ) ).

% minf(9)
tff(fact_3757_minf_I10_J,axiom,
    ! [D: code_integer,S3: code_integer] :
    ? [Z3: code_integer] :
    ! [X3: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X3),Z3)
     => ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,X3),S3))
      <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,X3),S3)) ) ) ).

% minf(10)
tff(fact_3758_minf_I10_J,axiom,
    ! [D: real,S3: real] :
    ? [Z3: real] :
    ! [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Z3)
     => ( ~ aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X3),S3))
      <=> ~ aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X3),S3)) ) ) ).

% minf(10)
tff(fact_3759_minf_I10_J,axiom,
    ! [D: rat,S3: rat] :
    ? [Z3: rat] :
    ! [X3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Z3)
     => ( ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X3),S3))
      <=> ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(plus_plus_rat(X3),S3)) ) ) ).

% minf(10)
tff(fact_3760_minf_I10_J,axiom,
    ! [D: nat,S3: nat] :
    ? [Z3: nat] :
    ! [X3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Z3)
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X3),S3))
      <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X3),S3)) ) ) ).

% minf(10)
tff(fact_3761_minf_I10_J,axiom,
    ! [D: int,S3: int] :
    ? [Z3: int] :
    ! [X3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Z3)
     => ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),S3))
      <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),S3)) ) ) ).

% minf(10)
tff(fact_3762_dvd__div__eq__0__iff,axiom,
    ! [Ba: rat,Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Ba),Aa)
     => ( ( aa_rat_rat(divide_divide_rat(Aa),Ba) = zero_zero_rat )
      <=> ( Aa = zero_zero_rat ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_3763_dvd__div__eq__0__iff,axiom,
    ! [Ba: nat,Aa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Ba),Aa)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Aa),Ba) = zero_zero_nat )
      <=> ( Aa = zero_zero_nat ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_3764_dvd__div__eq__0__iff,axiom,
    ! [Ba: int,Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Ba),Aa)
     => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),Ba) = zero_zero_int )
      <=> ( Aa = zero_zero_int ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_3765_dvd__div__eq__0__iff,axiom,
    ! [Ba: real,Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Ba),Aa)
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Aa),Ba) = zero_zero_real )
      <=> ( Aa = zero_zero_real ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_3766_dvd__div__eq__0__iff,axiom,
    ! [Ba: code_integer,Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Ba),Aa)
     => ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Aa),Ba) = zero_z3403309356797280102nteger )
      <=> ( Aa = zero_z3403309356797280102nteger ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_3767_dvd__div__unit__iff,axiom,
    ! [Ba: nat,Aa: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Ba),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),Ba))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),C) ) ) ).

% dvd_div_unit_iff
tff(fact_3768_dvd__div__unit__iff,axiom,
    ! [Ba: int,Aa: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Ba),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),aa_int_int(aa_int_fun_int_int(divide_divide_int,C),Ba))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),C) ) ) ).

% dvd_div_unit_iff
tff(fact_3769_dvd__div__unit__iff,axiom,
    ! [Ba: code_integer,Aa: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Ba),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,C),Ba))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),C) ) ) ).

% dvd_div_unit_iff
tff(fact_3770_div__unit__dvd__iff,axiom,
    ! [Ba: nat,Aa: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Ba),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,Aa),Ba)),C)
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),C) ) ) ).

% div_unit_dvd_iff
tff(fact_3771_div__unit__dvd__iff,axiom,
    ! [Ba: int,Aa: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Ba),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,Aa),Ba)),C)
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),C) ) ) ).

% div_unit_dvd_iff
tff(fact_3772_div__unit__dvd__iff,axiom,
    ! [Ba: code_integer,Aa: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Ba),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Aa),Ba)),C)
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),C) ) ) ).

% div_unit_dvd_iff
tff(fact_3773_unit__div__cancel,axiom,
    ! [Aa: nat,Ba: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),one_one_nat)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Ba),Aa) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),Aa) )
      <=> ( Ba = C ) ) ) ).

% unit_div_cancel
tff(fact_3774_unit__div__cancel,axiom,
    ! [Aa: int,Ba: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),one_one_int)
     => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,Ba),Aa) = aa_int_int(aa_int_fun_int_int(divide_divide_int,C),Aa) )
      <=> ( Ba = C ) ) ) ).

% unit_div_cancel
tff(fact_3775_unit__div__cancel,axiom,
    ! [Aa: code_integer,Ba: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),one_one_Code_integer)
     => ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Ba),Aa) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,C),Aa) )
      <=> ( Ba = C ) ) ) ).

% unit_div_cancel
tff(fact_3776_div__plus__div__distrib__dvd__right,axiom,
    ! [C: nat,Ba: nat,Aa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),Ba)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),Ba)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Aa),C)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Ba),C)) ) ) ).

% div_plus_div_distrib_dvd_right
tff(fact_3777_div__plus__div__distrib__dvd__right,axiom,
    ! [C: int,Ba: int,Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),Ba)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),Ba)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),C)),aa_int_int(aa_int_fun_int_int(divide_divide_int,Ba),C)) ) ) ).

% div_plus_div_distrib_dvd_right
tff(fact_3778_div__plus__div__distrib__dvd__right,axiom,
    ! [C: code_integer,Ba: code_integer,Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),Ba)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Aa),Ba)),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Aa),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Ba),C)) ) ) ).

% div_plus_div_distrib_dvd_right
tff(fact_3779_div__plus__div__distrib__dvd__left,axiom,
    ! [C: nat,Aa: nat,Ba: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),Aa)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),Ba)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Aa),C)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Ba),C)) ) ) ).

% div_plus_div_distrib_dvd_left
tff(fact_3780_div__plus__div__distrib__dvd__left,axiom,
    ! [C: int,Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),Aa)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),Ba)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),C)),aa_int_int(aa_int_fun_int_int(divide_divide_int,Ba),C)) ) ) ).

% div_plus_div_distrib_dvd_left
tff(fact_3781_div__plus__div__distrib__dvd__left,axiom,
    ! [C: code_integer,Aa: code_integer,Ba: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),Aa)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Aa),Ba)),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Aa),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Ba),C)) ) ) ).

% div_plus_div_distrib_dvd_left
tff(fact_3782_dvd__neg__div,axiom,
    ! [Ba: real,Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Ba),Aa)
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,Aa)),Ba) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Aa),Ba)) ) ) ).

% dvd_neg_div
tff(fact_3783_dvd__neg__div,axiom,
    ! [Ba: int,Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Ba),Aa)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,Aa)),Ba) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),Ba)) ) ) ).

% dvd_neg_div
tff(fact_3784_dvd__neg__div,axiom,
    ! [Ba: code_integer,Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Ba),Aa)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa)),Ba) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Aa),Ba)) ) ) ).

% dvd_neg_div
tff(fact_3785_dvd__neg__div,axiom,
    ! [Ba: rat,Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Ba),Aa)
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(uminus_uminus_rat,Aa)),Ba) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(Aa),Ba)) ) ) ).

% dvd_neg_div
tff(fact_3786_dvd__div__neg,axiom,
    ! [Ba: real,Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Ba),Aa)
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Aa),aa_real_real(uminus_uminus_real,Ba)) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Aa),Ba)) ) ) ).

% dvd_div_neg
tff(fact_3787_dvd__div__neg,axiom,
    ! [Ba: int,Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Ba),Aa)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),aa_int_int(uminus_uminus_int,Ba)) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),Ba)) ) ) ).

% dvd_div_neg
tff(fact_3788_dvd__div__neg,axiom,
    ! [Ba: code_integer,Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Ba),Aa)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Aa),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Ba)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Aa),Ba)) ) ) ).

% dvd_div_neg
tff(fact_3789_dvd__div__neg,axiom,
    ! [Ba: rat,Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Ba),Aa)
     => ( aa_rat_rat(divide_divide_rat(Aa),aa_rat_rat(uminus_uminus_rat,Ba)) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(Aa),Ba)) ) ) ).

% dvd_div_neg
tff(fact_3790_le__imp__power__dvd,axiom,
    ! [M: nat,Na: nat,Aa: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(Aa),M)),aa_nat_Code_integer(power_8256067586552552935nteger(Aa),Na)) ) ).

% le_imp_power_dvd
tff(fact_3791_le__imp__power__dvd,axiom,
    ! [M: nat,Na: nat,Aa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Aa),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Aa),Na)) ) ).

% le_imp_power_dvd
tff(fact_3792_le__imp__power__dvd,axiom,
    ! [M: nat,Na: nat,Aa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),Na)) ) ).

% le_imp_power_dvd
tff(fact_3793_le__imp__power__dvd,axiom,
    ! [M: nat,Na: nat,Aa: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(Aa),M)),aa_nat_int(power_power_int(Aa),Na)) ) ).

% le_imp_power_dvd
tff(fact_3794_le__imp__power__dvd,axiom,
    ! [M: nat,Na: nat,Aa: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,aa_nat_complex(power_power_complex(Aa),M)),aa_nat_complex(power_power_complex(Aa),Na)) ) ).

% le_imp_power_dvd
tff(fact_3795_power__le__dvd,axiom,
    ! [Aa: code_integer,Na: nat,Ba: code_integer,M: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(Aa),Na)),Ba)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(Aa),M)),Ba) ) ) ).

% power_le_dvd
tff(fact_3796_power__le__dvd,axiom,
    ! [Aa: nat,Na: nat,Ba: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Aa),Na)),Ba)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Aa),M)),Ba) ) ) ).

% power_le_dvd
tff(fact_3797_power__le__dvd,axiom,
    ! [Aa: real,Na: nat,Ba: real,M: nat] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),Na)),Ba)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
       => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),M)),Ba) ) ) ).

% power_le_dvd
tff(fact_3798_power__le__dvd,axiom,
    ! [Aa: int,Na: nat,Ba: int,M: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(Aa),Na)),Ba)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(Aa),M)),Ba) ) ) ).

% power_le_dvd
tff(fact_3799_power__le__dvd,axiom,
    ! [Aa: complex,Na: nat,Ba: complex,M: nat] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,aa_nat_complex(power_power_complex(Aa),Na)),Ba)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
       => aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,aa_nat_complex(power_power_complex(Aa),M)),Ba) ) ) ).

% power_le_dvd
tff(fact_3800_dvd__power__le,axiom,
    ! [Xb: code_integer,Y: code_integer,Na: nat,M: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Xb),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(Xb),Na)),aa_nat_Code_integer(power_8256067586552552935nteger(Y),M)) ) ) ).

% dvd_power_le
tff(fact_3801_dvd__power__le,axiom,
    ! [Xb: nat,Y: nat,Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Xb),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),Na)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),M)) ) ) ).

% dvd_power_le
tff(fact_3802_dvd__power__le,axiom,
    ! [Xb: real,Y: real,Na: nat,M: nat] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Xb),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
       => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Na)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),M)) ) ) ).

% dvd_power_le
tff(fact_3803_dvd__power__le,axiom,
    ! [Xb: int,Y: int,Na: nat,M: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xb),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(Xb),Na)),aa_nat_int(power_power_int(Y),M)) ) ) ).

% dvd_power_le
tff(fact_3804_dvd__power__le,axiom,
    ! [Xb: complex,Y: complex,Na: nat,M: nat] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,Xb),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
       => aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,aa_nat_complex(power_power_complex(Xb),Na)),aa_nat_complex(power_power_complex(Y),M)) ) ) ).

% dvd_power_le
tff(fact_3805_dvd__pos__nat,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),Na)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M) ) ) ).

% dvd_pos_nat
tff(fact_3806_nat__dvd__not__less,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
       => ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Na),M) ) ) ).

% nat_dvd_not_less
tff(fact_3807_dvd__minus__self,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),M))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M)
        | aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),Na) ) ) ).

% dvd_minus_self
tff(fact_3808_zdvd__antisym__nonneg,axiom,
    ! [M: int,Na: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),M)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Na)
       => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,M),Na)
         => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Na),M)
           => ( M = Na ) ) ) ) ) ).

% zdvd_antisym_nonneg
tff(fact_3809_less__eq__dvd__minus,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),Na)
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),M)) ) ) ).

% less_eq_dvd_minus
tff(fact_3810_dvd__diffD1,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na))
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),M)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),Na) ) ) ) ).

% dvd_diffD1
tff(fact_3811_dvd__diffD,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na))
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),Na)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),M) ) ) ) ).

% dvd_diffD
tff(fact_3812_zdvd__not__zless,axiom,
    ! [M: int,Na: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),M)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,M),Na)
       => ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Na),M) ) ) ).

% zdvd_not_zless
tff(fact_3813_fact__dvd,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(semiri1406184849735516958ct_int,Na)),aa_nat_int(semiri1406184849735516958ct_int,M)) ) ).

% fact_dvd
tff(fact_3814_fact__dvd,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(semiri3624122377584611663nteger,Na)),aa_nat_Code_integer(semiri3624122377584611663nteger,M)) ) ).

% fact_dvd
tff(fact_3815_fact__dvd,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Na)),aa_nat_nat(semiri1408675320244567234ct_nat,M)) ) ).

% fact_dvd
tff(fact_3816_fact__dvd,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_nat_real(semiri2265585572941072030t_real,Na)),aa_nat_real(semiri2265585572941072030t_real,M)) ) ).

% fact_dvd
tff(fact_3817_div__neg__pos__less0,axiom,
    ! [Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Ba)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),Ba)),zero_zero_int) ) ) ).

% div_neg_pos_less0
tff(fact_3818_neg__imp__zdiv__neg__iff,axiom,
    ! [Ba: int,Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Ba),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,Aa),Ba)),zero_zero_int)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Aa) ) ) ).

% neg_imp_zdiv_neg_iff
tff(fact_3819_pos__imp__zdiv__neg__iff,axiom,
    ! [Ba: int,Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Ba)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),Ba)),zero_zero_int)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),zero_zero_int) ) ) ).

% pos_imp_zdiv_neg_iff
tff(fact_3820_dvd__euclidean__size__eq__imp__dvd,axiom,
    ! [Aa: code_integer,Ba: code_integer] :
      ( ( Aa != zero_z3403309356797280102nteger )
     => ( ( aa_Code_integer_nat(euclid6377331345833325938nteger,Aa) = aa_Code_integer_nat(euclid6377331345833325938nteger,Ba) )
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Ba),Aa)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),Ba) ) ) ) ).

% dvd_euclidean_size_eq_imp_dvd
tff(fact_3821_dvd__euclidean__size__eq__imp__dvd,axiom,
    ! [Aa: int,Ba: int] :
      ( ( Aa != zero_zero_int )
     => ( ( aa_int_nat(euclid4774559944035922753ze_int,Aa) = aa_int_nat(euclid4774559944035922753ze_int,Ba) )
       => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Ba),Aa)
         => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),Ba) ) ) ) ).

% dvd_euclidean_size_eq_imp_dvd
tff(fact_3822_dvd__euclidean__size__eq__imp__dvd,axiom,
    ! [Aa: nat,Ba: nat] :
      ( ( Aa != zero_zero_nat )
     => ( ( aa_nat_nat(euclid4777050414544973029ze_nat,Aa) = aa_nat_nat(euclid4777050414544973029ze_nat,Ba) )
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Ba),Aa)
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),Ba) ) ) ) ).

% dvd_euclidean_size_eq_imp_dvd
tff(fact_3823_euclidean__size__unit,axiom,
    ! [Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),one_one_Code_integer)
     => ( aa_Code_integer_nat(euclid6377331345833325938nteger,Aa) = aa_Code_integer_nat(euclid6377331345833325938nteger,one_one_Code_integer) ) ) ).

% euclidean_size_unit
tff(fact_3824_euclidean__size__unit,axiom,
    ! [Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),one_one_int)
     => ( aa_int_nat(euclid4774559944035922753ze_int,Aa) = aa_int_nat(euclid4774559944035922753ze_int,one_one_int) ) ) ).

% euclidean_size_unit
tff(fact_3825_euclidean__size__unit,axiom,
    ! [Aa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),one_one_nat)
     => ( aa_nat_nat(euclid4777050414544973029ze_nat,Aa) = aa_nat_nat(euclid4777050414544973029ze_nat,one_one_nat) ) ) ).

% euclidean_size_unit
tff(fact_3826_unit__div__eq__0__iff,axiom,
    ! [Ba: nat,Aa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Ba),one_one_nat)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Aa),Ba) = zero_zero_nat )
      <=> ( Aa = zero_zero_nat ) ) ) ).

% unit_div_eq_0_iff
tff(fact_3827_unit__div__eq__0__iff,axiom,
    ! [Ba: int,Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Ba),one_one_int)
     => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),Ba) = zero_zero_int )
      <=> ( Aa = zero_zero_int ) ) ) ).

% unit_div_eq_0_iff
tff(fact_3828_unit__div__eq__0__iff,axiom,
    ! [Ba: code_integer,Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Ba),one_one_Code_integer)
     => ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Aa),Ba) = zero_z3403309356797280102nteger )
      <=> ( Aa = zero_z3403309356797280102nteger ) ) ) ).

% unit_div_eq_0_iff
tff(fact_3829_is__unit__power__iff,axiom,
    ! [Aa: code_integer,Na: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(Aa),Na)),one_one_Code_integer)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),one_one_Code_integer)
        | ( Na = zero_zero_nat ) ) ) ).

% is_unit_power_iff
tff(fact_3830_is__unit__power__iff,axiom,
    ! [Aa: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Aa),Na)),one_one_nat)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),one_one_nat)
        | ( Na = zero_zero_nat ) ) ) ).

% is_unit_power_iff
tff(fact_3831_is__unit__power__iff,axiom,
    ! [Aa: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(Aa),Na)),one_one_int)
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),one_one_int)
        | ( Na = zero_zero_nat ) ) ) ).

% is_unit_power_iff
tff(fact_3832_dvd__imp__le,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na) ) ) ).

% dvd_imp_le
tff(fact_3833_unit__iff__euclidean__size,axiom,
    ! [Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),one_one_Code_integer)
    <=> ( ( aa_Code_integer_nat(euclid6377331345833325938nteger,Aa) = aa_Code_integer_nat(euclid6377331345833325938nteger,one_one_Code_integer) )
        & ( Aa != zero_z3403309356797280102nteger ) ) ) ).

% unit_iff_euclidean_size
tff(fact_3834_unit__iff__euclidean__size,axiom,
    ! [Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),one_one_int)
    <=> ( ( aa_int_nat(euclid4774559944035922753ze_int,Aa) = aa_int_nat(euclid4774559944035922753ze_int,one_one_int) )
        & ( Aa != zero_zero_int ) ) ) ).

% unit_iff_euclidean_size
tff(fact_3835_unit__iff__euclidean__size,axiom,
    ! [Aa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),one_one_nat)
    <=> ( ( aa_nat_nat(euclid4777050414544973029ze_nat,Aa) = aa_nat_nat(euclid4777050414544973029ze_nat,one_one_nat) )
        & ( Aa != zero_zero_nat ) ) ) ).

% unit_iff_euclidean_size
tff(fact_3836_dvd__proper__imp__size__less,axiom,
    ! [Aa: code_integer,Ba: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),Ba)
     => ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Ba),Aa)
       => ( ( Ba != zero_z3403309356797280102nteger )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_Code_integer_nat(euclid6377331345833325938nteger,Aa)),aa_Code_integer_nat(euclid6377331345833325938nteger,Ba)) ) ) ) ).

% dvd_proper_imp_size_less
tff(fact_3837_dvd__proper__imp__size__less,axiom,
    ! [Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),Ba)
     => ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Ba),Aa)
       => ( ( Ba != zero_zero_int )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(euclid4774559944035922753ze_int,Aa)),aa_int_nat(euclid4774559944035922753ze_int,Ba)) ) ) ) ).

% dvd_proper_imp_size_less
tff(fact_3838_dvd__proper__imp__size__less,axiom,
    ! [Aa: nat,Ba: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),Ba)
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Ba),Aa)
       => ( ( Ba != zero_zero_nat )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(euclid4777050414544973029ze_nat,Aa)),aa_nat_nat(euclid4777050414544973029ze_nat,Ba)) ) ) ) ).

% dvd_proper_imp_size_less
tff(fact_3839_dvd__imp__size__le,axiom,
    ! [Aa: code_integer,Ba: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),Ba)
     => ( ( Ba != zero_z3403309356797280102nteger )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_Code_integer_nat(euclid6377331345833325938nteger,Aa)),aa_Code_integer_nat(euclid6377331345833325938nteger,Ba)) ) ) ).

% dvd_imp_size_le
tff(fact_3840_dvd__imp__size__le,axiom,
    ! [Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),Ba)
     => ( ( Ba != zero_zero_int )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(euclid4774559944035922753ze_int,Aa)),aa_int_nat(euclid4774559944035922753ze_int,Ba)) ) ) ).

% dvd_imp_size_le
tff(fact_3841_dvd__imp__size__le,axiom,
    ! [Aa: nat,Ba: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),Ba)
     => ( ( Ba != zero_zero_nat )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(euclid4777050414544973029ze_nat,Aa)),aa_nat_nat(euclid4777050414544973029ze_nat,Ba)) ) ) ).

% dvd_imp_size_le
tff(fact_3842_zdvd__imp__le,axiom,
    ! [Z: int,Na: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Z),Na)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Na)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),Na) ) ) ).

% zdvd_imp_le
tff(fact_3843_zdiv__mono1,axiom,
    ! [Aa: int,A5: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),A5)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Ba)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),Ba)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A5),Ba)) ) ) ).

% zdiv_mono1
tff(fact_3844_zdiv__mono2,axiom,
    ! [Aa: int,B5: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Aa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B5)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B5),Ba)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),Ba)),aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),B5)) ) ) ) ).

% zdiv_mono2
tff(fact_3845_zdiv__eq__0__iff,axiom,
    ! [I: int,K: int] :
      ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,I),K) = zero_zero_int )
    <=> ( ( K = zero_zero_int )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),I)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,I),K) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,K),I) ) ) ) ).

% zdiv_eq_0_iff
tff(fact_3846_zdiv__mono1__neg,axiom,
    ! [Aa: int,A5: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),A5)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Ba),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,A5),Ba)),aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),Ba)) ) ) ).

% zdiv_mono1_neg
tff(fact_3847_zdiv__mono2__neg,axiom,
    ! [Aa: int,B5: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B5)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B5),Ba)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),B5)),aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),Ba)) ) ) ) ).

% zdiv_mono2_neg
tff(fact_3848_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_3849_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_3850_div__nonneg__neg__le0,axiom,
    ! [Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Aa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Ba),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,Aa),Ba)),zero_zero_int) ) ) ).

% div_nonneg_neg_le0
tff(fact_3851_div__nonpos__pos__le0,axiom,
    ! [Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Ba)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),Ba)),zero_zero_int) ) ) ).

% div_nonpos_pos_le0
tff(fact_3852_pos__imp__zdiv__pos__iff,axiom,
    ! [K: int,I: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,I),K))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),I) ) ) ).

% pos_imp_zdiv_pos_iff
tff(fact_3853_neg__imp__zdiv__nonneg__iff,axiom,
    ! [Ba: int,Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Ba),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,Aa),Ba))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),zero_zero_int) ) ) ).

% neg_imp_zdiv_nonneg_iff
tff(fact_3854_pos__imp__zdiv__nonneg__iff,axiom,
    ! [Ba: int,Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Ba)
     => ( 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,Aa),Ba))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Aa) ) ) ).

% pos_imp_zdiv_nonneg_iff
tff(fact_3855_nonneg1__imp__zdiv__pos__iff,axiom,
    ! [Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Aa)
     => ( 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,Aa),Ba))
      <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ba),Aa)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Ba) ) ) ) ).

% nonneg1_imp_zdiv_pos_iff
tff(fact_3856_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_3857_dvd__fact,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),aa_nat_nat(semiri1408675320244567234ct_nat,Na)) ) ) ).

% dvd_fact
tff(fact_3858_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_3859_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_3860_even__zero,axiom,
    aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),zero_z3403309356797280102nteger) ).

% even_zero
tff(fact_3861_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_3862_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_3863_odd__one,axiom,
    ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),one_one_Code_integer) ).

% odd_one
tff(fact_3864_dvd__power__iff,axiom,
    ! [Xb: code_integer,M: nat,Na: nat] :
      ( ( Xb != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(Xb),M)),aa_nat_Code_integer(power_8256067586552552935nteger(Xb),Na))
      <=> ( 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,M),Na) ) ) ) ).

% dvd_power_iff
tff(fact_3865_dvd__power__iff,axiom,
    ! [Xb: nat,M: nat,Na: 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),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),Na))
      <=> ( 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,M),Na) ) ) ) ).

% dvd_power_iff
tff(fact_3866_dvd__power__iff,axiom,
    ! [Xb: int,M: nat,Na: nat] :
      ( ( Xb != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(Xb),M)),aa_nat_int(power_power_int(Xb),Na))
      <=> ( 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,M),Na) ) ) ) ).

% dvd_power_iff
tff(fact_3867_dvd__power,axiom,
    ! [Na: nat,Xb: code_integer] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
        | ( Xb = one_one_Code_integer ) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Xb),aa_nat_Code_integer(power_8256067586552552935nteger(Xb),Na)) ) ).

% dvd_power
tff(fact_3868_dvd__power,axiom,
    ! [Na: nat,Xb: rat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
        | ( Xb = one_one_rat ) )
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Xb),aa_nat_rat(power_power_rat(Xb),Na)) ) ).

% dvd_power
tff(fact_3869_dvd__power,axiom,
    ! [Na: nat,Xb: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
        | ( 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),Na)) ) ).

% dvd_power
tff(fact_3870_dvd__power,axiom,
    ! [Na: nat,Xb: real] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
        | ( 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),Na)) ) ).

% dvd_power
tff(fact_3871_dvd__power,axiom,
    ! [Na: nat,Xb: int] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
        | ( Xb = one_one_int ) )
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xb),aa_nat_int(power_power_int(Xb),Na)) ) ).

% dvd_power
tff(fact_3872_dvd__power,axiom,
    ! [Na: nat,Xb: complex] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
        | ( Xb = one_one_complex ) )
     => aa_complex_o(aa_com5808801942499186519plex_o(dvd_dvd_complex,Xb),aa_nat_complex(power_power_complex(Xb),Na)) ) ).

% dvd_power
tff(fact_3873_power__dvd__imp__le,axiom,
    ! [I: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),Na))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),I)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ) ).

% power_dvd_imp_le
tff(fact_3874_verit__less__mono__div__int2,axiom,
    ! [A3: int,B4: int,Na: 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,Na))
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,B4),Na)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A3),Na)) ) ) ).

% verit_less_mono_div_int2
tff(fact_3875_div__eq__minus1,axiom,
    ! [Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Ba)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,one_one_int)),Ba) = aa_int_int(uminus_uminus_int,one_one_int) ) ) ).

% div_eq_minus1
tff(fact_3876_aset_I7_J,axiom,
    ! [D6: int,A3: set_int,Ta: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D6)
     => ! [X3: int] :
          ( ! [Xa2: int] :
              ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D6))
             => ! [Xb2: int] :
                  ( aa_set_int_o(member_int(Xb2),A3)
                 => ( X3 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa2) ) ) )
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Ta),X3)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,Ta),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D6)) ) ) ) ).

% aset(7)
tff(fact_3877_aset_I5_J,axiom,
    ! [D6: int,Ta: int,A3: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D6)
     => ( aa_set_int_o(member_int(Ta),A3)
       => ! [X3: int] :
            ( ! [Xa2: int] :
                ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D6))
               => ! [Xb2: int] :
                    ( aa_set_int_o(member_int(Xb2),A3)
                   => ( X3 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa2) ) ) )
           => ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Ta)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D6)),Ta) ) ) ) ) ).

% aset(5)
tff(fact_3878_aset_I4_J,axiom,
    ! [D6: int,Ta: int,A3: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D6)
     => ( aa_set_int_o(member_int(Ta),A3)
       => ! [X3: int] :
            ( ! [Xa2: int] :
                ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D6))
               => ! [Xb2: int] :
                    ( aa_set_int_o(member_int(Xb2),A3)
                   => ( X3 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa2) ) ) )
           => ( ( X3 != Ta )
             => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D6) != Ta ) ) ) ) ) ).

% aset(4)
tff(fact_3879_aset_I3_J,axiom,
    ! [D6: int,Ta: int,A3: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D6)
     => ( aa_set_int_o(member_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,Ta),one_one_int)),A3)
       => ! [X3: int] :
            ( ! [Xa2: int] :
                ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D6))
               => ! [Xb2: int] :
                    ( aa_set_int_o(member_int(Xb2),A3)
                   => ( X3 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa2) ) ) )
           => ( ( X3 = Ta )
             => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D6) = Ta ) ) ) ) ) ).

% aset(3)
tff(fact_3880_bset_I7_J,axiom,
    ! [D6: int,Ta: int,B4: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D6)
     => ( aa_set_int_o(member_int(Ta),B4)
       => ! [X3: int] :
            ( ! [Xa2: int] :
                ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D6))
               => ! [Xb2: int] :
                    ( aa_set_int_o(member_int(Xb2),B4)
                   => ( X3 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa2) ) ) )
           => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Ta),X3)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,Ta),aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D6)) ) ) ) ) ).

% bset(7)
tff(fact_3881_bset_I5_J,axiom,
    ! [D6: int,B4: set_int,Ta: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D6)
     => ! [X3: int] :
          ( ! [Xa2: int] :
              ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D6))
             => ! [Xb2: int] :
                  ( aa_set_int_o(member_int(Xb2),B4)
                 => ( X3 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa2) ) ) )
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Ta)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D6)),Ta) ) ) ) ).

% bset(5)
tff(fact_3882_bset_I4_J,axiom,
    ! [D6: int,Ta: int,B4: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D6)
     => ( aa_set_int_o(member_int(Ta),B4)
       => ! [X3: int] :
            ( ! [Xa2: int] :
                ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D6))
               => ! [Xb2: int] :
                    ( aa_set_int_o(member_int(Xb2),B4)
                   => ( X3 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa2) ) ) )
           => ( ( X3 != Ta )
             => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D6) != Ta ) ) ) ) ) ).

% bset(4)
tff(fact_3883_bset_I3_J,axiom,
    ! [D6: int,Ta: int,B4: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D6)
     => ( aa_set_int_o(member_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,Ta),one_one_int)),B4)
       => ! [X3: int] :
            ( ! [Xa2: int] :
                ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D6))
               => ! [Xb2: int] :
                    ( aa_set_int_o(member_int(Xb2),B4)
                   => ( X3 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa2) ) ) )
           => ( ( X3 = Ta )
             => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D6) = Ta ) ) ) ) ) ).

% bset(3)
tff(fact_3884_power__mono__odd,axiom,
    ! [Na: nat,Aa: real,Ba: real] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),Ba)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),Na)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ba),Na)) ) ) ).

% power_mono_odd
tff(fact_3885_power__mono__odd,axiom,
    ! [Na: nat,Aa: rat,Ba: rat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),Ba)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(Aa),Na)),aa_nat_rat(power_power_rat(Ba),Na)) ) ) ).

% power_mono_odd
tff(fact_3886_power__mono__odd,axiom,
    ! [Na: nat,Aa: int,Ba: int] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),Ba)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(Aa),Na)),aa_nat_int(power_power_int(Ba),Na)) ) ) ).

% power_mono_odd
tff(fact_3887_odd__pos,axiom,
    ! [Na: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ).

% odd_pos
tff(fact_3888_dvd__power__iff__le,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K),Na))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ) ).

% dvd_power_iff_le
tff(fact_3889_num_Osize__gen_I1_J,axiom,
    aa_num_nat(size_num,one) = zero_zero_nat ).

% num.size_gen(1)
tff(fact_3890_aset_I8_J,axiom,
    ! [D6: int,A3: set_int,Ta: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D6)
     => ! [X3: int] :
          ( ! [Xa2: int] :
              ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D6))
             => ! [Xb2: int] :
                  ( aa_set_int_o(member_int(Xb2),A3)
                 => ( X3 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa2) ) ) )
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ta),X3)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ta),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D6)) ) ) ) ).

% aset(8)
tff(fact_3891_aset_I6_J,axiom,
    ! [D6: int,Ta: int,A3: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D6)
     => ( aa_set_int_o(member_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,Ta),one_one_int)),A3)
       => ! [X3: int] :
            ( ! [Xa2: int] :
                ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D6))
               => ! [Xb2: int] :
                    ( aa_set_int_o(member_int(Xb2),A3)
                   => ( X3 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa2) ) ) )
           => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),Ta)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D6)),Ta) ) ) ) ) ).

% aset(6)
tff(fact_3892_bset_I8_J,axiom,
    ! [D6: int,Ta: int,B4: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D6)
     => ( aa_set_int_o(member_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,Ta),one_one_int)),B4)
       => ! [X3: int] :
            ( ! [Xa2: int] :
                ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D6))
               => ! [Xb2: int] :
                    ( aa_set_int_o(member_int(Xb2),B4)
                   => ( X3 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa2) ) ) )
           => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ta),X3)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ta),aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D6)) ) ) ) ) ).

% bset(8)
tff(fact_3893_bset_I6_J,axiom,
    ! [D6: int,B4: set_int,Ta: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D6)
     => ! [X3: int] :
          ( ! [Xa2: int] :
              ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D6))
             => ! [Xb2: int] :
                  ( aa_set_int_o(member_int(Xb2),B4)
                 => ( X3 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa2) ) ) )
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),Ta)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D6)),Ta) ) ) ) ).

% bset(6)
tff(fact_3894_zero__le__power__eq,axiom,
    ! [Aa: real,Na: 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),Na))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Aa) ) ) ) ).

% zero_le_power_eq
tff(fact_3895_zero__le__power__eq,axiom,
    ! [Aa: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(Aa),Na))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Aa) ) ) ) ).

% zero_le_power_eq
tff(fact_3896_zero__le__power__eq,axiom,
    ! [Aa: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(Aa),Na))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Aa) ) ) ) ).

% zero_le_power_eq
tff(fact_3897_zero__le__odd__power,axiom,
    ! [Na: nat,Aa: real] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),Na))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Aa) ) ) ).

% zero_le_odd_power
tff(fact_3898_zero__le__odd__power,axiom,
    ! [Na: nat,Aa: rat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(Aa),Na))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Aa) ) ) ).

% zero_le_odd_power
tff(fact_3899_zero__le__odd__power,axiom,
    ! [Na: nat,Aa: int] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(Aa),Na))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Aa) ) ) ).

% zero_le_odd_power
tff(fact_3900_zero__le__even__power,axiom,
    ! [Na: nat,Aa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),Na)) ) ).

% zero_le_even_power
tff(fact_3901_zero__le__even__power,axiom,
    ! [Na: nat,Aa: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(Aa),Na)) ) ).

% zero_le_even_power
tff(fact_3902_zero__le__even__power,axiom,
    ! [Na: nat,Aa: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(Aa),Na)) ) ).

% zero_le_even_power
tff(fact_3903_minus__one__power__iff,axiom,
    ! [Na: nat] :
      aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Na) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na),one_one_complex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) ).

% minus_one_power_iff
tff(fact_3904_minus__one__power__iff,axiom,
    ! [Na: nat] :
      aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Na) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na),one_one_real,aa_real_real(uminus_uminus_real,one_one_real)) ).

% minus_one_power_iff
tff(fact_3905_minus__one__power__iff,axiom,
    ! [Na: nat] :
      aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),Na) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na),one_one_int,aa_int_int(uminus_uminus_int,one_one_int)) ).

% minus_one_power_iff
tff(fact_3906_minus__one__power__iff,axiom,
    ! [Na: nat] :
      aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Na) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na),one_one_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).

% minus_one_power_iff
tff(fact_3907_minus__one__power__iff,axiom,
    ! [Na: nat] :
      aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Na) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na),one_one_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)) ).

% minus_one_power_iff
tff(fact_3908_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_3909_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_3910_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_3911_VEBT__internal_Oexp__split__high__low_I1_J,axiom,
    ! [Xb: nat,Na: nat,M: 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,Na),M)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),Na)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) ) ) ) ).

% VEBT_internal.exp_split_high_low(1)
tff(fact_3912_zero__less__power__eq,axiom,
    ! [Aa: real,Na: 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),Na))
    <=> ( ( Na = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
          & ( Aa != zero_zero_real ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Aa) ) ) ) ).

% zero_less_power_eq
tff(fact_3913_zero__less__power__eq,axiom,
    ! [Aa: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(Aa),Na))
    <=> ( ( Na = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
          & ( Aa != zero_zero_rat ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Aa) ) ) ) ).

% zero_less_power_eq
tff(fact_3914_zero__less__power__eq,axiom,
    ! [Aa: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(Aa),Na))
    <=> ( ( Na = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
          & ( Aa != zero_zero_int ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Aa) ) ) ) ).

% zero_less_power_eq
tff(fact_3915_int__power__div__base,axiom,
    ! [M: nat,K: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(power_power_int(K),M)),K) = aa_nat_int(power_power_int(K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(suc,zero_zero_nat))) ) ) ) ).

% int_power_div_base
tff(fact_3916_even__mask__div__iff_H,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ).

% even_mask_div_iff'
tff(fact_3917_even__mask__div__iff_H,axiom,
    ! [M: nat,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M)),one_one_int)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ).

% even_mask_div_iff'
tff(fact_3918_even__mask__div__iff_H,axiom,
    ! [M: nat,Na: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),M)),one_one_Code_integer)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Na)))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ).

% even_mask_div_iff'
tff(fact_3919_power__le__zero__eq,axiom,
    ! [Aa: real,Na: 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),Na)),zero_zero_real)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
        & ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),zero_zero_real) )
          | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
            & ( Aa = zero_zero_real ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_3920_power__le__zero__eq,axiom,
    ! [Aa: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(Aa),Na)),zero_zero_rat)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
        & ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),zero_zero_rat) )
          | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
            & ( Aa = zero_zero_rat ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_3921_power__le__zero__eq,axiom,
    ! [Aa: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(Aa),Na)),zero_zero_int)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
        & ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
            & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),zero_zero_int) )
          | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
            & ( Aa = zero_zero_int ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_3922_even__mask__div__iff,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)))
    <=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na) = zero_zero_nat )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ) ).

% even_mask_div_iff
tff(fact_3923_even__mask__div__iff,axiom,
    ! [M: nat,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M)),one_one_int)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)))
    <=> ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na) = zero_zero_int )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ) ).

% even_mask_div_iff
tff(fact_3924_even__mask__div__iff,axiom,
    ! [M: nat,Na: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),M)),one_one_Code_integer)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Na)))
    <=> ( ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Na) = zero_z3403309356797280102nteger )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ) ).

% even_mask_div_iff
tff(fact_3925_summaxma,axiom,
    ! [Mi2: nat,Ma2: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( aa_nat_o(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Deg,TreeList,Summary)),Deg)
     => ( ( Mi2 != Ma2 )
       => ( the_nat(vEBT_vebt_maxt(Summary)) = aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ) ).

% summaxma
tff(fact_3926_both__member__options__from__chilf__to__complete__tree,axiom,
    ! [Xb: nat,Deg: nat,TreeList: list_VEBT_VEBT,Mi2: nat,Ma2: nat,Summary: 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,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Deg)
       => ( aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
         => aa_nat_o(vEBT_V8194947554948674370ptions(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Deg,TreeList,Summary)),Xb) ) ) ) ).

% both_member_options_from_chilf_to_complete_tree
tff(fact_3927_member__inv,axiom,
    ! [Mi2: nat,Ma2: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Xb: nat] :
      ( aa_nat_o(vEBT_vebt_member(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Deg,TreeList,Summary)),Xb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)
        & ( ( Xb = Mi2 )
          | ( Xb = Ma2 )
          | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Ma2)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi2),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,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))
            & aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ) ) ).

% member_inv
tff(fact_3928_both__member__options__from__complete__tree__to__child,axiom,
    ! [Deg: nat,Mi2: nat,Ma2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Deg)
     => ( aa_nat_o(vEBT_V8194947554948674370ptions(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Deg,TreeList,Summary)),Xb)
       => ( aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
          | ( Xb = Mi2 )
          | ( Xb = Ma2 ) ) ) ) ).

% both_member_options_from_complete_tree_to_child
tff(fact_3929_both__member__options__ding,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Na: nat,Xb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(vEBT_Node(Info,Deg,TreeList,Summary)),Na)
     => ( 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))),Deg))
       => ( aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
         => aa_nat_o(vEBT_V8194947554948674370ptions(vEBT_Node(Info,Deg,TreeList,Summary)),Xb) ) ) ) ).

% both_member_options_ding
tff(fact_3930_high__inv,axiom,
    ! [Xb: nat,Na: nat,Y: 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))),Na))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))),Xb)),Na) = Y ) ) ).

% high_inv
tff(fact_3931_invar__vebt_Ocases,axiom,
    ! [A1: vEBT_VEBT,A22: nat] :
      ( aa_nat_o(vEBT_invar_vebt(A1),A22)
     => ( ( ? [A2: $o,B2: $o] : A1 = vEBT_Leaf((A2),(B2))
         => ( A22 != aa_nat_nat(suc,zero_zero_nat) ) )
       => ( ! [TreeList2: list_VEBT_VEBT,N: nat,Summary2: vEBT_VEBT,M3: nat,Deg2: nat] :
              ( ( A1 = vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2,Summary2) )
             => ( ( A22 = Deg2 )
               => ( ! [X3: vEBT_VEBT] :
                      ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeList2))
                     => aa_nat_o(vEBT_invar_vebt(X3),N) )
                 => ( aa_nat_o(vEBT_invar_vebt(Summary2),M3)
                   => ( ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M3) )
                     => ( ( M3 = N )
                       => ( ( Deg2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M3) )
                         => ( ~ ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary2),X_13)
                           => ~ ! [X3: vEBT_VEBT] :
                                  ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeList2))
                                 => ~ ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X3),X_13) ) ) ) ) ) ) ) ) )
         => ( ! [TreeList2: list_VEBT_VEBT,N: nat,Summary2: vEBT_VEBT,M3: nat,Deg2: nat] :
                ( ( A1 = vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2,Summary2) )
               => ( ( A22 = Deg2 )
                 => ( ! [X3: vEBT_VEBT] :
                        ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeList2))
                       => aa_nat_o(vEBT_invar_vebt(X3),N) )
                   => ( aa_nat_o(vEBT_invar_vebt(Summary2),M3)
                     => ( ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M3) )
                       => ( ( M3 = aa_nat_nat(suc,N) )
                         => ( ( Deg2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M3) )
                           => ( ~ ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary2),X_13)
                             => ~ ! [X3: vEBT_VEBT] :
                                    ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeList2))
                                   => ~ ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X3),X_13) ) ) ) ) ) ) ) ) )
           => ( ! [TreeList2: list_VEBT_VEBT,N: nat,Summary2: vEBT_VEBT,M3: nat,Deg2: nat,Mi: nat,Ma: nat] :
                  ( ( A1 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg2,TreeList2,Summary2) )
                 => ( ( A22 = Deg2 )
                   => ( ! [X3: vEBT_VEBT] :
                          ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeList2))
                         => aa_nat_o(vEBT_invar_vebt(X3),N) )
                     => ( aa_nat_o(vEBT_invar_vebt(Summary2),M3)
                       => ( ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M3) )
                         => ( ( M3 = N )
                           => ( ( Deg2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M3) )
                             => ( ! [I4: nat] :
                                    ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M3))
                                   => ( ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList2,I4)),X_12)
                                    <=> aa_nat_o(vEBT_V8194947554948674370ptions(Summary2),I4) ) )
                               => ( ( ( Mi = Ma )
                                   => ! [X3: vEBT_VEBT] :
                                        ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeList2))
                                       => ~ ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X3),X_13) ) )
                                 => ( 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))),Deg2))
                                     => ~ ( ( Mi != Ma )
                                         => ! [I4: nat] :
                                              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M3))
                                             => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma),N) = I4 )
                                                 => aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList2,I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma),N)) )
                                                & ! [X3: nat] :
                                                    ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),N) = I4 )
                                                      & aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList2,I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),N)) )
                                                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi),X3)
                                                      & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Ma) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
             => ~ ! [TreeList2: list_VEBT_VEBT,N: nat,Summary2: vEBT_VEBT,M3: nat,Deg2: nat,Mi: nat,Ma: nat] :
                    ( ( A1 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg2,TreeList2,Summary2) )
                   => ( ( A22 = Deg2 )
                     => ( ! [X3: vEBT_VEBT] :
                            ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeList2))
                           => aa_nat_o(vEBT_invar_vebt(X3),N) )
                       => ( aa_nat_o(vEBT_invar_vebt(Summary2),M3)
                         => ( ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M3) )
                           => ( ( M3 = aa_nat_nat(suc,N) )
                             => ( ( Deg2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M3) )
                               => ( ! [I4: nat] :
                                      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M3))
                                     => ( ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList2,I4)),X_12)
                                      <=> aa_nat_o(vEBT_V8194947554948674370ptions(Summary2),I4) ) )
                                 => ( ( ( Mi = Ma )
                                     => ! [X3: vEBT_VEBT] :
                                          ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeList2))
                                         => ~ ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X3),X_13) ) )
                                   => ( 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))),Deg2))
                                       => ~ ( ( Mi != Ma )
                                           => ! [I4: nat] :
                                                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M3))
                                               => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma),N) = I4 )
                                                   => aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList2,I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma),N)) )
                                                  & ! [X3: nat] :
                                                      ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),N) = I4 )
                                                        & aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList2,I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),N)) )
                                                     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi),X3)
                                                        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Ma) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.cases
tff(fact_3932_invar__vebt_Osimps,axiom,
    ! [A1: vEBT_VEBT,A22: nat] :
      ( aa_nat_o(vEBT_invar_vebt(A1),A22)
    <=> ( ( ? [A: $o,B: $o] : A1 = vEBT_Leaf((A),(B))
          & ( A22 = aa_nat_nat(suc,zero_zero_nat) ) )
        | ? [TreeList3: list_VEBT_VEBT,N4: nat,Summary3: vEBT_VEBT] :
            ( ( A1 = vEBT_Node(none_P5556105721700978146at_nat,A22,TreeList3,Summary3) )
            & ! [X4: vEBT_VEBT] :
                ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList3))
               => aa_nat_o(vEBT_invar_vebt(X4),N4) )
            & aa_nat_o(vEBT_invar_vebt(Summary3),N4)
            & ( size_s6755466524823107622T_VEBT(TreeList3) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N4) )
            & ( A22 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N4),N4) )
            & ~ ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary3),X_12)
            & ! [X4: vEBT_VEBT] :
                ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList3))
               => ~ ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X4),X_12) ) )
        | ? [TreeList3: list_VEBT_VEBT,N4: nat,Summary3: vEBT_VEBT] :
            ( ( A1 = vEBT_Node(none_P5556105721700978146at_nat,A22,TreeList3,Summary3) )
            & ! [X4: vEBT_VEBT] :
                ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList3))
               => aa_nat_o(vEBT_invar_vebt(X4),N4) )
            & aa_nat_o(vEBT_invar_vebt(Summary3),aa_nat_nat(suc,N4))
            & ( size_s6755466524823107622T_VEBT(TreeList3) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,N4)) )
            & ( A22 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N4),aa_nat_nat(suc,N4)) )
            & ~ ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary3),X_12)
            & ! [X4: vEBT_VEBT] :
                ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList3))
               => ~ ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X4),X_12) ) )
        | ? [TreeList3: list_VEBT_VEBT,N4: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
            ( ( A1 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),A22,TreeList3,Summary3) )
            & ! [X4: vEBT_VEBT] :
                ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList3))
               => aa_nat_o(vEBT_invar_vebt(X4),N4) )
            & aa_nat_o(vEBT_invar_vebt(Summary3),N4)
            & ( size_s6755466524823107622T_VEBT(TreeList3) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N4) )
            & ( A22 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N4),N4) )
            & ! [I3: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N4))
               => ( ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList3,I3)),X_12)
                <=> aa_nat_o(vEBT_V8194947554948674370ptions(Summary3),I3) ) )
            & ( ( Mi3 = Ma3 )
             => ! [X4: vEBT_VEBT] :
                  ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList3))
                 => ~ ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X4),X_12) ) )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mi3),Ma3)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A22))
            & ( ( Mi3 != Ma3 )
             => ! [I3: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N4))
                 => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma3),N4) = I3 )
                     => aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList3,I3)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma3),N4)) )
                    & ! [X4: nat] :
                        ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X4),N4) = I3 )
                          & aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList3,I3)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X4),N4)) )
                       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi3),X4)
                          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X4),Ma3) ) ) ) ) ) )
        | ? [TreeList3: list_VEBT_VEBT,N4: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
            ( ( A1 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi3),Ma3)),A22,TreeList3,Summary3) )
            & ! [X4: vEBT_VEBT] :
                ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList3))
               => aa_nat_o(vEBT_invar_vebt(X4),N4) )
            & aa_nat_o(vEBT_invar_vebt(Summary3),aa_nat_nat(suc,N4))
            & ( size_s6755466524823107622T_VEBT(TreeList3) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,N4)) )
            & ( A22 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N4),aa_nat_nat(suc,N4)) )
            & ! [I3: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,N4)))
               => ( ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList3,I3)),X_12)
                <=> aa_nat_o(vEBT_V8194947554948674370ptions(Summary3),I3) ) )
            & ( ( Mi3 = Ma3 )
             => ! [X4: vEBT_VEBT] :
                  ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList3))
                 => ~ ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X4),X_12) ) )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mi3),Ma3)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A22))
            & ( ( Mi3 != Ma3 )
             => ! [I3: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,N4)))
                 => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma3),N4) = I3 )
                     => aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList3,I3)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma3),N4)) )
                    & ! [X4: nat] :
                        ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X4),N4) = I3 )
                          & aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList3,I3)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X4),N4)) )
                       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi3),X4)
                          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X4),Ma3) ) ) ) ) ) ) ) ) ).

% invar_vebt.simps
tff(fact_3933_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_3934_finite__atLeastAtMost__int,axiom,
    ! [L: int,U: int] : aa_set_int_o(finite_finite_int,set_or1266510415728281911st_int(L,U)) ).

% finite_atLeastAtMost_int
tff(fact_3935_mult__cancel__right,axiom,
    ! [Aa: real,C: real,Ba: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),C) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Ba),C) )
    <=> ( ( C = zero_zero_real )
        | ( Aa = Ba ) ) ) ).

% mult_cancel_right
tff(fact_3936_mult__cancel__right,axiom,
    ! [Aa: rat,C: rat,Ba: rat] :
      ( ( aa_rat_rat(times_times_rat(Aa),C) = aa_rat_rat(times_times_rat(Ba),C) )
    <=> ( ( C = zero_zero_rat )
        | ( Aa = Ba ) ) ) ).

% mult_cancel_right
tff(fact_3937_mult__cancel__right,axiom,
    ! [Aa: nat,C: nat,Ba: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),C) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ba),C) )
    <=> ( ( C = zero_zero_nat )
        | ( Aa = Ba ) ) ) ).

% mult_cancel_right
tff(fact_3938_mult__cancel__right,axiom,
    ! [Aa: int,C: int,Ba: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,Aa),C) = aa_int_int(aa_int_fun_int_int(times_times_int,Ba),C) )
    <=> ( ( C = zero_zero_int )
        | ( Aa = Ba ) ) ) ).

% mult_cancel_right
tff(fact_3939_mult__cancel__left,axiom,
    ! [C: real,Aa: real,Ba: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),Aa) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),Ba) )
    <=> ( ( C = zero_zero_real )
        | ( Aa = Ba ) ) ) ).

% mult_cancel_left
tff(fact_3940_mult__cancel__left,axiom,
    ! [C: rat,Aa: rat,Ba: rat] :
      ( ( aa_rat_rat(times_times_rat(C),Aa) = aa_rat_rat(times_times_rat(C),Ba) )
    <=> ( ( C = zero_zero_rat )
        | ( Aa = Ba ) ) ) ).

% mult_cancel_left
tff(fact_3941_mult__cancel__left,axiom,
    ! [C: nat,Aa: nat,Ba: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),Aa) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),Ba) )
    <=> ( ( C = zero_zero_nat )
        | ( Aa = Ba ) ) ) ).

% mult_cancel_left
tff(fact_3942_mult__cancel__left,axiom,
    ! [C: int,Aa: int,Ba: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,C),Aa) = aa_int_int(aa_int_fun_int_int(times_times_int,C),Ba) )
    <=> ( ( C = zero_zero_int )
        | ( Aa = Ba ) ) ) ).

% mult_cancel_left
tff(fact_3943_mult__eq__0__iff,axiom,
    ! [Aa: real,Ba: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),Ba) = zero_zero_real )
    <=> ( ( Aa = zero_zero_real )
        | ( Ba = zero_zero_real ) ) ) ).

% mult_eq_0_iff
tff(fact_3944_mult__eq__0__iff,axiom,
    ! [Aa: rat,Ba: rat] :
      ( ( aa_rat_rat(times_times_rat(Aa),Ba) = zero_zero_rat )
    <=> ( ( Aa = zero_zero_rat )
        | ( Ba = zero_zero_rat ) ) ) ).

% mult_eq_0_iff
tff(fact_3945_mult__eq__0__iff,axiom,
    ! [Aa: nat,Ba: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),Ba) = zero_zero_nat )
    <=> ( ( Aa = zero_zero_nat )
        | ( Ba = zero_zero_nat ) ) ) ).

% mult_eq_0_iff
tff(fact_3946_mult__eq__0__iff,axiom,
    ! [Aa: int,Ba: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,Aa),Ba) = zero_zero_int )
    <=> ( ( Aa = zero_zero_int )
        | ( Ba = zero_zero_int ) ) ) ).

% mult_eq_0_iff
tff(fact_3947_mult__zero__right,axiom,
    ! [Aa: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),zero_zero_real) = zero_zero_real ).

% mult_zero_right
tff(fact_3948_mult__zero__right,axiom,
    ! [Aa: rat] : aa_rat_rat(times_times_rat(Aa),zero_zero_rat) = zero_zero_rat ).

% mult_zero_right
tff(fact_3949_mult__zero__right,axiom,
    ! [Aa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),zero_zero_nat) = zero_zero_nat ).

% mult_zero_right
tff(fact_3950_mult__zero__right,axiom,
    ! [Aa: int] : aa_int_int(aa_int_fun_int_int(times_times_int,Aa),zero_zero_int) = zero_zero_int ).

% mult_zero_right
tff(fact_3951_mult__zero__left,axiom,
    ! [Aa: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,zero_zero_real),Aa) = zero_zero_real ).

% mult_zero_left
tff(fact_3952_mult__zero__left,axiom,
    ! [Aa: rat] : aa_rat_rat(times_times_rat(zero_zero_rat),Aa) = zero_zero_rat ).

% mult_zero_left
tff(fact_3953_mult__zero__left,axiom,
    ! [Aa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,zero_zero_nat),Aa) = zero_zero_nat ).

% mult_zero_left
tff(fact_3954_mult__zero__left,axiom,
    ! [Aa: int] : aa_int_int(aa_int_fun_int_int(times_times_int,zero_zero_int),Aa) = zero_zero_int ).

% mult_zero_left
tff(fact_3955_numeral__times__numeral,axiom,
    ! [M: num,Na: num] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,M)),aa_num_rat(numeral_numeral_rat,Na)) = aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(times_times_num,M),Na)) ).

% numeral_times_numeral
tff(fact_3956_numeral__times__numeral,axiom,
    ! [M: num,Na: num] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,Na)) = aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(times_times_num,M),Na)) ).

% numeral_times_numeral
tff(fact_3957_numeral__times__numeral,axiom,
    ! [M: num,Na: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,M)),aa_num_real(numeral_numeral_real,Na)) = aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(times_times_num,M),Na)) ).

% numeral_times_numeral
tff(fact_3958_numeral__times__numeral,axiom,
    ! [M: num,Na: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,Na)) = aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(times_times_num,M),Na)) ).

% numeral_times_numeral
tff(fact_3959_numeral__times__numeral,axiom,
    ! [M: num,Na: num] : times_7803423173614009249d_enat(numera1916890842035813515d_enat(M),numera1916890842035813515d_enat(Na)) = numera1916890842035813515d_enat(aa_num_num(aa_num_fun_num_num(times_times_num,M),Na)) ).

% numeral_times_numeral
tff(fact_3960_numeral__times__numeral,axiom,
    ! [M: num,Na: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)),aa_num_Code_integer(numera6620942414471956472nteger,Na)) = aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(aa_num_fun_num_num(times_times_num,M),Na)) ).

% numeral_times_numeral
tff(fact_3961_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W2: 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,W2)),Z)) = aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(times_times_num,V),W2))),Z) ).

% mult_numeral_left_semiring_numeral
tff(fact_3962_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W2: 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,W2)),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(times_times_num,V),W2))),Z) ).

% mult_numeral_left_semiring_numeral
tff(fact_3963_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W2: num,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,V)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W2)),Z)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(times_times_num,V),W2))),Z) ).

% mult_numeral_left_semiring_numeral
tff(fact_3964_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W2: 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,W2)),Z)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(times_times_num,V),W2))),Z) ).

% mult_numeral_left_semiring_numeral
tff(fact_3965_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W2: num,Z: extended_enat] : times_7803423173614009249d_enat(numera1916890842035813515d_enat(V),times_7803423173614009249d_enat(numera1916890842035813515d_enat(W2),Z)) = times_7803423173614009249d_enat(numera1916890842035813515d_enat(aa_num_num(aa_num_fun_num_num(times_times_num,V),W2)),Z) ).

% mult_numeral_left_semiring_numeral
tff(fact_3966_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W2: num,Z: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,V)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,W2)),Z)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(aa_num_fun_num_num(times_times_num,V),W2))),Z) ).

% mult_numeral_left_semiring_numeral
tff(fact_3967_mult_Oright__neutral,axiom,
    ! [Aa: complex] : aa_complex_complex(times_times_complex(Aa),one_one_complex) = Aa ).

% mult.right_neutral
tff(fact_3968_mult_Oright__neutral,axiom,
    ! [Aa: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),one_one_real) = Aa ).

% mult.right_neutral
tff(fact_3969_mult_Oright__neutral,axiom,
    ! [Aa: rat] : aa_rat_rat(times_times_rat(Aa),one_one_rat) = Aa ).

% mult.right_neutral
tff(fact_3970_mult_Oright__neutral,axiom,
    ! [Aa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),one_one_nat) = Aa ).

% mult.right_neutral
tff(fact_3971_mult_Oright__neutral,axiom,
    ! [Aa: int] : aa_int_int(aa_int_fun_int_int(times_times_int,Aa),one_one_int) = Aa ).

% mult.right_neutral
tff(fact_3972_mult__1,axiom,
    ! [Aa: complex] : aa_complex_complex(times_times_complex(one_one_complex),Aa) = Aa ).

% mult_1
tff(fact_3973_mult__1,axiom,
    ! [Aa: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,one_one_real),Aa) = Aa ).

% mult_1
tff(fact_3974_mult__1,axiom,
    ! [Aa: rat] : aa_rat_rat(times_times_rat(one_one_rat),Aa) = Aa ).

% mult_1
tff(fact_3975_mult__1,axiom,
    ! [Aa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,one_one_nat),Aa) = Aa ).

% mult_1
tff(fact_3976_mult__1,axiom,
    ! [Aa: int] : aa_int_int(aa_int_fun_int_int(times_times_int,one_one_int),Aa) = Aa ).

% mult_1
tff(fact_3977_low__inv,axiom,
    ! [Xb: nat,Na: nat,Y: 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))),Na))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))),Xb)),Na) = Xb ) ) ).

% low_inv
tff(fact_3978_mult__minus__left,axiom,
    ! [Aa: real,Ba: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,Aa)),Ba) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),Ba)) ).

% mult_minus_left
tff(fact_3979_mult__minus__left,axiom,
    ! [Aa: int,Ba: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,Aa)),Ba) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Aa),Ba)) ).

% mult_minus_left
tff(fact_3980_mult__minus__left,axiom,
    ! [Aa: code_integer,Ba: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa)),Ba) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Aa),Ba)) ).

% mult_minus_left
tff(fact_3981_mult__minus__left,axiom,
    ! [Aa: rat,Ba: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,Aa)),Ba) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(times_times_rat(Aa),Ba)) ).

% mult_minus_left
tff(fact_3982_minus__mult__minus,axiom,
    ! [Aa: real,Ba: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,Aa)),aa_real_real(uminus_uminus_real,Ba)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),Ba) ).

% minus_mult_minus
tff(fact_3983_minus__mult__minus,axiom,
    ! [Aa: int,Ba: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,Aa)),aa_int_int(uminus_uminus_int,Ba)) = aa_int_int(aa_int_fun_int_int(times_times_int,Aa),Ba) ).

% minus_mult_minus
tff(fact_3984_minus__mult__minus,axiom,
    ! [Aa: code_integer,Ba: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Ba)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Aa),Ba) ).

% minus_mult_minus
tff(fact_3985_minus__mult__minus,axiom,
    ! [Aa: rat,Ba: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,Aa)),aa_rat_rat(uminus_uminus_rat,Ba)) = aa_rat_rat(times_times_rat(Aa),Ba) ).

% minus_mult_minus
tff(fact_3986_mult__minus__right,axiom,
    ! [Aa: real,Ba: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),aa_real_real(uminus_uminus_real,Ba)) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),Ba)) ).

% mult_minus_right
tff(fact_3987_mult__minus__right,axiom,
    ! [Aa: int,Ba: int] : aa_int_int(aa_int_fun_int_int(times_times_int,Aa),aa_int_int(uminus_uminus_int,Ba)) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Aa),Ba)) ).

% mult_minus_right
tff(fact_3988_mult__minus__right,axiom,
    ! [Aa: code_integer,Ba: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Aa),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Ba)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Aa),Ba)) ).

% mult_minus_right
tff(fact_3989_mult__minus__right,axiom,
    ! [Aa: rat,Ba: rat] : aa_rat_rat(times_times_rat(Aa),aa_rat_rat(uminus_uminus_rat,Ba)) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(times_times_rat(Aa),Ba)) ).

% mult_minus_right
tff(fact_3990_mult__cancel2,axiom,
    ! [M: nat,K: nat,Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),K) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),K) )
    <=> ( ( M = Na )
        | ( K = zero_zero_nat ) ) ) ).

% mult_cancel2
tff(fact_3991_mult__cancel1,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Na) )
    <=> ( ( M = Na )
        | ( K = zero_zero_nat ) ) ) ).

% mult_cancel1
tff(fact_3992_mult__0__right,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),zero_zero_nat) = zero_zero_nat ).

% mult_0_right
tff(fact_3993_mult__is__0,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na) = zero_zero_nat )
    <=> ( ( M = zero_zero_nat )
        | ( Na = zero_zero_nat ) ) ) ).

% mult_is_0
tff(fact_3994_nat__1__eq__mult__iff,axiom,
    ! [M: nat,Na: nat] :
      ( ( one_one_nat = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na) )
    <=> ( ( M = one_one_nat )
        & ( Na = one_one_nat ) ) ) ).

% nat_1_eq_mult_iff
tff(fact_3995_nat__mult__eq__1__iff,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na) = one_one_nat )
    <=> ( ( M = one_one_nat )
        & ( Na = one_one_nat ) ) ) ).

% nat_mult_eq_1_iff
tff(fact_3996_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_3997_mult__cancel__left1,axiom,
    ! [C: complex,Ba: complex] :
      ( ( C = aa_complex_complex(times_times_complex(C),Ba) )
    <=> ( ( C = zero_zero_complex )
        | ( Ba = one_one_complex ) ) ) ).

% mult_cancel_left1
tff(fact_3998_mult__cancel__left1,axiom,
    ! [C: real,Ba: real] :
      ( ( C = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),Ba) )
    <=> ( ( C = zero_zero_real )
        | ( Ba = one_one_real ) ) ) ).

% mult_cancel_left1
tff(fact_3999_mult__cancel__left1,axiom,
    ! [C: rat,Ba: rat] :
      ( ( C = aa_rat_rat(times_times_rat(C),Ba) )
    <=> ( ( C = zero_zero_rat )
        | ( Ba = one_one_rat ) ) ) ).

% mult_cancel_left1
tff(fact_4000_mult__cancel__left1,axiom,
    ! [C: int,Ba: int] :
      ( ( C = aa_int_int(aa_int_fun_int_int(times_times_int,C),Ba) )
    <=> ( ( C = zero_zero_int )
        | ( Ba = one_one_int ) ) ) ).

% mult_cancel_left1
tff(fact_4001_mult__cancel__left2,axiom,
    ! [C: complex,Aa: complex] :
      ( ( aa_complex_complex(times_times_complex(C),Aa) = C )
    <=> ( ( C = zero_zero_complex )
        | ( Aa = one_one_complex ) ) ) ).

% mult_cancel_left2
tff(fact_4002_mult__cancel__left2,axiom,
    ! [C: real,Aa: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),Aa) = C )
    <=> ( ( C = zero_zero_real )
        | ( Aa = one_one_real ) ) ) ).

% mult_cancel_left2
tff(fact_4003_mult__cancel__left2,axiom,
    ! [C: rat,Aa: rat] :
      ( ( aa_rat_rat(times_times_rat(C),Aa) = C )
    <=> ( ( C = zero_zero_rat )
        | ( Aa = one_one_rat ) ) ) ).

% mult_cancel_left2
tff(fact_4004_mult__cancel__left2,axiom,
    ! [C: int,Aa: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,C),Aa) = C )
    <=> ( ( C = zero_zero_int )
        | ( Aa = one_one_int ) ) ) ).

% mult_cancel_left2
tff(fact_4005_mult__cancel__right1,axiom,
    ! [C: complex,Ba: complex] :
      ( ( C = aa_complex_complex(times_times_complex(Ba),C) )
    <=> ( ( C = zero_zero_complex )
        | ( Ba = one_one_complex ) ) ) ).

% mult_cancel_right1
tff(fact_4006_mult__cancel__right1,axiom,
    ! [C: real,Ba: real] :
      ( ( C = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Ba),C) )
    <=> ( ( C = zero_zero_real )
        | ( Ba = one_one_real ) ) ) ).

% mult_cancel_right1
tff(fact_4007_mult__cancel__right1,axiom,
    ! [C: rat,Ba: rat] :
      ( ( C = aa_rat_rat(times_times_rat(Ba),C) )
    <=> ( ( C = zero_zero_rat )
        | ( Ba = one_one_rat ) ) ) ).

% mult_cancel_right1
tff(fact_4008_mult__cancel__right1,axiom,
    ! [C: int,Ba: int] :
      ( ( C = aa_int_int(aa_int_fun_int_int(times_times_int,Ba),C) )
    <=> ( ( C = zero_zero_int )
        | ( Ba = one_one_int ) ) ) ).

% mult_cancel_right1
tff(fact_4009_mult__cancel__right2,axiom,
    ! [Aa: complex,C: complex] :
      ( ( aa_complex_complex(times_times_complex(Aa),C) = C )
    <=> ( ( C = zero_zero_complex )
        | ( Aa = one_one_complex ) ) ) ).

% mult_cancel_right2
tff(fact_4010_mult__cancel__right2,axiom,
    ! [Aa: real,C: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),C) = C )
    <=> ( ( C = zero_zero_real )
        | ( Aa = one_one_real ) ) ) ).

% mult_cancel_right2
tff(fact_4011_mult__cancel__right2,axiom,
    ! [Aa: rat,C: rat] :
      ( ( aa_rat_rat(times_times_rat(Aa),C) = C )
    <=> ( ( C = zero_zero_rat )
        | ( Aa = one_one_rat ) ) ) ).

% mult_cancel_right2
tff(fact_4012_mult__cancel__right2,axiom,
    ! [Aa: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,Aa),C) = C )
    <=> ( ( C = zero_zero_int )
        | ( Aa = one_one_int ) ) ) ).

% mult_cancel_right2
tff(fact_4013_sum__squares__eq__zero__iff,axiom,
    ! [Xb: real,Y: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Xb)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Y)) = zero_zero_real )
    <=> ( ( Xb = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_4014_sum__squares__eq__zero__iff,axiom,
    ! [Xb: rat,Y: rat] :
      ( ( aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(Xb),Xb)),aa_rat_rat(times_times_rat(Y),Y)) = zero_zero_rat )
    <=> ( ( Xb = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_4015_sum__squares__eq__zero__iff,axiom,
    ! [Xb: int,Y: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Xb),Xb)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y)) = zero_zero_int )
    <=> ( ( Xb = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_4016_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C: rat,Aa: rat,Ba: rat] :
      ( ( C != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(Aa),C)),aa_rat_rat(times_times_rat(C),Ba)) = aa_rat_rat(divide_divide_rat(Aa),Ba) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
tff(fact_4017_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C: real,Aa: real,Ba: real] :
      ( ( C != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),Ba)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Aa),Ba) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
tff(fact_4018_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C: rat,Aa: rat,Ba: rat] :
      ( ( C != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(Aa),C)),aa_rat_rat(times_times_rat(Ba),C)) = aa_rat_rat(divide_divide_rat(Aa),Ba) ) ) ).

% nonzero_mult_divide_mult_cancel_right
tff(fact_4019_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C: real,Aa: real,Ba: real] :
      ( ( C != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Ba),C)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Aa),Ba) ) ) ).

% nonzero_mult_divide_mult_cancel_right
tff(fact_4020_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C: rat,Aa: rat,Ba: rat] :
      ( ( C != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(C),Aa)),aa_rat_rat(times_times_rat(Ba),C)) = aa_rat_rat(divide_divide_rat(Aa),Ba) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
tff(fact_4021_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C: real,Aa: real,Ba: real] :
      ( ( C != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),Aa)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Ba),C)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Aa),Ba) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
tff(fact_4022_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C: rat,Aa: rat,Ba: rat] :
      ( ( C != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(C),Aa)),aa_rat_rat(times_times_rat(C),Ba)) = aa_rat_rat(divide_divide_rat(Aa),Ba) ) ) ).

% nonzero_mult_divide_mult_cancel_left
tff(fact_4023_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C: real,Aa: real,Ba: real] :
      ( ( C != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),Aa)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),Ba)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Aa),Ba) ) ) ).

% nonzero_mult_divide_mult_cancel_left
tff(fact_4024_mult__divide__mult__cancel__left__if,axiom,
    ! [C: rat,Aa: rat,Ba: rat] :
      aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(C),Aa)),aa_rat_rat(times_times_rat(C),Ba)) = $ite(C = zero_zero_rat,zero_zero_rat,aa_rat_rat(divide_divide_rat(Aa),Ba)) ).

% mult_divide_mult_cancel_left_if
tff(fact_4025_mult__divide__mult__cancel__left__if,axiom,
    ! [C: real,Aa: real,Ba: real] :
      aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),Aa)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),Ba)) = $ite(C = zero_zero_real,zero_zero_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Aa),Ba)) ).

% mult_divide_mult_cancel_left_if
tff(fact_4026_div__mult__mult1,axiom,
    ! [C: nat,Aa: nat,Ba: 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),Aa)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),Ba)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Aa),Ba) ) ) ).

% div_mult_mult1
tff(fact_4027_div__mult__mult1,axiom,
    ! [C: int,Aa: int,Ba: 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),Aa)),aa_int_int(aa_int_fun_int_int(times_times_int,C),Ba)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),Ba) ) ) ).

% div_mult_mult1
tff(fact_4028_div__mult__mult1,axiom,
    ! [C: code_integer,Aa: code_integer,Ba: code_integer] :
      ( ( C != zero_z3403309356797280102nteger )
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),Aa)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),Ba)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Aa),Ba) ) ) ).

% div_mult_mult1
tff(fact_4029_div__mult__mult2,axiom,
    ! [C: nat,Aa: nat,Ba: 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,Aa),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ba),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Aa),Ba) ) ) ).

% div_mult_mult2
tff(fact_4030_div__mult__mult2,axiom,
    ! [C: int,Aa: int,Ba: 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,Aa),C)),aa_int_int(aa_int_fun_int_int(times_times_int,Ba),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),Ba) ) ) ).

% div_mult_mult2
tff(fact_4031_div__mult__mult2,axiom,
    ! [C: code_integer,Aa: code_integer,Ba: code_integer] :
      ( ( C != zero_z3403309356797280102nteger )
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Aa),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Ba),C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Aa),Ba) ) ) ).

% div_mult_mult2
tff(fact_4032_div__mult__mult1__if,axiom,
    ! [C: nat,Aa: nat,Ba: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),Aa)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),Ba)) = $ite(C = zero_zero_nat,zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Aa),Ba)) ).

% div_mult_mult1_if
tff(fact_4033_div__mult__mult1__if,axiom,
    ! [C: int,Aa: int,Ba: int] :
      aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),Aa)),aa_int_int(aa_int_fun_int_int(times_times_int,C),Ba)) = $ite(C = zero_zero_int,zero_zero_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),Ba)) ).

% div_mult_mult1_if
tff(fact_4034_div__mult__mult1__if,axiom,
    ! [C: code_integer,Aa: code_integer,Ba: code_integer] :
      aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),Aa)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),Ba)) = $ite(C = zero_z3403309356797280102nteger,zero_z3403309356797280102nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Aa),Ba)) ).

% div_mult_mult1_if
tff(fact_4035_nonzero__mult__div__cancel__left,axiom,
    ! [Aa: rat,Ba: rat] :
      ( ( Aa != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(Aa),Ba)),Aa) = Ba ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_4036_nonzero__mult__div__cancel__left,axiom,
    ! [Aa: nat,Ba: nat] :
      ( ( Aa != 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,Aa),Ba)),Aa) = Ba ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_4037_nonzero__mult__div__cancel__left,axiom,
    ! [Aa: int,Ba: int] :
      ( ( Aa != 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),Ba)),Aa) = Ba ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_4038_nonzero__mult__div__cancel__left,axiom,
    ! [Aa: real,Ba: real] :
      ( ( Aa != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),Ba)),Aa) = Ba ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_4039_nonzero__mult__div__cancel__left,axiom,
    ! [Aa: code_integer,Ba: code_integer] :
      ( ( Aa != zero_z3403309356797280102nteger )
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Aa),Ba)),Aa) = Ba ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_4040_nonzero__mult__div__cancel__right,axiom,
    ! [Ba: rat,Aa: rat] :
      ( ( Ba != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(Aa),Ba)),Ba) = Aa ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_4041_nonzero__mult__div__cancel__right,axiom,
    ! [Ba: nat,Aa: nat] :
      ( ( Ba != 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,Aa),Ba)),Ba) = Aa ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_4042_nonzero__mult__div__cancel__right,axiom,
    ! [Ba: int,Aa: int] :
      ( ( Ba != 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),Ba)),Ba) = Aa ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_4043_nonzero__mult__div__cancel__right,axiom,
    ! [Ba: real,Aa: real] :
      ( ( Ba != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),Ba)),Ba) = Aa ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_4044_nonzero__mult__div__cancel__right,axiom,
    ! [Ba: code_integer,Aa: code_integer] :
      ( ( Ba != zero_z3403309356797280102nteger )
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Aa),Ba)),Ba) = Aa ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_4045_distrib__left__numeral,axiom,
    ! [V: num,Ba: rat,C: rat] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,V)),aa_rat_rat(plus_plus_rat(Ba),C)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,V)),Ba)),aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,V)),C)) ).

% distrib_left_numeral
tff(fact_4046_distrib__left__numeral,axiom,
    ! [V: num,Ba: 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,Ba),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)),Ba)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,V)),C)) ).

% distrib_left_numeral
tff(fact_4047_distrib__left__numeral,axiom,
    ! [V: num,Ba: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,V)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Ba),C)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,V)),Ba)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,V)),C)) ).

% distrib_left_numeral
tff(fact_4048_distrib__left__numeral,axiom,
    ! [V: num,Ba: 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,Ba),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)),Ba)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,V)),C)) ).

% distrib_left_numeral
tff(fact_4049_distrib__left__numeral,axiom,
    ! [V: num,Ba: extended_enat,C: extended_enat] : times_7803423173614009249d_enat(numera1916890842035813515d_enat(V),plus_p3455044024723400733d_enat(Ba,C)) = plus_p3455044024723400733d_enat(times_7803423173614009249d_enat(numera1916890842035813515d_enat(V),Ba),times_7803423173614009249d_enat(numera1916890842035813515d_enat(V),C)) ).

% distrib_left_numeral
tff(fact_4050_distrib__left__numeral,axiom,
    ! [V: num,Ba: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,V)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Ba),C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,V)),Ba)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,V)),C)) ).

% distrib_left_numeral
tff(fact_4051_distrib__right__numeral,axiom,
    ! [Aa: rat,Ba: rat,V: num] : aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(Aa),Ba)),aa_num_rat(numeral_numeral_rat,V)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(Aa),aa_num_rat(numeral_numeral_rat,V))),aa_rat_rat(times_times_rat(Ba),aa_num_rat(numeral_numeral_rat,V))) ).

% distrib_right_numeral
tff(fact_4052_distrib__right__numeral,axiom,
    ! [Aa: nat,Ba: 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,Aa),Ba)),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,Aa),aa_num_nat(numeral_numeral_nat,V))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ba),aa_num_nat(numeral_numeral_nat,V))) ).

% distrib_right_numeral
tff(fact_4053_distrib__right__numeral,axiom,
    ! [Aa: real,Ba: real,V: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),Ba)),aa_num_real(numeral_numeral_real,V)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),aa_num_real(numeral_numeral_real,V))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Ba),aa_num_real(numeral_numeral_real,V))) ).

% distrib_right_numeral
tff(fact_4054_distrib__right__numeral,axiom,
    ! [Aa: int,Ba: 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,Aa),Ba)),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,Aa),aa_num_int(numeral_numeral_int,V))),aa_int_int(aa_int_fun_int_int(times_times_int,Ba),aa_num_int(numeral_numeral_int,V))) ).

% distrib_right_numeral
tff(fact_4055_distrib__right__numeral,axiom,
    ! [Aa: extended_enat,Ba: extended_enat,V: num] : times_7803423173614009249d_enat(plus_p3455044024723400733d_enat(Aa,Ba),numera1916890842035813515d_enat(V)) = plus_p3455044024723400733d_enat(times_7803423173614009249d_enat(Aa,numera1916890842035813515d_enat(V)),times_7803423173614009249d_enat(Ba,numera1916890842035813515d_enat(V))) ).

% distrib_right_numeral
tff(fact_4056_distrib__right__numeral,axiom,
    ! [Aa: code_integer,Ba: code_integer,V: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Aa),Ba)),aa_num_Code_integer(numera6620942414471956472nteger,V)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Aa),aa_num_Code_integer(numera6620942414471956472nteger,V))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Ba),aa_num_Code_integer(numera6620942414471956472nteger,V))) ).

% distrib_right_numeral
tff(fact_4057_left__diff__distrib__numeral,axiom,
    ! [Aa: rat,Ba: rat,V: num] : aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(Aa),Ba)),aa_num_rat(numeral_numeral_rat,V)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(Aa),aa_num_rat(numeral_numeral_rat,V))),aa_rat_rat(times_times_rat(Ba),aa_num_rat(numeral_numeral_rat,V))) ).

% left_diff_distrib_numeral
tff(fact_4058_left__diff__distrib__numeral,axiom,
    ! [Aa: real,Ba: real,V: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Aa),Ba)),aa_num_real(numeral_numeral_real,V)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),aa_num_real(numeral_numeral_real,V))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Ba),aa_num_real(numeral_numeral_real,V))) ).

% left_diff_distrib_numeral
tff(fact_4059_left__diff__distrib__numeral,axiom,
    ! [Aa: int,Ba: 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,Aa),Ba)),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,Aa),aa_num_int(numeral_numeral_int,V))),aa_int_int(aa_int_fun_int_int(times_times_int,Ba),aa_num_int(numeral_numeral_int,V))) ).

% left_diff_distrib_numeral
tff(fact_4060_left__diff__distrib__numeral,axiom,
    ! [Aa: code_integer,Ba: code_integer,V: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,Aa),Ba)),aa_num_Code_integer(numera6620942414471956472nteger,V)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Aa),aa_num_Code_integer(numera6620942414471956472nteger,V))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Ba),aa_num_Code_integer(numera6620942414471956472nteger,V))) ).

% left_diff_distrib_numeral
tff(fact_4061_right__diff__distrib__numeral,axiom,
    ! [V: num,Ba: rat,C: rat] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,V)),aa_rat_rat(minus_minus_rat(Ba),C)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,V)),Ba)),aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,V)),C)) ).

% right_diff_distrib_numeral
tff(fact_4062_right__diff__distrib__numeral,axiom,
    ! [V: num,Ba: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,V)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Ba),C)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,V)),Ba)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,V)),C)) ).

% right_diff_distrib_numeral
tff(fact_4063_right__diff__distrib__numeral,axiom,
    ! [V: num,Ba: 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,Ba),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)),Ba)),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_4064_right__diff__distrib__numeral,axiom,
    ! [V: num,Ba: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,V)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,Ba),C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,V)),Ba)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,V)),C)) ).

% right_diff_distrib_numeral
tff(fact_4065_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,Na: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,M)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Na))) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(times_times_num,M),Na))) ).

% mult_neg_numeral_simps(3)
tff(fact_4066_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,Na: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Na))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(times_times_num,M),Na))) ).

% mult_neg_numeral_simps(3)
tff(fact_4067_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,Na: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Na))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(aa_num_fun_num_num(times_times_num,M),Na))) ).

% mult_neg_numeral_simps(3)
tff(fact_4068_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,Na: num] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,M)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Na))) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(times_times_num,M),Na))) ).

% mult_neg_numeral_simps(3)
tff(fact_4069_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,Na: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M))),aa_num_real(numeral_numeral_real,Na)) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(times_times_num,M),Na))) ).

% mult_neg_numeral_simps(2)
tff(fact_4070_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,Na: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,Na)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(times_times_num,M),Na))) ).

% mult_neg_numeral_simps(2)
tff(fact_4071_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,Na: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M))),aa_num_Code_integer(numera6620942414471956472nteger,Na)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(aa_num_fun_num_num(times_times_num,M),Na))) ).

% mult_neg_numeral_simps(2)
tff(fact_4072_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,Na: num] : aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))),aa_num_rat(numeral_numeral_rat,Na)) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(times_times_num,M),Na))) ).

% mult_neg_numeral_simps(2)
tff(fact_4073_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,Na: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M))),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Na))) = aa_num_real(numeral_numeral_real,aa_num_num(aa_num_fun_num_num(times_times_num,M),Na)) ).

% mult_neg_numeral_simps(1)
tff(fact_4074_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,Na: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Na))) = aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(times_times_num,M),Na)) ).

% mult_neg_numeral_simps(1)
tff(fact_4075_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,Na: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,M))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Na))) = aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(aa_num_fun_num_num(times_times_num,M),Na)) ).

% mult_neg_numeral_simps(1)
tff(fact_4076_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,Na: num] : aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Na))) = aa_num_rat(numeral_numeral_rat,aa_num_num(aa_num_fun_num_num(times_times_num,M),Na)) ).

% mult_neg_numeral_simps(1)
tff(fact_4077_dvd__mult__cancel__left,axiom,
    ! [C: code_integer,Aa: code_integer,Ba: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),Aa)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),Ba))
    <=> ( ( C = zero_z3403309356797280102nteger )
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),Ba) ) ) ).

% dvd_mult_cancel_left
tff(fact_4078_dvd__mult__cancel__left,axiom,
    ! [C: real,Aa: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),Aa)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),Ba))
    <=> ( ( C = zero_zero_real )
        | aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Aa),Ba) ) ) ).

% dvd_mult_cancel_left
tff(fact_4079_dvd__mult__cancel__left,axiom,
    ! [C: rat,Aa: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(times_times_rat(C),Aa)),aa_rat_rat(times_times_rat(C),Ba))
    <=> ( ( C = zero_zero_rat )
        | aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Aa),Ba) ) ) ).

% dvd_mult_cancel_left
tff(fact_4080_dvd__mult__cancel__left,axiom,
    ! [C: int,Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),Aa)),aa_int_int(aa_int_fun_int_int(times_times_int,C),Ba))
    <=> ( ( C = zero_zero_int )
        | aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),Ba) ) ) ).

% dvd_mult_cancel_left
tff(fact_4081_dvd__mult__cancel__right,axiom,
    ! [Aa: code_integer,C: code_integer,Ba: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Aa),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Ba),C))
    <=> ( ( C = zero_z3403309356797280102nteger )
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),Ba) ) ) ).

% dvd_mult_cancel_right
tff(fact_4082_dvd__mult__cancel__right,axiom,
    ! [Aa: real,C: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Ba),C))
    <=> ( ( C = zero_zero_real )
        | aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Aa),Ba) ) ) ).

% dvd_mult_cancel_right
tff(fact_4083_dvd__mult__cancel__right,axiom,
    ! [Aa: rat,C: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(times_times_rat(Aa),C)),aa_rat_rat(times_times_rat(Ba),C))
    <=> ( ( C = zero_zero_rat )
        | aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Aa),Ba) ) ) ).

% dvd_mult_cancel_right
tff(fact_4084_dvd__mult__cancel__right,axiom,
    ! [Aa: int,C: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,Aa),C)),aa_int_int(aa_int_fun_int_int(times_times_int,Ba),C))
    <=> ( ( C = zero_zero_int )
        | aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),Ba) ) ) ).

% dvd_mult_cancel_right
tff(fact_4085_dvd__times__left__cancel__iff,axiom,
    ! [Aa: code_integer,Ba: code_integer,C: code_integer] :
      ( ( Aa != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Aa),Ba)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Aa),C))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Ba),C) ) ) ).

% dvd_times_left_cancel_iff
tff(fact_4086_dvd__times__left__cancel__iff,axiom,
    ! [Aa: nat,Ba: nat,C: nat] :
      ( ( Aa != 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,Aa),Ba)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),C))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Ba),C) ) ) ).

% dvd_times_left_cancel_iff
tff(fact_4087_dvd__times__left__cancel__iff,axiom,
    ! [Aa: int,Ba: int,C: int] :
      ( ( Aa != 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,Aa),Ba)),aa_int_int(aa_int_fun_int_int(times_times_int,Aa),C))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Ba),C) ) ) ).

% dvd_times_left_cancel_iff
tff(fact_4088_dvd__times__right__cancel__iff,axiom,
    ! [Aa: code_integer,Ba: code_integer,C: code_integer] :
      ( ( Aa != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Ba),Aa)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),Aa))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Ba),C) ) ) ).

% dvd_times_right_cancel_iff
tff(fact_4089_dvd__times__right__cancel__iff,axiom,
    ! [Aa: nat,Ba: nat,C: nat] :
      ( ( Aa != 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,Ba),Aa)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),Aa))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Ba),C) ) ) ).

% dvd_times_right_cancel_iff
tff(fact_4090_dvd__times__right__cancel__iff,axiom,
    ! [Aa: int,Ba: int,C: int] :
      ( ( Aa != 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,Ba),Aa)),aa_int_int(aa_int_fun_int_int(times_times_int,C),Aa))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Ba),C) ) ) ).

% dvd_times_right_cancel_iff
tff(fact_4091_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_4092_mult__minus1,axiom,
    ! [Z: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,one_one_real)),Z) = aa_real_real(uminus_uminus_real,Z) ).

% mult_minus1
tff(fact_4093_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_4094_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_4095_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_4096_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_4097_mult__minus1__right,axiom,
    ! [Z: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,Z) ).

% mult_minus1_right
tff(fact_4098_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_4099_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_4100_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_4101_unit__prod,axiom,
    ! [Aa: code_integer,Ba: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Ba),one_one_Code_integer)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Aa),Ba)),one_one_Code_integer) ) ) ).

% unit_prod
tff(fact_4102_unit__prod,axiom,
    ! [Aa: nat,Ba: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Ba),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,Aa),Ba)),one_one_nat) ) ) ).

% unit_prod
tff(fact_4103_unit__prod,axiom,
    ! [Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Ba),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,Aa),Ba)),one_one_int) ) ) ).

% unit_prod
tff(fact_4104_dvd__add__times__triv__right__iff,axiom,
    ! [Aa: code_integer,Ba: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Ba),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),Aa)))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),Ba) ) ).

% dvd_add_times_triv_right_iff
tff(fact_4105_dvd__add__times__triv__right__iff,axiom,
    ! [Aa: real,Ba: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Aa),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Ba),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),Aa)))
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Aa),Ba) ) ).

% dvd_add_times_triv_right_iff
tff(fact_4106_dvd__add__times__triv__right__iff,axiom,
    ! [Aa: rat,Ba: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Aa),aa_rat_rat(plus_plus_rat(Ba),aa_rat_rat(times_times_rat(C),Aa)))
    <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Aa),Ba) ) ).

% dvd_add_times_triv_right_iff
tff(fact_4107_dvd__add__times__triv__right__iff,axiom,
    ! [Aa: nat,Ba: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ba),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),Aa)))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),Ba) ) ).

% dvd_add_times_triv_right_iff
tff(fact_4108_dvd__add__times__triv__right__iff,axiom,
    ! [Aa: int,Ba: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),aa_int_int(aa_int_fun_int_int(plus_plus_int,Ba),aa_int_int(aa_int_fun_int_int(times_times_int,C),Aa)))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),Ba) ) ).

% dvd_add_times_triv_right_iff
tff(fact_4109_dvd__add__times__triv__left__iff,axiom,
    ! [Aa: code_integer,C: code_integer,Ba: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),Aa)),Ba))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),Ba) ) ).

% dvd_add_times_triv_left_iff
tff(fact_4110_dvd__add__times__triv__left__iff,axiom,
    ! [Aa: real,C: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Aa),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),Aa)),Ba))
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Aa),Ba) ) ).

% dvd_add_times_triv_left_iff
tff(fact_4111_dvd__add__times__triv__left__iff,axiom,
    ! [Aa: rat,C: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Aa),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(C),Aa)),Ba))
    <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Aa),Ba) ) ).

% dvd_add_times_triv_left_iff
tff(fact_4112_dvd__add__times__triv__left__iff,axiom,
    ! [Aa: nat,C: nat,Ba: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),Aa)),Ba))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),Ba) ) ).

% dvd_add_times_triv_left_iff
tff(fact_4113_dvd__add__times__triv__left__iff,axiom,
    ! [Aa: int,C: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),Aa)),Ba))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),Ba) ) ).

% dvd_add_times_triv_left_iff
tff(fact_4114_dvd__mult__div__cancel,axiom,
    ! [Aa: nat,Ba: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),Ba)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Ba),Aa)) = Ba ) ) ).

% dvd_mult_div_cancel
tff(fact_4115_dvd__mult__div__cancel,axiom,
    ! [Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),Ba)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,Aa),aa_int_int(aa_int_fun_int_int(divide_divide_int,Ba),Aa)) = Ba ) ) ).

% dvd_mult_div_cancel
tff(fact_4116_dvd__mult__div__cancel,axiom,
    ! [Aa: code_integer,Ba: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),Ba)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Aa),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Ba),Aa)) = Ba ) ) ).

% dvd_mult_div_cancel
tff(fact_4117_dvd__div__mult__self,axiom,
    ! [Aa: nat,Ba: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),Ba)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Ba),Aa)),Aa) = Ba ) ) ).

% dvd_div_mult_self
tff(fact_4118_dvd__div__mult__self,axiom,
    ! [Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),Ba)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,Ba),Aa)),Aa) = Ba ) ) ).

% dvd_div_mult_self
tff(fact_4119_dvd__div__mult__self,axiom,
    ! [Aa: code_integer,Ba: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),Ba)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Ba),Aa)),Aa) = Ba ) ) ).

% dvd_div_mult_self
tff(fact_4120_mult__eq__1__iff,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ( ( M = aa_nat_nat(suc,zero_zero_nat) )
        & ( Na = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% mult_eq_1_iff
tff(fact_4121_one__eq__mult__iff,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa_nat_nat(suc,zero_zero_nat) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na) )
    <=> ( ( M = aa_nat_nat(suc,zero_zero_nat) )
        & ( Na = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% one_eq_mult_iff
tff(fact_4122_nat__mult__less__cancel__disj,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Na))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ) ).

% nat_mult_less_cancel_disj
tff(fact_4123_mult__less__cancel2,axiom,
    ! [M: nat,K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),K))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ) ).

% mult_less_cancel2
tff(fact_4124_nat__0__less__mult__iff,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ) ).

% nat_0_less_mult_iff
tff(fact_4125_mult__Suc__right,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(suc,Na)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na)) ).

% mult_Suc_right
tff(fact_4126_nat__mult__div__cancel__disj,axiom,
    ! [K: nat,M: nat,Na: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Na)) = $ite(K = zero_zero_nat,zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na)) ).

% nat_mult_div_cancel_disj
tff(fact_4127_nat__mult__dvd__cancel__disj,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Na))
    <=> ( ( K = zero_zero_nat )
        | aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),Na) ) ) ).

% nat_mult_dvd_cancel_disj
tff(fact_4128_option_Ocollapse,axiom,
    ! [Option: option_nat] :
      ( ( Option != none_nat )
     => ( some_nat(the_nat(Option)) = Option ) ) ).

% option.collapse
tff(fact_4129_option_Ocollapse,axiom,
    ! [Option: option4927543243414619207at_nat] :
      ( ( Option != none_P5556105721700978146at_nat )
     => ( some_P7363390416028606310at_nat(the_Pr8591224930841456533at_nat(Option)) = Option ) ) ).

% option.collapse
tff(fact_4130_option_Ocollapse,axiom,
    ! [Option: option_num] :
      ( ( Option != none_num )
     => ( some_num(the_num(Option)) = Option ) ) ).

% option.collapse
tff(fact_4131_le__divide__eq__numeral1_I1_J,axiom,
    ! [Aa: real,Ba: real,W2: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Ba),aa_num_real(numeral_numeral_real,W2)))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),aa_num_real(numeral_numeral_real,W2))),Ba) ) ).

% le_divide_eq_numeral1(1)
tff(fact_4132_le__divide__eq__numeral1_I1_J,axiom,
    ! [Aa: rat,Ba: rat,W2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),aa_rat_rat(divide_divide_rat(Ba),aa_num_rat(numeral_numeral_rat,W2)))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(Aa),aa_num_rat(numeral_numeral_rat,W2))),Ba) ) ).

% le_divide_eq_numeral1(1)
tff(fact_4133_divide__le__eq__numeral1_I1_J,axiom,
    ! [Ba: real,W2: num,Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Ba),aa_num_real(numeral_numeral_real,W2))),Aa)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ba),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),aa_num_real(numeral_numeral_real,W2))) ) ).

% divide_le_eq_numeral1(1)
tff(fact_4134_divide__le__eq__numeral1_I1_J,axiom,
    ! [Ba: rat,W2: num,Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(Ba),aa_num_rat(numeral_numeral_rat,W2))),Aa)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ba),aa_rat_rat(times_times_rat(Aa),aa_num_rat(numeral_numeral_rat,W2))) ) ).

% divide_le_eq_numeral1(1)
tff(fact_4135_divide__eq__eq__numeral1_I1_J,axiom,
    ! [Ba: rat,W2: num,Aa: rat] :
      ( ( aa_rat_rat(divide_divide_rat(Ba),aa_num_rat(numeral_numeral_rat,W2)) = Aa )
    <=> $ite(aa_num_rat(numeral_numeral_rat,W2) != zero_zero_rat,Ba = aa_rat_rat(times_times_rat(Aa),aa_num_rat(numeral_numeral_rat,W2)),Aa = zero_zero_rat) ) ).

% divide_eq_eq_numeral1(1)
tff(fact_4136_divide__eq__eq__numeral1_I1_J,axiom,
    ! [Ba: real,W2: num,Aa: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Ba),aa_num_real(numeral_numeral_real,W2)) = Aa )
    <=> $ite(aa_num_real(numeral_numeral_real,W2) != zero_zero_real,Ba = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),aa_num_real(numeral_numeral_real,W2)),Aa = zero_zero_real) ) ).

% divide_eq_eq_numeral1(1)
tff(fact_4137_eq__divide__eq__numeral1_I1_J,axiom,
    ! [Aa: rat,Ba: rat,W2: num] :
      ( ( Aa = aa_rat_rat(divide_divide_rat(Ba),aa_num_rat(numeral_numeral_rat,W2)) )
    <=> $ite(aa_num_rat(numeral_numeral_rat,W2) != zero_zero_rat,aa_rat_rat(times_times_rat(Aa),aa_num_rat(numeral_numeral_rat,W2)) = Ba,Aa = zero_zero_rat) ) ).

% eq_divide_eq_numeral1(1)
tff(fact_4138_eq__divide__eq__numeral1_I1_J,axiom,
    ! [Aa: real,Ba: real,W2: num] :
      ( ( Aa = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Ba),aa_num_real(numeral_numeral_real,W2)) )
    <=> $ite(aa_num_real(numeral_numeral_real,W2) != zero_zero_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),aa_num_real(numeral_numeral_real,W2)) = Ba,Aa = zero_zero_real) ) ).

% eq_divide_eq_numeral1(1)
tff(fact_4139_less__divide__eq__numeral1_I1_J,axiom,
    ! [Aa: rat,Ba: rat,W2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),aa_rat_rat(divide_divide_rat(Ba),aa_num_rat(numeral_numeral_rat,W2)))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(Aa),aa_num_rat(numeral_numeral_rat,W2))),Ba) ) ).

% less_divide_eq_numeral1(1)
tff(fact_4140_less__divide__eq__numeral1_I1_J,axiom,
    ! [Aa: real,Ba: real,W2: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Ba),aa_num_real(numeral_numeral_real,W2)))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),aa_num_real(numeral_numeral_real,W2))),Ba) ) ).

% less_divide_eq_numeral1(1)
tff(fact_4141_divide__less__eq__numeral1_I1_J,axiom,
    ! [Ba: rat,W2: num,Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(Ba),aa_num_rat(numeral_numeral_rat,W2))),Aa)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ba),aa_rat_rat(times_times_rat(Aa),aa_num_rat(numeral_numeral_rat,W2))) ) ).

% divide_less_eq_numeral1(1)
tff(fact_4142_divide__less__eq__numeral1_I1_J,axiom,
    ! [Ba: real,W2: num,Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Ba),aa_num_real(numeral_numeral_real,W2))),Aa)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Ba),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),aa_num_real(numeral_numeral_real,W2))) ) ).

% divide_less_eq_numeral1(1)
tff(fact_4143_nonzero__divide__mult__cancel__left,axiom,
    ! [Aa: complex,Ba: complex] :
      ( ( Aa != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(Aa),aa_complex_complex(times_times_complex(Aa),Ba)) = aa_complex_complex(divide1717551699836669952omplex(one_one_complex),Ba) ) ) ).

% nonzero_divide_mult_cancel_left
tff(fact_4144_nonzero__divide__mult__cancel__left,axiom,
    ! [Aa: rat,Ba: rat] :
      ( ( Aa != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(Aa),aa_rat_rat(times_times_rat(Aa),Ba)) = aa_rat_rat(divide_divide_rat(one_one_rat),Ba) ) ) ).

% nonzero_divide_mult_cancel_left
tff(fact_4145_nonzero__divide__mult__cancel__left,axiom,
    ! [Aa: real,Ba: real] :
      ( ( Aa != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Aa),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),Ba)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),Ba) ) ) ).

% nonzero_divide_mult_cancel_left
tff(fact_4146_nonzero__divide__mult__cancel__right,axiom,
    ! [Ba: complex,Aa: complex] :
      ( ( Ba != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(Ba),aa_complex_complex(times_times_complex(Aa),Ba)) = aa_complex_complex(divide1717551699836669952omplex(one_one_complex),Aa) ) ) ).

% nonzero_divide_mult_cancel_right
tff(fact_4147_nonzero__divide__mult__cancel__right,axiom,
    ! [Ba: rat,Aa: rat] :
      ( ( Ba != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(Ba),aa_rat_rat(times_times_rat(Aa),Ba)) = aa_rat_rat(divide_divide_rat(one_one_rat),Aa) ) ) ).

% nonzero_divide_mult_cancel_right
tff(fact_4148_nonzero__divide__mult__cancel__right,axiom,
    ! [Ba: real,Aa: real] :
      ( ( Ba != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Ba),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),Ba)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),Aa) ) ) ).

% nonzero_divide_mult_cancel_right
tff(fact_4149_div__mult__self1,axiom,
    ! [Ba: nat,Aa: nat,C: nat] :
      ( ( Ba != 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),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),Ba))),Ba) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Aa),Ba)) ) ) ).

% div_mult_self1
tff(fact_4150_div__mult__self1,axiom,
    ! [Ba: int,Aa: int,C: int] :
      ( ( Ba != 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),aa_int_int(aa_int_fun_int_int(times_times_int,C),Ba))),Ba) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),Ba)) ) ) ).

% div_mult_self1
tff(fact_4151_div__mult__self1,axiom,
    ! [Ba: code_integer,Aa: code_integer,C: code_integer] :
      ( ( Ba != zero_z3403309356797280102nteger )
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Aa),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),Ba))),Ba) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Aa),Ba)) ) ) ).

% div_mult_self1
tff(fact_4152_div__mult__self2,axiom,
    ! [Ba: nat,Aa: nat,C: nat] :
      ( ( Ba != 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),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ba),C))),Ba) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Aa),Ba)) ) ) ).

% div_mult_self2
tff(fact_4153_div__mult__self2,axiom,
    ! [Ba: int,Aa: int,C: int] :
      ( ( Ba != 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),aa_int_int(aa_int_fun_int_int(times_times_int,Ba),C))),Ba) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),Ba)) ) ) ).

% div_mult_self2
tff(fact_4154_div__mult__self2,axiom,
    ! [Ba: code_integer,Aa: code_integer,C: code_integer] :
      ( ( Ba != zero_z3403309356797280102nteger )
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Aa),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Ba),C))),Ba) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Aa),Ba)) ) ) ).

% div_mult_self2
tff(fact_4155_div__mult__self3,axiom,
    ! [Ba: nat,C: nat,Aa: nat] :
      ( ( Ba != 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),Ba)),Aa)),Ba) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Aa),Ba)) ) ) ).

% div_mult_self3
tff(fact_4156_div__mult__self3,axiom,
    ! [Ba: int,C: int,Aa: int] :
      ( ( Ba != 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),Ba)),Aa)),Ba) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),Ba)) ) ) ).

% div_mult_self3
tff(fact_4157_div__mult__self3,axiom,
    ! [Ba: code_integer,C: code_integer,Aa: code_integer] :
      ( ( Ba != zero_z3403309356797280102nteger )
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),Ba)),Aa)),Ba) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Aa),Ba)) ) ) ).

% div_mult_self3
tff(fact_4158_div__mult__self4,axiom,
    ! [Ba: nat,C: nat,Aa: nat] :
      ( ( Ba != 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,Ba),C)),Aa)),Ba) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Aa),Ba)) ) ) ).

% div_mult_self4
tff(fact_4159_div__mult__self4,axiom,
    ! [Ba: int,C: int,Aa: int] :
      ( ( Ba != 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,Ba),C)),Aa)),Ba) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),Ba)) ) ) ).

% div_mult_self4
tff(fact_4160_div__mult__self4,axiom,
    ! [Ba: code_integer,C: code_integer,Aa: code_integer] :
      ( ( Ba != zero_z3403309356797280102nteger )
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Ba),C)),Aa)),Ba) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Aa),Ba)) ) ) ).

% div_mult_self4
tff(fact_4161_minus__one__mult__self,axiom,
    ! [Na: nat] : aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Na)),aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Na)) = one_one_complex ).

% minus_one_mult_self
tff(fact_4162_minus__one__mult__self,axiom,
    ! [Na: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Na)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Na)) = one_one_real ).

% minus_one_mult_self
tff(fact_4163_minus__one__mult__self,axiom,
    ! [Na: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),Na)),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),Na)) = one_one_int ).

% minus_one_mult_self
tff(fact_4164_minus__one__mult__self,axiom,
    ! [Na: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Na)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Na)) = one_one_Code_integer ).

% minus_one_mult_self
tff(fact_4165_minus__one__mult__self,axiom,
    ! [Na: nat] : aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Na)),aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Na)) = one_one_rat ).

% minus_one_mult_self
tff(fact_4166_left__minus__one__mult__self,axiom,
    ! [Na: nat,Aa: complex] : aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Na)),aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Na)),Aa)) = Aa ).

% left_minus_one_mult_self
tff(fact_4167_left__minus__one__mult__self,axiom,
    ! [Na: nat,Aa: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Na)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Na)),Aa)) = Aa ).

% left_minus_one_mult_self
tff(fact_4168_left__minus__one__mult__self,axiom,
    ! [Na: nat,Aa: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),Na)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),Na)),Aa)) = Aa ).

% left_minus_one_mult_self
tff(fact_4169_left__minus__one__mult__self,axiom,
    ! [Na: nat,Aa: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Na)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Na)),Aa)) = Aa ).

% left_minus_one_mult_self
tff(fact_4170_left__minus__one__mult__self,axiom,
    ! [Na: nat,Aa: rat] : aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Na)),aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Na)),Aa)) = Aa ).

% left_minus_one_mult_self
tff(fact_4171_unit__mult__div__div,axiom,
    ! [Aa: nat,Ba: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),one_one_nat)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ba),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),Aa)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Ba),Aa) ) ) ).

% unit_mult_div_div
tff(fact_4172_unit__mult__div__div,axiom,
    ! [Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),one_one_int)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,Ba),aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),Aa)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,Ba),Aa) ) ) ).

% unit_mult_div_div
tff(fact_4173_unit__mult__div__div,axiom,
    ! [Aa: code_integer,Ba: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),one_one_Code_integer)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Ba),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,one_one_Code_integer),Aa)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Ba),Aa) ) ) ).

% unit_mult_div_div
tff(fact_4174_unit__div__mult__self,axiom,
    ! [Aa: nat,Ba: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),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,Ba),Aa)),Aa) = Ba ) ) ).

% unit_div_mult_self
tff(fact_4175_unit__div__mult__self,axiom,
    ! [Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),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,Ba),Aa)),Aa) = Ba ) ) ).

% unit_div_mult_self
tff(fact_4176_unit__div__mult__self,axiom,
    ! [Aa: code_integer,Ba: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),one_one_Code_integer)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Ba),Aa)),Aa) = Ba ) ) ).

% unit_div_mult_self
tff(fact_4177_one__le__mult__iff,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),M)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),Na) ) ) ).

% one_le_mult_iff
tff(fact_4178_nat__mult__le__cancel__disj,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Na))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ) ).

% nat_mult_le_cancel_disj
tff(fact_4179_mult__le__cancel2,axiom,
    ! [M: nat,K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),K))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ) ).

% mult_le_cancel2
tff(fact_4180_div__mult__self1__is__m,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),M)),Na) = M ) ) ).

% div_mult_self1_is_m
tff(fact_4181_div__mult__self__is__m,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na)),Na) = M ) ) ).

% div_mult_self_is_m
tff(fact_4182_le__divide__eq__numeral1_I2_J,axiom,
    ! [Aa: real,Ba: real,W2: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Ba),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ba),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)))) ) ).

% le_divide_eq_numeral1(2)
tff(fact_4183_le__divide__eq__numeral1_I2_J,axiom,
    ! [Aa: rat,Ba: rat,W2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),aa_rat_rat(divide_divide_rat(Ba),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ba),aa_rat_rat(times_times_rat(Aa),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)))) ) ).

% le_divide_eq_numeral1(2)
tff(fact_4184_divide__le__eq__numeral1_I2_J,axiom,
    ! [Ba: real,W2: num,Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Ba),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)))),Aa)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)))),Ba) ) ).

% divide_le_eq_numeral1(2)
tff(fact_4185_divide__le__eq__numeral1_I2_J,axiom,
    ! [Ba: rat,W2: num,Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(Ba),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)))),Aa)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(Aa),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)))),Ba) ) ).

% divide_le_eq_numeral1(2)
tff(fact_4186_divide__eq__eq__numeral1_I2_J,axiom,
    ! [Ba: real,W2: num,Aa: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Ba),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))) = Aa )
    <=> $ite(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)) != zero_zero_real,Ba = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))),Aa = zero_zero_real) ) ).

% divide_eq_eq_numeral1(2)
tff(fact_4187_divide__eq__eq__numeral1_I2_J,axiom,
    ! [Ba: rat,W2: num,Aa: rat] :
      ( ( aa_rat_rat(divide_divide_rat(Ba),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))) = Aa )
    <=> $ite(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)) != zero_zero_rat,Ba = aa_rat_rat(times_times_rat(Aa),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),Aa = zero_zero_rat) ) ).

% divide_eq_eq_numeral1(2)
tff(fact_4188_eq__divide__eq__numeral1_I2_J,axiom,
    ! [Aa: real,Ba: real,W2: num] :
      ( ( Aa = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Ba),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))) )
    <=> $ite(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)) != zero_zero_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))) = Ba,Aa = zero_zero_real) ) ).

% eq_divide_eq_numeral1(2)
tff(fact_4189_eq__divide__eq__numeral1_I2_J,axiom,
    ! [Aa: rat,Ba: rat,W2: num] :
      ( ( Aa = aa_rat_rat(divide_divide_rat(Ba),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))) )
    <=> $ite(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)) != zero_zero_rat,aa_rat_rat(times_times_rat(Aa),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))) = Ba,Aa = zero_zero_rat) ) ).

% eq_divide_eq_numeral1(2)
tff(fact_4190_less__divide__eq__numeral1_I2_J,axiom,
    ! [Aa: real,Ba: real,W2: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Ba),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Ba),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)))) ) ).

% less_divide_eq_numeral1(2)
tff(fact_4191_less__divide__eq__numeral1_I2_J,axiom,
    ! [Aa: rat,Ba: rat,W2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),aa_rat_rat(divide_divide_rat(Ba),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ba),aa_rat_rat(times_times_rat(Aa),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)))) ) ).

% less_divide_eq_numeral1(2)
tff(fact_4192_divide__less__eq__numeral1_I2_J,axiom,
    ! [Ba: real,W2: num,Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Ba),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)))),Aa)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)))),Ba) ) ).

% divide_less_eq_numeral1(2)
tff(fact_4193_divide__less__eq__numeral1_I2_J,axiom,
    ! [Ba: rat,W2: num,Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(Ba),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)))),Aa)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(Aa),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)))),Ba) ) ).

% divide_less_eq_numeral1(2)
tff(fact_4194_nested__mint,axiom,
    ! [Mi2: nat,Ma2: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Na: nat,Va: nat] :
      ( aa_nat_o(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Deg,TreeList,Summary)),Na)
     => ( ( Na = aa_nat_nat(suc,aa_nat_nat(suc,Va)) )
       => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),Mi2)
         => ( ( Ma2 != Mi2 )
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Va),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),the_nat(vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(vEBT_vebt_mint(Summary))))))),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Va),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),size_s6755466524823107622T_VEBT(TreeList)) ) ) ) ) ).

% nested_mint
tff(fact_4195_power__minus1__even,axiom,
    ! [Na: nat] : aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) = one_one_complex ).

% power_minus1_even
tff(fact_4196_power__minus1__even,axiom,
    ! [Na: 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))),Na)) = one_one_real ).

% power_minus1_even
tff(fact_4197_power__minus1__even,axiom,
    ! [Na: nat] : aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) = one_one_int ).

% power_minus1_even
tff(fact_4198_power__minus1__even,axiom,
    ! [Na: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) = one_one_Code_integer ).

% power_minus1_even
tff(fact_4199_power__minus1__even,axiom,
    ! [Na: 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))),Na)) = one_one_rat ).

% power_minus1_even
tff(fact_4200_odd__two__times__div__two__succ,axiom,
    ! [Aa: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Aa)
     => ( 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,Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),one_one_nat) = Aa ) ) ).

% odd_two_times_div_two_succ
tff(fact_4201_odd__two__times__div__two__succ,axiom,
    ! [Aa: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Aa)
     => ( 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,Aa),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),one_one_int) = Aa ) ) ).

% odd_two_times_div_two_succ
tff(fact_4202_odd__two__times__div__two__succ,axiom,
    ! [Aa: code_integer] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Aa)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Aa),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))))),one_one_Code_integer) = Aa ) ) ).

% odd_two_times_div_two_succ
tff(fact_4203_odd__two__times__div__two__nat,axiom,
    ! [Na: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat) ) ) ).

% odd_two_times_div_two_nat
tff(fact_4204_dvd__antisym,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Na),M)
       => ( M = Na ) ) ) ).

% dvd_antisym
tff(fact_4205_euclidean__size__mult,axiom,
    ! [Aa: code_integer,Ba: code_integer] : aa_Code_integer_nat(euclid6377331345833325938nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Aa),Ba)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_Code_integer_nat(euclid6377331345833325938nteger,Aa)),aa_Code_integer_nat(euclid6377331345833325938nteger,Ba)) ).

% euclidean_size_mult
tff(fact_4206_euclidean__size__mult,axiom,
    ! [Aa: int,Ba: int] : aa_int_nat(euclid4774559944035922753ze_int,aa_int_int(aa_int_fun_int_int(times_times_int,Aa),Ba)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_int_nat(euclid4774559944035922753ze_int,Aa)),aa_int_nat(euclid4774559944035922753ze_int,Ba)) ).

% euclidean_size_mult
tff(fact_4207_euclidean__size__mult,axiom,
    ! [Aa: nat,Ba: nat] : aa_nat_nat(euclid4777050414544973029ze_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),Ba)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(euclid4777050414544973029ze_nat,Aa)),aa_nat_nat(euclid4777050414544973029ze_nat,Ba)) ).

% euclidean_size_mult
tff(fact_4208_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [Aa: real,Ba: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),Ba)),C) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Ba),C)) ).

% ab_semigroup_mult_class.mult_ac(1)
tff(fact_4209_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [Aa: rat,Ba: rat,C: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(Aa),Ba)),C) = aa_rat_rat(times_times_rat(Aa),aa_rat_rat(times_times_rat(Ba),C)) ).

% ab_semigroup_mult_class.mult_ac(1)
tff(fact_4210_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [Aa: nat,Ba: 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,Aa),Ba)),C) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ba),C)) ).

% ab_semigroup_mult_class.mult_ac(1)
tff(fact_4211_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [Aa: int,Ba: 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,Aa),Ba)),C) = aa_int_int(aa_int_fun_int_int(times_times_int,Aa),aa_int_int(aa_int_fun_int_int(times_times_int,Ba),C)) ).

% ab_semigroup_mult_class.mult_ac(1)
tff(fact_4212_mult_Oassoc,axiom,
    ! [Aa: real,Ba: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),Ba)),C) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Ba),C)) ).

% mult.assoc
tff(fact_4213_mult_Oassoc,axiom,
    ! [Aa: rat,Ba: rat,C: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(Aa),Ba)),C) = aa_rat_rat(times_times_rat(Aa),aa_rat_rat(times_times_rat(Ba),C)) ).

% mult.assoc
tff(fact_4214_mult_Oassoc,axiom,
    ! [Aa: nat,Ba: 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,Aa),Ba)),C) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ba),C)) ).

% mult.assoc
tff(fact_4215_mult_Oassoc,axiom,
    ! [Aa: int,Ba: 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,Aa),Ba)),C) = aa_int_int(aa_int_fun_int_int(times_times_int,Aa),aa_int_int(aa_int_fun_int_int(times_times_int,Ba),C)) ).

% mult.assoc
tff(fact_4216_mult_Ocommute,axiom,
    ! [Aa: real,Ba: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),Ba) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Ba),Aa) ).

% mult.commute
tff(fact_4217_mult_Ocommute,axiom,
    ! [Aa: rat,Ba: rat] : aa_rat_rat(times_times_rat(Aa),Ba) = aa_rat_rat(times_times_rat(Ba),Aa) ).

% mult.commute
tff(fact_4218_mult_Ocommute,axiom,
    ! [Aa: nat,Ba: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),Ba) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ba),Aa) ).

% mult.commute
tff(fact_4219_mult_Ocommute,axiom,
    ! [Aa: int,Ba: int] : aa_int_int(aa_int_fun_int_int(times_times_int,Aa),Ba) = aa_int_int(aa_int_fun_int_int(times_times_int,Ba),Aa) ).

% mult.commute
tff(fact_4220_mult_Oleft__commute,axiom,
    ! [Ba: real,Aa: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Ba),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),C)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Ba),C)) ).

% mult.left_commute
tff(fact_4221_mult_Oleft__commute,axiom,
    ! [Ba: rat,Aa: rat,C: rat] : aa_rat_rat(times_times_rat(Ba),aa_rat_rat(times_times_rat(Aa),C)) = aa_rat_rat(times_times_rat(Aa),aa_rat_rat(times_times_rat(Ba),C)) ).

% mult.left_commute
tff(fact_4222_mult_Oleft__commute,axiom,
    ! [Ba: nat,Aa: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ba),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),C)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ba),C)) ).

% mult.left_commute
tff(fact_4223_mult_Oleft__commute,axiom,
    ! [Ba: int,Aa: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,Ba),aa_int_int(aa_int_fun_int_int(times_times_int,Aa),C)) = aa_int_int(aa_int_fun_int_int(times_times_int,Aa),aa_int_int(aa_int_fun_int_int(times_times_int,Ba),C)) ).

% mult.left_commute
tff(fact_4224_mult__right__cancel,axiom,
    ! [C: real,Aa: real,Ba: real] :
      ( ( C != zero_zero_real )
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),C) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Ba),C) )
      <=> ( Aa = Ba ) ) ) ).

% mult_right_cancel
tff(fact_4225_mult__right__cancel,axiom,
    ! [C: rat,Aa: rat,Ba: rat] :
      ( ( C != zero_zero_rat )
     => ( ( aa_rat_rat(times_times_rat(Aa),C) = aa_rat_rat(times_times_rat(Ba),C) )
      <=> ( Aa = Ba ) ) ) ).

% mult_right_cancel
tff(fact_4226_mult__right__cancel,axiom,
    ! [C: nat,Aa: nat,Ba: nat] :
      ( ( C != zero_zero_nat )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),C) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ba),C) )
      <=> ( Aa = Ba ) ) ) ).

% mult_right_cancel
tff(fact_4227_mult__right__cancel,axiom,
    ! [C: int,Aa: int,Ba: int] :
      ( ( C != zero_zero_int )
     => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,Aa),C) = aa_int_int(aa_int_fun_int_int(times_times_int,Ba),C) )
      <=> ( Aa = Ba ) ) ) ).

% mult_right_cancel
tff(fact_4228_mult__left__cancel,axiom,
    ! [C: real,Aa: real,Ba: real] :
      ( ( C != zero_zero_real )
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),Aa) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),Ba) )
      <=> ( Aa = Ba ) ) ) ).

% mult_left_cancel
tff(fact_4229_mult__left__cancel,axiom,
    ! [C: rat,Aa: rat,Ba: rat] :
      ( ( C != zero_zero_rat )
     => ( ( aa_rat_rat(times_times_rat(C),Aa) = aa_rat_rat(times_times_rat(C),Ba) )
      <=> ( Aa = Ba ) ) ) ).

% mult_left_cancel
tff(fact_4230_mult__left__cancel,axiom,
    ! [C: nat,Aa: nat,Ba: nat] :
      ( ( C != zero_zero_nat )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),Aa) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),Ba) )
      <=> ( Aa = Ba ) ) ) ).

% mult_left_cancel
tff(fact_4231_mult__left__cancel,axiom,
    ! [C: int,Aa: int,Ba: int] :
      ( ( C != zero_zero_int )
     => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,C),Aa) = aa_int_int(aa_int_fun_int_int(times_times_int,C),Ba) )
      <=> ( Aa = Ba ) ) ) ).

% mult_left_cancel
tff(fact_4232_no__zero__divisors,axiom,
    ! [Aa: real,Ba: real] :
      ( ( Aa != zero_zero_real )
     => ( ( Ba != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),Ba) != zero_zero_real ) ) ) ).

% no_zero_divisors
tff(fact_4233_no__zero__divisors,axiom,
    ! [Aa: rat,Ba: rat] :
      ( ( Aa != zero_zero_rat )
     => ( ( Ba != zero_zero_rat )
       => ( aa_rat_rat(times_times_rat(Aa),Ba) != zero_zero_rat ) ) ) ).

% no_zero_divisors
tff(fact_4234_no__zero__divisors,axiom,
    ! [Aa: nat,Ba: nat] :
      ( ( Aa != zero_zero_nat )
     => ( ( Ba != zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),Ba) != zero_zero_nat ) ) ) ).

% no_zero_divisors
tff(fact_4235_no__zero__divisors,axiom,
    ! [Aa: int,Ba: int] :
      ( ( Aa != zero_zero_int )
     => ( ( Ba != zero_zero_int )
       => ( aa_int_int(aa_int_fun_int_int(times_times_int,Aa),Ba) != zero_zero_int ) ) ) ).

% no_zero_divisors
tff(fact_4236_divisors__zero,axiom,
    ! [Aa: real,Ba: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),Ba) = zero_zero_real )
     => ( ( Aa = zero_zero_real )
        | ( Ba = zero_zero_real ) ) ) ).

% divisors_zero
tff(fact_4237_divisors__zero,axiom,
    ! [Aa: rat,Ba: rat] :
      ( ( aa_rat_rat(times_times_rat(Aa),Ba) = zero_zero_rat )
     => ( ( Aa = zero_zero_rat )
        | ( Ba = zero_zero_rat ) ) ) ).

% divisors_zero
tff(fact_4238_divisors__zero,axiom,
    ! [Aa: nat,Ba: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),Ba) = zero_zero_nat )
     => ( ( Aa = zero_zero_nat )
        | ( Ba = zero_zero_nat ) ) ) ).

% divisors_zero
tff(fact_4239_divisors__zero,axiom,
    ! [Aa: int,Ba: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,Aa),Ba) = zero_zero_int )
     => ( ( Aa = zero_zero_int )
        | ( Ba = zero_zero_int ) ) ) ).

% divisors_zero
tff(fact_4240_mult__not__zero,axiom,
    ! [Aa: real,Ba: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),Ba) != zero_zero_real )
     => ( ( Aa != zero_zero_real )
        & ( Ba != zero_zero_real ) ) ) ).

% mult_not_zero
tff(fact_4241_mult__not__zero,axiom,
    ! [Aa: rat,Ba: rat] :
      ( ( aa_rat_rat(times_times_rat(Aa),Ba) != zero_zero_rat )
     => ( ( Aa != zero_zero_rat )
        & ( Ba != zero_zero_rat ) ) ) ).

% mult_not_zero
tff(fact_4242_mult__not__zero,axiom,
    ! [Aa: nat,Ba: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),Ba) != zero_zero_nat )
     => ( ( Aa != zero_zero_nat )
        & ( Ba != zero_zero_nat ) ) ) ).

% mult_not_zero
tff(fact_4243_mult__not__zero,axiom,
    ! [Aa: int,Ba: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,Aa),Ba) != zero_zero_int )
     => ( ( Aa != zero_zero_int )
        & ( Ba != zero_zero_int ) ) ) ).

% mult_not_zero
tff(fact_4244_comm__monoid__mult__class_Omult__1,axiom,
    ! [Aa: complex] : aa_complex_complex(times_times_complex(one_one_complex),Aa) = Aa ).

% comm_monoid_mult_class.mult_1
tff(fact_4245_comm__monoid__mult__class_Omult__1,axiom,
    ! [Aa: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,one_one_real),Aa) = Aa ).

% comm_monoid_mult_class.mult_1
tff(fact_4246_comm__monoid__mult__class_Omult__1,axiom,
    ! [Aa: rat] : aa_rat_rat(times_times_rat(one_one_rat),Aa) = Aa ).

% comm_monoid_mult_class.mult_1
tff(fact_4247_comm__monoid__mult__class_Omult__1,axiom,
    ! [Aa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,one_one_nat),Aa) = Aa ).

% comm_monoid_mult_class.mult_1
tff(fact_4248_comm__monoid__mult__class_Omult__1,axiom,
    ! [Aa: int] : aa_int_int(aa_int_fun_int_int(times_times_int,one_one_int),Aa) = Aa ).

% comm_monoid_mult_class.mult_1
tff(fact_4249_mult_Ocomm__neutral,axiom,
    ! [Aa: complex] : aa_complex_complex(times_times_complex(Aa),one_one_complex) = Aa ).

% mult.comm_neutral
tff(fact_4250_mult_Ocomm__neutral,axiom,
    ! [Aa: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),one_one_real) = Aa ).

% mult.comm_neutral
tff(fact_4251_mult_Ocomm__neutral,axiom,
    ! [Aa: rat] : aa_rat_rat(times_times_rat(Aa),one_one_rat) = Aa ).

% mult.comm_neutral
tff(fact_4252_mult_Ocomm__neutral,axiom,
    ! [Aa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),one_one_nat) = Aa ).

% mult.comm_neutral
tff(fact_4253_mult_Ocomm__neutral,axiom,
    ! [Aa: int] : aa_int_int(aa_int_fun_int_int(times_times_int,Aa),one_one_int) = Aa ).

% mult.comm_neutral
tff(fact_4254_ring__class_Oring__distribs_I2_J,axiom,
    ! [Aa: real,Ba: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),Ba)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Ba),C)) ).

% ring_class.ring_distribs(2)
tff(fact_4255_ring__class_Oring__distribs_I2_J,axiom,
    ! [Aa: rat,Ba: rat,C: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(Aa),Ba)),C) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(Aa),C)),aa_rat_rat(times_times_rat(Ba),C)) ).

% ring_class.ring_distribs(2)
tff(fact_4256_ring__class_Oring__distribs_I2_J,axiom,
    ! [Aa: int,Ba: 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,Aa),Ba)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Aa),C)),aa_int_int(aa_int_fun_int_int(times_times_int,Ba),C)) ).

% ring_class.ring_distribs(2)
tff(fact_4257_ring__class_Oring__distribs_I1_J,axiom,
    ! [Aa: real,Ba: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Ba),C)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),Ba)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),C)) ).

% ring_class.ring_distribs(1)
tff(fact_4258_ring__class_Oring__distribs_I1_J,axiom,
    ! [Aa: rat,Ba: rat,C: rat] : aa_rat_rat(times_times_rat(Aa),aa_rat_rat(plus_plus_rat(Ba),C)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(Aa),Ba)),aa_rat_rat(times_times_rat(Aa),C)) ).

% ring_class.ring_distribs(1)
tff(fact_4259_ring__class_Oring__distribs_I1_J,axiom,
    ! [Aa: int,Ba: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,Aa),aa_int_int(aa_int_fun_int_int(plus_plus_int,Ba),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Aa),Ba)),aa_int_int(aa_int_fun_int_int(times_times_int,Aa),C)) ).

% ring_class.ring_distribs(1)
tff(fact_4260_comm__semiring__class_Odistrib,axiom,
    ! [Aa: real,Ba: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),Ba)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Ba),C)) ).

% comm_semiring_class.distrib
tff(fact_4261_comm__semiring__class_Odistrib,axiom,
    ! [Aa: rat,Ba: rat,C: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(Aa),Ba)),C) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(Aa),C)),aa_rat_rat(times_times_rat(Ba),C)) ).

% comm_semiring_class.distrib
tff(fact_4262_comm__semiring__class_Odistrib,axiom,
    ! [Aa: nat,Ba: 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,Aa),Ba)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ba),C)) ).

% comm_semiring_class.distrib
tff(fact_4263_comm__semiring__class_Odistrib,axiom,
    ! [Aa: int,Ba: 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,Aa),Ba)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Aa),C)),aa_int_int(aa_int_fun_int_int(times_times_int,Ba),C)) ).

% comm_semiring_class.distrib
tff(fact_4264_distrib__left,axiom,
    ! [Aa: real,Ba: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Ba),C)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),Ba)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),C)) ).

% distrib_left
tff(fact_4265_distrib__left,axiom,
    ! [Aa: rat,Ba: rat,C: rat] : aa_rat_rat(times_times_rat(Aa),aa_rat_rat(plus_plus_rat(Ba),C)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(Aa),Ba)),aa_rat_rat(times_times_rat(Aa),C)) ).

% distrib_left
tff(fact_4266_distrib__left,axiom,
    ! [Aa: nat,Ba: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ba),C)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),Ba)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),C)) ).

% distrib_left
tff(fact_4267_distrib__left,axiom,
    ! [Aa: int,Ba: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,Aa),aa_int_int(aa_int_fun_int_int(plus_plus_int,Ba),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Aa),Ba)),aa_int_int(aa_int_fun_int_int(times_times_int,Aa),C)) ).

% distrib_left
tff(fact_4268_distrib__right,axiom,
    ! [Aa: real,Ba: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),Ba)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Ba),C)) ).

% distrib_right
tff(fact_4269_distrib__right,axiom,
    ! [Aa: rat,Ba: rat,C: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(Aa),Ba)),C) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(Aa),C)),aa_rat_rat(times_times_rat(Ba),C)) ).

% distrib_right
tff(fact_4270_distrib__right,axiom,
    ! [Aa: nat,Ba: 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,Aa),Ba)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ba),C)) ).

% distrib_right
tff(fact_4271_distrib__right,axiom,
    ! [Aa: int,Ba: 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,Aa),Ba)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Aa),C)),aa_int_int(aa_int_fun_int_int(times_times_int,Ba),C)) ).

% distrib_right
tff(fact_4272_combine__common__factor,axiom,
    ! [Aa: real,E2: real,Ba: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),E2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Ba),E2)),C)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),Ba)),E2)),C) ).

% combine_common_factor
tff(fact_4273_combine__common__factor,axiom,
    ! [Aa: rat,E2: rat,Ba: rat,C: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(Aa),E2)),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(Ba),E2)),C)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(Aa),Ba)),E2)),C) ).

% combine_common_factor
tff(fact_4274_combine__common__factor,axiom,
    ! [Aa: nat,E2: nat,Ba: 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,Aa),E2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ba),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,Aa),Ba)),E2)),C) ).

% combine_common_factor
tff(fact_4275_combine__common__factor,axiom,
    ! [Aa: int,E2: int,Ba: 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,Aa),E2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Ba),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,Aa),Ba)),E2)),C) ).

% combine_common_factor
tff(fact_4276_left__diff__distrib,axiom,
    ! [Aa: real,Ba: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Aa),Ba)),C) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Ba),C)) ).

% left_diff_distrib
tff(fact_4277_left__diff__distrib,axiom,
    ! [Aa: rat,Ba: rat,C: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(Aa),Ba)),C) = aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(Aa),C)),aa_rat_rat(times_times_rat(Ba),C)) ).

% left_diff_distrib
tff(fact_4278_left__diff__distrib,axiom,
    ! [Aa: int,Ba: 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,Aa),Ba)),C) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Aa),C)),aa_int_int(aa_int_fun_int_int(times_times_int,Ba),C)) ).

% left_diff_distrib
tff(fact_4279_right__diff__distrib,axiom,
    ! [Aa: real,Ba: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Ba),C)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),Ba)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),C)) ).

% right_diff_distrib
tff(fact_4280_right__diff__distrib,axiom,
    ! [Aa: rat,Ba: rat,C: rat] : aa_rat_rat(times_times_rat(Aa),aa_rat_rat(minus_minus_rat(Ba),C)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(Aa),Ba)),aa_rat_rat(times_times_rat(Aa),C)) ).

% right_diff_distrib
tff(fact_4281_right__diff__distrib,axiom,
    ! [Aa: int,Ba: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,Aa),aa_int_int(aa_int_fun_int_int(minus_minus_int,Ba),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Aa),Ba)),aa_int_int(aa_int_fun_int_int(times_times_int,Aa),C)) ).

% right_diff_distrib
tff(fact_4282_left__diff__distrib_H,axiom,
    ! [Ba: real,C: real,Aa: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Ba),C)),Aa) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Ba),Aa)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),Aa)) ).

% left_diff_distrib'
tff(fact_4283_left__diff__distrib_H,axiom,
    ! [Ba: rat,C: rat,Aa: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(Ba),C)),Aa) = aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(Ba),Aa)),aa_rat_rat(times_times_rat(C),Aa)) ).

% left_diff_distrib'
tff(fact_4284_left__diff__distrib_H,axiom,
    ! [Ba: nat,C: nat,Aa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ba),C)),Aa) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ba),Aa)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),Aa)) ).

% left_diff_distrib'
tff(fact_4285_left__diff__distrib_H,axiom,
    ! [Ba: int,C: int,Aa: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Ba),C)),Aa) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Ba),Aa)),aa_int_int(aa_int_fun_int_int(times_times_int,C),Aa)) ).

% left_diff_distrib'
tff(fact_4286_right__diff__distrib_H,axiom,
    ! [Aa: real,Ba: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Ba),C)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),Ba)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),C)) ).

% right_diff_distrib'
tff(fact_4287_right__diff__distrib_H,axiom,
    ! [Aa: rat,Ba: rat,C: rat] : aa_rat_rat(times_times_rat(Aa),aa_rat_rat(minus_minus_rat(Ba),C)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(Aa),Ba)),aa_rat_rat(times_times_rat(Aa),C)) ).

% right_diff_distrib'
tff(fact_4288_right__diff__distrib_H,axiom,
    ! [Aa: nat,Ba: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ba),C)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),Ba)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),C)) ).

% right_diff_distrib'
tff(fact_4289_right__diff__distrib_H,axiom,
    ! [Aa: int,Ba: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,Aa),aa_int_int(aa_int_fun_int_int(minus_minus_int,Ba),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Aa),Ba)),aa_int_int(aa_int_fun_int_int(times_times_int,Aa),C)) ).

% right_diff_distrib'
tff(fact_4290_square__eq__iff,axiom,
    ! [Aa: real,Ba: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),Aa) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Ba),Ba) )
    <=> ( ( Aa = Ba )
        | ( Aa = aa_real_real(uminus_uminus_real,Ba) ) ) ) ).

% square_eq_iff
tff(fact_4291_square__eq__iff,axiom,
    ! [Aa: int,Ba: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,Aa),Aa) = aa_int_int(aa_int_fun_int_int(times_times_int,Ba),Ba) )
    <=> ( ( Aa = Ba )
        | ( Aa = aa_int_int(uminus_uminus_int,Ba) ) ) ) ).

% square_eq_iff
tff(fact_4292_square__eq__iff,axiom,
    ! [Aa: code_integer,Ba: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Aa),Aa) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Ba),Ba) )
    <=> ( ( Aa = Ba )
        | ( Aa = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Ba) ) ) ) ).

% square_eq_iff
tff(fact_4293_square__eq__iff,axiom,
    ! [Aa: rat,Ba: rat] :
      ( ( aa_rat_rat(times_times_rat(Aa),Aa) = aa_rat_rat(times_times_rat(Ba),Ba) )
    <=> ( ( Aa = Ba )
        | ( Aa = aa_rat_rat(uminus_uminus_rat,Ba) ) ) ) ).

% square_eq_iff
tff(fact_4294_minus__mult__commute,axiom,
    ! [Aa: real,Ba: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,Aa)),Ba) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),aa_real_real(uminus_uminus_real,Ba)) ).

% minus_mult_commute
tff(fact_4295_minus__mult__commute,axiom,
    ! [Aa: int,Ba: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,Aa)),Ba) = aa_int_int(aa_int_fun_int_int(times_times_int,Aa),aa_int_int(uminus_uminus_int,Ba)) ).

% minus_mult_commute
tff(fact_4296_minus__mult__commute,axiom,
    ! [Aa: code_integer,Ba: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa)),Ba) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Aa),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Ba)) ).

% minus_mult_commute
tff(fact_4297_minus__mult__commute,axiom,
    ! [Aa: rat,Ba: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,Aa)),Ba) = aa_rat_rat(times_times_rat(Aa),aa_rat_rat(uminus_uminus_rat,Ba)) ).

% minus_mult_commute
tff(fact_4298_Suc__mult__cancel1,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),M) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),Na) )
    <=> ( M = Na ) ) ).

% Suc_mult_cancel1
tff(fact_4299_dvdE,axiom,
    ! [Ba: code_integer,Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Ba),Aa)
     => ~ ! [K2: code_integer] : Aa != aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Ba),K2) ) ).

% dvdE
tff(fact_4300_dvdE,axiom,
    ! [Ba: real,Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Ba),Aa)
     => ~ ! [K2: real] : Aa != aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Ba),K2) ) ).

% dvdE
tff(fact_4301_dvdE,axiom,
    ! [Ba: rat,Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Ba),Aa)
     => ~ ! [K2: rat] : Aa != aa_rat_rat(times_times_rat(Ba),K2) ) ).

% dvdE
tff(fact_4302_dvdE,axiom,
    ! [Ba: nat,Aa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Ba),Aa)
     => ~ ! [K2: nat] : Aa != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ba),K2) ) ).

% dvdE
tff(fact_4303_dvdE,axiom,
    ! [Ba: int,Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Ba),Aa)
     => ~ ! [K2: int] : Aa != aa_int_int(aa_int_fun_int_int(times_times_int,Ba),K2) ) ).

% dvdE
tff(fact_4304_dvdI,axiom,
    ! [Aa: code_integer,Ba: code_integer,K: code_integer] :
      ( ( Aa = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Ba),K) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Ba),Aa) ) ).

% dvdI
tff(fact_4305_dvdI,axiom,
    ! [Aa: real,Ba: real,K: real] :
      ( ( Aa = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Ba),K) )
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Ba),Aa) ) ).

% dvdI
tff(fact_4306_dvdI,axiom,
    ! [Aa: rat,Ba: rat,K: rat] :
      ( ( Aa = aa_rat_rat(times_times_rat(Ba),K) )
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Ba),Aa) ) ).

% dvdI
tff(fact_4307_dvdI,axiom,
    ! [Aa: nat,Ba: nat,K: nat] :
      ( ( Aa = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ba),K) )
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Ba),Aa) ) ).

% dvdI
tff(fact_4308_dvdI,axiom,
    ! [Aa: int,Ba: int,K: int] :
      ( ( Aa = aa_int_int(aa_int_fun_int_int(times_times_int,Ba),K) )
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Ba),Aa) ) ).

% dvdI
tff(fact_4309_dvd__def,axiom,
    ! [Ba: code_integer,Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Ba),Aa)
    <=> ? [K3: code_integer] : Aa = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Ba),K3) ) ).

% dvd_def
tff(fact_4310_dvd__def,axiom,
    ! [Ba: real,Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Ba),Aa)
    <=> ? [K3: real] : Aa = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Ba),K3) ) ).

% dvd_def
tff(fact_4311_dvd__def,axiom,
    ! [Ba: rat,Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Ba),Aa)
    <=> ? [K3: rat] : Aa = aa_rat_rat(times_times_rat(Ba),K3) ) ).

% dvd_def
tff(fact_4312_dvd__def,axiom,
    ! [Ba: nat,Aa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Ba),Aa)
    <=> ? [K3: nat] : Aa = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ba),K3) ) ).

% dvd_def
tff(fact_4313_dvd__def,axiom,
    ! [Ba: int,Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Ba),Aa)
    <=> ? [K3: int] : Aa = aa_int_int(aa_int_fun_int_int(times_times_int,Ba),K3) ) ).

% dvd_def
tff(fact_4314_dvd__mult,axiom,
    ! [Aa: code_integer,C: code_integer,Ba: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),C)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Ba),C)) ) ).

% dvd_mult
tff(fact_4315_dvd__mult,axiom,
    ! [Aa: real,C: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Aa),C)
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Aa),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Ba),C)) ) ).

% dvd_mult
tff(fact_4316_dvd__mult,axiom,
    ! [Aa: rat,C: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Aa),C)
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Aa),aa_rat_rat(times_times_rat(Ba),C)) ) ).

% dvd_mult
tff(fact_4317_dvd__mult,axiom,
    ! [Aa: nat,C: nat,Ba: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),C)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ba),C)) ) ).

% dvd_mult
tff(fact_4318_dvd__mult,axiom,
    ! [Aa: int,C: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),C)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),aa_int_int(aa_int_fun_int_int(times_times_int,Ba),C)) ) ).

% dvd_mult
tff(fact_4319_dvd__mult2,axiom,
    ! [Aa: code_integer,Ba: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),Ba)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Ba),C)) ) ).

% dvd_mult2
tff(fact_4320_dvd__mult2,axiom,
    ! [Aa: real,Ba: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Aa),Ba)
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Aa),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Ba),C)) ) ).

% dvd_mult2
tff(fact_4321_dvd__mult2,axiom,
    ! [Aa: rat,Ba: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Aa),Ba)
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Aa),aa_rat_rat(times_times_rat(Ba),C)) ) ).

% dvd_mult2
tff(fact_4322_dvd__mult2,axiom,
    ! [Aa: nat,Ba: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),Ba)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ba),C)) ) ).

% dvd_mult2
tff(fact_4323_dvd__mult2,axiom,
    ! [Aa: int,Ba: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),Ba)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),aa_int_int(aa_int_fun_int_int(times_times_int,Ba),C)) ) ).

% dvd_mult2
tff(fact_4324_dvd__mult__left,axiom,
    ! [Aa: code_integer,Ba: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Aa),Ba)),C)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),C) ) ).

% dvd_mult_left
tff(fact_4325_dvd__mult__left,axiom,
    ! [Aa: real,Ba: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),Ba)),C)
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Aa),C) ) ).

% dvd_mult_left
tff(fact_4326_dvd__mult__left,axiom,
    ! [Aa: rat,Ba: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(times_times_rat(Aa),Ba)),C)
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Aa),C) ) ).

% dvd_mult_left
tff(fact_4327_dvd__mult__left,axiom,
    ! [Aa: nat,Ba: 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,Aa),Ba)),C)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),C) ) ).

% dvd_mult_left
tff(fact_4328_dvd__mult__left,axiom,
    ! [Aa: int,Ba: 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,Aa),Ba)),C)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),C) ) ).

% dvd_mult_left
tff(fact_4329_dvd__triv__left,axiom,
    ! [Aa: code_integer,Ba: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Aa),Ba)) ).

% dvd_triv_left
tff(fact_4330_dvd__triv__left,axiom,
    ! [Aa: real,Ba: real] : aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Aa),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),Ba)) ).

% dvd_triv_left
tff(fact_4331_dvd__triv__left,axiom,
    ! [Aa: rat,Ba: rat] : aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Aa),aa_rat_rat(times_times_rat(Aa),Ba)) ).

% dvd_triv_left
tff(fact_4332_dvd__triv__left,axiom,
    ! [Aa: nat,Ba: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),Ba)) ).

% dvd_triv_left
tff(fact_4333_dvd__triv__left,axiom,
    ! [Aa: int,Ba: int] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),aa_int_int(aa_int_fun_int_int(times_times_int,Aa),Ba)) ).

% dvd_triv_left
tff(fact_4334_mult__dvd__mono,axiom,
    ! [Aa: code_integer,Ba: code_integer,C: code_integer,D: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),Ba)
     => ( 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,Aa),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Ba),D)) ) ) ).

% mult_dvd_mono
tff(fact_4335_mult__dvd__mono,axiom,
    ! [Aa: real,Ba: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Aa),Ba)
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,C),D)
       => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Ba),D)) ) ) ).

% mult_dvd_mono
tff(fact_4336_mult__dvd__mono,axiom,
    ! [Aa: rat,Ba: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Aa),Ba)
     => ( 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(Aa),C)),aa_rat_rat(times_times_rat(Ba),D)) ) ) ).

% mult_dvd_mono
tff(fact_4337_mult__dvd__mono,axiom,
    ! [Aa: nat,Ba: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),Ba)
     => ( 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,Aa),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ba),D)) ) ) ).

% mult_dvd_mono
tff(fact_4338_mult__dvd__mono,axiom,
    ! [Aa: int,Ba: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),Ba)
     => ( 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,Aa),C)),aa_int_int(aa_int_fun_int_int(times_times_int,Ba),D)) ) ) ).

% mult_dvd_mono
tff(fact_4339_dvd__mult__right,axiom,
    ! [Aa: code_integer,Ba: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Aa),Ba)),C)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Ba),C) ) ).

% dvd_mult_right
tff(fact_4340_dvd__mult__right,axiom,
    ! [Aa: real,Ba: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),Ba)),C)
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Ba),C) ) ).

% dvd_mult_right
tff(fact_4341_dvd__mult__right,axiom,
    ! [Aa: rat,Ba: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(times_times_rat(Aa),Ba)),C)
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Ba),C) ) ).

% dvd_mult_right
tff(fact_4342_dvd__mult__right,axiom,
    ! [Aa: nat,Ba: 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,Aa),Ba)),C)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Ba),C) ) ).

% dvd_mult_right
tff(fact_4343_dvd__mult__right,axiom,
    ! [Aa: int,Ba: 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,Aa),Ba)),C)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Ba),C) ) ).

% dvd_mult_right
tff(fact_4344_dvd__triv__right,axiom,
    ! [Aa: code_integer,Ba: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Ba),Aa)) ).

% dvd_triv_right
tff(fact_4345_dvd__triv__right,axiom,
    ! [Aa: real,Ba: real] : aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Aa),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Ba),Aa)) ).

% dvd_triv_right
tff(fact_4346_dvd__triv__right,axiom,
    ! [Aa: rat,Ba: rat] : aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Aa),aa_rat_rat(times_times_rat(Ba),Aa)) ).

% dvd_triv_right
tff(fact_4347_dvd__triv__right,axiom,
    ! [Aa: nat,Ba: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ba),Aa)) ).

% dvd_triv_right
tff(fact_4348_dvd__triv__right,axiom,
    ! [Aa: int,Ba: int] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),aa_int_int(aa_int_fun_int_int(times_times_int,Ba),Aa)) ).

% dvd_triv_right
tff(fact_4349_nat__mult__eq__cancel__disj,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Na) )
    <=> ( ( K = zero_zero_nat )
        | ( M = Na ) ) ) ).

% nat_mult_eq_cancel_disj
tff(fact_4350_mult__0,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,zero_zero_nat),Na) = zero_zero_nat ).

% mult_0
tff(fact_4351_mult__le__mono2,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),I)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),J)) ) ).

% mult_le_mono2
tff(fact_4352_mult__le__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),K)) ) ).

% mult_le_mono1
tff(fact_4353_mult__le__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),L)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),L)) ) ) ).

% mult_le_mono
tff(fact_4354_le__square,axiom,
    ! [M: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),M)) ).

% le_square
tff(fact_4355_le__cube,axiom,
    ! [M: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),M))) ).

% le_cube
tff(fact_4356_add__mult__distrib,axiom,
    ! [M: nat,Na: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na)),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),K)) ).

% add_mult_distrib
tff(fact_4357_add__mult__distrib2,axiom,
    ! [K: nat,M: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Na)) ).

% add_mult_distrib2
tff(fact_4358_diff__mult__distrib,axiom,
    ! [M: nat,Na: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na)),K) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),K)) ).

% diff_mult_distrib
tff(fact_4359_diff__mult__distrib2,axiom,
    ! [K: nat,M: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Na)) ).

% diff_mult_distrib2
tff(fact_4360_nat__mult__1,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,one_one_nat),Na) = Na ).

% nat_mult_1
tff(fact_4361_nat__mult__1__right,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),one_one_nat) = Na ).

% nat_mult_1_right
tff(fact_4362_div__mult2__eq,axiom,
    ! [M: nat,Na: nat,Q4: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),Q4)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na)),Q4) ).

% div_mult2_eq
tff(fact_4363_option_Osel,axiom,
    ! [X2: nat] : the_nat(some_nat(X2)) = X2 ).

% option.sel
tff(fact_4364_option_Osel,axiom,
    ! [X2: product_prod_nat_nat] : the_Pr8591224930841456533at_nat(some_P7363390416028606310at_nat(X2)) = X2 ).

% option.sel
tff(fact_4365_option_Osel,axiom,
    ! [X2: num] : the_num(some_num(X2)) = X2 ).

% option.sel
tff(fact_4366_option_Oexpand,axiom,
    ! [Option: option_nat,Option2: option_nat] :
      ( ( ( Option = none_nat )
      <=> ( Option2 = none_nat ) )
     => ( ( ( Option != none_nat )
         => ( ( Option2 != none_nat )
           => ( the_nat(Option) = the_nat(Option2) ) ) )
       => ( Option = Option2 ) ) ) ).

% option.expand
tff(fact_4367_option_Oexpand,axiom,
    ! [Option: option4927543243414619207at_nat,Option2: option4927543243414619207at_nat] :
      ( ( ( Option = none_P5556105721700978146at_nat )
      <=> ( Option2 = none_P5556105721700978146at_nat ) )
     => ( ( ( Option != none_P5556105721700978146at_nat )
         => ( ( Option2 != none_P5556105721700978146at_nat )
           => ( the_Pr8591224930841456533at_nat(Option) = the_Pr8591224930841456533at_nat(Option2) ) ) )
       => ( Option = Option2 ) ) ) ).

% option.expand
tff(fact_4368_option_Oexpand,axiom,
    ! [Option: option_num,Option2: option_num] :
      ( ( ( Option = none_num )
      <=> ( Option2 = none_num ) )
     => ( ( ( Option != none_num )
         => ( ( Option2 != none_num )
           => ( the_num(Option) = the_num(Option2) ) ) )
       => ( Option = Option2 ) ) ) ).

% option.expand
tff(fact_4369_mult__ceiling__le,axiom,
    ! [Aa: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Aa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Ba)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),Ba))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_real_int(archim7802044766580827645g_real,Aa)),aa_real_int(archim7802044766580827645g_real,Ba))) ) ) ).

% mult_ceiling_le
tff(fact_4370_mult__ceiling__le,axiom,
    ! [Aa: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Aa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Ba)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(times_times_rat(Aa),Ba))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_rat_int(archim2889992004027027881ng_rat,Aa)),aa_rat_int(archim2889992004027027881ng_rat,Ba))) ) ) ).

% mult_ceiling_le
tff(fact_4371_mult__mono,axiom,
    ! [Aa: real,Ba: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),Ba)
     => ( 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),Ba)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Ba),D)) ) ) ) ) ).

% mult_mono
tff(fact_4372_mult__mono,axiom,
    ! [Aa: rat,Ba: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),Ba)
     => ( 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),Ba)
         => ( 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(Aa),C)),aa_rat_rat(times_times_rat(Ba),D)) ) ) ) ) ).

% mult_mono
tff(fact_4373_mult__mono,axiom,
    ! [Aa: nat,Ba: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),Ba)
     => ( 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),Ba)
         => ( 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,Aa),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ba),D)) ) ) ) ) ).

% mult_mono
tff(fact_4374_mult__mono,axiom,
    ! [Aa: int,Ba: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),Ba)
     => ( 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),Ba)
         => ( 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,Aa),C)),aa_int_int(aa_int_fun_int_int(times_times_int,Ba),D)) ) ) ) ) ).

% mult_mono
tff(fact_4375_mult__mono_H,axiom,
    ! [Aa: real,Ba: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),Ba)
     => ( 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),Aa)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Ba),D)) ) ) ) ) ).

% mult_mono'
tff(fact_4376_mult__mono_H,axiom,
    ! [Aa: rat,Ba: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),Ba)
     => ( 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),Aa)
         => ( 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(Aa),C)),aa_rat_rat(times_times_rat(Ba),D)) ) ) ) ) ).

% mult_mono'
tff(fact_4377_mult__mono_H,axiom,
    ! [Aa: nat,Ba: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),Ba)
     => ( 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),Aa)
         => ( 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,Aa),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ba),D)) ) ) ) ) ).

% mult_mono'
tff(fact_4378_mult__mono_H,axiom,
    ! [Aa: int,Ba: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),Ba)
     => ( 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),Aa)
         => ( 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,Aa),C)),aa_int_int(aa_int_fun_int_int(times_times_int,Ba),D)) ) ) ) ) ).

% mult_mono'
tff(fact_4379_zero__le__square,axiom,
    ! [Aa: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),Aa)) ).

% zero_le_square
tff(fact_4380_zero__le__square,axiom,
    ! [Aa: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(times_times_rat(Aa),Aa)) ).

% zero_le_square
tff(fact_4381_zero__le__square,axiom,
    ! [Aa: 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,Aa),Aa)) ).

% zero_le_square
tff(fact_4382_split__mult__pos__le,axiom,
    ! [Aa: real,Ba: real] :
      ( ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Aa)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Ba) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ba),zero_zero_real) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),Ba)) ) ).

% split_mult_pos_le
tff(fact_4383_split__mult__pos__le,axiom,
    ! [Aa: rat,Ba: rat] :
      ( ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Aa)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Ba) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ba),zero_zero_rat) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(times_times_rat(Aa),Ba)) ) ).

% split_mult_pos_le
tff(fact_4384_split__mult__pos__le,axiom,
    ! [Aa: int,Ba: int] :
      ( ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Aa)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Ba) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ba),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,Aa),Ba)) ) ).

% split_mult_pos_le
tff(fact_4385_mult__left__mono__neg,axiom,
    ! [Ba: real,Aa: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ba),Aa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),Aa)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),Ba)) ) ) ).

% mult_left_mono_neg
tff(fact_4386_mult__left__mono__neg,axiom,
    ! [Ba: rat,Aa: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ba),Aa)
     => ( 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),Aa)),aa_rat_rat(times_times_rat(C),Ba)) ) ) ).

% mult_left_mono_neg
tff(fact_4387_mult__left__mono__neg,axiom,
    ! [Ba: int,Aa: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ba),Aa)
     => ( 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),Aa)),aa_int_int(aa_int_fun_int_int(times_times_int,C),Ba)) ) ) ).

% mult_left_mono_neg
tff(fact_4388_mult__nonpos__nonpos,axiom,
    ! [Aa: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ba),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),Ba)) ) ) ).

% mult_nonpos_nonpos
tff(fact_4389_mult__nonpos__nonpos,axiom,
    ! [Aa: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ba),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(times_times_rat(Aa),Ba)) ) ) ).

% mult_nonpos_nonpos
tff(fact_4390_mult__nonpos__nonpos,axiom,
    ! [Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ba),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,Aa),Ba)) ) ) ).

% mult_nonpos_nonpos
tff(fact_4391_mult__left__mono,axiom,
    ! [Aa: real,Ba: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),Ba)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),Aa)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),Ba)) ) ) ).

% mult_left_mono
tff(fact_4392_mult__left__mono,axiom,
    ! [Aa: rat,Ba: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),Ba)
     => ( 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),Aa)),aa_rat_rat(times_times_rat(C),Ba)) ) ) ).

% mult_left_mono
tff(fact_4393_mult__left__mono,axiom,
    ! [Aa: nat,Ba: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),Ba)
     => ( 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),Aa)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),Ba)) ) ) ).

% mult_left_mono
tff(fact_4394_mult__left__mono,axiom,
    ! [Aa: int,Ba: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),Ba)
     => ( 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),Aa)),aa_int_int(aa_int_fun_int_int(times_times_int,C),Ba)) ) ) ).

% mult_left_mono
tff(fact_4395_mult__right__mono__neg,axiom,
    ! [Ba: real,Aa: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ba),Aa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Ba),C)) ) ) ).

% mult_right_mono_neg
tff(fact_4396_mult__right__mono__neg,axiom,
    ! [Ba: rat,Aa: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ba),Aa)
     => ( 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(Aa),C)),aa_rat_rat(times_times_rat(Ba),C)) ) ) ).

% mult_right_mono_neg
tff(fact_4397_mult__right__mono__neg,axiom,
    ! [Ba: int,Aa: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ba),Aa)
     => ( 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,Aa),C)),aa_int_int(aa_int_fun_int_int(times_times_int,Ba),C)) ) ) ).

% mult_right_mono_neg
tff(fact_4398_mult__right__mono,axiom,
    ! [Aa: real,Ba: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),Ba)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Ba),C)) ) ) ).

% mult_right_mono
tff(fact_4399_mult__right__mono,axiom,
    ! [Aa: rat,Ba: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),Ba)
     => ( 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(Aa),C)),aa_rat_rat(times_times_rat(Ba),C)) ) ) ).

% mult_right_mono
tff(fact_4400_mult__right__mono,axiom,
    ! [Aa: nat,Ba: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),Ba)
     => ( 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,Aa),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ba),C)) ) ) ).

% mult_right_mono
tff(fact_4401_mult__right__mono,axiom,
    ! [Aa: int,Ba: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),Ba)
     => ( 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,Aa),C)),aa_int_int(aa_int_fun_int_int(times_times_int,Ba),C)) ) ) ).

% mult_right_mono
tff(fact_4402_mult__le__0__iff,axiom,
    ! [Aa: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),Ba)),zero_zero_real)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Aa)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ba),zero_zero_real) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Ba) ) ) ) ).

% mult_le_0_iff
tff(fact_4403_mult__le__0__iff,axiom,
    ! [Aa: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(Aa),Ba)),zero_zero_rat)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Aa)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ba),zero_zero_rat) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Ba) ) ) ) ).

% mult_le_0_iff
tff(fact_4404_mult__le__0__iff,axiom,
    ! [Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,Aa),Ba)),zero_zero_int)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Aa)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ba),zero_zero_int) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Ba) ) ) ) ).

% mult_le_0_iff
tff(fact_4405_split__mult__neg__le,axiom,
    ! [Aa: real,Ba: real] :
      ( ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Aa)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ba),zero_zero_real) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Ba) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),Ba)),zero_zero_real) ) ).

% split_mult_neg_le
tff(fact_4406_split__mult__neg__le,axiom,
    ! [Aa: rat,Ba: rat] :
      ( ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Aa)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ba),zero_zero_rat) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Ba) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(Aa),Ba)),zero_zero_rat) ) ).

% split_mult_neg_le
tff(fact_4407_split__mult__neg__le,axiom,
    ! [Aa: nat,Ba: nat] :
      ( ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Aa)
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ba),zero_zero_nat) )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),zero_zero_nat)
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Ba) ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),Ba)),zero_zero_nat) ) ).

% split_mult_neg_le
tff(fact_4408_split__mult__neg__le,axiom,
    ! [Aa: int,Ba: int] :
      ( ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Aa)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ba),zero_zero_int) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Ba) ) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,Aa),Ba)),zero_zero_int) ) ).

% split_mult_neg_le
tff(fact_4409_mult__nonneg__nonneg,axiom,
    ! [Aa: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Aa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Ba)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),Ba)) ) ) ).

% mult_nonneg_nonneg
tff(fact_4410_mult__nonneg__nonneg,axiom,
    ! [Aa: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Aa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Ba)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(times_times_rat(Aa),Ba)) ) ) ).

% mult_nonneg_nonneg
tff(fact_4411_mult__nonneg__nonneg,axiom,
    ! [Aa: nat,Ba: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Aa)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Ba)
       => 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,Aa),Ba)) ) ) ).

% mult_nonneg_nonneg
tff(fact_4412_mult__nonneg__nonneg,axiom,
    ! [Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Aa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Ba)
       => 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,Aa),Ba)) ) ) ).

% mult_nonneg_nonneg
tff(fact_4413_mult__nonneg__nonpos,axiom,
    ! [Aa: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Aa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ba),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),Ba)),zero_zero_real) ) ) ).

% mult_nonneg_nonpos
tff(fact_4414_mult__nonneg__nonpos,axiom,
    ! [Aa: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Aa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ba),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(Aa),Ba)),zero_zero_rat) ) ) ).

% mult_nonneg_nonpos
tff(fact_4415_mult__nonneg__nonpos,axiom,
    ! [Aa: nat,Ba: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Aa)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ba),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,Aa),Ba)),zero_zero_nat) ) ) ).

% mult_nonneg_nonpos
tff(fact_4416_mult__nonneg__nonpos,axiom,
    ! [Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Aa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ba),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,Aa),Ba)),zero_zero_int) ) ) ).

% mult_nonneg_nonpos
tff(fact_4417_mult__nonpos__nonneg,axiom,
    ! [Aa: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Ba)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),Ba)),zero_zero_real) ) ) ).

% mult_nonpos_nonneg
tff(fact_4418_mult__nonpos__nonneg,axiom,
    ! [Aa: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Ba)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(Aa),Ba)),zero_zero_rat) ) ) ).

% mult_nonpos_nonneg
tff(fact_4419_mult__nonpos__nonneg,axiom,
    ! [Aa: nat,Ba: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Ba)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),Ba)),zero_zero_nat) ) ) ).

% mult_nonpos_nonneg
tff(fact_4420_mult__nonpos__nonneg,axiom,
    ! [Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Ba)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,Aa),Ba)),zero_zero_int) ) ) ).

% mult_nonpos_nonneg
tff(fact_4421_mult__nonneg__nonpos2,axiom,
    ! [Aa: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Aa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ba),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Ba),Aa)),zero_zero_real) ) ) ).

% mult_nonneg_nonpos2
tff(fact_4422_mult__nonneg__nonpos2,axiom,
    ! [Aa: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Aa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ba),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(Ba),Aa)),zero_zero_rat) ) ) ).

% mult_nonneg_nonpos2
tff(fact_4423_mult__nonneg__nonpos2,axiom,
    ! [Aa: nat,Ba: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Aa)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ba),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,Ba),Aa)),zero_zero_nat) ) ) ).

% mult_nonneg_nonpos2
tff(fact_4424_mult__nonneg__nonpos2,axiom,
    ! [Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Aa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ba),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,Ba),Aa)),zero_zero_int) ) ) ).

% mult_nonneg_nonpos2
tff(fact_4425_zero__le__mult__iff,axiom,
    ! [Aa: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),Ba))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Aa)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Ba) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ba),zero_zero_real) ) ) ) ).

% zero_le_mult_iff
tff(fact_4426_zero__le__mult__iff,axiom,
    ! [Aa: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(times_times_rat(Aa),Ba))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Aa)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Ba) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ba),zero_zero_rat) ) ) ) ).

% zero_le_mult_iff
tff(fact_4427_zero__le__mult__iff,axiom,
    ! [Aa: int,Ba: 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,Aa),Ba))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Aa)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Ba) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ba),zero_zero_int) ) ) ) ).

% zero_le_mult_iff
tff(fact_4428_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [Aa: real,Ba: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),Ba)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),Aa)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),Ba)) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_4429_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [Aa: rat,Ba: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),Ba)
     => ( 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),Aa)),aa_rat_rat(times_times_rat(C),Ba)) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_4430_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [Aa: nat,Ba: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),Ba)
     => ( 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),Aa)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),Ba)) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_4431_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [Aa: int,Ba: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),Ba)
     => ( 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),Aa)),aa_int_int(aa_int_fun_int_int(times_times_int,C),Ba)) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_4432_mult__less__iff1,axiom,
    ! [Z: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Z)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y) ) ) ).

% mult_less_iff1
tff(fact_4433_mult__less__iff1,axiom,
    ! [Z: rat,Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Z)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(Xb),Z)),aa_rat_rat(times_times_rat(Y),Z))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y) ) ) ).

% mult_less_iff1
tff(fact_4434_mult__less__iff1,axiom,
    ! [Z: int,Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,Xb),Z)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Z))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y) ) ) ).

% mult_less_iff1
tff(fact_4435_mult__neg__neg,axiom,
    ! [Aa: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Ba),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),Ba)) ) ) ).

% mult_neg_neg
tff(fact_4436_mult__neg__neg,axiom,
    ! [Aa: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ba),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(Aa),Ba)) ) ) ).

% mult_neg_neg
tff(fact_4437_mult__neg__neg,axiom,
    ! [Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Ba),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,Aa),Ba)) ) ) ).

% mult_neg_neg
tff(fact_4438_not__square__less__zero,axiom,
    ! [Aa: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),Aa)),zero_zero_real) ).

% not_square_less_zero
tff(fact_4439_not__square__less__zero,axiom,
    ! [Aa: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(Aa),Aa)),zero_zero_rat) ).

% not_square_less_zero
tff(fact_4440_not__square__less__zero,axiom,
    ! [Aa: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,Aa),Aa)),zero_zero_int) ).

% not_square_less_zero
tff(fact_4441_mult__less__0__iff,axiom,
    ! [Aa: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),Ba)),zero_zero_real)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Aa)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,Ba),zero_zero_real) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Ba) ) ) ) ).

% mult_less_0_iff
tff(fact_4442_mult__less__0__iff,axiom,
    ! [Aa: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(Aa),Ba)),zero_zero_rat)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Aa)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ba),zero_zero_rat) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Ba) ) ) ) ).

% mult_less_0_iff
tff(fact_4443_mult__less__0__iff,axiom,
    ! [Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,Aa),Ba)),zero_zero_int)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Aa)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,Ba),zero_zero_int) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Ba) ) ) ) ).

% mult_less_0_iff
tff(fact_4444_mult__neg__pos,axiom,
    ! [Aa: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Ba)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),Ba)),zero_zero_real) ) ) ).

% mult_neg_pos
tff(fact_4445_mult__neg__pos,axiom,
    ! [Aa: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Ba)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(Aa),Ba)),zero_zero_rat) ) ) ).

% mult_neg_pos
tff(fact_4446_mult__neg__pos,axiom,
    ! [Aa: nat,Ba: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Ba)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),Ba)),zero_zero_nat) ) ) ).

% mult_neg_pos
tff(fact_4447_mult__neg__pos,axiom,
    ! [Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Ba)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,Aa),Ba)),zero_zero_int) ) ) ).

% mult_neg_pos
tff(fact_4448_mult__pos__neg,axiom,
    ! [Aa: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Aa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Ba),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),Ba)),zero_zero_real) ) ) ).

% mult_pos_neg
tff(fact_4449_mult__pos__neg,axiom,
    ! [Aa: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Aa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ba),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(Aa),Ba)),zero_zero_rat) ) ) ).

% mult_pos_neg
tff(fact_4450_mult__pos__neg,axiom,
    ! [Aa: nat,Ba: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Aa)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ba),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,Aa),Ba)),zero_zero_nat) ) ) ).

% mult_pos_neg
tff(fact_4451_mult__pos__neg,axiom,
    ! [Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Aa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Ba),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,Aa),Ba)),zero_zero_int) ) ) ).

% mult_pos_neg
tff(fact_4452_mult__pos__pos,axiom,
    ! [Aa: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Aa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Ba)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),Ba)) ) ) ).

% mult_pos_pos
tff(fact_4453_mult__pos__pos,axiom,
    ! [Aa: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Aa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Ba)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(Aa),Ba)) ) ) ).

% mult_pos_pos
tff(fact_4454_mult__pos__pos,axiom,
    ! [Aa: nat,Ba: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Aa)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Ba)
       => 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,Aa),Ba)) ) ) ).

% mult_pos_pos
tff(fact_4455_mult__pos__pos,axiom,
    ! [Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Aa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Ba)
       => 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,Aa),Ba)) ) ) ).

% mult_pos_pos
tff(fact_4456_mult__pos__neg2,axiom,
    ! [Aa: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Aa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Ba),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Ba),Aa)),zero_zero_real) ) ) ).

% mult_pos_neg2
tff(fact_4457_mult__pos__neg2,axiom,
    ! [Aa: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Aa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ba),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(Ba),Aa)),zero_zero_rat) ) ) ).

% mult_pos_neg2
tff(fact_4458_mult__pos__neg2,axiom,
    ! [Aa: nat,Ba: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Aa)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ba),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,Ba),Aa)),zero_zero_nat) ) ) ).

% mult_pos_neg2
tff(fact_4459_mult__pos__neg2,axiom,
    ! [Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Aa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Ba),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,Ba),Aa)),zero_zero_int) ) ) ).

% mult_pos_neg2
tff(fact_4460_zero__less__mult__iff,axiom,
    ! [Aa: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),Ba))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Aa)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Ba) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,Ba),zero_zero_real) ) ) ) ).

% zero_less_mult_iff
tff(fact_4461_zero__less__mult__iff,axiom,
    ! [Aa: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(Aa),Ba))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Aa)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Ba) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ba),zero_zero_rat) ) ) ) ).

% zero_less_mult_iff
tff(fact_4462_zero__less__mult__iff,axiom,
    ! [Aa: int,Ba: 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,Aa),Ba))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Aa)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Ba) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,Ba),zero_zero_int) ) ) ) ).

% zero_less_mult_iff
tff(fact_4463_zero__less__mult__pos,axiom,
    ! [Aa: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),Ba))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Aa)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Ba) ) ) ).

% zero_less_mult_pos
tff(fact_4464_zero__less__mult__pos,axiom,
    ! [Aa: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(Aa),Ba))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Aa)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Ba) ) ) ).

% zero_less_mult_pos
tff(fact_4465_zero__less__mult__pos,axiom,
    ! [Aa: nat,Ba: 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,Aa),Ba))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Aa)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Ba) ) ) ).

% zero_less_mult_pos
tff(fact_4466_zero__less__mult__pos,axiom,
    ! [Aa: int,Ba: 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,Aa),Ba))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Aa)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Ba) ) ) ).

% zero_less_mult_pos
tff(fact_4467_zero__less__mult__pos2,axiom,
    ! [Ba: real,Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Ba),Aa))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Aa)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Ba) ) ) ).

% zero_less_mult_pos2
tff(fact_4468_zero__less__mult__pos2,axiom,
    ! [Ba: rat,Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(Ba),Aa))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Aa)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Ba) ) ) ).

% zero_less_mult_pos2
tff(fact_4469_zero__less__mult__pos2,axiom,
    ! [Ba: nat,Aa: 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,Ba),Aa))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Aa)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Ba) ) ) ).

% zero_less_mult_pos2
tff(fact_4470_zero__less__mult__pos2,axiom,
    ! [Ba: int,Aa: 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,Ba),Aa))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Aa)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Ba) ) ) ).

% zero_less_mult_pos2
tff(fact_4471_mult__less__cancel__left__neg,axiom,
    ! [C: real,Aa: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),Aa)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),Ba))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Ba),Aa) ) ) ).

% mult_less_cancel_left_neg
tff(fact_4472_mult__less__cancel__left__neg,axiom,
    ! [C: rat,Aa: rat,Ba: 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),Aa)),aa_rat_rat(times_times_rat(C),Ba))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ba),Aa) ) ) ).

% mult_less_cancel_left_neg
tff(fact_4473_mult__less__cancel__left__neg,axiom,
    ! [C: int,Aa: int,Ba: 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),Aa)),aa_int_int(aa_int_fun_int_int(times_times_int,C),Ba))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Ba),Aa) ) ) ).

% mult_less_cancel_left_neg
tff(fact_4474_mult__less__cancel__left__pos,axiom,
    ! [C: real,Aa: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),Aa)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),Ba))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),Ba) ) ) ).

% mult_less_cancel_left_pos
tff(fact_4475_mult__less__cancel__left__pos,axiom,
    ! [C: rat,Aa: rat,Ba: 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),Aa)),aa_rat_rat(times_times_rat(C),Ba))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),Ba) ) ) ).

% mult_less_cancel_left_pos
tff(fact_4476_mult__less__cancel__left__pos,axiom,
    ! [C: int,Aa: int,Ba: 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),Aa)),aa_int_int(aa_int_fun_int_int(times_times_int,C),Ba))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),Ba) ) ) ).

% mult_less_cancel_left_pos
tff(fact_4477_mult__strict__left__mono__neg,axiom,
    ! [Ba: real,Aa: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Ba),Aa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),Aa)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),Ba)) ) ) ).

% mult_strict_left_mono_neg
tff(fact_4478_mult__strict__left__mono__neg,axiom,
    ! [Ba: rat,Aa: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ba),Aa)
     => ( 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),Aa)),aa_rat_rat(times_times_rat(C),Ba)) ) ) ).

% mult_strict_left_mono_neg
tff(fact_4479_mult__strict__left__mono__neg,axiom,
    ! [Ba: int,Aa: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Ba),Aa)
     => ( 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),Aa)),aa_int_int(aa_int_fun_int_int(times_times_int,C),Ba)) ) ) ).

% mult_strict_left_mono_neg
tff(fact_4480_mult__strict__left__mono,axiom,
    ! [Aa: real,Ba: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),Ba)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),Aa)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),Ba)) ) ) ).

% mult_strict_left_mono
tff(fact_4481_mult__strict__left__mono,axiom,
    ! [Aa: rat,Ba: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),Ba)
     => ( 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),Aa)),aa_rat_rat(times_times_rat(C),Ba)) ) ) ).

% mult_strict_left_mono
tff(fact_4482_mult__strict__left__mono,axiom,
    ! [Aa: nat,Ba: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),Ba)
     => ( 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),Aa)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),Ba)) ) ) ).

% mult_strict_left_mono
tff(fact_4483_mult__strict__left__mono,axiom,
    ! [Aa: int,Ba: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),Ba)
     => ( 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),Aa)),aa_int_int(aa_int_fun_int_int(times_times_int,C),Ba)) ) ) ).

% mult_strict_left_mono
tff(fact_4484_mult__less__cancel__left__disj,axiom,
    ! [C: real,Aa: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),Aa)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),Ba))
    <=> ( ( 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),Ba) )
        | ( 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,Ba),Aa) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_4485_mult__less__cancel__left__disj,axiom,
    ! [C: rat,Aa: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(C),Aa)),aa_rat_rat(times_times_rat(C),Ba))
    <=> ( ( 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),Ba) )
        | ( 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,Ba),Aa) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_4486_mult__less__cancel__left__disj,axiom,
    ! [C: int,Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),Aa)),aa_int_int(aa_int_fun_int_int(times_times_int,C),Ba))
    <=> ( ( 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),Ba) )
        | ( 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,Ba),Aa) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_4487_mult__strict__right__mono__neg,axiom,
    ! [Ba: real,Aa: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Ba),Aa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Ba),C)) ) ) ).

% mult_strict_right_mono_neg
tff(fact_4488_mult__strict__right__mono__neg,axiom,
    ! [Ba: rat,Aa: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ba),Aa)
     => ( 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),C)),aa_rat_rat(times_times_rat(Ba),C)) ) ) ).

% mult_strict_right_mono_neg
tff(fact_4489_mult__strict__right__mono__neg,axiom,
    ! [Ba: int,Aa: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Ba),Aa)
     => ( 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,Aa),C)),aa_int_int(aa_int_fun_int_int(times_times_int,Ba),C)) ) ) ).

% mult_strict_right_mono_neg
tff(fact_4490_mult__strict__right__mono,axiom,
    ! [Aa: real,Ba: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),Ba)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Ba),C)) ) ) ).

% mult_strict_right_mono
tff(fact_4491_mult__strict__right__mono,axiom,
    ! [Aa: rat,Ba: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),Ba)
     => ( 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),C)),aa_rat_rat(times_times_rat(Ba),C)) ) ) ).

% mult_strict_right_mono
tff(fact_4492_mult__strict__right__mono,axiom,
    ! [Aa: nat,Ba: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),Ba)
     => ( 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,Aa),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ba),C)) ) ) ).

% mult_strict_right_mono
tff(fact_4493_mult__strict__right__mono,axiom,
    ! [Aa: int,Ba: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),Ba)
     => ( 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,Aa),C)),aa_int_int(aa_int_fun_int_int(times_times_int,Ba),C)) ) ) ).

% mult_strict_right_mono
tff(fact_4494_mult__less__cancel__right__disj,axiom,
    ! [Aa: real,C: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Ba),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,Aa),Ba) )
        | ( 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,Ba),Aa) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_4495_mult__less__cancel__right__disj,axiom,
    ! [Aa: rat,C: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(Aa),C)),aa_rat_rat(times_times_rat(Ba),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,Aa),Ba) )
        | ( 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,Ba),Aa) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_4496_mult__less__cancel__right__disj,axiom,
    ! [Aa: int,C: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,Aa),C)),aa_int_int(aa_int_fun_int_int(times_times_int,Ba),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,Aa),Ba) )
        | ( 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,Ba),Aa) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_4497_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [Aa: real,Ba: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),Ba)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),Aa)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),Ba)) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_4498_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [Aa: rat,Ba: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),Ba)
     => ( 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),Aa)),aa_rat_rat(times_times_rat(C),Ba)) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_4499_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [Aa: nat,Ba: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),Ba)
     => ( 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),Aa)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),Ba)) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_4500_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [Aa: int,Ba: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),Ba)
     => ( 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),Aa)),aa_int_int(aa_int_fun_int_int(times_times_int,C),Ba)) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_4501_add__scale__eq__noteq,axiom,
    ! [R2: real,Aa: real,Ba: real,C: real,D: real] :
      ( ( R2 != zero_zero_real )
     => ( ( ( Aa = Ba )
          & ( C != D ) )
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),C)) != aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Ba),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_4502_add__scale__eq__noteq,axiom,
    ! [R2: rat,Aa: rat,Ba: rat,C: rat,D: rat] :
      ( ( R2 != zero_zero_rat )
     => ( ( ( Aa = Ba )
          & ( C != D ) )
       => ( aa_rat_rat(plus_plus_rat(Aa),aa_rat_rat(times_times_rat(R2),C)) != aa_rat_rat(plus_plus_rat(Ba),aa_rat_rat(times_times_rat(R2),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_4503_add__scale__eq__noteq,axiom,
    ! [R2: nat,Aa: nat,Ba: nat,C: nat,D: nat] :
      ( ( R2 != zero_zero_nat )
     => ( ( ( Aa = Ba )
          & ( C != D ) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R2),C)) != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ba),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R2),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_4504_add__scale__eq__noteq,axiom,
    ! [R2: int,Aa: int,Ba: int,C: int,D: int] :
      ( ( R2 != zero_zero_int )
     => ( ( ( Aa = Ba )
          & ( C != D ) )
       => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),aa_int_int(aa_int_fun_int_int(times_times_int,R2),C)) != aa_int_int(aa_int_fun_int_int(plus_plus_int,Ba),aa_int_int(aa_int_fun_int_int(times_times_int,R2),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_4505_less__1__mult,axiom,
    ! [M: real,Na: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),M)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Na)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,M),Na)) ) ) ).

% less_1_mult
tff(fact_4506_less__1__mult,axiom,
    ! [M: rat,Na: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),M)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),Na)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_rat_rat(times_times_rat(M),Na)) ) ) ).

% less_1_mult
tff(fact_4507_less__1__mult,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),Na)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na)) ) ) ).

% less_1_mult
tff(fact_4508_less__1__mult,axiom,
    ! [M: int,Na: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),M)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),Na)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,M),Na)) ) ) ).

% less_1_mult
tff(fact_4509_nonzero__eq__divide__eq,axiom,
    ! [C: rat,Aa: rat,Ba: rat] :
      ( ( C != zero_zero_rat )
     => ( ( Aa = aa_rat_rat(divide_divide_rat(Ba),C) )
      <=> ( aa_rat_rat(times_times_rat(Aa),C) = Ba ) ) ) ).

% nonzero_eq_divide_eq
tff(fact_4510_nonzero__eq__divide__eq,axiom,
    ! [C: real,Aa: real,Ba: real] :
      ( ( C != zero_zero_real )
     => ( ( Aa = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Ba),C) )
      <=> ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),C) = Ba ) ) ) ).

% nonzero_eq_divide_eq
tff(fact_4511_nonzero__divide__eq__eq,axiom,
    ! [C: rat,Ba: rat,Aa: rat] :
      ( ( C != zero_zero_rat )
     => ( ( aa_rat_rat(divide_divide_rat(Ba),C) = Aa )
      <=> ( Ba = aa_rat_rat(times_times_rat(Aa),C) ) ) ) ).

% nonzero_divide_eq_eq
tff(fact_4512_nonzero__divide__eq__eq,axiom,
    ! [C: real,Ba: real,Aa: real] :
      ( ( C != zero_zero_real )
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Ba),C) = Aa )
      <=> ( Ba = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),C) ) ) ) ).

% nonzero_divide_eq_eq
tff(fact_4513_eq__divide__imp,axiom,
    ! [C: rat,Aa: rat,Ba: rat] :
      ( ( C != zero_zero_rat )
     => ( ( aa_rat_rat(times_times_rat(Aa),C) = Ba )
       => ( Aa = aa_rat_rat(divide_divide_rat(Ba),C) ) ) ) ).

% eq_divide_imp
tff(fact_4514_eq__divide__imp,axiom,
    ! [C: real,Aa: real,Ba: real] :
      ( ( C != zero_zero_real )
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),C) = Ba )
       => ( Aa = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Ba),C) ) ) ) ).

% eq_divide_imp
tff(fact_4515_divide__eq__imp,axiom,
    ! [C: rat,Ba: rat,Aa: rat] :
      ( ( C != zero_zero_rat )
     => ( ( Ba = aa_rat_rat(times_times_rat(Aa),C) )
       => ( aa_rat_rat(divide_divide_rat(Ba),C) = Aa ) ) ) ).

% divide_eq_imp
tff(fact_4516_divide__eq__imp,axiom,
    ! [C: real,Ba: real,Aa: real] :
      ( ( C != zero_zero_real )
     => ( ( Ba = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),C) )
       => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Ba),C) = Aa ) ) ) ).

% divide_eq_imp
tff(fact_4517_eq__divide__eq,axiom,
    ! [Aa: rat,Ba: rat,C: rat] :
      ( ( Aa = aa_rat_rat(divide_divide_rat(Ba),C) )
    <=> $ite(C != zero_zero_rat,aa_rat_rat(times_times_rat(Aa),C) = Ba,Aa = zero_zero_rat) ) ).

% eq_divide_eq
tff(fact_4518_eq__divide__eq,axiom,
    ! [Aa: real,Ba: real,C: real] :
      ( ( Aa = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Ba),C) )
    <=> $ite(C != zero_zero_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),C) = Ba,Aa = zero_zero_real) ) ).

% eq_divide_eq
tff(fact_4519_divide__eq__eq,axiom,
    ! [Ba: rat,C: rat,Aa: rat] :
      ( ( aa_rat_rat(divide_divide_rat(Ba),C) = Aa )
    <=> $ite(C != zero_zero_rat,Ba = aa_rat_rat(times_times_rat(Aa),C),Aa = zero_zero_rat) ) ).

% divide_eq_eq
tff(fact_4520_divide__eq__eq,axiom,
    ! [Ba: real,C: real,Aa: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Ba),C) = Aa )
    <=> $ite(C != zero_zero_real,Ba = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),C),Aa = zero_zero_real) ) ).

% divide_eq_eq
tff(fact_4521_frac__eq__eq,axiom,
    ! [Y: rat,Z: rat,Xb: rat,W2: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( ( aa_rat_rat(divide_divide_rat(Xb),Y) = aa_rat_rat(divide_divide_rat(W2),Z) )
        <=> ( aa_rat_rat(times_times_rat(Xb),Z) = aa_rat_rat(times_times_rat(W2),Y) ) ) ) ) ).

% frac_eq_eq
tff(fact_4522_frac__eq__eq,axiom,
    ! [Y: real,Z: real,Xb: real,W2: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,W2),Z) )
        <=> ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Z) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W2),Y) ) ) ) ) ).

% frac_eq_eq
tff(fact_4523_mult__numeral__1__right,axiom,
    ! [Aa: rat] : aa_rat_rat(times_times_rat(Aa),aa_num_rat(numeral_numeral_rat,one)) = Aa ).

% mult_numeral_1_right
tff(fact_4524_mult__numeral__1__right,axiom,
    ! [Aa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),aa_num_nat(numeral_numeral_nat,one)) = Aa ).

% mult_numeral_1_right
tff(fact_4525_mult__numeral__1__right,axiom,
    ! [Aa: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),aa_num_real(numeral_numeral_real,one)) = Aa ).

% mult_numeral_1_right
tff(fact_4526_mult__numeral__1__right,axiom,
    ! [Aa: int] : aa_int_int(aa_int_fun_int_int(times_times_int,Aa),aa_num_int(numeral_numeral_int,one)) = Aa ).

% mult_numeral_1_right
tff(fact_4527_mult__numeral__1__right,axiom,
    ! [Aa: extended_enat] : times_7803423173614009249d_enat(Aa,numera1916890842035813515d_enat(one)) = Aa ).

% mult_numeral_1_right
tff(fact_4528_mult__numeral__1__right,axiom,
    ! [Aa: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Aa),aa_num_Code_integer(numera6620942414471956472nteger,one)) = Aa ).

% mult_numeral_1_right
tff(fact_4529_mult__numeral__1,axiom,
    ! [Aa: rat] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,one)),Aa) = Aa ).

% mult_numeral_1
tff(fact_4530_mult__numeral__1,axiom,
    ! [Aa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,one)),Aa) = Aa ).

% mult_numeral_1
tff(fact_4531_mult__numeral__1,axiom,
    ! [Aa: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,one)),Aa) = Aa ).

% mult_numeral_1
tff(fact_4532_mult__numeral__1,axiom,
    ! [Aa: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,one)),Aa) = Aa ).

% mult_numeral_1
tff(fact_4533_mult__numeral__1,axiom,
    ! [Aa: extended_enat] : times_7803423173614009249d_enat(numera1916890842035813515d_enat(one),Aa) = Aa ).

% mult_numeral_1
tff(fact_4534_mult__numeral__1,axiom,
    ! [Aa: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,one)),Aa) = Aa ).

% mult_numeral_1
tff(fact_4535_eq__add__iff1,axiom,
    ! [Aa: real,E2: real,C: real,Ba: real,D: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),E2)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Ba),E2)),D) )
    <=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Aa),Ba)),E2)),C) = D ) ) ).

% eq_add_iff1
tff(fact_4536_eq__add__iff1,axiom,
    ! [Aa: rat,E2: rat,C: rat,Ba: rat,D: rat] :
      ( ( aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(Aa),E2)),C) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(Ba),E2)),D) )
    <=> ( aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(Aa),Ba)),E2)),C) = D ) ) ).

% eq_add_iff1
tff(fact_4537_eq__add__iff1,axiom,
    ! [Aa: int,E2: int,C: int,Ba: 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,Aa),E2)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Ba),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,Aa),Ba)),E2)),C) = D ) ) ).

% eq_add_iff1
tff(fact_4538_eq__add__iff2,axiom,
    ! [Aa: real,E2: real,C: real,Ba: real,D: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),E2)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Ba),E2)),D) )
    <=> ( C = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Ba),Aa)),E2)),D) ) ) ).

% eq_add_iff2
tff(fact_4539_eq__add__iff2,axiom,
    ! [Aa: rat,E2: rat,C: rat,Ba: rat,D: rat] :
      ( ( aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(Aa),E2)),C) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(Ba),E2)),D) )
    <=> ( C = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(Ba),Aa)),E2)),D) ) ) ).

% eq_add_iff2
tff(fact_4540_eq__add__iff2,axiom,
    ! [Aa: int,E2: int,C: int,Ba: 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,Aa),E2)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Ba),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,Ba),Aa)),E2)),D) ) ) ).

% eq_add_iff2
tff(fact_4541_square__diff__square__factored,axiom,
    ! [Xb: real,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Xb)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Y)) ).

% square_diff_square_factored
tff(fact_4542_square__diff__square__factored,axiom,
    ! [Xb: rat,Y: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(Xb),Xb)),aa_rat_rat(times_times_rat(Y),Y)) = aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(Xb),Y)),aa_rat_rat(minus_minus_rat(Xb),Y)) ).

% square_diff_square_factored
tff(fact_4543_square__diff__square__factored,axiom,
    ! [Xb: int,Y: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Xb),Xb)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Y)),aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),Y)) ).

% square_diff_square_factored
tff(fact_4544_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_4545_square__eq__1__iff,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_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_4546_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_4547_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_4548_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_4549_left__right__inverse__power,axiom,
    ! [Xb: complex,Y: complex,Na: nat] :
      ( ( aa_complex_complex(times_times_complex(Xb),Y) = one_one_complex )
     => ( aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(Xb),Na)),aa_nat_complex(power_power_complex(Y),Na)) = one_one_complex ) ) ).

% left_right_inverse_power
tff(fact_4550_left__right__inverse__power,axiom,
    ! [Xb: real,Y: real,Na: nat] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Y) = one_one_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Na)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),Na)) = one_one_real ) ) ).

% left_right_inverse_power
tff(fact_4551_left__right__inverse__power,axiom,
    ! [Xb: rat,Y: rat,Na: nat] :
      ( ( aa_rat_rat(times_times_rat(Xb),Y) = one_one_rat )
     => ( aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(Xb),Na)),aa_nat_rat(power_power_rat(Y),Na)) = one_one_rat ) ) ).

% left_right_inverse_power
tff(fact_4552_left__right__inverse__power,axiom,
    ! [Xb: nat,Y: nat,Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Xb),Y) = one_one_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),Na)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),Na)) = one_one_nat ) ) ).

% left_right_inverse_power
tff(fact_4553_left__right__inverse__power,axiom,
    ! [Xb: int,Y: int,Na: nat] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,Xb),Y) = one_one_int )
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(Xb),Na)),aa_nat_int(power_power_int(Y),Na)) = one_one_int ) ) ).

% left_right_inverse_power
tff(fact_4554_unit__mult__right__cancel,axiom,
    ! [Aa: code_integer,Ba: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),one_one_Code_integer)
     => ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Ba),Aa) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),Aa) )
      <=> ( Ba = C ) ) ) ).

% unit_mult_right_cancel
tff(fact_4555_unit__mult__right__cancel,axiom,
    ! [Aa: nat,Ba: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),one_one_nat)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ba),Aa) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),Aa) )
      <=> ( Ba = C ) ) ) ).

% unit_mult_right_cancel
tff(fact_4556_unit__mult__right__cancel,axiom,
    ! [Aa: int,Ba: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),one_one_int)
     => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,Ba),Aa) = aa_int_int(aa_int_fun_int_int(times_times_int,C),Aa) )
      <=> ( Ba = C ) ) ) ).

% unit_mult_right_cancel
tff(fact_4557_unit__mult__left__cancel,axiom,
    ! [Aa: code_integer,Ba: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),one_one_Code_integer)
     => ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Aa),Ba) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Aa),C) )
      <=> ( Ba = C ) ) ) ).

% unit_mult_left_cancel
tff(fact_4558_unit__mult__left__cancel,axiom,
    ! [Aa: nat,Ba: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),one_one_nat)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),Ba) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),C) )
      <=> ( Ba = C ) ) ) ).

% unit_mult_left_cancel
tff(fact_4559_unit__mult__left__cancel,axiom,
    ! [Aa: int,Ba: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),one_one_int)
     => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,Aa),Ba) = aa_int_int(aa_int_fun_int_int(times_times_int,Aa),C) )
      <=> ( Ba = C ) ) ) ).

% unit_mult_left_cancel
tff(fact_4560_mult__unit__dvd__iff_H,axiom,
    ! [Aa: code_integer,Ba: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Aa),Ba)),C)
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Ba),C) ) ) ).

% mult_unit_dvd_iff'
tff(fact_4561_mult__unit__dvd__iff_H,axiom,
    ! [Aa: nat,Ba: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),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,Aa),Ba)),C)
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Ba),C) ) ) ).

% mult_unit_dvd_iff'
tff(fact_4562_mult__unit__dvd__iff_H,axiom,
    ! [Aa: int,Ba: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),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,Aa),Ba)),C)
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Ba),C) ) ) ).

% mult_unit_dvd_iff'
tff(fact_4563_dvd__mult__unit__iff_H,axiom,
    ! [Ba: code_integer,Aa: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Ba),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Ba),C))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),C) ) ) ).

% dvd_mult_unit_iff'
tff(fact_4564_dvd__mult__unit__iff_H,axiom,
    ! [Ba: nat,Aa: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Ba),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ba),C))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),C) ) ) ).

% dvd_mult_unit_iff'
tff(fact_4565_dvd__mult__unit__iff_H,axiom,
    ! [Ba: int,Aa: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Ba),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),aa_int_int(aa_int_fun_int_int(times_times_int,Ba),C))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),C) ) ) ).

% dvd_mult_unit_iff'
tff(fact_4566_mult__unit__dvd__iff,axiom,
    ! [Ba: code_integer,Aa: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Ba),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Aa),Ba)),C)
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),C) ) ) ).

% mult_unit_dvd_iff
tff(fact_4567_mult__unit__dvd__iff,axiom,
    ! [Ba: nat,Aa: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Ba),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,Aa),Ba)),C)
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),C) ) ) ).

% mult_unit_dvd_iff
tff(fact_4568_mult__unit__dvd__iff,axiom,
    ! [Ba: int,Aa: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Ba),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,Aa),Ba)),C)
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),C) ) ) ).

% mult_unit_dvd_iff
tff(fact_4569_dvd__mult__unit__iff,axiom,
    ! [Ba: code_integer,Aa: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Ba),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),Ba))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),C) ) ) ).

% dvd_mult_unit_iff
tff(fact_4570_dvd__mult__unit__iff,axiom,
    ! [Ba: nat,Aa: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Ba),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),Ba))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),C) ) ) ).

% dvd_mult_unit_iff
tff(fact_4571_dvd__mult__unit__iff,axiom,
    ! [Ba: int,Aa: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Ba),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),aa_int_int(aa_int_fun_int_int(times_times_int,C),Ba))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),C) ) ) ).

% dvd_mult_unit_iff
tff(fact_4572_is__unit__mult__iff,axiom,
    ! [Aa: code_integer,Ba: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Aa),Ba)),one_one_Code_integer)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),one_one_Code_integer)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Ba),one_one_Code_integer) ) ) ).

% is_unit_mult_iff
tff(fact_4573_is__unit__mult__iff,axiom,
    ! [Aa: nat,Ba: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),Ba)),one_one_nat)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),one_one_nat)
        & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Ba),one_one_nat) ) ) ).

% is_unit_mult_iff
tff(fact_4574_is__unit__mult__iff,axiom,
    ! [Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,Aa),Ba)),one_one_int)
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),one_one_int)
        & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Ba),one_one_int) ) ) ).

% is_unit_mult_iff
tff(fact_4575_power__Suc,axiom,
    ! [Aa: complex,Na: nat] : aa_nat_complex(power_power_complex(Aa),aa_nat_nat(suc,Na)) = aa_complex_complex(times_times_complex(Aa),aa_nat_complex(power_power_complex(Aa),Na)) ).

% power_Suc
tff(fact_4576_power__Suc,axiom,
    ! [Aa: real,Na: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),aa_nat_nat(suc,Na)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),Na)) ).

% power_Suc
tff(fact_4577_power__Suc,axiom,
    ! [Aa: rat,Na: nat] : aa_nat_rat(power_power_rat(Aa),aa_nat_nat(suc,Na)) = aa_rat_rat(times_times_rat(Aa),aa_nat_rat(power_power_rat(Aa),Na)) ).

% power_Suc
tff(fact_4578_power__Suc,axiom,
    ! [Aa: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Aa),aa_nat_nat(suc,Na)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Aa),Na)) ).

% power_Suc
tff(fact_4579_power__Suc,axiom,
    ! [Aa: int,Na: nat] : aa_nat_int(power_power_int(Aa),aa_nat_nat(suc,Na)) = aa_int_int(aa_int_fun_int_int(times_times_int,Aa),aa_nat_int(power_power_int(Aa),Na)) ).

% power_Suc
tff(fact_4580_power__Suc2,axiom,
    ! [Aa: complex,Na: nat] : aa_nat_complex(power_power_complex(Aa),aa_nat_nat(suc,Na)) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(Aa),Na)),Aa) ).

% power_Suc2
tff(fact_4581_power__Suc2,axiom,
    ! [Aa: real,Na: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),aa_nat_nat(suc,Na)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),Na)),Aa) ).

% power_Suc2
tff(fact_4582_power__Suc2,axiom,
    ! [Aa: rat,Na: nat] : aa_nat_rat(power_power_rat(Aa),aa_nat_nat(suc,Na)) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(Aa),Na)),Aa) ).

% power_Suc2
tff(fact_4583_power__Suc2,axiom,
    ! [Aa: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Aa),aa_nat_nat(suc,Na)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Aa),Na)),Aa) ).

% power_Suc2
tff(fact_4584_power__Suc2,axiom,
    ! [Aa: int,Na: nat] : aa_nat_int(power_power_int(Aa),aa_nat_nat(suc,Na)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(Aa),Na)),Aa) ).

% power_Suc2
tff(fact_4585_div__mult__div__if__dvd,axiom,
    ! [Ba: nat,Aa: nat,D: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Ba),Aa)
     => ( 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,Aa),Ba)),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,Aa),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ba),D)) ) ) ) ).

% div_mult_div_if_dvd
tff(fact_4586_div__mult__div__if__dvd,axiom,
    ! [Ba: int,Aa: int,D: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Ba),Aa)
     => ( 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,Aa),Ba)),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,Aa),C)),aa_int_int(aa_int_fun_int_int(times_times_int,Ba),D)) ) ) ) ).

% div_mult_div_if_dvd
tff(fact_4587_div__mult__div__if__dvd,axiom,
    ! [Ba: code_integer,Aa: code_integer,D: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Ba),Aa)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),C)
       => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Aa),Ba)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,C),D)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Aa),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Ba),D)) ) ) ) ).

% div_mult_div_if_dvd
tff(fact_4588_dvd__mult__imp__div,axiom,
    ! [Aa: nat,C: nat,Ba: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),C)),Ba)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Ba),C)) ) ).

% dvd_mult_imp_div
tff(fact_4589_dvd__mult__imp__div,axiom,
    ! [Aa: int,C: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,Aa),C)),Ba)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),aa_int_int(aa_int_fun_int_int(divide_divide_int,Ba),C)) ) ).

% dvd_mult_imp_div
tff(fact_4590_dvd__mult__imp__div,axiom,
    ! [Aa: code_integer,C: code_integer,Ba: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Aa),C)),Ba)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Ba),C)) ) ).

% dvd_mult_imp_div
tff(fact_4591_dvd__div__mult2__eq,axiom,
    ! [Ba: nat,C: nat,Aa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ba),C)),Aa)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ba),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Aa),Ba)),C) ) ) ).

% dvd_div_mult2_eq
tff(fact_4592_dvd__div__mult2__eq,axiom,
    ! [Ba: int,C: int,Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,Ba),C)),Aa)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),aa_int_int(aa_int_fun_int_int(times_times_int,Ba),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),Ba)),C) ) ) ).

% dvd_div_mult2_eq
tff(fact_4593_dvd__div__mult2__eq,axiom,
    ! [Ba: code_integer,C: code_integer,Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Ba),C)),Aa)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Aa),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Ba),C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Aa),Ba)),C) ) ) ).

% dvd_div_mult2_eq
tff(fact_4594_div__div__eq__right,axiom,
    ! [C: nat,Ba: nat,Aa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),Ba)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Ba),Aa)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Ba),C)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Aa),Ba)),C) ) ) ) ).

% div_div_eq_right
tff(fact_4595_div__div__eq__right,axiom,
    ! [C: int,Ba: int,Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),Ba)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Ba),Aa)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),aa_int_int(aa_int_fun_int_int(divide_divide_int,Ba),C)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),Ba)),C) ) ) ) ).

% div_div_eq_right
tff(fact_4596_div__div__eq__right,axiom,
    ! [C: code_integer,Ba: code_integer,Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),Ba)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Ba),Aa)
       => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Aa),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Ba),C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Aa),Ba)),C) ) ) ) ).

% div_div_eq_right
tff(fact_4597_div__mult__swap,axiom,
    ! [C: nat,Ba: nat,Aa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),Ba)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Ba),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),Ba)),C) ) ) ).

% div_mult_swap
tff(fact_4598_div__mult__swap,axiom,
    ! [C: int,Ba: int,Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),Ba)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,Aa),aa_int_int(aa_int_fun_int_int(divide_divide_int,Ba),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,Aa),Ba)),C) ) ) ).

% div_mult_swap
tff(fact_4599_div__mult__swap,axiom,
    ! [C: code_integer,Ba: code_integer,Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),Ba)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Aa),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Ba),C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Aa),Ba)),C) ) ) ).

% div_mult_swap
tff(fact_4600_dvd__div__mult,axiom,
    ! [C: nat,Ba: nat,Aa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),Ba)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Ba),C)),Aa) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ba),Aa)),C) ) ) ).

% dvd_div_mult
tff(fact_4601_dvd__div__mult,axiom,
    ! [C: int,Ba: int,Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),Ba)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,Ba),C)),Aa) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,Ba),Aa)),C) ) ) ).

% dvd_div_mult
tff(fact_4602_dvd__div__mult,axiom,
    ! [C: code_integer,Ba: code_integer,Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),Ba)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Ba),C)),Aa) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Ba),Aa)),C) ) ) ).

% dvd_div_mult
tff(fact_4603_Suc__mult__less__cancel1,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ).

% Suc_mult_less_cancel1
tff(fact_4604_nat__mult__eq__cancel1,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Na) )
      <=> ( M = Na ) ) ) ).

% nat_mult_eq_cancel1
tff(fact_4605_nat__mult__less__cancel1,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Na))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ) ).

% nat_mult_less_cancel1
tff(fact_4606_mult__less__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),K)) ) ) ).

% mult_less_mono1
tff(fact_4607_mult__less__mono2,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),I)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),J)) ) ) ).

% mult_less_mono2
tff(fact_4608_nat__mult__dvd__cancel1,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Na))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),Na) ) ) ).

% nat_mult_dvd_cancel1
tff(fact_4609_dvd__mult__cancel,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Na))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),Na) ) ) ).

% dvd_mult_cancel
tff(fact_4610_Suc__mult__le__cancel1,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ).

% Suc_mult_le_cancel1
tff(fact_4611_mult__Suc,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,M)),Na) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na)) ).

% mult_Suc
tff(fact_4612_bezout__add__strong__nat,axiom,
    ! [Aa: nat,Ba: nat] :
      ( ( Aa != zero_zero_nat )
     => ? [D4: nat,X: nat,Y3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D4),Aa)
          & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D4),Ba)
          & ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),X) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ba),Y3)),D4) ) ) ) ).

% bezout_add_strong_nat
tff(fact_4613_mult__eq__self__implies__10,axiom,
    ! [M: nat,Na: nat] :
      ( ( M = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na) )
     => ( ( Na = one_one_nat )
        | ( M = zero_zero_nat ) ) ) ).

% mult_eq_self_implies_10
tff(fact_4614_less__mult__imp__div__less,axiom,
    ! [M: nat,I: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),Na))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na)),I) ) ).

% less_mult_imp_div_less
tff(fact_4615_times__div__less__eq__dividend,axiom,
    ! [Na: nat,M: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na))),M) ).

% times_div_less_eq_dividend
tff(fact_4616_div__times__less__eq__dividend,axiom,
    ! [M: nat,Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na)),Na)),M) ).

% div_times_less_eq_dividend
tff(fact_4617_power__odd__eq,axiom,
    ! [Aa: complex,Na: nat] : aa_nat_complex(power_power_complex(Aa),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))) = aa_complex_complex(times_times_complex(Aa),aa_nat_complex(power_power_complex(aa_nat_complex(power_power_complex(Aa),Na)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_4618_power__odd__eq,axiom,
    ! [Aa: real,Na: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),Na)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_4619_power__odd__eq,axiom,
    ! [Aa: rat,Na: nat] : aa_nat_rat(power_power_rat(Aa),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))) = aa_rat_rat(times_times_rat(Aa),aa_nat_rat(power_power_rat(aa_nat_rat(power_power_rat(Aa),Na)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_4620_power__odd__eq,axiom,
    ! [Aa: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Aa),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Aa),Na)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_4621_power__odd__eq,axiom,
    ! [Aa: int,Na: nat] : aa_nat_int(power_power_int(Aa),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))) = aa_int_int(aa_int_fun_int_int(times_times_int,Aa),aa_nat_int(power_power_int(aa_nat_int(power_power_int(Aa),Na)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_4622_option_Oexhaust__sel,axiom,
    ! [Option: option_nat] :
      ( ( Option != none_nat )
     => ( Option = some_nat(the_nat(Option)) ) ) ).

% option.exhaust_sel
tff(fact_4623_option_Oexhaust__sel,axiom,
    ! [Option: option4927543243414619207at_nat] :
      ( ( Option != none_P5556105721700978146at_nat )
     => ( Option = some_P7363390416028606310at_nat(the_Pr8591224930841456533at_nat(Option)) ) ) ).

% option.exhaust_sel
tff(fact_4624_option_Oexhaust__sel,axiom,
    ! [Option: option_num] :
      ( ( Option != none_num )
     => ( Option = some_num(the_num(Option)) ) ) ).

% option.exhaust_sel
tff(fact_4625_mult__le__cancel__left,axiom,
    ! [C: real,Aa: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),Aa)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),Ba))
    <=> ( ( 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),Ba) )
        & ( 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,Ba),Aa) ) ) ) ).

% mult_le_cancel_left
tff(fact_4626_mult__le__cancel__left,axiom,
    ! [C: rat,Aa: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(C),Aa)),aa_rat_rat(times_times_rat(C),Ba))
    <=> ( ( 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),Ba) )
        & ( 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,Ba),Aa) ) ) ) ).

% mult_le_cancel_left
tff(fact_4627_mult__le__cancel__left,axiom,
    ! [C: int,Aa: int,Ba: 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),Aa)),aa_int_int(aa_int_fun_int_int(times_times_int,C),Ba))
    <=> ( ( 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),Ba) )
        & ( 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,Ba),Aa) ) ) ) ).

% mult_le_cancel_left
tff(fact_4628_mult__le__cancel__right,axiom,
    ! [Aa: real,C: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Ba),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,Aa),Ba) )
        & ( 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,Ba),Aa) ) ) ) ).

% mult_le_cancel_right
tff(fact_4629_mult__le__cancel__right,axiom,
    ! [Aa: rat,C: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(Aa),C)),aa_rat_rat(times_times_rat(Ba),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,Aa),Ba) )
        & ( 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,Ba),Aa) ) ) ) ).

% mult_le_cancel_right
tff(fact_4630_mult__le__cancel__right,axiom,
    ! [Aa: int,C: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,Aa),C)),aa_int_int(aa_int_fun_int_int(times_times_int,Ba),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,Aa),Ba) )
        & ( 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,Ba),Aa) ) ) ) ).

% mult_le_cancel_right
tff(fact_4631_mult__left__less__imp__less,axiom,
    ! [C: real,Aa: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),Aa)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),Ba))
     => ( 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),Ba) ) ) ).

% mult_left_less_imp_less
tff(fact_4632_mult__left__less__imp__less,axiom,
    ! [C: rat,Aa: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(C),Aa)),aa_rat_rat(times_times_rat(C),Ba))
     => ( 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),Ba) ) ) ).

% mult_left_less_imp_less
tff(fact_4633_mult__left__less__imp__less,axiom,
    ! [C: nat,Aa: nat,Ba: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),Aa)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),Ba))
     => ( 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),Ba) ) ) ).

% mult_left_less_imp_less
tff(fact_4634_mult__left__less__imp__less,axiom,
    ! [C: int,Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),Aa)),aa_int_int(aa_int_fun_int_int(times_times_int,C),Ba))
     => ( 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),Ba) ) ) ).

% mult_left_less_imp_less
tff(fact_4635_mult__strict__mono,axiom,
    ! [Aa: real,Ba: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),Ba)
     => ( 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),Ba)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Ba),D)) ) ) ) ) ).

% mult_strict_mono
tff(fact_4636_mult__strict__mono,axiom,
    ! [Aa: rat,Ba: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),Ba)
     => ( 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),Ba)
         => ( 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(Aa),C)),aa_rat_rat(times_times_rat(Ba),D)) ) ) ) ) ).

% mult_strict_mono
tff(fact_4637_mult__strict__mono,axiom,
    ! [Aa: nat,Ba: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),Ba)
     => ( 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),Ba)
         => ( 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,Aa),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ba),D)) ) ) ) ) ).

% mult_strict_mono
tff(fact_4638_mult__strict__mono,axiom,
    ! [Aa: int,Ba: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),Ba)
     => ( 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),Ba)
         => ( 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,Aa),C)),aa_int_int(aa_int_fun_int_int(times_times_int,Ba),D)) ) ) ) ) ).

% mult_strict_mono
tff(fact_4639_mult__less__cancel__left,axiom,
    ! [C: real,Aa: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),Aa)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),Ba))
    <=> ( ( 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),Ba) )
        & ( 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,Ba),Aa) ) ) ) ).

% mult_less_cancel_left
tff(fact_4640_mult__less__cancel__left,axiom,
    ! [C: rat,Aa: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(C),Aa)),aa_rat_rat(times_times_rat(C),Ba))
    <=> ( ( 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),Ba) )
        & ( 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,Ba),Aa) ) ) ) ).

% mult_less_cancel_left
tff(fact_4641_mult__less__cancel__left,axiom,
    ! [C: int,Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),Aa)),aa_int_int(aa_int_fun_int_int(times_times_int,C),Ba))
    <=> ( ( 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),Ba) )
        & ( 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,Ba),Aa) ) ) ) ).

% mult_less_cancel_left
tff(fact_4642_mult__right__less__imp__less,axiom,
    ! [Aa: real,C: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Ba),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,Aa),Ba) ) ) ).

% mult_right_less_imp_less
tff(fact_4643_mult__right__less__imp__less,axiom,
    ! [Aa: rat,C: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(Aa),C)),aa_rat_rat(times_times_rat(Ba),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,Aa),Ba) ) ) ).

% mult_right_less_imp_less
tff(fact_4644_mult__right__less__imp__less,axiom,
    ! [Aa: nat,C: nat,Ba: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ba),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,Aa),Ba) ) ) ).

% mult_right_less_imp_less
tff(fact_4645_mult__right__less__imp__less,axiom,
    ! [Aa: int,C: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,Aa),C)),aa_int_int(aa_int_fun_int_int(times_times_int,Ba),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,Aa),Ba) ) ) ).

% mult_right_less_imp_less
tff(fact_4646_mult__strict__mono_H,axiom,
    ! [Aa: real,Ba: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),Ba)
     => ( 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),Aa)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Ba),D)) ) ) ) ) ).

% mult_strict_mono'
tff(fact_4647_mult__strict__mono_H,axiom,
    ! [Aa: rat,Ba: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),Ba)
     => ( 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),Aa)
         => ( 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(Aa),C)),aa_rat_rat(times_times_rat(Ba),D)) ) ) ) ) ).

% mult_strict_mono'
tff(fact_4648_mult__strict__mono_H,axiom,
    ! [Aa: nat,Ba: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),Ba)
     => ( 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),Aa)
         => ( 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,Aa),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ba),D)) ) ) ) ) ).

% mult_strict_mono'
tff(fact_4649_mult__strict__mono_H,axiom,
    ! [Aa: int,Ba: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),Ba)
     => ( 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),Aa)
         => ( 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,Aa),C)),aa_int_int(aa_int_fun_int_int(times_times_int,Ba),D)) ) ) ) ) ).

% mult_strict_mono'
tff(fact_4650_mult__less__cancel__right,axiom,
    ! [Aa: real,C: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Ba),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,Aa),Ba) )
        & ( 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,Ba),Aa) ) ) ) ).

% mult_less_cancel_right
tff(fact_4651_mult__less__cancel__right,axiom,
    ! [Aa: rat,C: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(Aa),C)),aa_rat_rat(times_times_rat(Ba),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,Aa),Ba) )
        & ( 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,Ba),Aa) ) ) ) ).

% mult_less_cancel_right
tff(fact_4652_mult__less__cancel__right,axiom,
    ! [Aa: int,C: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,Aa),C)),aa_int_int(aa_int_fun_int_int(times_times_int,Ba),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,Aa),Ba) )
        & ( 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,Ba),Aa) ) ) ) ).

% mult_less_cancel_right
tff(fact_4653_mult__le__cancel__left__neg,axiom,
    ! [C: real,Aa: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),Aa)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),Ba))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ba),Aa) ) ) ).

% mult_le_cancel_left_neg
tff(fact_4654_mult__le__cancel__left__neg,axiom,
    ! [C: rat,Aa: rat,Ba: 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),Aa)),aa_rat_rat(times_times_rat(C),Ba))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ba),Aa) ) ) ).

% mult_le_cancel_left_neg
tff(fact_4655_mult__le__cancel__left__neg,axiom,
    ! [C: int,Aa: int,Ba: 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),Aa)),aa_int_int(aa_int_fun_int_int(times_times_int,C),Ba))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ba),Aa) ) ) ).

% mult_le_cancel_left_neg
tff(fact_4656_mult__le__cancel__left__pos,axiom,
    ! [C: real,Aa: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),Aa)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),Ba))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),Ba) ) ) ).

% mult_le_cancel_left_pos
tff(fact_4657_mult__le__cancel__left__pos,axiom,
    ! [C: rat,Aa: rat,Ba: 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),Aa)),aa_rat_rat(times_times_rat(C),Ba))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),Ba) ) ) ).

% mult_le_cancel_left_pos
tff(fact_4658_mult__le__cancel__left__pos,axiom,
    ! [C: int,Aa: int,Ba: 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),Aa)),aa_int_int(aa_int_fun_int_int(times_times_int,C),Ba))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),Ba) ) ) ).

% mult_le_cancel_left_pos
tff(fact_4659_mult__left__le__imp__le,axiom,
    ! [C: real,Aa: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),Aa)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),Ba))
     => ( 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),Ba) ) ) ).

% mult_left_le_imp_le
tff(fact_4660_mult__left__le__imp__le,axiom,
    ! [C: rat,Aa: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(C),Aa)),aa_rat_rat(times_times_rat(C),Ba))
     => ( 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),Ba) ) ) ).

% mult_left_le_imp_le
tff(fact_4661_mult__left__le__imp__le,axiom,
    ! [C: nat,Aa: nat,Ba: 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),Aa)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),Ba))
     => ( 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,Aa),Ba) ) ) ).

% mult_left_le_imp_le
tff(fact_4662_mult__left__le__imp__le,axiom,
    ! [C: int,Aa: int,Ba: 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),Aa)),aa_int_int(aa_int_fun_int_int(times_times_int,C),Ba))
     => ( 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),Ba) ) ) ).

% mult_left_le_imp_le
tff(fact_4663_mult__right__le__imp__le,axiom,
    ! [Aa: real,C: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Ba),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,Aa),Ba) ) ) ).

% mult_right_le_imp_le
tff(fact_4664_mult__right__le__imp__le,axiom,
    ! [Aa: rat,C: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(Aa),C)),aa_rat_rat(times_times_rat(Ba),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,Aa),Ba) ) ) ).

% mult_right_le_imp_le
tff(fact_4665_mult__right__le__imp__le,axiom,
    ! [Aa: nat,C: nat,Ba: 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),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ba),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,Aa),Ba) ) ) ).

% mult_right_le_imp_le
tff(fact_4666_mult__right__le__imp__le,axiom,
    ! [Aa: int,C: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,Aa),C)),aa_int_int(aa_int_fun_int_int(times_times_int,Ba),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,Aa),Ba) ) ) ).

% mult_right_le_imp_le
tff(fact_4667_mult__le__less__imp__less,axiom,
    ! [Aa: real,Ba: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),Ba)
     => ( 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),Aa)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Ba),D)) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_4668_mult__le__less__imp__less,axiom,
    ! [Aa: rat,Ba: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),Ba)
     => ( 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),Aa)
         => ( 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(Aa),C)),aa_rat_rat(times_times_rat(Ba),D)) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_4669_mult__le__less__imp__less,axiom,
    ! [Aa: nat,Ba: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),Ba)
     => ( 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),Aa)
         => ( 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,Aa),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ba),D)) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_4670_mult__le__less__imp__less,axiom,
    ! [Aa: int,Ba: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),Ba)
     => ( 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),Aa)
         => ( 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,Aa),C)),aa_int_int(aa_int_fun_int_int(times_times_int,Ba),D)) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_4671_mult__less__le__imp__less,axiom,
    ! [Aa: real,Ba: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),Ba)
     => ( 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),Aa)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),C)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Ba),D)) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_4672_mult__less__le__imp__less,axiom,
    ! [Aa: rat,Ba: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),Ba)
     => ( 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),Aa)
         => ( 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),C)),aa_rat_rat(times_times_rat(Ba),D)) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_4673_mult__less__le__imp__less,axiom,
    ! [Aa: nat,Ba: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),Ba)
     => ( 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),Aa)
         => ( 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,Aa),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ba),D)) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_4674_mult__less__le__imp__less,axiom,
    ! [Aa: int,Ba: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),Ba)
     => ( 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),Aa)
         => ( 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,Aa),C)),aa_int_int(aa_int_fun_int_int(times_times_int,Ba),D)) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_4675_mult__le__cancel__iff2,axiom,
    ! [Z: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Z)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Xb)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Y))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y) ) ) ).

% mult_le_cancel_iff2
tff(fact_4676_mult__le__cancel__iff2,axiom,
    ! [Z: rat,Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Z)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(Z),Xb)),aa_rat_rat(times_times_rat(Z),Y))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y) ) ) ).

% mult_le_cancel_iff2
tff(fact_4677_mult__le__cancel__iff2,axiom,
    ! [Z: int,Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,Z),Xb)),aa_int_int(aa_int_fun_int_int(times_times_int,Z),Y))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y) ) ) ).

% mult_le_cancel_iff2
tff(fact_4678_mult__le__cancel__iff1,axiom,
    ! [Z: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Z)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y) ) ) ).

% mult_le_cancel_iff1
tff(fact_4679_mult__le__cancel__iff1,axiom,
    ! [Z: rat,Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Z)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(Xb),Z)),aa_rat_rat(times_times_rat(Y),Z))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y) ) ) ).

% mult_le_cancel_iff1
tff(fact_4680_mult__le__cancel__iff1,axiom,
    ! [Z: int,Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,Xb),Z)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Z))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y) ) ) ).

% mult_le_cancel_iff1
tff(fact_4681_mult__left__le__one__le,axiom,
    ! [Xb: real,Y: 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),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Xb)),Xb) ) ) ) ).

% mult_left_le_one_le
tff(fact_4682_mult__left__le__one__le,axiom,
    ! [Xb: rat,Y: 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),Y)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),one_one_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(Y),Xb)),Xb) ) ) ) ).

% mult_left_le_one_le
tff(fact_4683_mult__left__le__one__le,axiom,
    ! [Xb: int,Y: 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),Y)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),one_one_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,Y),Xb)),Xb) ) ) ) ).

% mult_left_le_one_le
tff(fact_4684_mult__right__le__one__le,axiom,
    ! [Xb: real,Y: 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),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Y)),Xb) ) ) ) ).

% mult_right_le_one_le
tff(fact_4685_mult__right__le__one__le,axiom,
    ! [Xb: rat,Y: 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),Y)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),one_one_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(Xb),Y)),Xb) ) ) ) ).

% mult_right_le_one_le
tff(fact_4686_mult__right__le__one__le,axiom,
    ! [Xb: int,Y: 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),Y)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),one_one_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,Xb),Y)),Xb) ) ) ) ).

% mult_right_le_one_le
tff(fact_4687_mult__le__one,axiom,
    ! [Aa: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Ba)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ba),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),Ba)),one_one_real) ) ) ) ).

% mult_le_one
tff(fact_4688_mult__le__one,axiom,
    ! [Aa: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),one_one_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Ba)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ba),one_one_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(Aa),Ba)),one_one_rat) ) ) ) ).

% mult_le_one
tff(fact_4689_mult__le__one,axiom,
    ! [Aa: nat,Ba: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Ba)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ba),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,Aa),Ba)),one_one_nat) ) ) ) ).

% mult_le_one
tff(fact_4690_mult__le__one,axiom,
    ! [Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Ba)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ba),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,Aa),Ba)),one_one_int) ) ) ) ).

% mult_le_one
tff(fact_4691_mult__left__le,axiom,
    ! [C: real,Aa: 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),Aa)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),C)),Aa) ) ) ).

% mult_left_le
tff(fact_4692_mult__left__le,axiom,
    ! [C: rat,Aa: 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),Aa)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(Aa),C)),Aa) ) ) ).

% mult_left_le
tff(fact_4693_mult__left__le,axiom,
    ! [C: nat,Aa: 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),Aa)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),C)),Aa) ) ) ).

% mult_left_le
tff(fact_4694_mult__left__le,axiom,
    ! [C: int,Aa: 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),Aa)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,Aa),C)),Aa) ) ) ).

% mult_left_le
tff(fact_4695_sum__squares__ge__zero,axiom,
    ! [Xb: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Xb)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Y))) ).

% sum_squares_ge_zero
tff(fact_4696_sum__squares__ge__zero,axiom,
    ! [Xb: rat,Y: 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(Y),Y))) ).

% sum_squares_ge_zero
tff(fact_4697_sum__squares__ge__zero,axiom,
    ! [Xb: int,Y: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Xb),Xb)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y))) ).

% sum_squares_ge_zero
tff(fact_4698_sum__squares__le__zero__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Xb)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Y))),zero_zero_real)
    <=> ( ( Xb = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_4699_sum__squares__le__zero__iff,axiom,
    ! [Xb: rat,Y: 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(Y),Y))),zero_zero_rat)
    <=> ( ( Xb = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_4700_sum__squares__le__zero__iff,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Xb),Xb)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y))),zero_zero_int)
    <=> ( ( Xb = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_4701_not__sum__squares__lt__zero,axiom,
    ! [Xb: real,Y: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Xb)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Y))),zero_zero_real) ).

% not_sum_squares_lt_zero
tff(fact_4702_not__sum__squares__lt__zero,axiom,
    ! [Xb: rat,Y: 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(Y),Y))),zero_zero_rat) ).

% not_sum_squares_lt_zero
tff(fact_4703_not__sum__squares__lt__zero,axiom,
    ! [Xb: int,Y: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Xb),Xb)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y))),zero_zero_int) ).

% not_sum_squares_lt_zero
tff(fact_4704_sum__squares__gt__zero__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Xb)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Y)))
    <=> ( ( Xb != zero_zero_real )
        | ( Y != zero_zero_real ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_4705_sum__squares__gt__zero__iff,axiom,
    ! [Xb: rat,Y: 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(Y),Y)))
    <=> ( ( Xb != zero_zero_rat )
        | ( Y != zero_zero_rat ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_4706_sum__squares__gt__zero__iff,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Xb),Xb)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y)))
    <=> ( ( Xb != zero_zero_int )
        | ( Y != zero_zero_int ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_4707_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [C: code_integer,Aa: code_integer,Ba: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Aa),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Ba),C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Aa),Ba)),C) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
tff(fact_4708_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [C: nat,Aa: nat,Ba: 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,Aa),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ba),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Aa),Ba)),C) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
tff(fact_4709_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [C: int,Aa: int,Ba: 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,Aa),aa_int_int(aa_int_fun_int_int(times_times_int,Ba),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),Ba)),C) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
tff(fact_4710_divide__less__eq,axiom,
    ! [Ba: rat,C: rat,Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(Ba),C)),Aa)
    <=> $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,Ba),aa_rat_rat(times_times_rat(Aa),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),C)),Ba),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Aa)) ) ) ).

% divide_less_eq
tff(fact_4711_divide__less__eq,axiom,
    ! [Ba: real,C: real,Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Ba),C)),Aa)
    <=> $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,Ba),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),C)),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),C)),Ba),aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Aa)) ) ) ).

% divide_less_eq
tff(fact_4712_less__divide__eq,axiom,
    ! [Aa: rat,Ba: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),aa_rat_rat(divide_divide_rat(Ba),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),C)),Ba),
          $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,Ba),aa_rat_rat(times_times_rat(Aa),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),zero_zero_rat)) ) ) ).

% less_divide_eq
tff(fact_4713_less__divide__eq,axiom,
    ! [Aa: real,Ba: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Ba),C))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),C)),Ba),
          $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,Ba),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),C)),aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),zero_zero_real)) ) ) ).

% less_divide_eq
tff(fact_4714_neg__divide__less__eq,axiom,
    ! [C: rat,Ba: rat,Aa: 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(Ba),C)),Aa)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(Aa),C)),Ba) ) ) ).

% neg_divide_less_eq
tff(fact_4715_neg__divide__less__eq,axiom,
    ! [C: real,Ba: real,Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Ba),C)),Aa)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),C)),Ba) ) ) ).

% neg_divide_less_eq
tff(fact_4716_neg__less__divide__eq,axiom,
    ! [C: rat,Aa: rat,Ba: 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),aa_rat_rat(divide_divide_rat(Ba),C))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ba),aa_rat_rat(times_times_rat(Aa),C)) ) ) ).

% neg_less_divide_eq
tff(fact_4717_neg__less__divide__eq,axiom,
    ! [C: real,Aa: real,Ba: 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),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Ba),C))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Ba),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),C)) ) ) ).

% neg_less_divide_eq
tff(fact_4718_pos__divide__less__eq,axiom,
    ! [C: rat,Ba: rat,Aa: 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(Ba),C)),Aa)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ba),aa_rat_rat(times_times_rat(Aa),C)) ) ) ).

% pos_divide_less_eq
tff(fact_4719_pos__divide__less__eq,axiom,
    ! [C: real,Ba: real,Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Ba),C)),Aa)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Ba),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),C)) ) ) ).

% pos_divide_less_eq
tff(fact_4720_pos__less__divide__eq,axiom,
    ! [C: rat,Aa: rat,Ba: 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),aa_rat_rat(divide_divide_rat(Ba),C))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(Aa),C)),Ba) ) ) ).

% pos_less_divide_eq
tff(fact_4721_pos__less__divide__eq,axiom,
    ! [C: real,Aa: real,Ba: 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),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Ba),C))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),C)),Ba) ) ) ).

% pos_less_divide_eq
tff(fact_4722_mult__imp__div__pos__less,axiom,
    ! [Y: rat,Xb: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),aa_rat_rat(times_times_rat(Z),Y))
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(Xb),Y)),Z) ) ) ).

% mult_imp_div_pos_less
tff(fact_4723_mult__imp__div__pos__less,axiom,
    ! [Y: real,Xb: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Y))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)),Z) ) ) ).

% mult_imp_div_pos_less
tff(fact_4724_mult__imp__less__div__pos,axiom,
    ! [Y: rat,Z: rat,Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(Z),Y)),Xb)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),aa_rat_rat(divide_divide_rat(Xb),Y)) ) ) ).

% mult_imp_less_div_pos
tff(fact_4725_mult__imp__less__div__pos,axiom,
    ! [Y: real,Z: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Y)),Xb)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Z),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)) ) ) ).

% mult_imp_less_div_pos
tff(fact_4726_divide__strict__left__mono,axiom,
    ! [Ba: rat,Aa: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ba),Aa)
     => ( 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(Aa),Ba))
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(C),Aa)),aa_rat_rat(divide_divide_rat(C),Ba)) ) ) ) ).

% divide_strict_left_mono
tff(fact_4727_divide__strict__left__mono,axiom,
    ! [Ba: real,Aa: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Ba),Aa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),Ba))
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),Aa)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),Ba)) ) ) ) ).

% divide_strict_left_mono
tff(fact_4728_divide__strict__left__mono__neg,axiom,
    ! [Aa: rat,Ba: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),Ba)
     => ( 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(Aa),Ba))
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(C),Aa)),aa_rat_rat(divide_divide_rat(C),Ba)) ) ) ) ).

% divide_strict_left_mono_neg
tff(fact_4729_divide__strict__left__mono__neg,axiom,
    ! [Aa: real,Ba: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),Ba)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),Ba))
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),Aa)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),Ba)) ) ) ) ).

% divide_strict_left_mono_neg
tff(fact_4730_divide__eq__eq__numeral_I1_J,axiom,
    ! [Ba: rat,C: rat,W2: num] :
      ( ( aa_rat_rat(divide_divide_rat(Ba),C) = aa_num_rat(numeral_numeral_rat,W2) )
    <=> $ite(C != zero_zero_rat,Ba = aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W2)),C),aa_num_rat(numeral_numeral_rat,W2) = zero_zero_rat) ) ).

% divide_eq_eq_numeral(1)
tff(fact_4731_divide__eq__eq__numeral_I1_J,axiom,
    ! [Ba: real,C: real,W2: num] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Ba),C) = aa_num_real(numeral_numeral_real,W2) )
    <=> $ite(C != zero_zero_real,Ba = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W2)),C),aa_num_real(numeral_numeral_real,W2) = zero_zero_real) ) ).

% divide_eq_eq_numeral(1)
tff(fact_4732_eq__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,Ba: rat,C: rat] :
      ( ( aa_num_rat(numeral_numeral_rat,W2) = aa_rat_rat(divide_divide_rat(Ba),C) )
    <=> $ite(C != zero_zero_rat,aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W2)),C) = Ba,aa_num_rat(numeral_numeral_rat,W2) = zero_zero_rat) ) ).

% eq_divide_eq_numeral(1)
tff(fact_4733_eq__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,Ba: real,C: real] :
      ( ( aa_num_real(numeral_numeral_real,W2) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Ba),C) )
    <=> $ite(C != zero_zero_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W2)),C) = Ba,aa_num_real(numeral_numeral_real,W2) = zero_zero_real) ) ).

% eq_divide_eq_numeral(1)
tff(fact_4734_ordered__ring__class_Ole__add__iff1,axiom,
    ! [Aa: real,E2: real,C: real,Ba: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),E2)),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Ba),E2)),D))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Aa),Ba)),E2)),C)),D) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_4735_ordered__ring__class_Ole__add__iff1,axiom,
    ! [Aa: rat,E2: rat,C: rat,Ba: 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(Aa),E2)),C)),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(Ba),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(Aa),Ba)),E2)),C)),D) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_4736_ordered__ring__class_Ole__add__iff1,axiom,
    ! [Aa: int,E2: int,C: int,Ba: 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,Aa),E2)),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Ba),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,Aa),Ba)),E2)),C)),D) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_4737_ordered__ring__class_Ole__add__iff2,axiom,
    ! [Aa: real,E2: real,C: real,Ba: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),E2)),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Ba),E2)),D))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Ba),Aa)),E2)),D)) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_4738_ordered__ring__class_Ole__add__iff2,axiom,
    ! [Aa: rat,E2: rat,C: rat,Ba: 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(Aa),E2)),C)),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(Ba),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(Ba),Aa)),E2)),D)) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_4739_ordered__ring__class_Ole__add__iff2,axiom,
    ! [Aa: int,E2: int,C: int,Ba: 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,Aa),E2)),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Ba),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,Ba),Aa)),E2)),D)) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_4740_divide__add__eq__iff,axiom,
    ! [Z: rat,Xb: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(plus_plus_rat(aa_rat_rat(divide_divide_rat(Xb),Z)),Y) = aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(Xb),aa_rat_rat(times_times_rat(Y),Z))),Z) ) ) ).

% divide_add_eq_iff
tff(fact_4741_divide__add__eq__iff,axiom,
    ! [Z: real,Xb: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Z)),Y) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z))),Z) ) ) ).

% divide_add_eq_iff
tff(fact_4742_add__divide__eq__iff,axiom,
    ! [Z: rat,Xb: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(plus_plus_rat(Xb),aa_rat_rat(divide_divide_rat(Y),Z)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(Xb),Z)),Y)),Z) ) ) ).

% add_divide_eq_iff
tff(fact_4743_add__divide__eq__iff,axiom,
    ! [Z: real,Xb: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Y),Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Z)),Y)),Z) ) ) ).

% add_divide_eq_iff
tff(fact_4744_add__num__frac,axiom,
    ! [Y: rat,Z: rat,Xb: rat] :
      ( ( Y != zero_zero_rat )
     => ( aa_rat_rat(plus_plus_rat(Z),aa_rat_rat(divide_divide_rat(Xb),Y)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(Xb),aa_rat_rat(times_times_rat(Z),Y))),Y) ) ) ).

% add_num_frac
tff(fact_4745_add__num__frac,axiom,
    ! [Y: real,Z: real,Xb: real] :
      ( ( Y != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Y))),Y) ) ) ).

% add_num_frac
tff(fact_4746_add__frac__num,axiom,
    ! [Y: rat,Xb: rat,Z: rat] :
      ( ( Y != zero_zero_rat )
     => ( aa_rat_rat(plus_plus_rat(aa_rat_rat(divide_divide_rat(Xb),Y)),Z) = aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(Xb),aa_rat_rat(times_times_rat(Z),Y))),Y) ) ) ).

% add_frac_num
tff(fact_4747_add__frac__num,axiom,
    ! [Y: real,Xb: real,Z: real] :
      ( ( Y != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)),Z) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Y))),Y) ) ) ).

% add_frac_num
tff(fact_4748_add__frac__eq,axiom,
    ! [Y: rat,Z: rat,Xb: rat,W2: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( aa_rat_rat(plus_plus_rat(aa_rat_rat(divide_divide_rat(Xb),Y)),aa_rat_rat(divide_divide_rat(W2),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(W2),Y))),aa_rat_rat(times_times_rat(Y),Z)) ) ) ) ).

% add_frac_eq
tff(fact_4749_add__frac__eq,axiom,
    ! [Y: real,Z: real,Xb: real,W2: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,W2),Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W2),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z)) ) ) ) ).

% add_frac_eq
tff(fact_4750_add__divide__eq__if__simps_I1_J,axiom,
    ! [Aa: rat,Ba: rat,Z: rat] :
      aa_rat_rat(plus_plus_rat(Aa),aa_rat_rat(divide_divide_rat(Ba),Z)) = $ite(Z = zero_zero_rat,Aa,aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(Aa),Z)),Ba)),Z)) ).

% add_divide_eq_if_simps(1)
tff(fact_4751_add__divide__eq__if__simps_I1_J,axiom,
    ! [Aa: real,Ba: real,Z: real] :
      aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Ba),Z)) = $ite(Z = zero_zero_real,Aa,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),Z)),Ba)),Z)) ).

% add_divide_eq_if_simps(1)
tff(fact_4752_add__divide__eq__if__simps_I2_J,axiom,
    ! [Aa: rat,Z: rat,Ba: rat] :
      aa_rat_rat(plus_plus_rat(aa_rat_rat(divide_divide_rat(Aa),Z)),Ba) = $ite(Z = zero_zero_rat,Ba,aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(Aa),aa_rat_rat(times_times_rat(Ba),Z))),Z)) ).

% add_divide_eq_if_simps(2)
tff(fact_4753_add__divide__eq__if__simps_I2_J,axiom,
    ! [Aa: real,Z: real,Ba: real] :
      aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Aa),Z)),Ba) = $ite(Z = zero_zero_real,Ba,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Ba),Z))),Z)) ).

% add_divide_eq_if_simps(2)
tff(fact_4754_less__add__iff2,axiom,
    ! [Aa: real,E2: real,C: real,Ba: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),E2)),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Ba),E2)),D))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,C),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Ba),Aa)),E2)),D)) ) ).

% less_add_iff2
tff(fact_4755_less__add__iff2,axiom,
    ! [Aa: rat,E2: rat,C: rat,Ba: 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(Aa),E2)),C)),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(Ba),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(Ba),Aa)),E2)),D)) ) ).

% less_add_iff2
tff(fact_4756_less__add__iff2,axiom,
    ! [Aa: int,E2: int,C: int,Ba: 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,Aa),E2)),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Ba),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,Ba),Aa)),E2)),D)) ) ).

% less_add_iff2
tff(fact_4757_less__add__iff1,axiom,
    ! [Aa: real,E2: real,C: real,Ba: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),E2)),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Ba),E2)),D))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Aa),Ba)),E2)),C)),D) ) ).

% less_add_iff1
tff(fact_4758_less__add__iff1,axiom,
    ! [Aa: rat,E2: rat,C: rat,Ba: 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(Aa),E2)),C)),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(Ba),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(Aa),Ba)),E2)),C)),D) ) ).

% less_add_iff1
tff(fact_4759_less__add__iff1,axiom,
    ! [Aa: int,E2: int,C: int,Ba: 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,Aa),E2)),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Ba),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,Aa),Ba)),E2)),C)),D) ) ).

% less_add_iff1
tff(fact_4760_divide__diff__eq__iff,axiom,
    ! [Z: rat,Xb: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(minus_minus_rat(aa_rat_rat(divide_divide_rat(Xb),Z)),Y) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(Xb),aa_rat_rat(times_times_rat(Y),Z))),Z) ) ) ).

% divide_diff_eq_iff
tff(fact_4761_divide__diff__eq__iff,axiom,
    ! [Z: real,Xb: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Z)),Y) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z))),Z) ) ) ).

% divide_diff_eq_iff
tff(fact_4762_diff__divide__eq__iff,axiom,
    ! [Z: rat,Xb: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(minus_minus_rat(Xb),aa_rat_rat(divide_divide_rat(Y),Z)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(Xb),Z)),Y)),Z) ) ) ).

% diff_divide_eq_iff
tff(fact_4763_diff__divide__eq__iff,axiom,
    ! [Z: real,Xb: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Y),Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Z)),Y)),Z) ) ) ).

% diff_divide_eq_iff
tff(fact_4764_diff__frac__eq,axiom,
    ! [Y: rat,Z: rat,Xb: rat,W2: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( aa_rat_rat(minus_minus_rat(aa_rat_rat(divide_divide_rat(Xb),Y)),aa_rat_rat(divide_divide_rat(W2),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(W2),Y))),aa_rat_rat(times_times_rat(Y),Z)) ) ) ) ).

% diff_frac_eq
tff(fact_4765_diff__frac__eq,axiom,
    ! [Y: real,Z: real,Xb: real,W2: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,W2),Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W2),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z)) ) ) ) ).

% diff_frac_eq
tff(fact_4766_add__divide__eq__if__simps_I4_J,axiom,
    ! [Aa: rat,Ba: rat,Z: rat] :
      aa_rat_rat(minus_minus_rat(Aa),aa_rat_rat(divide_divide_rat(Ba),Z)) = $ite(Z = zero_zero_rat,Aa,aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(Aa),Z)),Ba)),Z)) ).

% add_divide_eq_if_simps(4)
tff(fact_4767_add__divide__eq__if__simps_I4_J,axiom,
    ! [Aa: real,Ba: real,Z: real] :
      aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Aa),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Ba),Z)) = $ite(Z = zero_zero_real,Aa,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),Z)),Ba)),Z)) ).

% add_divide_eq_if_simps(4)
tff(fact_4768_eq__minus__divide__eq,axiom,
    ! [Aa: real,Ba: real,C: real] :
      ( ( Aa = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Ba),C)) )
    <=> $ite(C != zero_zero_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),C) = aa_real_real(uminus_uminus_real,Ba),Aa = zero_zero_real) ) ).

% eq_minus_divide_eq
tff(fact_4769_eq__minus__divide__eq,axiom,
    ! [Aa: rat,Ba: rat,C: rat] :
      ( ( Aa = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(Ba),C)) )
    <=> $ite(C != zero_zero_rat,aa_rat_rat(times_times_rat(Aa),C) = aa_rat_rat(uminus_uminus_rat,Ba),Aa = zero_zero_rat) ) ).

% eq_minus_divide_eq
tff(fact_4770_minus__divide__eq__eq,axiom,
    ! [Ba: real,C: real,Aa: real] :
      ( ( aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Ba),C)) = Aa )
    <=> $ite(C != zero_zero_real,aa_real_real(uminus_uminus_real,Ba) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),C),Aa = zero_zero_real) ) ).

% minus_divide_eq_eq
tff(fact_4771_minus__divide__eq__eq,axiom,
    ! [Ba: rat,C: rat,Aa: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(Ba),C)) = Aa )
    <=> $ite(C != zero_zero_rat,aa_rat_rat(uminus_uminus_rat,Ba) = aa_rat_rat(times_times_rat(Aa),C),Aa = zero_zero_rat) ) ).

% minus_divide_eq_eq
tff(fact_4772_nonzero__neg__divide__eq__eq,axiom,
    ! [Ba: real,Aa: real,C: real] :
      ( ( Ba != zero_zero_real )
     => ( ( aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Aa),Ba)) = C )
      <=> ( aa_real_real(uminus_uminus_real,Aa) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),Ba) ) ) ) ).

% nonzero_neg_divide_eq_eq
tff(fact_4773_nonzero__neg__divide__eq__eq,axiom,
    ! [Ba: rat,Aa: rat,C: rat] :
      ( ( Ba != zero_zero_rat )
     => ( ( aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(Aa),Ba)) = C )
      <=> ( aa_rat_rat(uminus_uminus_rat,Aa) = aa_rat_rat(times_times_rat(C),Ba) ) ) ) ).

% nonzero_neg_divide_eq_eq
tff(fact_4774_nonzero__neg__divide__eq__eq2,axiom,
    ! [Ba: real,C: real,Aa: real] :
      ( ( Ba != zero_zero_real )
     => ( ( C = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Aa),Ba)) )
      <=> ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),Ba) = aa_real_real(uminus_uminus_real,Aa) ) ) ) ).

% nonzero_neg_divide_eq_eq2
tff(fact_4775_nonzero__neg__divide__eq__eq2,axiom,
    ! [Ba: rat,C: rat,Aa: rat] :
      ( ( Ba != zero_zero_rat )
     => ( ( C = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(Aa),Ba)) )
      <=> ( aa_rat_rat(times_times_rat(C),Ba) = aa_rat_rat(uminus_uminus_rat,Aa) ) ) ) ).

% nonzero_neg_divide_eq_eq2
tff(fact_4776_power__gt1__lemma,axiom,
    ! [Aa: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Aa)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),Na))) ) ).

% power_gt1_lemma
tff(fact_4777_power__gt1__lemma,axiom,
    ! [Aa: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),Aa)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_rat_rat(times_times_rat(Aa),aa_nat_rat(power_power_rat(Aa),Na))) ) ).

% power_gt1_lemma
tff(fact_4778_power__gt1__lemma,axiom,
    ! [Aa: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),Aa)
     => 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,Aa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Aa),Na))) ) ).

% power_gt1_lemma
tff(fact_4779_power__gt1__lemma,axiom,
    ! [Aa: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),Aa)
     => 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,Aa),aa_nat_int(power_power_int(Aa),Na))) ) ).

% power_gt1_lemma
tff(fact_4780_power__less__power__Suc,axiom,
    ! [Aa: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Aa)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),Na)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),Na))) ) ).

% power_less_power_Suc
tff(fact_4781_power__less__power__Suc,axiom,
    ! [Aa: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),Aa)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(Aa),Na)),aa_rat_rat(times_times_rat(Aa),aa_nat_rat(power_power_rat(Aa),Na))) ) ).

% power_less_power_Suc
tff(fact_4782_power__less__power__Suc,axiom,
    ! [Aa: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),Aa)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Aa),Na)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Aa),Na))) ) ).

% power_less_power_Suc
tff(fact_4783_power__less__power__Suc,axiom,
    ! [Aa: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),Aa)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(Aa),Na)),aa_int_int(aa_int_fun_int_int(times_times_int,Aa),aa_nat_int(power_power_int(Aa),Na))) ) ).

% power_less_power_Suc
tff(fact_4784_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_4785_square__diff__one__factored,axiom,
    ! [Xb: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Xb)),one_one_real) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),one_one_real)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),one_one_real)) ).

% square_diff_one_factored
tff(fact_4786_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_4787_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_4788_mult__1s__ring__1_I2_J,axiom,
    ! [Ba: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Ba),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,one))) = aa_real_real(uminus_uminus_real,Ba) ).

% mult_1s_ring_1(2)
tff(fact_4789_mult__1s__ring__1_I2_J,axiom,
    ! [Ba: int] : aa_int_int(aa_int_fun_int_int(times_times_int,Ba),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,one))) = aa_int_int(uminus_uminus_int,Ba) ).

% mult_1s_ring_1(2)
tff(fact_4790_mult__1s__ring__1_I2_J,axiom,
    ! [Ba: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Ba),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,one))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Ba) ).

% mult_1s_ring_1(2)
tff(fact_4791_mult__1s__ring__1_I2_J,axiom,
    ! [Ba: rat] : aa_rat_rat(times_times_rat(Ba),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,one))) = aa_rat_rat(uminus_uminus_rat,Ba) ).

% mult_1s_ring_1(2)
tff(fact_4792_mult__1s__ring__1_I1_J,axiom,
    ! [Ba: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,one))),Ba) = aa_real_real(uminus_uminus_real,Ba) ).

% mult_1s_ring_1(1)
tff(fact_4793_mult__1s__ring__1_I1_J,axiom,
    ! [Ba: 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))),Ba) = aa_int_int(uminus_uminus_int,Ba) ).

% mult_1s_ring_1(1)
tff(fact_4794_mult__1s__ring__1_I1_J,axiom,
    ! [Ba: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,one))),Ba) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Ba) ).

% mult_1s_ring_1(1)
tff(fact_4795_mult__1s__ring__1_I1_J,axiom,
    ! [Ba: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,one))),Ba) = aa_rat_rat(uminus_uminus_rat,Ba) ).

% mult_1s_ring_1(1)
tff(fact_4796_unit__dvdE,axiom,
    ! [Aa: code_integer,Ba: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),one_one_Code_integer)
     => ~ ( ( Aa != zero_z3403309356797280102nteger )
         => ! [C3: code_integer] : Ba != aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Aa),C3) ) ) ).

% unit_dvdE
tff(fact_4797_unit__dvdE,axiom,
    ! [Aa: nat,Ba: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),one_one_nat)
     => ~ ( ( Aa != zero_zero_nat )
         => ! [C3: nat] : Ba != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),C3) ) ) ).

% unit_dvdE
tff(fact_4798_unit__dvdE,axiom,
    ! [Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),one_one_int)
     => ~ ( ( Aa != zero_zero_int )
         => ! [C3: int] : Ba != aa_int_int(aa_int_fun_int_int(times_times_int,Aa),C3) ) ) ).

% unit_dvdE
tff(fact_4799_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),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,X4),zero_z3403309356797280102nteger))
          & aa_Code_integer_o(P,X4) ) ) ).

% unity_coeff_ex
tff(fact_4800_unity__coeff__ex,axiom,
    ! [P: fun_real_o,L: real] :
      ( ? [X4: real] : aa_real_o(P,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,L),X4))
    <=> ? [X4: real] :
          ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,L),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X4),zero_zero_real))
          & aa_real_o(P,X4) ) ) ).

% unity_coeff_ex
tff(fact_4801_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_4802_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_4803_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_4804_dvd__div__eq__mult,axiom,
    ! [Aa: nat,Ba: nat,C: nat] :
      ( ( Aa != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),Ba)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Ba),Aa) = C )
        <=> ( Ba = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),Aa) ) ) ) ) ).

% dvd_div_eq_mult
tff(fact_4805_dvd__div__eq__mult,axiom,
    ! [Aa: int,Ba: int,C: int] :
      ( ( Aa != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),Ba)
       => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,Ba),Aa) = C )
        <=> ( Ba = aa_int_int(aa_int_fun_int_int(times_times_int,C),Aa) ) ) ) ) ).

% dvd_div_eq_mult
tff(fact_4806_dvd__div__eq__mult,axiom,
    ! [Aa: code_integer,Ba: code_integer,C: code_integer] :
      ( ( Aa != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),Ba)
       => ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Ba),Aa) = C )
        <=> ( Ba = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),Aa) ) ) ) ) ).

% dvd_div_eq_mult
tff(fact_4807_div__dvd__iff__mult,axiom,
    ! [Ba: nat,Aa: nat,C: nat] :
      ( ( Ba != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Ba),Aa)
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Aa),Ba)),C)
        <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),Ba)) ) ) ) ).

% div_dvd_iff_mult
tff(fact_4808_div__dvd__iff__mult,axiom,
    ! [Ba: int,Aa: int,C: int] :
      ( ( Ba != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Ba),Aa)
       => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),Ba)),C)
        <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),aa_int_int(aa_int_fun_int_int(times_times_int,C),Ba)) ) ) ) ).

% div_dvd_iff_mult
tff(fact_4809_div__dvd__iff__mult,axiom,
    ! [Ba: code_integer,Aa: code_integer,C: code_integer] :
      ( ( Ba != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Ba),Aa)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Aa),Ba)),C)
        <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),Ba)) ) ) ) ).

% div_dvd_iff_mult
tff(fact_4810_dvd__div__iff__mult,axiom,
    ! [C: nat,Ba: nat,Aa: nat] :
      ( ( C != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),Ba)
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Ba),C))
        <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),C)),Ba) ) ) ) ).

% dvd_div_iff_mult
tff(fact_4811_dvd__div__iff__mult,axiom,
    ! [C: int,Ba: int,Aa: int] :
      ( ( C != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),Ba)
       => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),aa_int_int(aa_int_fun_int_int(divide_divide_int,Ba),C))
        <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,Aa),C)),Ba) ) ) ) ).

% dvd_div_iff_mult
tff(fact_4812_dvd__div__iff__mult,axiom,
    ! [C: code_integer,Ba: code_integer,Aa: code_integer] :
      ( ( C != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),Ba)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Ba),C))
        <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Aa),C)),Ba) ) ) ) ).

% dvd_div_iff_mult
tff(fact_4813_dvd__div__div__eq__mult,axiom,
    ! [Aa: nat,C: nat,Ba: nat,D: nat] :
      ( ( Aa != zero_zero_nat )
     => ( ( C != zero_zero_nat )
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),Ba)
         => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),D)
           => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Ba),Aa) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,D),C) )
            <=> ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ba),C) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),D) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
tff(fact_4814_dvd__div__div__eq__mult,axiom,
    ! [Aa: int,C: int,Ba: int,D: int] :
      ( ( Aa != zero_zero_int )
     => ( ( C != zero_zero_int )
       => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),Ba)
         => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),D)
           => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,Ba),Aa) = aa_int_int(aa_int_fun_int_int(divide_divide_int,D),C) )
            <=> ( aa_int_int(aa_int_fun_int_int(times_times_int,Ba),C) = aa_int_int(aa_int_fun_int_int(times_times_int,Aa),D) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
tff(fact_4815_dvd__div__div__eq__mult,axiom,
    ! [Aa: code_integer,C: code_integer,Ba: code_integer,D: code_integer] :
      ( ( Aa != zero_z3403309356797280102nteger )
     => ( ( C != zero_z3403309356797280102nteger )
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),Ba)
         => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),D)
           => ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Ba),Aa) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,D),C) )
            <=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Ba),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Aa),D) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
tff(fact_4816_power__minus,axiom,
    ! [Aa: complex,Na: nat] : aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,Aa)),Na) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Na)),aa_nat_complex(power_power_complex(Aa),Na)) ).

% power_minus
tff(fact_4817_power__minus,axiom,
    ! [Aa: real,Na: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,Aa)),Na) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Na)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),Na)) ).

% power_minus
tff(fact_4818_power__minus,axiom,
    ! [Aa: int,Na: nat] : aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,Aa)),Na) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),Na)),aa_nat_int(power_power_int(Aa),Na)) ).

% power_minus
tff(fact_4819_power__minus,axiom,
    ! [Aa: code_integer,Na: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa)),Na) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Na)),aa_nat_Code_integer(power_8256067586552552935nteger(Aa),Na)) ).

% power_minus
tff(fact_4820_power__minus,axiom,
    ! [Aa: rat,Na: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,Aa)),Na) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Na)),aa_nat_rat(power_power_rat(Aa),Na)) ).

% power_minus
tff(fact_4821_unit__eq__div1,axiom,
    ! [Ba: nat,Aa: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Ba),one_one_nat)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Aa),Ba) = C )
      <=> ( Aa = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),Ba) ) ) ) ).

% unit_eq_div1
tff(fact_4822_unit__eq__div1,axiom,
    ! [Ba: int,Aa: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Ba),one_one_int)
     => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),Ba) = C )
      <=> ( Aa = aa_int_int(aa_int_fun_int_int(times_times_int,C),Ba) ) ) ) ).

% unit_eq_div1
tff(fact_4823_unit__eq__div1,axiom,
    ! [Ba: code_integer,Aa: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Ba),one_one_Code_integer)
     => ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Aa),Ba) = C )
      <=> ( Aa = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),Ba) ) ) ) ).

% unit_eq_div1
tff(fact_4824_unit__eq__div2,axiom,
    ! [Ba: nat,Aa: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Ba),one_one_nat)
     => ( ( Aa = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),Ba) )
      <=> ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),Ba) = C ) ) ) ).

% unit_eq_div2
tff(fact_4825_unit__eq__div2,axiom,
    ! [Ba: int,Aa: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Ba),one_one_int)
     => ( ( Aa = aa_int_int(aa_int_fun_int_int(divide_divide_int,C),Ba) )
      <=> ( aa_int_int(aa_int_fun_int_int(times_times_int,Aa),Ba) = C ) ) ) ).

% unit_eq_div2
tff(fact_4826_unit__eq__div2,axiom,
    ! [Ba: code_integer,Aa: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Ba),one_one_Code_integer)
     => ( ( Aa = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,C),Ba) )
      <=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Aa),Ba) = C ) ) ) ).

% unit_eq_div2
tff(fact_4827_div__mult__unit2,axiom,
    ! [C: nat,Ba: nat,Aa: 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,Ba),Aa)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ba),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Aa),Ba)),C) ) ) ) ).

% div_mult_unit2
tff(fact_4828_div__mult__unit2,axiom,
    ! [C: int,Ba: int,Aa: 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,Ba),Aa)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),aa_int_int(aa_int_fun_int_int(times_times_int,Ba),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),Ba)),C) ) ) ) ).

% div_mult_unit2
tff(fact_4829_div__mult__unit2,axiom,
    ! [C: code_integer,Ba: code_integer,Aa: 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,Ba),Aa)
       => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Aa),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Ba),C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Aa),Ba)),C) ) ) ) ).

% div_mult_unit2
tff(fact_4830_unit__div__commute,axiom,
    ! [Ba: nat,Aa: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Ba),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,Aa),Ba)),C) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),C)),Ba) ) ) ).

% unit_div_commute
tff(fact_4831_unit__div__commute,axiom,
    ! [Ba: int,Aa: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Ba),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,Aa),Ba)),C) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,Aa),C)),Ba) ) ) ).

% unit_div_commute
tff(fact_4832_unit__div__commute,axiom,
    ! [Ba: code_integer,Aa: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Ba),one_one_Code_integer)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Aa),Ba)),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Aa),C)),Ba) ) ) ).

% unit_div_commute
tff(fact_4833_unit__div__mult__swap,axiom,
    ! [C: nat,Aa: nat,Ba: 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,Aa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Ba),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),Ba)),C) ) ) ).

% unit_div_mult_swap
tff(fact_4834_unit__div__mult__swap,axiom,
    ! [C: int,Aa: int,Ba: 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,Aa),aa_int_int(aa_int_fun_int_int(divide_divide_int,Ba),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,Aa),Ba)),C) ) ) ).

% unit_div_mult_swap
tff(fact_4835_unit__div__mult__swap,axiom,
    ! [C: code_integer,Aa: code_integer,Ba: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),one_one_Code_integer)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Aa),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Ba),C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Aa),Ba)),C) ) ) ).

% unit_div_mult_swap
tff(fact_4836_is__unit__div__mult2__eq,axiom,
    ! [Ba: nat,C: nat,Aa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Ba),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,Aa),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ba),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Aa),Ba)),C) ) ) ) ).

% is_unit_div_mult2_eq
tff(fact_4837_is__unit__div__mult2__eq,axiom,
    ! [Ba: int,C: int,Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Ba),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,Aa),aa_int_int(aa_int_fun_int_int(times_times_int,Ba),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),Ba)),C) ) ) ) ).

% is_unit_div_mult2_eq
tff(fact_4838_is__unit__div__mult2__eq,axiom,
    ! [Ba: code_integer,C: code_integer,Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Ba),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),one_one_Code_integer)
       => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Aa),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Ba),C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Aa),Ba)),C) ) ) ) ).

% is_unit_div_mult2_eq
tff(fact_4839_one__less__mult,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),M)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na)) ) ) ).

% one_less_mult
tff(fact_4840_n__less__m__mult__n,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),M)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na)) ) ) ).

% n_less_m_mult_n
tff(fact_4841_n__less__n__mult__m,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),M)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),M)) ) ) ).

% n_less_n_mult_m
tff(fact_4842_nat__mult__le__cancel1,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Na))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ) ).

% nat_mult_le_cancel1
tff(fact_4843_dvd__mult__cancel1,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na)),M)
      <=> ( Na = one_one_nat ) ) ) ).

% dvd_mult_cancel1
tff(fact_4844_dvd__mult__cancel2,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),M)),M)
      <=> ( Na = one_one_nat ) ) ) ).

% dvd_mult_cancel2
tff(fact_4845_nat__mult__div__cancel1,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na) ) ) ).

% nat_mult_div_cancel1
tff(fact_4846_div__less__iff__less__mult,axiom,
    ! [Q4: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Q4)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Q4)),Na)
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),Q4)) ) ) ).

% div_less_iff_less_mult
tff(fact_4847_nat__eq__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),I)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Na) )
      <=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),J)),U)),M) = Na ) ) ) ).

% nat_eq_add_iff1
tff(fact_4848_nat__eq__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Na) )
      <=> ( M = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I)),U)),Na) ) ) ) ).

% nat_eq_add_iff2
tff(fact_4849_nat__le__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),I)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Na))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),J)),U)),M)),Na) ) ) ).

% nat_le_add_iff1
tff(fact_4850_nat__le__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Na))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I)),U)),Na)) ) ) ).

% nat_le_add_iff2
tff(fact_4851_nat__diff__add__eq1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),I)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),J)),U)),M)),Na) ) ) ).

% nat_diff_add_eq1
tff(fact_4852_nat__diff__add__eq2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I)),U)),Na)) ) ) ).

% nat_diff_add_eq2
tff(fact_4853_dvd__minus__add,axiom,
    ! [Q4: nat,Na: nat,R2: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Q4),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Q4),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R2),M))
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),Q4))
        <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R2),M)),Q4))) ) ) ) ).

% dvd_minus_add
tff(fact_4854_size__mult__mono,axiom,
    ! [Ba: code_integer,Aa: code_integer] :
      ( ( Ba != zero_z3403309356797280102nteger )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_Code_integer_nat(euclid6377331345833325938nteger,Aa)),aa_Code_integer_nat(euclid6377331345833325938nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Aa),Ba))) ) ).

% size_mult_mono
tff(fact_4855_size__mult__mono,axiom,
    ! [Ba: int,Aa: int] :
      ( ( Ba != zero_zero_int )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(euclid4774559944035922753ze_int,Aa)),aa_int_nat(euclid4774559944035922753ze_int,aa_int_int(aa_int_fun_int_int(times_times_int,Aa),Ba))) ) ).

% size_mult_mono
tff(fact_4856_size__mult__mono,axiom,
    ! [Ba: nat,Aa: nat] :
      ( ( Ba != zero_zero_nat )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(euclid4777050414544973029ze_nat,Aa)),aa_nat_nat(euclid4777050414544973029ze_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),Ba))) ) ).

% size_mult_mono
tff(fact_4857_size__mult__mono_H,axiom,
    ! [Ba: code_integer,Aa: code_integer] :
      ( ( Ba != zero_z3403309356797280102nteger )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_Code_integer_nat(euclid6377331345833325938nteger,Aa)),aa_Code_integer_nat(euclid6377331345833325938nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Ba),Aa))) ) ).

% size_mult_mono'
tff(fact_4858_size__mult__mono_H,axiom,
    ! [Ba: int,Aa: int] :
      ( ( Ba != zero_zero_int )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(euclid4774559944035922753ze_int,Aa)),aa_int_nat(euclid4774559944035922753ze_int,aa_int_int(aa_int_fun_int_int(times_times_int,Ba),Aa))) ) ).

% size_mult_mono'
tff(fact_4859_size__mult__mono_H,axiom,
    ! [Ba: nat,Aa: nat] :
      ( ( Ba != zero_zero_nat )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(euclid4777050414544973029ze_nat,Aa)),aa_nat_nat(euclid4777050414544973029ze_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ba),Aa))) ) ).

% size_mult_mono'
tff(fact_4860_euclidean__size__times__unit,axiom,
    ! [Aa: code_integer,Ba: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),one_one_Code_integer)
     => ( aa_Code_integer_nat(euclid6377331345833325938nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Aa),Ba)) = aa_Code_integer_nat(euclid6377331345833325938nteger,Ba) ) ) ).

% euclidean_size_times_unit
tff(fact_4861_euclidean__size__times__unit,axiom,
    ! [Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),one_one_int)
     => ( aa_int_nat(euclid4774559944035922753ze_int,aa_int_int(aa_int_fun_int_int(times_times_int,Aa),Ba)) = aa_int_nat(euclid4774559944035922753ze_int,Ba) ) ) ).

% euclidean_size_times_unit
tff(fact_4862_euclidean__size__times__unit,axiom,
    ! [Aa: nat,Ba: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),one_one_nat)
     => ( aa_nat_nat(euclid4777050414544973029ze_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),Ba)) = aa_nat_nat(euclid4777050414544973029ze_nat,Ba) ) ) ).

% euclidean_size_times_unit
tff(fact_4863_field__le__mult__one__interval,axiom,
    ! [Xb: real,Y: real] :
      ( ! [Z3: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Z3)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),one_one_real)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z3),Xb)),Y) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y) ) ).

% field_le_mult_one_interval
tff(fact_4864_field__le__mult__one__interval,axiom,
    ! [Xb: rat,Y: rat] :
      ( ! [Z3: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Z3)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z3),one_one_rat)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(Z3),Xb)),Y) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y) ) ).

% field_le_mult_one_interval
tff(fact_4865_mult__le__cancel__left1,axiom,
    ! [C: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),Ba))
    <=> ( ( 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),Ba) )
        & ( 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,Ba),one_one_real) ) ) ) ).

% mult_le_cancel_left1
tff(fact_4866_mult__le__cancel__left1,axiom,
    ! [C: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),aa_rat_rat(times_times_rat(C),Ba))
    <=> ( ( 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),Ba) )
        & ( 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,Ba),one_one_rat) ) ) ) ).

% mult_le_cancel_left1
tff(fact_4867_mult__le__cancel__left1,axiom,
    ! [C: int,Ba: 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),Ba))
    <=> ( ( 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),Ba) )
        & ( 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,Ba),one_one_int) ) ) ) ).

% mult_le_cancel_left1
tff(fact_4868_mult__le__cancel__left2,axiom,
    ! [C: real,Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),Aa)),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,Aa),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),Aa) ) ) ) ).

% mult_le_cancel_left2
tff(fact_4869_mult__le__cancel__left2,axiom,
    ! [C: rat,Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(C),Aa)),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,Aa),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),Aa) ) ) ) ).

% mult_le_cancel_left2
tff(fact_4870_mult__le__cancel__left2,axiom,
    ! [C: int,Aa: 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),Aa)),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,Aa),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),Aa) ) ) ) ).

% mult_le_cancel_left2
tff(fact_4871_mult__le__cancel__right1,axiom,
    ! [C: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Ba),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),Ba) )
        & ( 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,Ba),one_one_real) ) ) ) ).

% mult_le_cancel_right1
tff(fact_4872_mult__le__cancel__right1,axiom,
    ! [C: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),aa_rat_rat(times_times_rat(Ba),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),Ba) )
        & ( 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,Ba),one_one_rat) ) ) ) ).

% mult_le_cancel_right1
tff(fact_4873_mult__le__cancel__right1,axiom,
    ! [C: int,Ba: 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,Ba),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),Ba) )
        & ( 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,Ba),one_one_int) ) ) ) ).

% mult_le_cancel_right1
tff(fact_4874_mult__le__cancel__right2,axiom,
    ! [Aa: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),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,Aa),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),Aa) ) ) ) ).

% mult_le_cancel_right2
tff(fact_4875_mult__le__cancel__right2,axiom,
    ! [Aa: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(Aa),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,Aa),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),Aa) ) ) ) ).

% mult_le_cancel_right2
tff(fact_4876_mult__le__cancel__right2,axiom,
    ! [Aa: 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,Aa),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,Aa),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),Aa) ) ) ) ).

% mult_le_cancel_right2
tff(fact_4877_mult__less__cancel__left1,axiom,
    ! [C: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),Ba))
    <=> ( ( 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),Ba) )
        & ( 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,Ba),one_one_real) ) ) ) ).

% mult_less_cancel_left1
tff(fact_4878_mult__less__cancel__left1,axiom,
    ! [C: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),aa_rat_rat(times_times_rat(C),Ba))
    <=> ( ( 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),Ba) )
        & ( 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,Ba),one_one_rat) ) ) ) ).

% mult_less_cancel_left1
tff(fact_4879_mult__less__cancel__left1,axiom,
    ! [C: int,Ba: 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),Ba))
    <=> ( ( 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),Ba) )
        & ( 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,Ba),one_one_int) ) ) ) ).

% mult_less_cancel_left1
tff(fact_4880_mult__less__cancel__left2,axiom,
    ! [C: real,Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),Aa)),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,Aa),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),Aa) ) ) ) ).

% mult_less_cancel_left2
tff(fact_4881_mult__less__cancel__left2,axiom,
    ! [C: rat,Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(C),Aa)),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,Aa),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),Aa) ) ) ) ).

% mult_less_cancel_left2
tff(fact_4882_mult__less__cancel__left2,axiom,
    ! [C: int,Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),Aa)),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,Aa),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),Aa) ) ) ) ).

% mult_less_cancel_left2
tff(fact_4883_mult__less__cancel__right1,axiom,
    ! [C: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Ba),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),Ba) )
        & ( 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,Ba),one_one_real) ) ) ) ).

% mult_less_cancel_right1
tff(fact_4884_mult__less__cancel__right1,axiom,
    ! [C: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),aa_rat_rat(times_times_rat(Ba),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),Ba) )
        & ( 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,Ba),one_one_rat) ) ) ) ).

% mult_less_cancel_right1
tff(fact_4885_mult__less__cancel__right1,axiom,
    ! [C: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),aa_int_int(aa_int_fun_int_int(times_times_int,Ba),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),Ba) )
        & ( 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,Ba),one_one_int) ) ) ) ).

% mult_less_cancel_right1
tff(fact_4886_mult__less__cancel__right2,axiom,
    ! [Aa: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),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,Aa),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),Aa) ) ) ) ).

% mult_less_cancel_right2
tff(fact_4887_mult__less__cancel__right2,axiom,
    ! [Aa: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(Aa),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,Aa),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),Aa) ) ) ) ).

% mult_less_cancel_right2
tff(fact_4888_mult__less__cancel__right2,axiom,
    ! [Aa: 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,Aa),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,Aa),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),Aa) ) ) ) ).

% mult_less_cancel_right2
tff(fact_4889_divide__le__eq,axiom,
    ! [Ba: real,C: real,Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Ba),C)),Aa)
    <=> $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,Ba),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),C)),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),C)),Ba),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Aa)) ) ) ).

% divide_le_eq
tff(fact_4890_divide__le__eq,axiom,
    ! [Ba: rat,C: rat,Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(Ba),C)),Aa)
    <=> $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,Ba),aa_rat_rat(times_times_rat(Aa),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),C)),Ba),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Aa)) ) ) ).

% divide_le_eq
tff(fact_4891_le__divide__eq,axiom,
    ! [Aa: real,Ba: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Ba),C))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),C)),Ba),
          $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,Ba),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),C)),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),zero_zero_real)) ) ) ).

% le_divide_eq
tff(fact_4892_le__divide__eq,axiom,
    ! [Aa: rat,Ba: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),aa_rat_rat(divide_divide_rat(Ba),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),C)),Ba),
          $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,Ba),aa_rat_rat(times_times_rat(Aa),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),zero_zero_rat)) ) ) ).

% le_divide_eq
tff(fact_4893_divide__left__mono,axiom,
    ! [Ba: real,Aa: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ba),Aa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),Ba))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),Aa)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),Ba)) ) ) ) ).

% divide_left_mono
tff(fact_4894_divide__left__mono,axiom,
    ! [Ba: rat,Aa: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ba),Aa)
     => ( 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(Aa),Ba))
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(C),Aa)),aa_rat_rat(divide_divide_rat(C),Ba)) ) ) ) ).

% divide_left_mono
tff(fact_4895_neg__divide__le__eq,axiom,
    ! [C: real,Ba: real,Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Ba),C)),Aa)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),C)),Ba) ) ) ).

% neg_divide_le_eq
tff(fact_4896_neg__divide__le__eq,axiom,
    ! [C: rat,Ba: rat,Aa: 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(Ba),C)),Aa)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(Aa),C)),Ba) ) ) ).

% neg_divide_le_eq
tff(fact_4897_neg__le__divide__eq,axiom,
    ! [C: real,Aa: real,Ba: 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),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Ba),C))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ba),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),C)) ) ) ).

% neg_le_divide_eq
tff(fact_4898_neg__le__divide__eq,axiom,
    ! [C: rat,Aa: rat,Ba: 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),aa_rat_rat(divide_divide_rat(Ba),C))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ba),aa_rat_rat(times_times_rat(Aa),C)) ) ) ).

% neg_le_divide_eq
tff(fact_4899_pos__divide__le__eq,axiom,
    ! [C: real,Ba: real,Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Ba),C)),Aa)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ba),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),C)) ) ) ).

% pos_divide_le_eq
tff(fact_4900_pos__divide__le__eq,axiom,
    ! [C: rat,Ba: rat,Aa: 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(Ba),C)),Aa)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ba),aa_rat_rat(times_times_rat(Aa),C)) ) ) ).

% pos_divide_le_eq
tff(fact_4901_pos__le__divide__eq,axiom,
    ! [C: real,Aa: real,Ba: 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),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Ba),C))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),C)),Ba) ) ) ).

% pos_le_divide_eq
tff(fact_4902_pos__le__divide__eq,axiom,
    ! [C: rat,Aa: rat,Ba: 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),aa_rat_rat(divide_divide_rat(Ba),C))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(Aa),C)),Ba) ) ) ).

% pos_le_divide_eq
tff(fact_4903_mult__imp__div__pos__le,axiom,
    ! [Y: real,Xb: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Y))
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)),Z) ) ) ).

% mult_imp_div_pos_le
tff(fact_4904_mult__imp__div__pos__le,axiom,
    ! [Y: rat,Xb: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),aa_rat_rat(times_times_rat(Z),Y))
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(Xb),Y)),Z) ) ) ).

% mult_imp_div_pos_le
tff(fact_4905_mult__imp__le__div__pos,axiom,
    ! [Y: real,Z: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Y)),Xb)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Z),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)) ) ) ).

% mult_imp_le_div_pos
tff(fact_4906_mult__imp__le__div__pos,axiom,
    ! [Y: rat,Z: rat,Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(Z),Y)),Xb)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z),aa_rat_rat(divide_divide_rat(Xb),Y)) ) ) ).

% mult_imp_le_div_pos
tff(fact_4907_divide__left__mono__neg,axiom,
    ! [Aa: real,Ba: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),Ba)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),zero_zero_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),Ba))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),Aa)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),Ba)) ) ) ) ).

% divide_left_mono_neg
tff(fact_4908_divide__left__mono__neg,axiom,
    ! [Aa: rat,Ba: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),Ba)
     => ( 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(Aa),Ba))
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(C),Aa)),aa_rat_rat(divide_divide_rat(C),Ba)) ) ) ) ).

% divide_left_mono_neg
tff(fact_4909_convex__bound__le,axiom,
    ! [Xb: real,Aa: real,Y: real,U: real,V: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Aa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Aa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),U)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),V)
           => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,U),V) = one_one_real )
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,U),Xb)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,V),Y))),Aa) ) ) ) ) ) ).

% convex_bound_le
tff(fact_4910_convex__bound__le,axiom,
    ! [Xb: rat,Aa: rat,Y: rat,U: rat,V: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Aa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Aa)
       => ( 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),Y))),Aa) ) ) ) ) ) ).

% convex_bound_le
tff(fact_4911_convex__bound__le,axiom,
    ! [Xb: int,Aa: int,Y: int,U: int,V: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Aa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Aa)
       => ( 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),Y))),Aa) ) ) ) ) ) ).

% convex_bound_le
tff(fact_4912_less__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,Ba: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,W2)),aa_rat_rat(divide_divide_rat(Ba),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,W2)),C)),Ba),
          $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,Ba),aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W2)),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,W2)),zero_zero_rat)) ) ) ).

% less_divide_eq_numeral(1)
tff(fact_4913_less__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,Ba: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,W2)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Ba),C))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W2)),C)),Ba),
          $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,Ba),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W2)),C)),aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,W2)),zero_zero_real)) ) ) ).

% less_divide_eq_numeral(1)
tff(fact_4914_divide__less__eq__numeral_I1_J,axiom,
    ! [Ba: rat,C: rat,W2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(Ba),C)),aa_num_rat(numeral_numeral_rat,W2))
    <=> $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,Ba),aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W2)),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,W2)),C)),Ba),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_num_rat(numeral_numeral_rat,W2))) ) ) ).

% divide_less_eq_numeral(1)
tff(fact_4915_divide__less__eq__numeral_I1_J,axiom,
    ! [Ba: real,C: real,W2: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Ba),C)),aa_num_real(numeral_numeral_real,W2))
    <=> $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,Ba),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W2)),C)),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W2)),C)),Ba),aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_num_real(numeral_numeral_real,W2))) ) ) ).

% divide_less_eq_numeral(1)
tff(fact_4916_frac__le__eq,axiom,
    ! [Y: real,Z: real,Xb: real,W2: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,W2),Z))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W2),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z))),zero_zero_real) ) ) ) ).

% frac_le_eq
tff(fact_4917_frac__le__eq,axiom,
    ! [Y: rat,Z: rat,Xb: rat,W2: rat] :
      ( ( Y != 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),Y)),aa_rat_rat(divide_divide_rat(W2),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(W2),Y))),aa_rat_rat(times_times_rat(Y),Z))),zero_zero_rat) ) ) ) ).

% frac_le_eq
tff(fact_4918_frac__less__eq,axiom,
    ! [Y: rat,Z: rat,Xb: rat,W2: rat] :
      ( ( Y != 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),Y)),aa_rat_rat(divide_divide_rat(W2),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(W2),Y))),aa_rat_rat(times_times_rat(Y),Z))),zero_zero_rat) ) ) ) ).

% frac_less_eq
tff(fact_4919_frac__less__eq,axiom,
    ! [Y: real,Z: real,Xb: real,W2: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,W2),Z))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W2),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z))),zero_zero_real) ) ) ) ).

% frac_less_eq
tff(fact_4920_power__Suc__less,axiom,
    ! [Aa: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Aa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),Na)) ) ) ).

% power_Suc_less
tff(fact_4921_power__Suc__less,axiom,
    ! [Aa: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Aa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),one_one_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(Aa),aa_nat_rat(power_power_rat(Aa),Na))),aa_nat_rat(power_power_rat(Aa),Na)) ) ) ).

% power_Suc_less
tff(fact_4922_power__Suc__less,axiom,
    ! [Aa: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Aa)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),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,Aa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Aa),Na))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Aa),Na)) ) ) ).

% power_Suc_less
tff(fact_4923_power__Suc__less,axiom,
    ! [Aa: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Aa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),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,Aa),aa_nat_int(power_power_int(Aa),Na))),aa_nat_int(power_power_int(Aa),Na)) ) ) ).

% power_Suc_less
tff(fact_4924_pos__minus__divide__less__eq,axiom,
    ! [C: real,Ba: real,Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Ba),C))),Aa)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,Ba)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),C)) ) ) ).

% pos_minus_divide_less_eq
tff(fact_4925_pos__minus__divide__less__eq,axiom,
    ! [C: rat,Ba: rat,Aa: 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(Ba),C))),Aa)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,Ba)),aa_rat_rat(times_times_rat(Aa),C)) ) ) ).

% pos_minus_divide_less_eq
tff(fact_4926_pos__less__minus__divide__eq,axiom,
    ! [C: real,Aa: real,Ba: 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),aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Ba),C)))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),C)),aa_real_real(uminus_uminus_real,Ba)) ) ) ).

% pos_less_minus_divide_eq
tff(fact_4927_pos__less__minus__divide__eq,axiom,
    ! [C: rat,Aa: rat,Ba: 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),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(Ba),C)))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(Aa),C)),aa_rat_rat(uminus_uminus_rat,Ba)) ) ) ).

% pos_less_minus_divide_eq
tff(fact_4928_neg__minus__divide__less__eq,axiom,
    ! [C: real,Ba: real,Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Ba),C))),Aa)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),C)),aa_real_real(uminus_uminus_real,Ba)) ) ) ).

% neg_minus_divide_less_eq
tff(fact_4929_neg__minus__divide__less__eq,axiom,
    ! [C: rat,Ba: rat,Aa: 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(Ba),C))),Aa)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(Aa),C)),aa_rat_rat(uminus_uminus_rat,Ba)) ) ) ).

% neg_minus_divide_less_eq
tff(fact_4930_neg__less__minus__divide__eq,axiom,
    ! [C: real,Aa: real,Ba: 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),aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Ba),C)))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,Ba)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),C)) ) ) ).

% neg_less_minus_divide_eq
tff(fact_4931_neg__less__minus__divide__eq,axiom,
    ! [C: rat,Aa: rat,Ba: 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),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(Ba),C)))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,Ba)),aa_rat_rat(times_times_rat(Aa),C)) ) ) ).

% neg_less_minus_divide_eq
tff(fact_4932_minus__divide__less__eq,axiom,
    ! [Ba: real,C: real,Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Ba),C))),Aa)
    <=> $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,Ba)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),C)),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),C)),aa_real_real(uminus_uminus_real,Ba)),aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Aa)) ) ) ).

% minus_divide_less_eq
tff(fact_4933_minus__divide__less__eq,axiom,
    ! [Ba: rat,C: rat,Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(Ba),C))),Aa)
    <=> $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,Ba)),aa_rat_rat(times_times_rat(Aa),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),C)),aa_rat_rat(uminus_uminus_rat,Ba)),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Aa)) ) ) ).

% minus_divide_less_eq
tff(fact_4934_less__minus__divide__eq,axiom,
    ! [Aa: real,Ba: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Ba),C)))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),C)),aa_real_real(uminus_uminus_real,Ba)),
          $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,Ba)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),C)),aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),zero_zero_real)) ) ) ).

% less_minus_divide_eq
tff(fact_4935_less__minus__divide__eq,axiom,
    ! [Aa: rat,Ba: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(Ba),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),C)),aa_rat_rat(uminus_uminus_rat,Ba)),
          $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,Ba)),aa_rat_rat(times_times_rat(Aa),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),zero_zero_rat)) ) ) ).

% less_minus_divide_eq
tff(fact_4936_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_4937_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_4938_mult__2,axiom,
    ! [Z: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Z) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),Z) ).

% mult_2
tff(fact_4939_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_4940_mult__2,axiom,
    ! [Z: extended_enat] : times_7803423173614009249d_enat(numera1916890842035813515d_enat(aa_num_num(bit0,one)),Z) = plus_p3455044024723400733d_enat(Z,Z) ).

% mult_2
tff(fact_4941_mult__2,axiom,
    ! [Z: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Z) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Z),Z) ).

% mult_2
tff(fact_4942_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_4943_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_4944_mult__2__right,axiom,
    ! [Z: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),Z) ).

% mult_2_right
tff(fact_4945_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_4946_mult__2__right,axiom,
    ! [Z: extended_enat] : times_7803423173614009249d_enat(Z,numera1916890842035813515d_enat(aa_num_num(bit0,one))) = plus_p3455044024723400733d_enat(Z,Z) ).

% mult_2_right
tff(fact_4947_mult__2__right,axiom,
    ! [Z: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Z),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Z),Z) ).

% mult_2_right
tff(fact_4948_left__add__twice,axiom,
    ! [Aa: rat,Ba: rat] : aa_rat_rat(plus_plus_rat(Aa),aa_rat_rat(plus_plus_rat(Aa),Ba)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),Aa)),Ba) ).

% left_add_twice
tff(fact_4949_left__add__twice,axiom,
    ! [Aa: nat,Ba: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),Ba)) = 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)),Ba) ).

% left_add_twice
tff(fact_4950_left__add__twice,axiom,
    ! [Aa: real,Ba: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),Ba)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Aa)),Ba) ).

% left_add_twice
tff(fact_4951_left__add__twice,axiom,
    ! [Aa: int,Ba: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),Ba)) = 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)),Ba) ).

% left_add_twice
tff(fact_4952_left__add__twice,axiom,
    ! [Aa: extended_enat,Ba: extended_enat] : plus_p3455044024723400733d_enat(Aa,plus_p3455044024723400733d_enat(Aa,Ba)) = plus_p3455044024723400733d_enat(times_7803423173614009249d_enat(numera1916890842035813515d_enat(aa_num_num(bit0,one)),Aa),Ba) ).

% left_add_twice
tff(fact_4953_left__add__twice,axiom,
    ! [Aa: code_integer,Ba: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Aa),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Aa),Ba)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Aa)),Ba) ).

% left_add_twice
tff(fact_4954_divide__eq__eq__numeral_I2_J,axiom,
    ! [Ba: real,C: real,W2: num] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Ba),C) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)) )
    <=> $ite(C != zero_zero_real,Ba = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))),C),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)) = zero_zero_real) ) ).

% divide_eq_eq_numeral(2)
tff(fact_4955_divide__eq__eq__numeral_I2_J,axiom,
    ! [Ba: rat,C: rat,W2: num] :
      ( ( aa_rat_rat(divide_divide_rat(Ba),C) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)) )
    <=> $ite(C != zero_zero_rat,Ba = aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),C),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)) = zero_zero_rat) ) ).

% divide_eq_eq_numeral(2)
tff(fact_4956_eq__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,Ba: real,C: real] :
      ( ( aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Ba),C) )
    <=> $ite(C != zero_zero_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))),C) = Ba,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)) = zero_zero_real) ) ).

% eq_divide_eq_numeral(2)
tff(fact_4957_eq__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,Ba: rat,C: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)) = aa_rat_rat(divide_divide_rat(Ba),C) )
    <=> $ite(C != zero_zero_rat,aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),C) = Ba,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)) = zero_zero_rat) ) ).

% eq_divide_eq_numeral(2)
tff(fact_4958_add__divide__eq__if__simps_I3_J,axiom,
    ! [Aa: real,Z: real,Ba: real] :
      aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Aa),Z))),Ba) = $ite(Z = zero_zero_real,Ba,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,Aa)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Ba),Z))),Z)) ).

% add_divide_eq_if_simps(3)
tff(fact_4959_add__divide__eq__if__simps_I3_J,axiom,
    ! [Aa: rat,Z: rat,Ba: rat] :
      aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(Aa),Z))),Ba) = $ite(Z = zero_zero_rat,Ba,aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,Aa)),aa_rat_rat(times_times_rat(Ba),Z))),Z)) ).

% add_divide_eq_if_simps(3)
tff(fact_4960_minus__divide__add__eq__iff,axiom,
    ! [Z: real,Xb: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Z))),Y) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,Xb)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z))),Z) ) ) ).

% minus_divide_add_eq_iff
tff(fact_4961_minus__divide__add__eq__iff,axiom,
    ! [Z: rat,Xb: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(Xb),Z))),Y) = 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(Y),Z))),Z) ) ) ).

% minus_divide_add_eq_iff
tff(fact_4962_add__divide__eq__if__simps_I6_J,axiom,
    ! [Aa: real,Z: real,Ba: real] :
      aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Aa),Z))),Ba) = $ite(Z = zero_zero_real,aa_real_real(uminus_uminus_real,Ba),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,Aa)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Ba),Z))),Z)) ).

% add_divide_eq_if_simps(6)
tff(fact_4963_add__divide__eq__if__simps_I6_J,axiom,
    ! [Aa: rat,Z: rat,Ba: rat] :
      aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(Aa),Z))),Ba) = $ite(Z = zero_zero_rat,aa_rat_rat(uminus_uminus_rat,Ba),aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,Aa)),aa_rat_rat(times_times_rat(Ba),Z))),Z)) ).

% add_divide_eq_if_simps(6)
tff(fact_4964_add__divide__eq__if__simps_I5_J,axiom,
    ! [Aa: real,Z: real,Ba: real] :
      aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Aa),Z)),Ba) = $ite(Z = zero_zero_real,aa_real_real(uminus_uminus_real,Ba),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Aa),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Ba),Z))),Z)) ).

% add_divide_eq_if_simps(5)
tff(fact_4965_add__divide__eq__if__simps_I5_J,axiom,
    ! [Aa: rat,Z: rat,Ba: rat] :
      aa_rat_rat(minus_minus_rat(aa_rat_rat(divide_divide_rat(Aa),Z)),Ba) = $ite(Z = zero_zero_rat,aa_rat_rat(uminus_uminus_rat,Ba),aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(Aa),aa_rat_rat(times_times_rat(Ba),Z))),Z)) ).

% add_divide_eq_if_simps(5)
tff(fact_4966_minus__divide__diff__eq__iff,axiom,
    ! [Z: real,Xb: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Z))),Y) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,Xb)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Z))),Z) ) ) ).

% minus_divide_diff_eq_iff
tff(fact_4967_minus__divide__diff__eq__iff,axiom,
    ! [Z: rat,Xb: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(Xb),Z))),Y) = 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(Y),Z))),Z) ) ) ).

% minus_divide_diff_eq_iff
tff(fact_4968_is__unitE,axiom,
    ! [Aa: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),one_one_nat)
     => ~ ( ( Aa != zero_zero_nat )
         => ! [B2: nat] :
              ( ( B2 != zero_zero_nat )
             => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),one_one_nat)
               => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),Aa) = B2 )
                 => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),B2) = Aa )
                   => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),B2) = one_one_nat )
                     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),Aa) != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B2) ) ) ) ) ) ) ) ) ).

% is_unitE
tff(fact_4969_is__unitE,axiom,
    ! [Aa: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),one_one_int)
     => ~ ( ( Aa != zero_zero_int )
         => ! [B2: int] :
              ( ( B2 != zero_zero_int )
             => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),one_one_int)
               => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),Aa) = B2 )
                 => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),B2) = Aa )
                   => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,Aa),B2) = one_one_int )
                     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,C),Aa) != aa_int_int(aa_int_fun_int_int(times_times_int,C),B2) ) ) ) ) ) ) ) ) ).

% is_unitE
tff(fact_4970_is__unitE,axiom,
    ! [Aa: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),one_one_Code_integer)
     => ~ ( ( Aa != zero_z3403309356797280102nteger )
         => ! [B2: code_integer] :
              ( ( B2 != zero_z3403309356797280102nteger )
             => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B2),one_one_Code_integer)
               => ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,one_one_Code_integer),Aa) = B2 )
                 => ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,one_one_Code_integer),B2) = Aa )
                   => ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Aa),B2) = one_one_Code_integer )
                     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,C),Aa) != aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B2) ) ) ) ) ) ) ) ) ).

% is_unitE
tff(fact_4971_is__unit__div__mult__cancel__left,axiom,
    ! [Aa: nat,Ba: nat] :
      ( ( Aa != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Ba),one_one_nat)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),Ba)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),Ba) ) ) ) ).

% is_unit_div_mult_cancel_left
tff(fact_4972_is__unit__div__mult__cancel__left,axiom,
    ! [Aa: int,Ba: int] :
      ( ( Aa != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Ba),one_one_int)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),aa_int_int(aa_int_fun_int_int(times_times_int,Aa),Ba)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),Ba) ) ) ) ).

% is_unit_div_mult_cancel_left
tff(fact_4973_is__unit__div__mult__cancel__left,axiom,
    ! [Aa: code_integer,Ba: code_integer] :
      ( ( Aa != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Ba),one_one_Code_integer)
       => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Aa),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Aa),Ba)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,one_one_Code_integer),Ba) ) ) ) ).

% is_unit_div_mult_cancel_left
tff(fact_4974_is__unit__div__mult__cancel__right,axiom,
    ! [Aa: nat,Ba: nat] :
      ( ( Aa != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Ba),one_one_nat)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ba),Aa)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),Ba) ) ) ) ).

% is_unit_div_mult_cancel_right
tff(fact_4975_is__unit__div__mult__cancel__right,axiom,
    ! [Aa: int,Ba: int] :
      ( ( Aa != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Ba),one_one_int)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),aa_int_int(aa_int_fun_int_int(times_times_int,Ba),Aa)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),Ba) ) ) ) ).

% is_unit_div_mult_cancel_right
tff(fact_4976_is__unit__div__mult__cancel__right,axiom,
    ! [Aa: code_integer,Ba: code_integer] :
      ( ( Aa != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Ba),one_one_Code_integer)
       => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Aa),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Ba),Aa)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,one_one_Code_integer),Ba) ) ) ) ).

% is_unit_div_mult_cancel_right
tff(fact_4977_double__not__eq__Suc__double,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M) != aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) ).

% double_not_eq_Suc_double
tff(fact_4978_Suc__double__not__eq__double,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na) ).

% Suc_double_not_eq_double
tff(fact_4979_div__nat__eqI,axiom,
    ! [Na: nat,Q4: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),Q4)),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),aa_nat_nat(suc,Q4)))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na) = Q4 ) ) ) ).

% div_nat_eqI
tff(fact_4980_less__eq__div__iff__mult__less__eq,axiom,
    ! [Q4: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Q4)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),Q4))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Q4)),Na) ) ) ).

% less_eq_div_iff_mult_less_eq
tff(fact_4981_nat__less__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),I)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Na))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),J)),U)),M)),Na) ) ) ).

% nat_less_add_iff1
tff(fact_4982_nat__less__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Na))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I)),U)),Na)) ) ) ).

% nat_less_add_iff2
tff(fact_4983_dividend__less__times__div,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na)))) ) ).

% dividend_less_times_div
tff(fact_4984_dividend__less__div__times,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na)),Na))) ) ).

% dividend_less_div_times
tff(fact_4985_split__div,axiom,
    ! [P: fun_nat_o,M: nat,Na: nat] :
      ( aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na))
    <=> ( ( ( Na = zero_zero_nat )
         => aa_nat_o(P,zero_zero_nat) )
        & ( ( Na != zero_zero_nat )
         => ! [I3: nat,J3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),Na)
             => ( ( M = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),I3)),J3) )
               => aa_nat_o(P,I3) ) ) ) ) ) ).

% split_div
tff(fact_4986_mult__eq__if,axiom,
    ! [M: nat,Na: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na) = $ite(M = zero_zero_nat,zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)),Na))) ).

% mult_eq_if
tff(fact_4987_convex__bound__lt,axiom,
    ! [Xb: real,Aa: real,Y: real,U: real,V: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Aa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Aa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),U)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),V)
           => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,U),V) = one_one_real )
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,U),Xb)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,V),Y))),Aa) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_4988_convex__bound__lt,axiom,
    ! [Xb: rat,Aa: rat,Y: rat,U: rat,V: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Aa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Aa)
       => ( 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),Y))),Aa) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_4989_convex__bound__lt,axiom,
    ! [Xb: int,Aa: int,Y: int,U: int,V: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Aa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Aa)
       => ( 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),Y))),Aa) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_4990_divide__le__eq__numeral_I1_J,axiom,
    ! [Ba: real,C: real,W2: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Ba),C)),aa_num_real(numeral_numeral_real,W2))
    <=> $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,Ba),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W2)),C)),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W2)),C)),Ba),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_num_real(numeral_numeral_real,W2))) ) ) ).

% divide_le_eq_numeral(1)
tff(fact_4991_divide__le__eq__numeral_I1_J,axiom,
    ! [Ba: rat,C: rat,W2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(Ba),C)),aa_num_rat(numeral_numeral_rat,W2))
    <=> $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,Ba),aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W2)),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,W2)),C)),Ba),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_num_rat(numeral_numeral_rat,W2))) ) ) ).

% divide_le_eq_numeral(1)
tff(fact_4992_le__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,Ba: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,W2)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Ba),C))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W2)),C)),Ba),
          $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,Ba),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W2)),C)),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,W2)),zero_zero_real)) ) ) ).

% le_divide_eq_numeral(1)
tff(fact_4993_le__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,Ba: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,W2)),aa_rat_rat(divide_divide_rat(Ba),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,W2)),C)),Ba),
          $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,Ba),aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W2)),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,W2)),zero_zero_rat)) ) ) ).

% le_divide_eq_numeral(1)
tff(fact_4994_le__minus__divide__eq,axiom,
    ! [Aa: real,Ba: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Ba),C)))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),C)),aa_real_real(uminus_uminus_real,Ba)),
          $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,Ba)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),C)),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),zero_zero_real)) ) ) ).

% le_minus_divide_eq
tff(fact_4995_le__minus__divide__eq,axiom,
    ! [Aa: rat,Ba: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(Ba),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),C)),aa_rat_rat(uminus_uminus_rat,Ba)),
          $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,Ba)),aa_rat_rat(times_times_rat(Aa),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),zero_zero_rat)) ) ) ).

% le_minus_divide_eq
tff(fact_4996_minus__divide__le__eq,axiom,
    ! [Ba: real,C: real,Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Ba),C))),Aa)
    <=> $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,Ba)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),C)),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),C)),aa_real_real(uminus_uminus_real,Ba)),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Aa)) ) ) ).

% minus_divide_le_eq
tff(fact_4997_minus__divide__le__eq,axiom,
    ! [Ba: rat,C: rat,Aa: 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(Ba),C))),Aa)
    <=> $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,Ba)),aa_rat_rat(times_times_rat(Aa),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),C)),aa_rat_rat(uminus_uminus_rat,Ba)),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Aa)) ) ) ).

% minus_divide_le_eq
tff(fact_4998_neg__le__minus__divide__eq,axiom,
    ! [C: real,Aa: real,Ba: 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),aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Ba),C)))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,Ba)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),C)) ) ) ).

% neg_le_minus_divide_eq
tff(fact_4999_neg__le__minus__divide__eq,axiom,
    ! [C: rat,Aa: rat,Ba: 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),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(Ba),C)))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,Ba)),aa_rat_rat(times_times_rat(Aa),C)) ) ) ).

% neg_le_minus_divide_eq
tff(fact_5000_neg__minus__divide__le__eq,axiom,
    ! [C: real,Ba: real,Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Ba),C))),Aa)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),C)),aa_real_real(uminus_uminus_real,Ba)) ) ) ).

% neg_minus_divide_le_eq
tff(fact_5001_neg__minus__divide__le__eq,axiom,
    ! [C: rat,Ba: rat,Aa: 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(Ba),C))),Aa)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(Aa),C)),aa_rat_rat(uminus_uminus_rat,Ba)) ) ) ).

% neg_minus_divide_le_eq
tff(fact_5002_pos__le__minus__divide__eq,axiom,
    ! [C: real,Aa: real,Ba: 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),aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Ba),C)))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),C)),aa_real_real(uminus_uminus_real,Ba)) ) ) ).

% pos_le_minus_divide_eq
tff(fact_5003_pos__le__minus__divide__eq,axiom,
    ! [C: rat,Aa: rat,Ba: 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),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(Ba),C)))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(Aa),C)),aa_rat_rat(uminus_uminus_rat,Ba)) ) ) ).

% pos_le_minus_divide_eq
tff(fact_5004_pos__minus__divide__le__eq,axiom,
    ! [C: real,Ba: real,Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Ba),C))),Aa)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,Ba)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),C)) ) ) ).

% pos_minus_divide_le_eq
tff(fact_5005_pos__minus__divide__le__eq,axiom,
    ! [C: rat,Ba: rat,Aa: 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(Ba),C))),Aa)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,Ba)),aa_rat_rat(times_times_rat(Aa),C)) ) ) ).

% pos_minus_divide_le_eq
tff(fact_5006_scaling__mono,axiom,
    ! [U: real,V: real,R2: real,S3: 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),S3)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,U),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,V),U))),S3))),V) ) ) ) ).

% scaling_mono
tff(fact_5007_scaling__mono,axiom,
    ! [U: rat,V: rat,R2: rat,S3: 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),S3)
         => 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))),S3))),V) ) ) ) ).

% scaling_mono
tff(fact_5008_divide__less__eq__numeral_I2_J,axiom,
    ! [Ba: real,C: real,W2: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Ba),C)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)))
    <=> $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,Ba),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))),C)),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))),C)),Ba),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,W2)))) ) ) ).

% divide_less_eq_numeral(2)
tff(fact_5009_divide__less__eq__numeral_I2_J,axiom,
    ! [Ba: rat,C: rat,W2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(Ba),C)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)))
    <=> $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,Ba),aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),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,W2))),C)),Ba),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,W2)))) ) ) ).

% divide_less_eq_numeral(2)
tff(fact_5010_less__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,Ba: 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,W2))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Ba),C))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))),C)),Ba),
          $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,Ba),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))),C)),aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))),zero_zero_real)) ) ) ).

% less_divide_eq_numeral(2)
tff(fact_5011_less__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,Ba: 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,W2))),aa_rat_rat(divide_divide_rat(Ba),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,W2))),C)),Ba),
          $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,Ba),aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),zero_zero_rat)) ) ) ).

% less_divide_eq_numeral(2)
tff(fact_5012_power__eq__if,axiom,
    ! [P3: complex,M: nat] :
      aa_nat_complex(power_power_complex(P3),M) = $ite(M = zero_zero_nat,one_one_complex,aa_complex_complex(times_times_complex(P3),aa_nat_complex(power_power_complex(P3),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))) ).

% power_eq_if
tff(fact_5013_power__eq__if,axiom,
    ! [P3: real,M: nat] :
      aa_nat_real(aa_real_fun_nat_real(power_power_real,P3),M) = $ite(M = zero_zero_nat,one_one_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,P3),aa_nat_real(aa_real_fun_nat_real(power_power_real,P3),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))) ).

% power_eq_if
tff(fact_5014_power__eq__if,axiom,
    ! [P3: rat,M: nat] :
      aa_nat_rat(power_power_rat(P3),M) = $ite(M = zero_zero_nat,one_one_rat,aa_rat_rat(times_times_rat(P3),aa_nat_rat(power_power_rat(P3),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))) ).

% power_eq_if
tff(fact_5015_power__eq__if,axiom,
    ! [P3: nat,M: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,P3),M) = $ite(M = zero_zero_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,P3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,P3),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))) ).

% power_eq_if
tff(fact_5016_power__eq__if,axiom,
    ! [P3: int,M: nat] :
      aa_nat_int(power_power_int(P3),M) = $ite(M = zero_zero_nat,one_one_int,aa_int_int(aa_int_fun_int_int(times_times_int,P3),aa_nat_int(power_power_int(P3),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))) ).

% power_eq_if
tff(fact_5017_power__minus__mult,axiom,
    ! [Na: nat,Aa: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(Aa),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat))),Aa) = aa_nat_complex(power_power_complex(Aa),Na) ) ) ).

% power_minus_mult
tff(fact_5018_power__minus__mult,axiom,
    ! [Na: nat,Aa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat))),Aa) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),Na) ) ) ).

% power_minus_mult
tff(fact_5019_power__minus__mult,axiom,
    ! [Na: nat,Aa: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(Aa),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat))),Aa) = aa_nat_rat(power_power_rat(Aa),Na) ) ) ).

% power_minus_mult
tff(fact_5020_power__minus__mult,axiom,
    ! [Na: nat,Aa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat))),Aa) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Aa),Na) ) ) ).

% power_minus_mult
tff(fact_5021_power__minus__mult,axiom,
    ! [Na: nat,Aa: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(Aa),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat))),Aa) = aa_nat_int(power_power_int(Aa),Na) ) ) ).

% power_minus_mult
tff(fact_5022_split__div_H,axiom,
    ! [P: fun_nat_o,M: nat,Na: nat] :
      ( aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na))
    <=> ( ( ( Na = zero_zero_nat )
          & aa_nat_o(P,zero_zero_nat) )
        | ? [Q5: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),Q5)),M)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),aa_nat_nat(suc,Q5)))
            & aa_nat_o(P,Q5) ) ) ) ).

% split_div'
tff(fact_5023_euclidean__size__times__nonunit,axiom,
    ! [Aa: code_integer,Ba: code_integer] :
      ( ( Aa != zero_z3403309356797280102nteger )
     => ( ( Ba != zero_z3403309356797280102nteger )
       => ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),one_one_Code_integer)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_Code_integer_nat(euclid6377331345833325938nteger,Ba)),aa_Code_integer_nat(euclid6377331345833325938nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Aa),Ba))) ) ) ) ).

% euclidean_size_times_nonunit
tff(fact_5024_euclidean__size__times__nonunit,axiom,
    ! [Aa: int,Ba: int] :
      ( ( Aa != zero_zero_int )
     => ( ( Ba != zero_zero_int )
       => ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),one_one_int)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(euclid4774559944035922753ze_int,Ba)),aa_int_nat(euclid4774559944035922753ze_int,aa_int_int(aa_int_fun_int_int(times_times_int,Aa),Ba))) ) ) ) ).

% euclidean_size_times_nonunit
tff(fact_5025_euclidean__size__times__nonunit,axiom,
    ! [Aa: nat,Ba: nat] :
      ( ( Aa != zero_zero_nat )
     => ( ( Ba != zero_zero_nat )
       => ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),one_one_nat)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(euclid4777050414544973029ze_nat,Ba)),aa_nat_nat(euclid4777050414544973029ze_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),Ba))) ) ) ) ).

% euclidean_size_times_nonunit
tff(fact_5026_divide__le__eq__numeral_I2_J,axiom,
    ! [Ba: real,C: real,W2: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Ba),C)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2)))
    <=> $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,Ba),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))),C)),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))),C)),Ba),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,W2)))) ) ) ).

% divide_le_eq_numeral(2)
tff(fact_5027_divide__le__eq__numeral_I2_J,axiom,
    ! [Ba: rat,C: rat,W2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(Ba),C)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2)))
    <=> $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,Ba),aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),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,W2))),C)),Ba),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,W2)))) ) ) ).

% divide_le_eq_numeral(2)
tff(fact_5028_le__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,Ba: 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,W2))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Ba),C))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C),
          aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))),C)),Ba),
          $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,Ba),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W2))),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,W2))),zero_zero_real)) ) ) ).

% le_divide_eq_numeral(2)
tff(fact_5029_le__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,Ba: 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,W2))),aa_rat_rat(divide_divide_rat(Ba),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,W2))),C)),Ba),
          $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,Ba),aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W2))),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,W2))),zero_zero_rat)) ) ) ).

% le_divide_eq_numeral(2)
tff(fact_5030_oddE,axiom,
    ! [Aa: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Aa)
     => ~ ! [B2: nat] : Aa != 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))),B2)),one_one_nat) ) ).

% oddE
tff(fact_5031_oddE,axiom,
    ! [Aa: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Aa)
     => ~ ! [B2: int] : Aa != 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))),B2)),one_one_int) ) ).

% oddE
tff(fact_5032_oddE,axiom,
    ! [Aa: code_integer] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Aa)
     => ~ ! [B2: code_integer] : Aa != aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),B2)),one_one_Code_integer) ) ).

% oddE
tff(fact_5033_zero__le__even__power_H,axiom,
    ! [Aa: real,Na: 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),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))) ).

% zero_le_even_power'
tff(fact_5034_zero__le__even__power_H,axiom,
    ! [Aa: rat,Na: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(Aa),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))) ).

% zero_le_even_power'
tff(fact_5035_zero__le__even__power_H,axiom,
    ! [Aa: int,Na: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(Aa),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))) ).

% zero_le_even_power'
tff(fact_5036_nat__bit__induct,axiom,
    ! [P: fun_nat_o,Na: nat] :
      ( aa_nat_o(P,zero_zero_nat)
     => ( ! [N: nat] :
            ( aa_nat_o(P,N)
           => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
             => aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ) )
       => ( ! [N: nat] :
              ( aa_nat_o(P,N)
             => aa_nat_o(P,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) )
         => aa_nat_o(P,Na) ) ) ) ).

% nat_bit_induct
tff(fact_5037_triangle__def,axiom,
    ! [Na: nat] : aa_nat_nat(nat_triangle,Na) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),aa_nat_nat(suc,Na))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% triangle_def
tff(fact_5038_VEBT__internal_Oexp__split__high__low_I2_J,axiom,
    ! [Xb: nat,Na: nat,M: 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,Na),M)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),Na)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) ) ) ) ).

% VEBT_internal.exp_split_high_low(2)
tff(fact_5039_sum__squares__bound,axiom,
    ! [Xb: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Xb)),Y)),aa_real_real(aa_rea7340605660300375871l_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,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% sum_squares_bound
tff(fact_5040_sum__squares__bound,axiom,
    ! [Xb: rat,Y: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),Xb)),Y)),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(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% sum_squares_bound
tff(fact_5041_odd__0__le__power__imp__0__le,axiom,
    ! [Aa: real,Na: 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),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Aa) ) ).

% odd_0_le_power_imp_0_le
tff(fact_5042_odd__0__le__power__imp__0__le,axiom,
    ! [Aa: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(Aa),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Aa) ) ).

% odd_0_le_power_imp_0_le
tff(fact_5043_odd__0__le__power__imp__0__le,axiom,
    ! [Aa: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(Aa),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Aa) ) ).

% odd_0_le_power_imp_0_le
tff(fact_5044_odd__power__less__zero,axiom,
    ! [Aa: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),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,Aa),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)))),zero_zero_real) ) ).

% odd_power_less_zero
tff(fact_5045_odd__power__less__zero,axiom,
    ! [Aa: rat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),zero_zero_rat)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(Aa),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)))),zero_zero_rat) ) ).

% odd_power_less_zero
tff(fact_5046_odd__power__less__zero,axiom,
    ! [Aa: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),zero_zero_int)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(Aa),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)))),zero_zero_int) ) ).

% odd_power_less_zero
tff(fact_5047_power__minus1__odd,axiom,
    ! [Na: nat] : aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% power_minus1_odd
tff(fact_5048_power__minus1__odd,axiom,
    ! [Na: 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))),Na))) = aa_real_real(uminus_uminus_real,one_one_real) ).

% power_minus1_odd
tff(fact_5049_power__minus1__odd,axiom,
    ! [Na: nat] : aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))) = aa_int_int(uminus_uminus_int,one_one_int) ).

% power_minus1_odd
tff(fact_5050_power__minus1__odd,axiom,
    ! [Na: nat] : aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% power_minus1_odd
tff(fact_5051_power__minus1__odd,axiom,
    ! [Na: 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))),Na))) = aa_rat_rat(uminus_uminus_rat,one_one_rat) ).

% power_minus1_odd
tff(fact_5052_arith__geo__mean,axiom,
    ! [U: real,Xb: real,Y: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,U),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Y) )
     => ( 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),Y)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,U),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ) ).

% arith_geo_mean
tff(fact_5053_arith__geo__mean,axiom,
    ! [U: rat,Xb: rat,Y: rat] :
      ( ( aa_nat_rat(power_power_rat(U),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_rat_rat(times_times_rat(Xb),Y) )
     => ( 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),Y)
         => 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),Y)),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))) ) ) ) ).

% arith_geo_mean
tff(fact_5054_even__mult__exp__div__exp__iff,axiom,
    ! [Aa: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M)
        | ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na) = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
          & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),M)))) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_5055_even__mult__exp__div__exp__iff,axiom,
    ! [Aa: int,M: nat,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,Aa),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M)
        | ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na) = zero_zero_int )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
          & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),M)))) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_5056_even__mult__exp__div__exp__iff,axiom,
    ! [Aa: code_integer,M: nat,Na: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Aa),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),M))),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Na)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M)
        | ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Na) = zero_z3403309356797280102nteger )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Aa),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),M)))) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_5057_invar__vebt_Ointros_I4_J,axiom,
    ! [TreeList: list_VEBT_VEBT,Na: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi2: nat,Ma2: nat] :
      ( ! [X: vEBT_VEBT] :
          ( member_VEBT_VEBT(X,set_VEBT_VEBT2(TreeList))
         => aa_nat_o(vEBT_invar_vebt(X),Na) )
     => ( aa_nat_o(vEBT_invar_vebt(Summary),M)
       => ( ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M) )
         => ( ( M = Na )
           => ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),M) )
             => ( ! [I2: nat] :
                    ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))
                   => ( ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList,I2)),X_12)
                    <=> aa_nat_o(vEBT_V8194947554948674370ptions(Summary),I2) ) )
               => ( ( ( Mi2 = Ma2 )
                   => ! [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,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))),Deg))
                     => ( ( ( Mi2 != Ma2 )
                         => ! [I2: nat] :
                              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))
                             => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma2),Na) = I2 )
                                 => aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList,I2)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma2),Na)) )
                                & ! [X: nat] :
                                    ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),Na) = I2 )
                                      & aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList,I2)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),Na)) )
                                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi2),X)
                                      & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Ma2) ) ) ) ) )
                       => aa_nat_o(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Deg,TreeList,Summary)),Deg) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(4)
tff(fact_5058_invar__vebt_Ointros_I5_J,axiom,
    ! [TreeList: list_VEBT_VEBT,Na: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi2: nat,Ma2: nat] :
      ( ! [X: vEBT_VEBT] :
          ( member_VEBT_VEBT(X,set_VEBT_VEBT2(TreeList))
         => aa_nat_o(vEBT_invar_vebt(X),Na) )
     => ( aa_nat_o(vEBT_invar_vebt(Summary),M)
       => ( ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M) )
         => ( ( M = aa_nat_nat(suc,Na) )
           => ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),M) )
             => ( ! [I2: nat] :
                    ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))
                   => ( ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList,I2)),X_12)
                    <=> aa_nat_o(vEBT_V8194947554948674370ptions(Summary),I2) ) )
               => ( ( ( Mi2 = Ma2 )
                   => ! [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,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))),Deg))
                     => ( ( ( Mi2 != Ma2 )
                         => ! [I2: nat] :
                              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))
                             => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma2),Na) = I2 )
                                 => aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList,I2)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma2),Na)) )
                                & ! [X: nat] :
                                    ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),Na) = I2 )
                                      & aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList,I2)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),Na)) )
                                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi2),X)
                                      & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Ma2) ) ) ) ) )
                       => aa_nat_o(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Deg,TreeList,Summary)),Deg) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(5)
tff(fact_5059_mul__def,axiom,
    vEBT_VEBT_mul = vEBT_V4262088993061758097ft_nat(times_times_nat) ).

% mul_def
tff(fact_5060_mul__shift,axiom,
    ! [Xb: nat,Y: nat,Z: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Xb),Y) = Z )
    <=> ( aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(Xb)),some_nat(Y)) = some_nat(Z) ) ) ).

% mul_shift
tff(fact_5061_in__children__def,axiom,
    ! [Na: nat,TreeList: list_VEBT_VEBT,Xb: nat] :
      ( aa_nat_o(vEBT_V5917875025757280293ildren(Na,TreeList),Xb)
    <=> aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),Na))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),Na)) ) ).

% in_children_def
tff(fact_5062_divmod__step__eq,axiom,
    ! [L: num,Q4: code_integer,R2: code_integer] :
      unique4921790084139445826nteger(L,produc1086072967326762835nteger(Q4,R2)) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_num_Code_integer(numera6620942414471956472nteger,L)),R2),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Q4)),one_one_Code_integer),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,R2),aa_num_Code_integer(numera6620942414471956472nteger,L))),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Q4),R2)) ).

% divmod_step_eq
tff(fact_5063_divmod__step__eq,axiom,
    ! [L: num,Q4: nat,R2: nat] :
      unique5026877609467782581ep_nat(L,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Q4),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))),Q4)),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))),Q4)),R2)) ).

% divmod_step_eq
tff(fact_5064_divmod__step__eq,axiom,
    ! [L: num,Q4: int,R2: int] :
      unique5024387138958732305ep_int(L,product_Pair_int_int(Q4,R2)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,L)),R2),product_Pair_int_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Q4)),one_one_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,R2),aa_num_int(numeral_numeral_int,L))),product_Pair_int_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Q4),R2)) ).

% divmod_step_eq
tff(fact_5065_set__bit__0,axiom,
    ! [Aa: int] : aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,zero_zero_nat),Aa) = 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,Aa),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))) ).

% set_bit_0
tff(fact_5066_set__bit__0,axiom,
    ! [Aa: code_integer] : aa_Cod1783974652533396728nteger(aa_nat6383147911832638789nteger(bit_se2793503036327961859nteger,zero_zero_nat),Aa) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Aa),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))))) ).

% set_bit_0
tff(fact_5067_set__bit__0,axiom,
    ! [Aa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,zero_zero_nat),Aa) = 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,Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% set_bit_0
tff(fact_5068_fact__diff__Suc,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(suc,M))
     => ( aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,M)),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,M)),Na)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na))) ) ) ).

% fact_diff_Suc
tff(fact_5069_choose__dvd,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri3624122377584611663nteger,K)),aa_nat_Code_integer(semiri3624122377584611663nteger,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K)))),aa_nat_Code_integer(semiri3624122377584611663nteger,Na)) ) ).

% choose_dvd
tff(fact_5070_choose__dvd,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(times_times_rat(aa_nat_rat(semiri773545260158071498ct_rat,K)),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K)))),aa_nat_rat(semiri773545260158071498ct_rat,Na)) ) ).

% choose_dvd
tff(fact_5071_choose__dvd,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1406184849735516958ct_int,K)),aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K)))),aa_nat_int(semiri1406184849735516958ct_int,Na)) ) ).

% choose_dvd
tff(fact_5072_choose__dvd,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,K)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K)))),aa_nat_nat(semiri1408675320244567234ct_nat,Na)) ) ).

% choose_dvd
tff(fact_5073_choose__dvd,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri2265585572941072030t_real,K)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K)))),aa_nat_real(semiri2265585572941072030t_real,Na)) ) ).

% choose_dvd
tff(fact_5074_unset__bit__0,axiom,
    ! [Aa: int] : aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,zero_zero_nat),Aa) = 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,Aa),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ).

% unset_bit_0
tff(fact_5075_unset__bit__0,axiom,
    ! [Aa: code_integer] : aa_Cod1783974652533396728nteger(aa_nat6383147911832638789nteger(bit_se8260200283734997820nteger,zero_zero_nat),Aa) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Aa),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))) ).

% unset_bit_0
tff(fact_5076_unset__bit__0,axiom,
    ! [Aa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,zero_zero_nat),Aa) = 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,Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% unset_bit_0
tff(fact_5077_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(aa_rea7340605660300375871l_real(times_times_real,Xb),Xb))
    <=> ( Xb = zero_zero_real ) ) ).

% not_real_square_gt_zero
tff(fact_5078_unset__bit__nonnegative__int__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,Na),K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K) ) ).

% unset_bit_nonnegative_int_iff
tff(fact_5079_set__bit__nonnegative__int__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,Na),K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K) ) ).

% set_bit_nonnegative_int_iff
tff(fact_5080_unset__bit__negative__int__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,Na),K)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int) ) ).

% unset_bit_negative_int_iff
tff(fact_5081_set__bit__negative__int__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,Na),K)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int) ) ).

% set_bit_negative_int_iff
tff(fact_5082_num__double,axiom,
    ! [Na: num] : aa_num_num(aa_num_fun_num_num(times_times_num,aa_num_num(bit0,one)),Na) = aa_num_num(bit0,Na) ).

% num_double
tff(fact_5083_unset__bit__less__eq,axiom,
    ! [Na: nat,K: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,Na),K)),K) ).

% unset_bit_less_eq
tff(fact_5084_set__bit__greater__eq,axiom,
    ! [K: int,Na: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,Na),K)) ).

% set_bit_greater_eq
tff(fact_5085_div__mult2__numeral__eq,axiom,
    ! [Aa: 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,Aa),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,Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(aa_num_fun_num_num(times_times_num,K),L))) ).

% div_mult2_numeral_eq
tff(fact_5086_div__mult2__numeral__eq,axiom,
    ! [Aa: 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,Aa),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,Aa),aa_num_int(numeral_numeral_int,aa_num_num(aa_num_fun_num_num(times_times_num,K),L))) ).

% div_mult2_numeral_eq
tff(fact_5087_div__mult2__numeral__eq,axiom,
    ! [Aa: code_integer,K: num,L: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Aa),aa_num_Code_integer(numera6620942414471956472nteger,K))),aa_num_Code_integer(numera6620942414471956472nteger,L)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Aa),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(aa_num_fun_num_num(times_times_num,K),L))) ).

% div_mult2_numeral_eq
tff(fact_5088_zmult__zless__mono2,axiom,
    ! [I: int,J: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,I),J)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I)),aa_int_int(aa_int_fun_int_int(times_times_int,K),J)) ) ) ).

% zmult_zless_mono2
tff(fact_5089_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(aa_rea7340605660300375871l_real(times_times_real,U),U))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Xb)) ).

% real_minus_mult_self_le
tff(fact_5090_enat__0__less__mult__iff,axiom,
    ! [M: extended_enat,Na: extended_enat] :
      ( aa_Extended_enat_o(ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat),times_7803423173614009249d_enat(M,Na))
    <=> ( aa_Extended_enat_o(ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat),M)
        & aa_Extended_enat_o(ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat),Na) ) ) ).

% enat_0_less_mult_iff
tff(fact_5091_mult__inc,axiom,
    ! [Xb: num,Y: num] : aa_num_num(aa_num_fun_num_num(times_times_num,Xb),aa_num_num(inc,Y)) = aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(aa_num_fun_num_num(times_times_num,Xb),Y)),Xb) ).

% mult_inc
tff(fact_5092_pos__zmult__eq__1__iff,axiom,
    ! [M: int,Na: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),M)
     => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,M),Na) = one_one_int )
      <=> ( ( M = one_one_int )
          & ( Na = one_one_int ) ) ) ) ).

% pos_zmult_eq_1_iff
tff(fact_5093_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)
     => ( ! [X: int,K2: int] :
            ( aa_int_o(P1,X)
          <=> aa_int_o(P1,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D))) )
       => ( ? [Z4: int] :
            ! [X: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Z4)
             => ( aa_int_o(P,X)
              <=> aa_int_o(P1,X) ) )
         => ( ? [X_13: int] : aa_int_o(P1,X_13)
           => ? [X_1: int] : aa_int_o(P,X_1) ) ) ) ) ).

% minusinfinity
tff(fact_5094_plusinfinity,axiom,
    ! [D: int,P2: fun_int_o,P: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
     => ( ! [X: int,K2: int] :
            ( aa_int_o(P2,X)
          <=> aa_int_o(P2,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D))) )
       => ( ? [Z4: int] :
            ! [X: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z4),X)
             => ( aa_int_o(P,X)
              <=> aa_int_o(P2,X) ) )
         => ( ? [X_13: int] : aa_int_o(P2,X_13)
           => ? [X_1: int] : aa_int_o(P,X_1) ) ) ) ) ).

% plusinfinity
tff(fact_5095_zdiv__zmult2__eq,axiom,
    ! [C: int,Aa: int,Ba: 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,Aa),aa_int_int(aa_int_fun_int_int(times_times_int,Ba),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),Ba)),C) ) ) ).

% zdiv_zmult2_eq
tff(fact_5096_unique__quotient__lemma__neg,axiom,
    ! [Ba: int,Q6: int,R4: int,Q4: 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,Ba),Q6)),R4)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Ba),Q4)),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,Ba),R2)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Ba),R4)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Q4),Q6) ) ) ) ) ).

% unique_quotient_lemma_neg
tff(fact_5097_unique__quotient__lemma,axiom,
    ! [Ba: int,Q6: int,R4: int,Q4: 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,Ba),Q6)),R4)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Ba),Q4)),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),Ba)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R2),Ba)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Q6),Q4) ) ) ) ) ).

% unique_quotient_lemma
tff(fact_5098_zdiv__mono2__neg__lemma,axiom,
    ! [Ba: int,Q4: int,R2: int,B5: int,Q6: int,R4: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Ba),Q4)),R2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B5),Q6)),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,B5),Q6)),R4)),zero_zero_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R2),Ba)
         => ( 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),B5)
             => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B5),Ba)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Q6),Q4) ) ) ) ) ) ) ).

% zdiv_mono2_neg_lemma
tff(fact_5099_zdiv__mono2__lemma,axiom,
    ! [Ba: int,Q4: int,R2: int,B5: int,Q6: int,R4: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Ba),Q4)),R2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B5),Q6)),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,B5),Q6)),R4))
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R4),B5)
         => ( 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),B5)
             => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B5),Ba)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Q4),Q6) ) ) ) ) ) ) ).

% zdiv_mono2_lemma
tff(fact_5100_q__pos__lemma,axiom,
    ! [B5: int,Q6: 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,B5),Q6)),R4))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R4),B5)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B5)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Q6) ) ) ) ).

% q_pos_lemma
tff(fact_5101_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)
     => ( ! [X: int] :
            ( aa_int_o(P,X)
           => aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),D)) )
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
         => ! [X3: int] :
              ( aa_int_o(P,X3)
             => aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K),D))) ) ) ) ) ).

% incr_mult_lemma
tff(fact_5102_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)
     => ( ! [X: int] :
            ( aa_int_o(P,X)
           => aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),D)) )
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
         => ! [X3: int] :
              ( aa_int_o(P,X3)
             => aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K),D))) ) ) ) ) ).

% decr_mult_lemma
tff(fact_5103_int__div__pos__eq,axiom,
    ! [Aa: int,Ba: int,Q4: int,R2: int] :
      ( ( Aa = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Ba),Q4)),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),Ba)
         => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),Ba) = Q4 ) ) ) ) ).

% int_div_pos_eq
tff(fact_5104_int__div__neg__eq,axiom,
    ! [Aa: int,Ba: int,Q4: int,R2: int] :
      ( ( Aa = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Ba),Q4)),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,Ba),R2)
         => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),Ba) = Q4 ) ) ) ) ).

% int_div_neg_eq
tff(fact_5105_split__zdiv,axiom,
    ! [P: fun_int_o,Na: int,K: int] :
      ( aa_int_o(P,aa_int_int(aa_int_fun_int_int(divide_divide_int,Na),K))
    <=> ( ( ( K = zero_zero_int )
         => aa_int_o(P,zero_zero_int) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
         => ! [I3: int,J3: int] :
              ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),J3)
                & aa_int_o(aa_int_fun_int_o(ord_less_int,J3),K)
                & ( Na = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I3)),J3) ) )
             => aa_int_o(P,I3) ) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
         => ! [I3: int,J3: int] :
              ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),J3)
                & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J3),zero_zero_int)
                & ( Na = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I3)),J3) ) )
             => aa_int_o(P,I3) ) ) ) ) ).

% split_zdiv
tff(fact_5106_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)
     => ( ! [X: int,K2: 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,K2),D))) )
       => ( ? [X_12: int] : aa_int_o(P,X_12)
        <=> ? [X4: int] :
              ( aa_set_int_o(member_int(X4),set_or1266510415728281911st_int(one_one_int,D))
              & aa_int_o(P,X4) ) ) ) ) ).

% periodic_finite_ex
tff(fact_5107_cppi,axiom,
    ! [D6: int,P: fun_int_o,P2: fun_int_o,A3: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D6)
     => ( ? [Z4: int] :
          ! [X: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z4),X)
           => ( aa_int_o(P,X)
            <=> aa_int_o(P2,X) ) )
       => ( ! [X: int] :
              ( ! [Xa: int] :
                  ( aa_set_int_o(member_int(Xa),set_or1266510415728281911st_int(one_one_int,D6))
                 => ! [Xb3: int] :
                      ( aa_set_int_o(member_int(Xb3),A3)
                     => ( X != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa) ) ) )
             => ( aa_int_o(P,X)
               => aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),D6)) ) )
         => ( ! [X: int,K2: int] :
                ( aa_int_o(P2,X)
              <=> aa_int_o(P2,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D6))) )
           => ( ? [X_12: int] : aa_int_o(P,X_12)
            <=> ( ? [X4: int] :
                    ( aa_set_int_o(member_int(X4),set_or1266510415728281911st_int(one_one_int,D6))
                    & aa_int_o(P2,X4) )
                | ? [X4: int] :
                    ( aa_set_int_o(member_int(X4),set_or1266510415728281911st_int(one_one_int,D6))
                    & ? [Xa3: int] :
                        ( aa_set_int_o(member_int(Xa3),A3)
                        & aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xa3),X4)) ) ) ) ) ) ) ) ) ).

% cppi
tff(fact_5108_cpmi,axiom,
    ! [D6: int,P: fun_int_o,P2: fun_int_o,B4: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D6)
     => ( ? [Z4: int] :
          ! [X: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Z4)
           => ( aa_int_o(P,X)
            <=> aa_int_o(P2,X) ) )
       => ( ! [X: int] :
              ( ! [Xa: int] :
                  ( aa_set_int_o(member_int(Xa),set_or1266510415728281911st_int(one_one_int,D6))
                 => ! [Xb3: int] :
                      ( aa_set_int_o(member_int(Xb3),B4)
                     => ( X != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa) ) ) )
             => ( aa_int_o(P,X)
               => aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),D6)) ) )
         => ( ! [X: int,K2: int] :
                ( aa_int_o(P2,X)
              <=> aa_int_o(P2,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D6))) )
           => ( ? [X_12: int] : aa_int_o(P,X_12)
            <=> ( ? [X4: int] :
                    ( aa_set_int_o(member_int(X4),set_or1266510415728281911st_int(one_one_int,D6))
                    & aa_int_o(P2,X4) )
                | ? [X4: int] :
                    ( aa_set_int_o(member_int(X4),set_or1266510415728281911st_int(one_one_int,D6))
                    & ? [Xa3: int] :
                        ( aa_set_int_o(member_int(Xa3),B4)
                        & aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xa3),X4)) ) ) ) ) ) ) ) ) ).

% cpmi
tff(fact_5109_L2__set__mult__ineq__lemma,axiom,
    ! [Aa: real,C: real,Ba: real,D: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),C))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Ba),D))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),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(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Ba),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_5110_neg__zdiv__mult__2,axiom,
    ! [Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),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))),Ba))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Aa)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Ba),one_one_int)),Aa) ) ) ).

% neg_zdiv_mult_2
tff(fact_5111_pos__zdiv__mult__2,axiom,
    ! [Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Aa)
     => ( 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))),Ba))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Aa)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,Ba),Aa) ) ) ).

% pos_zdiv_mult_2
tff(fact_5112_even__unset__bit__iff,axiom,
    ! [M: nat,Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,M),Aa))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Aa)
        | ( M = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
tff(fact_5113_even__unset__bit__iff,axiom,
    ! [M: nat,Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_nat6383147911832638789nteger(bit_se8260200283734997820nteger,M),Aa))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Aa)
        | ( M = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
tff(fact_5114_even__unset__bit__iff,axiom,
    ! [M: nat,Aa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,M),Aa))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Aa)
        | ( M = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
tff(fact_5115_even__set__bit__iff,axiom,
    ! [M: nat,Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,M),Aa))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Aa)
        & ( M != zero_zero_nat ) ) ) ).

% even_set_bit_iff
tff(fact_5116_even__set__bit__iff,axiom,
    ! [M: nat,Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_nat6383147911832638789nteger(bit_se2793503036327961859nteger,M),Aa))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Aa)
        & ( M != zero_zero_nat ) ) ) ).

% even_set_bit_iff
tff(fact_5117_even__set__bit__iff,axiom,
    ! [M: nat,Aa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,M),Aa))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Aa)
        & ( M != zero_zero_nat ) ) ) ).

% even_set_bit_iff
tff(fact_5118_fact__fact__dvd__fact,axiom,
    ! [K: nat,Na: nat] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri3624122377584611663nteger,K)),aa_nat_Code_integer(semiri3624122377584611663nteger,Na))),aa_nat_Code_integer(semiri3624122377584611663nteger,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Na))) ).

% fact_fact_dvd_fact
tff(fact_5119_fact__fact__dvd__fact,axiom,
    ! [K: nat,Na: nat] : aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(times_times_rat(aa_nat_rat(semiri773545260158071498ct_rat,K)),aa_nat_rat(semiri773545260158071498ct_rat,Na))),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Na))) ).

% fact_fact_dvd_fact
tff(fact_5120_fact__fact__dvd__fact,axiom,
    ! [K: nat,Na: nat] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1406184849735516958ct_int,K)),aa_nat_int(semiri1406184849735516958ct_int,Na))),aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Na))) ).

% fact_fact_dvd_fact
tff(fact_5121_fact__fact__dvd__fact,axiom,
    ! [K: nat,Na: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,K)),aa_nat_nat(semiri1408675320244567234ct_nat,Na))),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Na))) ).

% fact_fact_dvd_fact
tff(fact_5122_fact__fact__dvd__fact,axiom,
    ! [K: nat,Na: nat] : aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri2265585572941072030t_real,K)),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Na))) ).

% fact_fact_dvd_fact
tff(fact_5123_square__fact__le__2__fact,axiom,
    ! [Na: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri2265585572941072030t_real,Na)),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))) ).

% square_fact_le_2_fact
tff(fact_5124_pred__less__length__list,axiom,
    ! [Deg: nat,Xb: nat,Ma2: nat,TreeList: list_VEBT_VEBT,Mi2: nat,Summary: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Ma2)
       => ( 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,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))
         => ( vEBT_vebt_pred(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Deg,TreeList,Summary),Xb) = $let(
                l: nat,
                l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                $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,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                  $let(
                    minlow: option_nat,
                    minlow:= vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,h)),
                    $ite(
                      ( ( minlow != none_nat )
                      & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_greater,some_nat(l)),minlow) ),
                      aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(h))),vEBT_vebt_pred(nth_VEBT_VEBT(TreeList,h),l)),
                      $let(
                        pr: option_nat,
                        pr:= vEBT_vebt_pred(Summary,h),
                        $ite(
                          pr = none_nat,
                          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi2),Xb),some_nat(Mi2),none_nat),
                          aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),pr)),vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList,the_nat(pr)))) ) ) ) ) ) ) ) ) ) ) ).

% pred_less_length_list
tff(fact_5125_pred__lesseq__max,axiom,
    ! [Deg: nat,Xb: nat,Ma2: nat,Mi2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Ma2)
       => ( vEBT_vebt_pred(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Deg,TreeList,Summary),Xb) = $let(
              l: nat,
              l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
              $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,Deg),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)),
                  $let(
                    minlow: option_nat,
                    minlow:= vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,h)),
                    $ite(
                      ( ( minlow != none_nat )
                      & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_greater,some_nat(l)),minlow) ),
                      aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(h))),vEBT_vebt_pred(nth_VEBT_VEBT(TreeList,h),l)),
                      $let(
                        pr: option_nat,
                        pr:= vEBT_vebt_pred(Summary,h),
                        $ite(
                          pr = none_nat,
                          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi2),Xb),some_nat(Mi2),none_nat),
                          aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),pr)),vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList,the_nat(pr)))) ) ) ) ),
                  none_nat ) ) ) ) ) ) ).

% pred_lesseq_max
tff(fact_5126_vebt__pred_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: option_nat] :
      ( ( vEBT_vebt_pred(Xb,Xaa) = Y )
     => ( ( ? [Uu2: $o,Uv2: $o] : Xb = vEBT_Leaf((Uu2),(Uv2))
         => ( ( Xaa = zero_zero_nat )
           => ( Y != none_nat ) ) )
       => ( ! [A2: $o] :
              ( ? [Uw2: $o] : Xb = vEBT_Leaf((A2),(Uw2))
             => ( ( Xaa = aa_nat_nat(suc,zero_zero_nat) )
               => ( Y != $ite((A2),some_nat(zero_zero_nat),none_nat) ) ) )
         => ( ! [A2: $o,B2: $o] :
                ( ( Xb = vEBT_Leaf((A2),(B2)) )
               => ( ? [Va2: nat] : Xaa = aa_nat_nat(suc,aa_nat_nat(suc,Va2))
                 => ( Y != $ite(
                        (B2),
                        some_nat(one_one_nat),
                        $ite((A2),some_nat(zero_zero_nat),none_nat) ) ) ) )
           => ( ( ? [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT] : Xb = vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2,Va3)
               => ( Y != none_nat ) )
             => ( ( ? [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vd2,Ve2)
                 => ( Y != none_nat ) )
               => ( ( ? [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vh2,Vi2)
                   => ( Y != none_nat ) )
                 => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary2: 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,Va2)),TreeList2,Summary2) )
                       => ( Y != $ite(
                              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xaa),
                              some_nat(Ma),
                              $let(
                                l: nat,
                                l:= 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,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                $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,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(TreeList2)),
                                    $let(
                                      minlow: option_nat,
                                      minlow:= vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,h)),
                                      $ite(
                                        ( ( minlow != none_nat )
                                        & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_greater,some_nat(l)),minlow) ),
                                        aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(h))),vEBT_vebt_pred(nth_VEBT_VEBT(TreeList2,h),l)),
                                        $let(
                                          pr: option_nat,
                                          pr:= vEBT_vebt_pred(Summary2,h),
                                          $ite(
                                            pr = none_nat,
                                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi),Xaa),some_nat(Mi),none_nat),
                                            aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),pr)),vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList2,the_nat(pr)))) ) ) ) ),
                                    none_nat ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_pred.elims
tff(fact_5127_neg__eucl__rel__int__mult__2,axiom,
    ! [Ba: int,Aa: int,Q4: int,R2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ba),zero_zero_int)
     => ( aa_Pro4592068669816069595_int_o(eucl_rel_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),one_one_int),Ba),product_Pair_int_int(Q4,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))),Aa)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Ba)),product_Pair_int_int(Q4,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_5128_low__def,axiom,
    ! [Xb: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),Na) = 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))),Na)) ).

% low_def
tff(fact_5129_sin__coeff__def,axiom,
    ! [X3: nat] :
      aa_nat_real(sin_coeff,X3) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),X3),zero_zero_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X3),aa_nat_nat(suc,zero_zero_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_real(semiri2265585572941072030t_real,X3))) ).

% sin_coeff_def
tff(fact_5130_set__vebt_H__def,axiom,
    ! [Ta: vEBT_VEBT] : vEBT_VEBT_set_vebt(Ta) = collect_nat(vEBT_vebt_member(Ta)) ).

% set_vebt'_def
tff(fact_5131_mod__mod__trivial,axiom,
    ! [Aa: nat,Ba: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Aa),Ba)),Ba) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Aa),Ba) ).

% mod_mod_trivial
tff(fact_5132_mod__mod__trivial,axiom,
    ! [Aa: int,Ba: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),Ba)),Ba) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),Ba) ).

% mod_mod_trivial
tff(fact_5133_mod__mod__trivial,axiom,
    ! [Aa: code_integer,Ba: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Aa),Ba)),Ba) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Aa),Ba) ).

% mod_mod_trivial
tff(fact_5134_finite__Collect__disjI,axiom,
    ! [P: fun_real_o,Q: fun_real_o] :
      ( finite_finite_real(collect_real(aa_fun5868358341814070780real_o(aTP_Lamm_af(P),Q)))
    <=> ( finite_finite_real(collect_real(P))
        & finite_finite_real(collect_real(Q)) ) ) ).

% finite_Collect_disjI
tff(fact_5135_finite__Collect__disjI,axiom,
    ! [P: fun_list_nat_o,Q: fun_list_nat_o] :
      ( finite8100373058378681591st_nat(collect_list_nat(aa_fun6155608546934989052_nat_o(aTP_Lamm_ag(P),Q)))
    <=> ( finite8100373058378681591st_nat(collect_list_nat(P))
        & finite8100373058378681591st_nat(collect_list_nat(Q)) ) ) ).

% finite_Collect_disjI
tff(fact_5136_finite__Collect__disjI,axiom,
    ! [P: fun_set_nat_o,Q: fun_set_nat_o] :
      ( finite1152437895449049373et_nat(collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_ah(P),Q)))
    <=> ( finite1152437895449049373et_nat(collect_set_nat(P))
        & finite1152437895449049373et_nat(collect_set_nat(Q)) ) ) ).

% finite_Collect_disjI
tff(fact_5137_finite__Collect__disjI,axiom,
    ! [P: fun_nat_o,Q: fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_ai(P),Q)))
    <=> ( aa_set_nat_o(finite_finite_nat,collect_nat(P))
        & aa_set_nat_o(finite_finite_nat,collect_nat(Q)) ) ) ).

% finite_Collect_disjI
tff(fact_5138_finite__Collect__disjI,axiom,
    ! [P: fun_int_o,Q: fun_int_o] :
      ( aa_set_int_o(finite_finite_int,collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_aj(P),Q)))
    <=> ( aa_set_int_o(finite_finite_int,collect_int(P))
        & aa_set_int_o(finite_finite_int,collect_int(Q)) ) ) ).

% finite_Collect_disjI
tff(fact_5139_finite__Collect__disjI,axiom,
    ! [P: fun_complex_o,Q: fun_complex_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun2514317545224092796plex_o(aTP_Lamm_ak(P),Q)))
    <=> ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(P))
        & aa_set_complex_o(finite3207457112153483333omplex,collect_complex(Q)) ) ) ).

% finite_Collect_disjI
tff(fact_5140_finite__Collect__conjI,axiom,
    ! [P: fun_real_o,Q: fun_real_o] :
      ( ( finite_finite_real(collect_real(P))
        | finite_finite_real(collect_real(Q)) )
     => finite_finite_real(collect_real(aa_fun5868358341814070780real_o(aTP_Lamm_al(P),Q))) ) ).

% finite_Collect_conjI
tff(fact_5141_finite__Collect__conjI,axiom,
    ! [P: fun_list_nat_o,Q: fun_list_nat_o] :
      ( ( finite8100373058378681591st_nat(collect_list_nat(P))
        | finite8100373058378681591st_nat(collect_list_nat(Q)) )
     => finite8100373058378681591st_nat(collect_list_nat(aa_fun6155608546934989052_nat_o(aTP_Lamm_am(P),Q))) ) ).

% finite_Collect_conjI
tff(fact_5142_finite__Collect__conjI,axiom,
    ! [P: fun_set_nat_o,Q: fun_set_nat_o] :
      ( ( finite1152437895449049373et_nat(collect_set_nat(P))
        | finite1152437895449049373et_nat(collect_set_nat(Q)) )
     => finite1152437895449049373et_nat(collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_an(P),Q))) ) ).

% finite_Collect_conjI
tff(fact_5143_finite__Collect__conjI,axiom,
    ! [P: fun_nat_o,Q: fun_nat_o] :
      ( ( aa_set_nat_o(finite_finite_nat,collect_nat(P))
        | aa_set_nat_o(finite_finite_nat,collect_nat(Q)) )
     => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_ao(P),Q))) ) ).

% finite_Collect_conjI
tff(fact_5144_finite__Collect__conjI,axiom,
    ! [P: fun_int_o,Q: fun_int_o] :
      ( ( aa_set_int_o(finite_finite_int,collect_int(P))
        | aa_set_int_o(finite_finite_int,collect_int(Q)) )
     => aa_set_int_o(finite_finite_int,collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_ap(P),Q))) ) ).

% finite_Collect_conjI
tff(fact_5145_finite__Collect__conjI,axiom,
    ! [P: fun_complex_o,Q: fun_complex_o] :
      ( ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(P))
        | aa_set_complex_o(finite3207457112153483333omplex,collect_complex(Q)) )
     => aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun2514317545224092796plex_o(aTP_Lamm_aq(P),Q))) ) ).

% finite_Collect_conjI
tff(fact_5146_bits__mod__0,axiom,
    ! [Aa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,zero_zero_nat),Aa) = zero_zero_nat ).

% bits_mod_0
tff(fact_5147_bits__mod__0,axiom,
    ! [Aa: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,zero_zero_int),Aa) = zero_zero_int ).

% bits_mod_0
tff(fact_5148_bits__mod__0,axiom,
    ! [Aa: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,zero_z3403309356797280102nteger),Aa) = zero_z3403309356797280102nteger ).

% bits_mod_0
tff(fact_5149_mod__self,axiom,
    ! [Aa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Aa),Aa) = zero_zero_nat ).

% mod_self
tff(fact_5150_mod__self,axiom,
    ! [Aa: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),Aa) = zero_zero_int ).

% mod_self
tff(fact_5151_mod__self,axiom,
    ! [Aa: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Aa),Aa) = zero_z3403309356797280102nteger ).

% mod_self
tff(fact_5152_mod__by__0,axiom,
    ! [Aa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Aa),zero_zero_nat) = Aa ).

% mod_by_0
tff(fact_5153_mod__by__0,axiom,
    ! [Aa: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),zero_zero_int) = Aa ).

% mod_by_0
tff(fact_5154_mod__by__0,axiom,
    ! [Aa: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Aa),zero_z3403309356797280102nteger) = Aa ).

% mod_by_0
tff(fact_5155_mod__0,axiom,
    ! [Aa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,zero_zero_nat),Aa) = zero_zero_nat ).

% mod_0
tff(fact_5156_mod__0,axiom,
    ! [Aa: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,zero_zero_int),Aa) = zero_zero_int ).

% mod_0
tff(fact_5157_mod__0,axiom,
    ! [Aa: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,zero_z3403309356797280102nteger),Aa) = zero_z3403309356797280102nteger ).

% mod_0
tff(fact_5158_mod__add__self1,axiom,
    ! [Ba: nat,Aa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ba),Aa)),Ba) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Aa),Ba) ).

% mod_add_self1
tff(fact_5159_mod__add__self1,axiom,
    ! [Ba: int,Aa: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Ba),Aa)),Ba) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),Ba) ).

% mod_add_self1
tff(fact_5160_mod__add__self1,axiom,
    ! [Ba: code_integer,Aa: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Ba),Aa)),Ba) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Aa),Ba) ).

% mod_add_self1
tff(fact_5161_mod__add__self2,axiom,
    ! [Aa: nat,Ba: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),Ba)),Ba) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Aa),Ba) ).

% mod_add_self2
tff(fact_5162_mod__add__self2,axiom,
    ! [Aa: int,Ba: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),Ba)),Ba) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),Ba) ).

% mod_add_self2
tff(fact_5163_mod__add__self2,axiom,
    ! [Aa: code_integer,Ba: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Aa),Ba)),Ba) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Aa),Ba) ).

% mod_add_self2
tff(fact_5164_minus__mod__self2,axiom,
    ! [Aa: int,Ba: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Aa),Ba)),Ba) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),Ba) ).

% minus_mod_self2
tff(fact_5165_minus__mod__self2,axiom,
    ! [Aa: code_integer,Ba: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,Aa),Ba)),Ba) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Aa),Ba) ).

% minus_mod_self2
tff(fact_5166_mod__minus__minus,axiom,
    ! [Aa: int,Ba: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,Aa)),aa_int_int(uminus_uminus_int,Ba)) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),Ba)) ).

% mod_minus_minus
tff(fact_5167_mod__minus__minus,axiom,
    ! [Aa: code_integer,Ba: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Ba)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Aa),Ba)) ).

% mod_minus_minus
tff(fact_5168_mod__less,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Na) = M ) ) ).

% mod_less
tff(fact_5169_finite__Collect__subsets,axiom,
    ! [A3: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => finite1152437895449049373et_nat(collect_set_nat(aa_set7861427066675100071_nat_o(aTP_Lamm_ar,A3))) ) ).

% finite_Collect_subsets
tff(fact_5170_finite__Collect__subsets,axiom,
    ! [A3: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A3)
     => finite6551019134538273531omplex(collect_set_complex(aTP_Lamm_as(A3))) ) ).

% finite_Collect_subsets
tff(fact_5171_finite__Collect__subsets,axiom,
    ! [A3: set_int] :
      ( aa_set_int_o(finite_finite_int,A3)
     => finite6197958912794628473et_int(collect_set_int(aTP_Lamm_at(A3))) ) ).

% finite_Collect_subsets
tff(fact_5172_finite__Collect__less__nat,axiom,
    ! [K: nat] : aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_au,K))) ).

% finite_Collect_less_nat
tff(fact_5173_finite__Collect__le__nat,axiom,
    ! [K: nat] : aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_av,K))) ).

% finite_Collect_le_nat
tff(fact_5174_finite__interval__int1,axiom,
    ! [Aa: int,Ba: int] : aa_set_int_o(finite_finite_int,collect_int(aa_int_fun_int_o(aTP_Lamm_aw(Aa),Ba))) ).

% finite_interval_int1
tff(fact_5175_finite__interval__int4,axiom,
    ! [Aa: int,Ba: int] : aa_set_int_o(finite_finite_int,collect_int(aa_int_fun_int_o(aTP_Lamm_ax(Aa),Ba))) ).

% finite_interval_int4
tff(fact_5176_mod__mult__self2__is__0,axiom,
    ! [Aa: nat,Ba: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),Ba)),Ba) = zero_zero_nat ).

% mod_mult_self2_is_0
tff(fact_5177_mod__mult__self2__is__0,axiom,
    ! [Aa: int,Ba: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,Aa),Ba)),Ba) = zero_zero_int ).

% mod_mult_self2_is_0
tff(fact_5178_mod__mult__self2__is__0,axiom,
    ! [Aa: code_integer,Ba: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Aa),Ba)),Ba) = zero_z3403309356797280102nteger ).

% mod_mult_self2_is_0
tff(fact_5179_mod__mult__self1__is__0,axiom,
    ! [Ba: nat,Aa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ba),Aa)),Ba) = zero_zero_nat ).

% mod_mult_self1_is_0
tff(fact_5180_mod__mult__self1__is__0,axiom,
    ! [Ba: int,Aa: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,Ba),Aa)),Ba) = zero_zero_int ).

% mod_mult_self1_is_0
tff(fact_5181_mod__mult__self1__is__0,axiom,
    ! [Ba: code_integer,Aa: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Ba),Aa)),Ba) = zero_z3403309356797280102nteger ).

% mod_mult_self1_is_0
tff(fact_5182_bits__mod__by__1,axiom,
    ! [Aa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Aa),one_one_nat) = zero_zero_nat ).

% bits_mod_by_1
tff(fact_5183_bits__mod__by__1,axiom,
    ! [Aa: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),one_one_int) = zero_zero_int ).

% bits_mod_by_1
tff(fact_5184_bits__mod__by__1,axiom,
    ! [Aa: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Aa),one_one_Code_integer) = zero_z3403309356797280102nteger ).

% bits_mod_by_1
tff(fact_5185_mod__by__1,axiom,
    ! [Aa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Aa),one_one_nat) = zero_zero_nat ).

% mod_by_1
tff(fact_5186_mod__by__1,axiom,
    ! [Aa: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),one_one_int) = zero_zero_int ).

% mod_by_1
tff(fact_5187_mod__by__1,axiom,
    ! [Aa: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Aa),one_one_Code_integer) = zero_z3403309356797280102nteger ).

% mod_by_1
tff(fact_5188_mod__div__trivial,axiom,
    ! [Aa: nat,Ba: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Aa),Ba)),Ba) = zero_zero_nat ).

% mod_div_trivial
tff(fact_5189_mod__div__trivial,axiom,
    ! [Aa: int,Ba: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),Ba)),Ba) = zero_zero_int ).

% mod_div_trivial
tff(fact_5190_mod__div__trivial,axiom,
    ! [Aa: code_integer,Ba: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Aa),Ba)),Ba) = zero_z3403309356797280102nteger ).

% mod_div_trivial
tff(fact_5191_bits__mod__div__trivial,axiom,
    ! [Aa: nat,Ba: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Aa),Ba)),Ba) = zero_zero_nat ).

% bits_mod_div_trivial
tff(fact_5192_bits__mod__div__trivial,axiom,
    ! [Aa: int,Ba: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),Ba)),Ba) = zero_zero_int ).

% bits_mod_div_trivial
tff(fact_5193_bits__mod__div__trivial,axiom,
    ! [Aa: code_integer,Ba: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Aa),Ba)),Ba) = zero_z3403309356797280102nteger ).

% bits_mod_div_trivial
tff(fact_5194_mod__mult__self1,axiom,
    ! [Aa: nat,C: nat,Ba: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),Ba))),Ba) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Aa),Ba) ).

% mod_mult_self1
tff(fact_5195_mod__mult__self1,axiom,
    ! [Aa: int,C: int,Ba: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),aa_int_int(aa_int_fun_int_int(times_times_int,C),Ba))),Ba) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),Ba) ).

% mod_mult_self1
tff(fact_5196_mod__mult__self1,axiom,
    ! [Aa: code_integer,C: code_integer,Ba: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Aa),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),Ba))),Ba) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Aa),Ba) ).

% mod_mult_self1
tff(fact_5197_mod__mult__self2,axiom,
    ! [Aa: nat,Ba: 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,Aa),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ba),C))),Ba) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Aa),Ba) ).

% mod_mult_self2
tff(fact_5198_mod__mult__self2,axiom,
    ! [Aa: int,Ba: 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,Aa),aa_int_int(aa_int_fun_int_int(times_times_int,Ba),C))),Ba) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),Ba) ).

% mod_mult_self2
tff(fact_5199_mod__mult__self2,axiom,
    ! [Aa: code_integer,Ba: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Aa),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Ba),C))),Ba) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Aa),Ba) ).

% mod_mult_self2
tff(fact_5200_mod__mult__self3,axiom,
    ! [C: nat,Ba: nat,Aa: 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),Ba)),Aa)),Ba) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Aa),Ba) ).

% mod_mult_self3
tff(fact_5201_mod__mult__self3,axiom,
    ! [C: int,Ba: int,Aa: 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),Ba)),Aa)),Ba) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),Ba) ).

% mod_mult_self3
tff(fact_5202_mod__mult__self3,axiom,
    ! [C: code_integer,Ba: code_integer,Aa: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),Ba)),Aa)),Ba) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Aa),Ba) ).

% mod_mult_self3
tff(fact_5203_mod__mult__self4,axiom,
    ! [Ba: nat,C: nat,Aa: 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,Ba),C)),Aa)),Ba) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Aa),Ba) ).

% mod_mult_self4
tff(fact_5204_mod__mult__self4,axiom,
    ! [Ba: int,C: int,Aa: 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,Ba),C)),Aa)),Ba) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),Ba) ).

% mod_mult_self4
tff(fact_5205_mod__mult__self4,axiom,
    ! [Ba: code_integer,C: code_integer,Aa: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Ba),C)),Aa)),Ba) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Aa),Ba) ).

% mod_mult_self4
tff(fact_5206_dvd__imp__mod__0,axiom,
    ! [Aa: nat,Ba: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),Ba)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Ba),Aa) = zero_zero_nat ) ) ).

% dvd_imp_mod_0
tff(fact_5207_dvd__imp__mod__0,axiom,
    ! [Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),Ba)
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Ba),Aa) = zero_zero_int ) ) ).

% dvd_imp_mod_0
tff(fact_5208_dvd__imp__mod__0,axiom,
    ! [Aa: code_integer,Ba: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),Ba)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Ba),Aa) = zero_z3403309356797280102nteger ) ) ).

% dvd_imp_mod_0
tff(fact_5209_minus__mod__self1,axiom,
    ! [Ba: int,Aa: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Ba),Aa)),Ba) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,Aa)),Ba) ).

% minus_mod_self1
tff(fact_5210_minus__mod__self1,axiom,
    ! [Ba: code_integer,Aa: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,Ba),Aa)),Ba) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa)),Ba) ).

% minus_mod_self1
tff(fact_5211_mod__by__Suc__0,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_nat_nat(suc,zero_zero_nat)) = zero_zero_nat ).

% mod_by_Suc_0
tff(fact_5212_sin__coeff__0,axiom,
    aa_nat_real(sin_coeff,zero_zero_nat) = zero_zero_real ).

% sin_coeff_0
tff(fact_5213_finite__interval__int3,axiom,
    ! [Aa: int,Ba: int] : aa_set_int_o(finite_finite_int,collect_int(aa_int_fun_int_o(aTP_Lamm_ay(Aa),Ba))) ).

% finite_interval_int3
tff(fact_5214_finite__interval__int2,axiom,
    ! [Aa: int,Ba: int] : aa_set_int_o(finite_finite_int,collect_int(aa_int_fun_int_o(aTP_Lamm_az(Aa),Ba))) ).

% finite_interval_int2
tff(fact_5215_mod__minus1__right,axiom,
    ! [Aa: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),aa_int_int(uminus_uminus_int,one_one_int)) = zero_zero_int ).

% mod_minus1_right
tff(fact_5216_mod__minus1__right,axiom,
    ! [Aa: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Aa),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = zero_z3403309356797280102nteger ).

% mod_minus1_right
tff(fact_5217_Suc__mod__mult__self4,axiom,
    ! [Na: nat,K: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),K)),M))),Na) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),Na) ).

% Suc_mod_mult_self4
tff(fact_5218_Suc__mod__mult__self3,axiom,
    ! [K: nat,Na: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Na)),M))),Na) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),Na) ).

% Suc_mod_mult_self3
tff(fact_5219_Suc__mod__mult__self2,axiom,
    ! [M: nat,Na: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),K)))),Na) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),Na) ).

% Suc_mod_mult_self2
tff(fact_5220_Suc__mod__mult__self1,axiom,
    ! [M: nat,K: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Na)))),Na) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),Na) ).

% Suc_mod_mult_self1
tff(fact_5221_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_5222_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_5223_bits__one__mod__two__eq__one,axiom,
    aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,one_one_Code_integer),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = one_one_Code_integer ).

% bits_one_mod_two_eq_one
tff(fact_5224_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_5225_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_5226_one__mod__two__eq__one,axiom,
    aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,one_one_Code_integer),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = one_one_Code_integer ).

% one_mod_two_eq_one
tff(fact_5227_mod2__Suc__Suc,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,M))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% mod2_Suc_Suc
tff(fact_5228_Suc__times__numeral__mod__eq,axiom,
    ! [K: num,Na: nat] :
      ( ( aa_num_nat(numeral_numeral_nat,K) != one_one_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,K)),Na))),aa_num_nat(numeral_numeral_nat,K)) = one_one_nat ) ) ).

% Suc_times_numeral_mod_eq
tff(fact_5229_not__mod__2__eq__0__eq__1,axiom,
    ! [Aa: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Aa),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,Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_nat ) ) ).

% not_mod_2_eq_0_eq_1
tff(fact_5230_not__mod__2__eq__0__eq__1,axiom,
    ! [Aa: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),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,Aa),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int ) ) ).

% not_mod_2_eq_0_eq_1
tff(fact_5231_not__mod__2__eq__0__eq__1,axiom,
    ! [Aa: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Aa),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) != zero_z3403309356797280102nteger )
    <=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Aa),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = one_one_Code_integer ) ) ).

% not_mod_2_eq_0_eq_1
tff(fact_5232_not__mod__2__eq__1__eq__0,axiom,
    ! [Aa: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Aa),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,Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ) ) ).

% not_mod_2_eq_1_eq_0
tff(fact_5233_not__mod__2__eq__1__eq__0,axiom,
    ! [Aa: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),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,Aa),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int ) ) ).

% not_mod_2_eq_1_eq_0
tff(fact_5234_not__mod__2__eq__1__eq__0,axiom,
    ! [Aa: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Aa),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) != one_one_Code_integer )
    <=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Aa),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = zero_z3403309356797280102nteger ) ) ).

% not_mod_2_eq_1_eq_0
tff(fact_5235_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_5236_bits__minus__1__mod__2__eq,axiom,
    aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = one_one_Code_integer ).

% bits_minus_1_mod_2_eq
tff(fact_5237_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_5238_minus__1__mod__2__eq,axiom,
    aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = one_one_Code_integer ).

% minus_1_mod_2_eq
tff(fact_5239_not__mod2__eq__Suc__0__eq__0,axiom,
    ! [Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) != aa_nat_nat(suc,zero_zero_nat) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ) ) ).

% not_mod2_eq_Suc_0_eq_0
tff(fact_5240_add__self__mod__2,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),M)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ).

% add_self_mod_2
tff(fact_5241_mod2__gr__0,axiom,
    ! [M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_nat ) ) ).

% mod2_gr_0
tff(fact_5242_even__succ__mod__exp,axiom,
    ! [Aa: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Aa)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),Aa)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))) ) ) ) ).

% even_succ_mod_exp
tff(fact_5243_even__succ__mod__exp,axiom,
    ! [Aa: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Aa)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => ( 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)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na))) ) ) ) ).

% even_succ_mod_exp
tff(fact_5244_even__succ__mod__exp,axiom,
    ! [Aa: code_integer,Na: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Aa)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,one_one_Code_integer),Aa)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Na)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Aa),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Na))) ) ) ) ).

% even_succ_mod_exp
tff(fact_5245_pigeonhole__infinite__rel,axiom,
    ! [A3: set_real,B4: set_nat,R: fun_real_fun_nat_o] :
      ( ~ finite_finite_real(A3)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( ! [X: real] :
              ( member_real(X,A3)
             => ? [Xa: nat] :
                  ( aa_set_nat_o(member_nat(Xa),B4)
                  & aa_nat_o(aa_real_fun_nat_o(R,X),Xa) ) )
         => ? [X: nat] :
              ( aa_set_nat_o(member_nat(X),B4)
              & ~ finite_finite_real(collect_real(aa_nat_fun_real_o(aa_fun6917832400337718588real_o(aTP_Lamm_ba(A3),R),X))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_5246_pigeonhole__infinite__rel,axiom,
    ! [A3: set_real,B4: set_int,R: fun_real_fun_int_o] :
      ( ~ finite_finite_real(A3)
     => ( aa_set_int_o(finite_finite_int,B4)
       => ( ! [X: real] :
              ( member_real(X,A3)
             => ? [Xa: int] :
                  ( aa_set_int_o(member_int(Xa),B4)
                  & aa_int_o(aa_real_fun_int_o(R,X),Xa) ) )
         => ? [X: int] :
              ( aa_set_int_o(member_int(X),B4)
              & ~ finite_finite_real(collect_real(aa_int_fun_real_o(aa_fun2391495421969208892real_o(aTP_Lamm_bb(A3),R),X))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_5247_pigeonhole__infinite__rel,axiom,
    ! [A3: set_real,B4: set_complex,R: fun_re7319081385389557872plex_o] :
      ( ~ finite_finite_real(A3)
     => ( aa_set_complex_o(finite3207457112153483333omplex,B4)
       => ( ! [X: real] :
              ( member_real(X,A3)
             => ? [Xa: complex] :
                  ( aa_set_complex_o(member_complex(Xa),B4)
                  & aa_complex_o(aa_rea2382570592068293209plex_o(R,X),Xa) ) )
         => ? [X: complex] :
              ( aa_set_complex_o(member_complex(X),B4)
              & ~ finite_finite_real(collect_real(aa_com1951324360330529881real_o(aa_fun7692540709613205436real_o(aTP_Lamm_bc(A3),R),X))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_5248_pigeonhole__infinite__rel,axiom,
    ! [A3: set_nat,B4: set_nat,R: fun_nat_fun_nat_o] :
      ( ~ aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( ! [X: nat] :
              ( aa_set_nat_o(member_nat(X),A3)
             => ? [Xa: nat] :
                  ( aa_set_nat_o(member_nat(Xa),B4)
                  & aa_nat_o(aa_nat_fun_nat_o(R,X),Xa) ) )
         => ? [X: nat] :
              ( aa_set_nat_o(member_nat(X),B4)
              & ~ aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aa_fun6699960423209655484_nat_o(aTP_Lamm_bd(A3),R),X))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_5249_pigeonhole__infinite__rel,axiom,
    ! [A3: set_nat,B4: set_int,R: fun_nat_fun_int_o] :
      ( ~ aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_int_o(finite_finite_int,B4)
       => ( ! [X: nat] :
              ( aa_set_nat_o(member_nat(X),A3)
             => ? [Xa: int] :
                  ( aa_set_int_o(member_int(Xa),B4)
                  & aa_int_o(aa_nat_fun_int_o(R,X),Xa) ) )
         => ? [X: int] :
              ( aa_set_int_o(member_int(X),B4)
              & ~ aa_set_nat_o(finite_finite_nat,collect_nat(aa_int_fun_nat_o(aa_fun7393466555336238524_nat_o(aTP_Lamm_be(A3),R),X))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_5250_pigeonhole__infinite__rel,axiom,
    ! [A3: set_nat,B4: set_complex,R: fun_na1554462459066888140plex_o] :
      ( ~ aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_complex_o(finite3207457112153483333omplex,B4)
       => ( ! [X: nat] :
              ( aa_set_nat_o(member_nat(X),A3)
             => ? [Xa: complex] :
                  ( aa_set_complex_o(member_complex(Xa),B4)
                  & aa_complex_o(aa_nat_fun_complex_o(R,X),Xa) ) )
         => ? [X: complex] :
              ( aa_set_complex_o(member_complex(X),B4)
              & ~ aa_set_nat_o(finite_finite_nat,collect_nat(aa_complex_fun_nat_o(aa_fun1708641150486296380_nat_o(aTP_Lamm_bf(A3),R),X))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_5251_pigeonhole__infinite__rel,axiom,
    ! [A3: set_int,B4: set_nat,R: fun_int_fun_nat_o] :
      ( ~ aa_set_int_o(finite_finite_int,A3)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( ! [X: int] :
              ( aa_set_int_o(member_int(X),A3)
             => ? [Xa: nat] :
                  ( aa_set_nat_o(member_nat(Xa),B4)
                  & aa_nat_o(aa_int_fun_nat_o(R,X),Xa) ) )
         => ? [X: nat] :
              ( aa_set_nat_o(member_nat(X),B4)
              & ~ aa_set_int_o(finite_finite_int,collect_int(aa_nat_fun_int_o(aa_fun5439836533389078972_int_o(aTP_Lamm_bg(A3),R),X))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_5252_pigeonhole__infinite__rel,axiom,
    ! [A3: set_int,B4: set_int,R: fun_int_fun_int_o] :
      ( ~ aa_set_int_o(finite_finite_int,A3)
     => ( aa_set_int_o(finite_finite_int,B4)
       => ( ! [X: int] :
              ( aa_set_int_o(member_int(X),A3)
             => ? [Xa: int] :
                  ( aa_set_int_o(member_int(Xa),B4)
                  & aa_int_o(aa_int_fun_int_o(R,X),Xa) ) )
         => ? [X: int] :
              ( aa_set_int_o(member_int(X),B4)
              & ~ aa_set_int_o(finite_finite_int,collect_int(aa_int_fun_int_o(aa_fun6133342665515662012_int_o(aTP_Lamm_bh(A3),R),X))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_5253_pigeonhole__infinite__rel,axiom,
    ! [A3: set_int,B4: set_complex,R: fun_in5065666251602127472plex_o] :
      ( ~ aa_set_int_o(finite_finite_int,A3)
     => ( aa_set_complex_o(finite3207457112153483333omplex,B4)
       => ( ! [X: int] :
              ( aa_set_int_o(member_int(X),A3)
             => ? [Xa: complex] :
                  ( aa_set_complex_o(member_complex(Xa),B4)
                  & aa_complex_o(aa_int_fun_complex_o(R,X),Xa) ) )
         => ? [X: complex] :
              ( aa_set_complex_o(member_complex(X),B4)
              & ~ aa_set_int_o(finite_finite_int,collect_int(aa_complex_fun_int_o(aa_fun7455058932602313788_int_o(aTP_Lamm_bi(A3),R),X))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_5254_pigeonhole__infinite__rel,axiom,
    ! [A3: set_complex,B4: set_nat,R: fun_co5676208753217392844_nat_o] :
      ( ~ aa_set_complex_o(finite3207457112153483333omplex,A3)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( ! [X: complex] :
              ( aa_set_complex_o(member_complex(X),A3)
             => ? [Xa: nat] :
                  ( aa_set_nat_o(member_nat(Xa),B4)
                  & aa_nat_o(aa_complex_fun_nat_o(R,X),Xa) ) )
         => ? [X: nat] :
              ( aa_set_nat_o(member_nat(X),B4)
              & ~ aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_nat_fun_complex_o(aa_fun4610599729497257276plex_o(aTP_Lamm_bj(A3),R),X))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_5255_not__finite__existsD,axiom,
    ! [P: fun_real_o] :
      ( ~ finite_finite_real(collect_real(P))
     => ? [X_1: real] : aa_real_o(P,X_1) ) ).

% not_finite_existsD
tff(fact_5256_not__finite__existsD,axiom,
    ! [P: fun_list_nat_o] :
      ( ~ finite8100373058378681591st_nat(collect_list_nat(P))
     => ? [X_1: list_nat] : aa_list_nat_o(P,X_1) ) ).

% not_finite_existsD
tff(fact_5257_not__finite__existsD,axiom,
    ! [P: fun_set_nat_o] :
      ( ~ finite1152437895449049373et_nat(collect_set_nat(P))
     => ? [X_1: set_nat] : aa_set_nat_o(P,X_1) ) ).

% not_finite_existsD
tff(fact_5258_not__finite__existsD,axiom,
    ! [P: fun_nat_o] :
      ( ~ aa_set_nat_o(finite_finite_nat,collect_nat(P))
     => ? [X_1: nat] : aa_nat_o(P,X_1) ) ).

% not_finite_existsD
tff(fact_5259_not__finite__existsD,axiom,
    ! [P: fun_int_o] :
      ( ~ aa_set_int_o(finite_finite_int,collect_int(P))
     => ? [X_1: int] : aa_int_o(P,X_1) ) ).

% not_finite_existsD
tff(fact_5260_not__finite__existsD,axiom,
    ! [P: fun_complex_o] :
      ( ~ aa_set_complex_o(finite3207457112153483333omplex,collect_complex(P))
     => ? [X_1: complex] : aa_complex_o(P,X_1) ) ).

% not_finite_existsD
tff(fact_5261_less__set__def,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ord_le7866589430770878221at_nat(A3,B4)
    <=> ord_le9073604597836486254_nat_o(aTP_Lamm_a(A3),aTP_Lamm_a(B4)) ) ).

% less_set_def
tff(fact_5262_less__set__def,axiom,
    ! [A3: set_real,B4: set_real] :
      ( ord_less_set_real(A3,B4)
    <=> ord_less_fun_real_o(aTP_Lamm_aa(A3),aTP_Lamm_aa(B4)) ) ).

% less_set_def
tff(fact_5263_less__set__def,axiom,
    ! [A3: set_set_nat,B4: set_set_nat] :
      ( ord_less_set_set_nat(A3,B4)
    <=> ord_le921202756137246487_nat_o(aTP_Lamm_ac(A3),aTP_Lamm_ac(B4)) ) ).

% less_set_def
tff(fact_5264_less__set__def,axiom,
    ! [A3: set_nat,B4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A3),B4)
    <=> ord_less_fun_nat_o(aTP_Lamm_ad(A3),aTP_Lamm_ad(B4)) ) ).

% less_set_def
tff(fact_5265_less__set__def,axiom,
    ! [A3: set_int,B4: set_int] :
      ( aa_set_int_o(ord_less_set_int(A3),B4)
    <=> ord_less_fun_int_o(aTP_Lamm_ae(A3),aTP_Lamm_ae(B4)) ) ).

% less_set_def
tff(fact_5266_subset__Collect__iff,axiom,
    ! [B4: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat,P: fun_Pr4796943543331071514_nat_o] :
      ( ord_le3146513528884898305at_nat(B4,A3)
     => ( ord_le3146513528884898305at_nat(B4,collec3392354462482085612at_nat(aa_fun359341089595923442_nat_o(aTP_Lamm_bk(A3),P)))
      <=> ! [X4: product_prod_nat_nat] :
            ( member8440522571783428010at_nat(X4,B4)
           => aa_Pro6150637604445964947_nat_o(P,X4) ) ) ) ).

% subset_Collect_iff
tff(fact_5267_subset__Collect__iff,axiom,
    ! [B4: set_real,A3: set_real,P: fun_real_o] :
      ( ord_less_eq_set_real(B4,A3)
     => ( ord_less_eq_set_real(B4,collect_real(aa_fun5868358341814070780real_o(aTP_Lamm_bl(A3),P)))
      <=> ! [X4: real] :
            ( member_real(X4,B4)
           => aa_real_o(P,X4) ) ) ) ).

% subset_Collect_iff
tff(fact_5268_subset__Collect__iff,axiom,
    ! [B4: set_list_nat,A3: set_list_nat,P: fun_list_nat_o] :
      ( ord_le6045566169113846134st_nat(B4,A3)
     => ( ord_le6045566169113846134st_nat(B4,collect_list_nat(aa_fun6155608546934989052_nat_o(aTP_Lamm_bm(A3),P)))
      <=> ! [X4: list_nat] :
            ( member_list_nat(X4,B4)
           => aa_list_nat_o(P,X4) ) ) ) ).

% subset_Collect_iff
tff(fact_5269_subset__Collect__iff,axiom,
    ! [B4: set_set_nat,A3: set_set_nat,P: fun_set_nat_o] :
      ( ord_le6893508408891458716et_nat(B4,A3)
     => ( ord_le6893508408891458716et_nat(B4,collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_bn(A3),P)))
      <=> ! [X4: set_nat] :
            ( member_set_nat(X4,B4)
           => aa_set_nat_o(P,X4) ) ) ) ).

% subset_Collect_iff
tff(fact_5270_subset__Collect__iff,axiom,
    ! [B4: set_nat,A3: set_nat,P: fun_nat_o] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),A3)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_bo(A3),P)))
      <=> ! [X4: nat] :
            ( aa_set_nat_o(member_nat(X4),B4)
           => aa_nat_o(P,X4) ) ) ) ).

% subset_Collect_iff
tff(fact_5271_subset__Collect__iff,axiom,
    ! [B4: set_int,A3: set_int,P: fun_int_o] :
      ( aa_set_int_o(ord_less_eq_set_int(B4),A3)
     => ( aa_set_int_o(ord_less_eq_set_int(B4),collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_bp(A3),P)))
      <=> ! [X4: int] :
            ( aa_set_int_o(member_int(X4),B4)
           => aa_int_o(P,X4) ) ) ) ).

% subset_Collect_iff
tff(fact_5272_subset__CollectI,axiom,
    ! [B4: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat,Q: fun_Pr4796943543331071514_nat_o,P: fun_Pr4796943543331071514_nat_o] :
      ( ord_le3146513528884898305at_nat(B4,A3)
     => ( ! [X: product_prod_nat_nat] :
            ( member8440522571783428010at_nat(X,B4)
           => ( aa_Pro6150637604445964947_nat_o(Q,X)
             => aa_Pro6150637604445964947_nat_o(P,X) ) )
       => ord_le3146513528884898305at_nat(collec3392354462482085612at_nat(aa_fun359341089595923442_nat_o(aTP_Lamm_bk(B4),Q)),collec3392354462482085612at_nat(aa_fun359341089595923442_nat_o(aTP_Lamm_bk(A3),P))) ) ) ).

% subset_CollectI
tff(fact_5273_subset__CollectI,axiom,
    ! [B4: set_real,A3: set_real,Q: fun_real_o,P: fun_real_o] :
      ( ord_less_eq_set_real(B4,A3)
     => ( ! [X: real] :
            ( member_real(X,B4)
           => ( aa_real_o(Q,X)
             => aa_real_o(P,X) ) )
       => ord_less_eq_set_real(collect_real(aa_fun5868358341814070780real_o(aTP_Lamm_bl(B4),Q)),collect_real(aa_fun5868358341814070780real_o(aTP_Lamm_bl(A3),P))) ) ) ).

% subset_CollectI
tff(fact_5274_subset__CollectI,axiom,
    ! [B4: set_list_nat,A3: set_list_nat,Q: fun_list_nat_o,P: fun_list_nat_o] :
      ( ord_le6045566169113846134st_nat(B4,A3)
     => ( ! [X: list_nat] :
            ( member_list_nat(X,B4)
           => ( aa_list_nat_o(Q,X)
             => aa_list_nat_o(P,X) ) )
       => ord_le6045566169113846134st_nat(collect_list_nat(aa_fun6155608546934989052_nat_o(aTP_Lamm_bm(B4),Q)),collect_list_nat(aa_fun6155608546934989052_nat_o(aTP_Lamm_bm(A3),P))) ) ) ).

% subset_CollectI
tff(fact_5275_subset__CollectI,axiom,
    ! [B4: set_set_nat,A3: set_set_nat,Q: fun_set_nat_o,P: fun_set_nat_o] :
      ( ord_le6893508408891458716et_nat(B4,A3)
     => ( ! [X: set_nat] :
            ( member_set_nat(X,B4)
           => ( aa_set_nat_o(Q,X)
             => aa_set_nat_o(P,X) ) )
       => ord_le6893508408891458716et_nat(collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_bn(B4),Q)),collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_bn(A3),P))) ) ) ).

% subset_CollectI
tff(fact_5276_subset__CollectI,axiom,
    ! [B4: set_nat,A3: set_nat,Q: fun_nat_o,P: fun_nat_o] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),A3)
     => ( ! [X: nat] :
            ( aa_set_nat_o(member_nat(X),B4)
           => ( aa_nat_o(Q,X)
             => aa_nat_o(P,X) ) )
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_bo(B4),Q))),collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_bo(A3),P))) ) ) ).

% subset_CollectI
tff(fact_5277_subset__CollectI,axiom,
    ! [B4: set_int,A3: set_int,Q: fun_int_o,P: fun_int_o] :
      ( aa_set_int_o(ord_less_eq_set_int(B4),A3)
     => ( ! [X: int] :
            ( aa_set_int_o(member_int(X),B4)
           => ( aa_int_o(Q,X)
             => aa_int_o(P,X) ) )
       => aa_set_int_o(ord_less_eq_set_int(collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_bp(B4),Q))),collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_bp(A3),P))) ) ) ).

% subset_CollectI
tff(fact_5278_pred__subset__eq,axiom,
    ! [R: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
      ( ord_le1572679713340831354_nat_o(aTP_Lamm_a(R),aTP_Lamm_a(S2))
    <=> ord_le3146513528884898305at_nat(R,S2) ) ).

% pred_subset_eq
tff(fact_5279_pred__subset__eq,axiom,
    ! [R: set_real,S2: set_real] :
      ( ord_le3605727316730466277real_o(aTP_Lamm_aa(R),aTP_Lamm_aa(S2))
    <=> ord_less_eq_set_real(R,S2) ) ).

% pred_subset_eq
tff(fact_5280_pred__subset__eq,axiom,
    ! [R: set_set_nat,S2: set_set_nat] :
      ( ord_le6468474429079055883_nat_o(aTP_Lamm_ac(R),aTP_Lamm_ac(S2))
    <=> ord_le6893508408891458716et_nat(R,S2) ) ).

% pred_subset_eq
tff(fact_5281_pred__subset__eq,axiom,
    ! [R: set_nat,S2: set_nat] :
      ( ord_le7240017405068453697_nat_o(aTP_Lamm_ad(R),aTP_Lamm_ad(S2))
    <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,R),S2) ) ).

% pred_subset_eq
tff(fact_5282_pred__subset__eq,axiom,
    ! [R: set_int,S2: set_int] :
      ( ord_le5418975059635765733_int_o(aTP_Lamm_ae(R),aTP_Lamm_ae(S2))
    <=> aa_set_int_o(ord_less_eq_set_int(R),S2) ) ).

% pred_subset_eq
tff(fact_5283_Collect__subset,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,P: fun_Pr4796943543331071514_nat_o] : ord_le3146513528884898305at_nat(collec3392354462482085612at_nat(aa_fun359341089595923442_nat_o(aTP_Lamm_bk(A3),P)),A3) ).

% Collect_subset
tff(fact_5284_Collect__subset,axiom,
    ! [A3: set_real,P: fun_real_o] : ord_less_eq_set_real(collect_real(aa_fun5868358341814070780real_o(aTP_Lamm_bl(A3),P)),A3) ).

% Collect_subset
tff(fact_5285_Collect__subset,axiom,
    ! [A3: set_list_nat,P: fun_list_nat_o] : ord_le6045566169113846134st_nat(collect_list_nat(aa_fun6155608546934989052_nat_o(aTP_Lamm_bm(A3),P)),A3) ).

% Collect_subset
tff(fact_5286_Collect__subset,axiom,
    ! [A3: set_set_nat,P: fun_set_nat_o] : ord_le6893508408891458716et_nat(collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_bn(A3),P)),A3) ).

% Collect_subset
tff(fact_5287_Collect__subset,axiom,
    ! [A3: set_nat,P: fun_nat_o] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_bo(A3),P))),A3) ).

% Collect_subset
tff(fact_5288_Collect__subset,axiom,
    ! [A3: set_int,P: fun_int_o] : aa_set_int_o(ord_less_eq_set_int(collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_bp(A3),P))),A3) ).

% Collect_subset
tff(fact_5289_less__eq__set__def,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ord_le3146513528884898305at_nat(A3,B4)
    <=> ord_le1572679713340831354_nat_o(aTP_Lamm_a(A3),aTP_Lamm_a(B4)) ) ).

% less_eq_set_def
tff(fact_5290_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_5291_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_5292_less__eq__set__def,axiom,
    ! [A3: set_nat,B4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_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_5293_less__eq__set__def,axiom,
    ! [A3: set_int,B4: set_int] :
      ( aa_set_int_o(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_5294_Collect__restrict,axiom,
    ! [X5: set_Pr1261947904930325089at_nat,P: fun_Pr4796943543331071514_nat_o] : ord_le3146513528884898305at_nat(collec3392354462482085612at_nat(aa_fun359341089595923442_nat_o(aTP_Lamm_bk(X5),P)),X5) ).

% Collect_restrict
tff(fact_5295_Collect__restrict,axiom,
    ! [X5: set_real,P: fun_real_o] : ord_less_eq_set_real(collect_real(aa_fun5868358341814070780real_o(aTP_Lamm_bl(X5),P)),X5) ).

% Collect_restrict
tff(fact_5296_Collect__restrict,axiom,
    ! [X5: set_list_nat,P: fun_list_nat_o] : ord_le6045566169113846134st_nat(collect_list_nat(aa_fun6155608546934989052_nat_o(aTP_Lamm_bm(X5),P)),X5) ).

% Collect_restrict
tff(fact_5297_Collect__restrict,axiom,
    ! [X5: set_set_nat,P: fun_set_nat_o] : ord_le6893508408891458716et_nat(collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_bn(X5),P)),X5) ).

% Collect_restrict
tff(fact_5298_Collect__restrict,axiom,
    ! [X5: set_nat,P: fun_nat_o] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_bo(X5),P))),X5) ).

% Collect_restrict
tff(fact_5299_Collect__restrict,axiom,
    ! [X5: set_int,P: fun_int_o] : aa_set_int_o(ord_less_eq_set_int(collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_bp(X5),P))),X5) ).

% Collect_restrict
tff(fact_5300_prop__restrict,axiom,
    ! [Xb: product_prod_nat_nat,Z5: set_Pr1261947904930325089at_nat,X5: set_Pr1261947904930325089at_nat,P: fun_Pr4796943543331071514_nat_o] :
      ( member8440522571783428010at_nat(Xb,Z5)
     => ( ord_le3146513528884898305at_nat(Z5,collec3392354462482085612at_nat(aa_fun359341089595923442_nat_o(aTP_Lamm_bk(X5),P)))
       => aa_Pro6150637604445964947_nat_o(P,Xb) ) ) ).

% prop_restrict
tff(fact_5301_prop__restrict,axiom,
    ! [Xb: real,Z5: set_real,X5: set_real,P: fun_real_o] :
      ( member_real(Xb,Z5)
     => ( ord_less_eq_set_real(Z5,collect_real(aa_fun5868358341814070780real_o(aTP_Lamm_bl(X5),P)))
       => aa_real_o(P,Xb) ) ) ).

% prop_restrict
tff(fact_5302_prop__restrict,axiom,
    ! [Xb: list_nat,Z5: set_list_nat,X5: set_list_nat,P: fun_list_nat_o] :
      ( member_list_nat(Xb,Z5)
     => ( ord_le6045566169113846134st_nat(Z5,collect_list_nat(aa_fun6155608546934989052_nat_o(aTP_Lamm_bm(X5),P)))
       => aa_list_nat_o(P,Xb) ) ) ).

% prop_restrict
tff(fact_5303_prop__restrict,axiom,
    ! [Xb: set_nat,Z5: set_set_nat,X5: set_set_nat,P: fun_set_nat_o] :
      ( member_set_nat(Xb,Z5)
     => ( ord_le6893508408891458716et_nat(Z5,collect_set_nat(aa_fun751667557118977660_nat_o(aTP_Lamm_bn(X5),P)))
       => aa_set_nat_o(P,Xb) ) ) ).

% prop_restrict
tff(fact_5304_prop__restrict,axiom,
    ! [Xb: nat,Z5: set_nat,X5: set_nat,P: fun_nat_o] :
      ( aa_set_nat_o(member_nat(Xb),Z5)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Z5),collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_bo(X5),P)))
       => aa_nat_o(P,Xb) ) ) ).

% prop_restrict
tff(fact_5305_prop__restrict,axiom,
    ! [Xb: int,Z5: set_int,X5: set_int,P: fun_int_o] :
      ( aa_set_int_o(member_int(Xb),Z5)
     => ( aa_set_int_o(ord_less_eq_set_int(Z5),collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_bp(X5),P)))
       => aa_int_o(P,Xb) ) ) ).

% prop_restrict
tff(fact_5306_empty__def,axiom,
    bot_bot_set_list_nat = collect_list_nat(aTP_Lamm_bq) ).

% empty_def
tff(fact_5307_empty__def,axiom,
    bot_bot_set_set_nat = collect_set_nat(aTP_Lamm_br) ).

% empty_def
tff(fact_5308_empty__def,axiom,
    bot_bot_set_real = collect_real(aTP_Lamm_bs) ).

% empty_def
tff(fact_5309_empty__def,axiom,
    bot_bot_set_nat = collect_nat(aTP_Lamm_bt) ).

% empty_def
tff(fact_5310_empty__def,axiom,
    bot_bot_set_int = collect_int(aTP_Lamm_bu) ).

% empty_def
tff(fact_5311_lambda__zero,axiom,
    aTP_Lamm_bv = aa_rea7340605660300375871l_real(times_times_real,zero_zero_real) ).

% lambda_zero
tff(fact_5312_lambda__zero,axiom,
    aTP_Lamm_bw = times_times_rat(zero_zero_rat) ).

% lambda_zero
tff(fact_5313_lambda__zero,axiom,
    aTP_Lamm_bx = aa_nat_fun_nat_nat(times_times_nat,zero_zero_nat) ).

% lambda_zero
tff(fact_5314_lambda__zero,axiom,
    aTP_Lamm_by = aa_int_fun_int_int(times_times_int,zero_zero_int) ).

% lambda_zero
tff(fact_5315_lambda__one,axiom,
    aTP_Lamm_bz = times_times_complex(one_one_complex) ).

% lambda_one
tff(fact_5316_lambda__one,axiom,
    aTP_Lamm_ca = aa_rea7340605660300375871l_real(times_times_real,one_one_real) ).

% lambda_one
tff(fact_5317_lambda__one,axiom,
    aTP_Lamm_cb = times_times_rat(one_one_rat) ).

% lambda_one
tff(fact_5318_lambda__one,axiom,
    aTP_Lamm_cc = aa_nat_fun_nat_nat(times_times_nat,one_one_nat) ).

% lambda_one
tff(fact_5319_lambda__one,axiom,
    aTP_Lamm_cd = aa_int_fun_int_int(times_times_int,one_one_int) ).

% lambda_one
tff(fact_5320_subset__divisors__dvd,axiom,
    ! [Aa: real,Ba: real] :
      ( ord_less_eq_set_real(collect_real(aa_real_fun_real_o(aTP_Lamm_ce,Aa)),collect_real(aa_real_fun_real_o(aTP_Lamm_ce,Ba)))
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Aa),Ba) ) ).

% subset_divisors_dvd
tff(fact_5321_subset__divisors__dvd,axiom,
    ! [Aa: nat,Ba: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_cf,Aa))),collect_nat(aa_nat_fun_nat_o(aTP_Lamm_cf,Ba)))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),Ba) ) ).

% subset_divisors_dvd
tff(fact_5322_subset__divisors__dvd,axiom,
    ! [Aa: code_integer,Ba: code_integer] :
      ( ord_le7084787975880047091nteger(collect_Code_integer(aa_Cod326357343816142457eger_o(aTP_Lamm_cg,Aa)),collect_Code_integer(aa_Cod326357343816142457eger_o(aTP_Lamm_cg,Ba)))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),Ba) ) ).

% subset_divisors_dvd
tff(fact_5323_subset__divisors__dvd,axiom,
    ! [Aa: int,Ba: int] :
      ( aa_set_int_o(ord_less_eq_set_int(collect_int(aa_int_fun_int_o(aTP_Lamm_ch,Aa))),collect_int(aa_int_fun_int_o(aTP_Lamm_ch,Ba)))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),Ba) ) ).

% subset_divisors_dvd
tff(fact_5324_pred__subset__eq2,axiom,
    ! [R: set_Pr4811707699266497531nteger,S2: set_Pr4811707699266497531nteger] :
      ( ord_le6737357399843569584eger_o(aTP_Lamm_ci(R),aTP_Lamm_ci(S2))
    <=> ord_le3725938330318615451nteger(R,S2) ) ).

% pred_subset_eq2
tff(fact_5325_pred__subset__eq2,axiom,
    ! [R: set_Pr448751882837621926eger_o,S2: set_Pr448751882837621926eger_o] :
      ( ord_le2909477399203599485un_o_o(aTP_Lamm_cj(R),aTP_Lamm_cj(S2))
    <=> ord_le8980329558974975238eger_o(R,S2) ) ).

% pred_subset_eq2
tff(fact_5326_pred__subset__eq2,axiom,
    ! [R: set_Pr8693737435421807431at_nat,S2: set_Pr8693737435421807431at_nat] :
      ( ord_le3650406611588909774_nat_o(aTP_Lamm_ck(R),aTP_Lamm_ck(S2))
    <=> ord_le3000389064537975527at_nat(R,S2) ) ).

% pred_subset_eq2
tff(fact_5327_pred__subset__eq2,axiom,
    ! [R: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
      ( ord_le768339454706049738_nat_o(aTP_Lamm_cl(R),aTP_Lamm_cl(S2))
    <=> ord_le3146513528884898305at_nat(R,S2) ) ).

% pred_subset_eq2
tff(fact_5328_pred__subset__eq2,axiom,
    ! [R: set_Pr958786334691620121nt_int,S2: set_Pr958786334691620121nt_int] :
      ( ord_le77980682512674322_int_o(aTP_Lamm_cm(R),aTP_Lamm_cm(S2))
    <=> ord_le2843351958646193337nt_int(R,S2) ) ).

% pred_subset_eq2
tff(fact_5329_strict__subset__divisors__dvd,axiom,
    ! [Aa: real,Ba: real] :
      ( ord_less_set_real(collect_real(aa_real_fun_real_o(aTP_Lamm_ce,Aa)),collect_real(aa_real_fun_real_o(aTP_Lamm_ce,Ba)))
    <=> ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Aa),Ba)
        & ~ aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Ba),Aa) ) ) ).

% strict_subset_divisors_dvd
tff(fact_5330_strict__subset__divisors__dvd,axiom,
    ! [Aa: nat,Ba: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_cf,Aa))),collect_nat(aa_nat_fun_nat_o(aTP_Lamm_cf,Ba)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),Ba)
        & ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Ba),Aa) ) ) ).

% strict_subset_divisors_dvd
tff(fact_5331_strict__subset__divisors__dvd,axiom,
    ! [Aa: int,Ba: int] :
      ( aa_set_int_o(ord_less_set_int(collect_int(aa_int_fun_int_o(aTP_Lamm_ch,Aa))),collect_int(aa_int_fun_int_o(aTP_Lamm_ch,Ba)))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),Ba)
        & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Ba),Aa) ) ) ).

% strict_subset_divisors_dvd
tff(fact_5332_strict__subset__divisors__dvd,axiom,
    ! [Aa: code_integer,Ba: code_integer] :
      ( ord_le1307284697595431911nteger(collect_Code_integer(aa_Cod326357343816142457eger_o(aTP_Lamm_cg,Aa)),collect_Code_integer(aa_Cod326357343816142457eger_o(aTP_Lamm_cg,Ba)))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),Ba)
        & ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Ba),Aa) ) ) ).

% strict_subset_divisors_dvd
tff(fact_5333_bot__empty__eq2,axiom,
    ! [X3: code_integer,Xa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(bot_bo8746478546042982524eger_o,X3),Xa)
    <=> member157494554546826820nteger(produc1086072967326762835nteger(X3,Xa),bot_bo4276436098303576167nteger) ) ).

% bot_empty_eq2
tff(fact_5334_bot__empty__eq2,axiom,
    ! [X3: code_integer,Xa: $o] :
      ( aa_o_o(aa_Cod6379977994291289310un_o_o(bot_bo897209107778963633un_o_o,X3),(Xa))
    <=> member1379723562493234055eger_o(produc6677183202524767010eger_o(X3,(Xa)),bot_bo5379713665208646970eger_o) ) ).

% bot_empty_eq2
tff(fact_5335_bot__empty__eq2,axiom,
    ! [X3: product_prod_nat_nat,Xa: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(bot_bo5970965355748268442_nat_o,X3),Xa)
    <=> member8206827879206165904at_nat(produc6161850002892822231at_nat(X3,Xa),bot_bo5327735625951526323at_nat) ) ).

% bot_empty_eq2
tff(fact_5336_bot__empty__eq2,axiom,
    ! [X3: nat,Xa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bot_bo1118997454112379798_nat_o,X3),Xa)
    <=> member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),Xa),bot_bo2099793752762293965at_nat) ) ).

% bot_empty_eq2
tff(fact_5337_bot__empty__eq2,axiom,
    ! [X3: int,Xa: int] :
      ( aa_int_o(aa_int_fun_int_o(bot_bo428638681919004382_int_o,X3),Xa)
    <=> member5262025264175285858nt_int(product_Pair_int_int(X3,Xa),bot_bo1796632182523588997nt_int) ) ).

% bot_empty_eq2
tff(fact_5338_finite__M__bounded__by__nat,axiom,
    ! [P: fun_nat_o,I: nat] : aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_cn(P),I))) ).

% finite_M_bounded_by_nat
tff(fact_5339_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_co(F),U))) ) ).

% finite_less_ub
tff(fact_5340_mod__mult__eq,axiom,
    ! [Aa: nat,C: nat,Ba: 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,Aa),C)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Ba),C))),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),Ba)),C) ).

% mod_mult_eq
tff(fact_5341_mod__mult__eq,axiom,
    ! [Aa: int,C: int,Ba: 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,Aa),C)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Ba),C))),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,Aa),Ba)),C) ).

% mod_mult_eq
tff(fact_5342_mod__mult__eq,axiom,
    ! [Aa: code_integer,C: code_integer,Ba: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Aa),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Ba),C))),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Aa),Ba)),C) ).

% mod_mult_eq
tff(fact_5343_mod__mult__cong,axiom,
    ! [Aa: nat,C: nat,A5: nat,Ba: nat,B5: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Aa),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A5),C) )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Ba),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B5),C) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),Ba)),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A5),B5)),C) ) ) ) ).

% mod_mult_cong
tff(fact_5344_mod__mult__cong,axiom,
    ! [Aa: int,C: int,A5: int,Ba: int,B5: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A5),C) )
     => ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Ba),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B5),C) )
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,Aa),Ba)),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A5),B5)),C) ) ) ) ).

% mod_mult_cong
tff(fact_5345_mod__mult__cong,axiom,
    ! [Aa: code_integer,C: code_integer,A5: code_integer,Ba: code_integer,B5: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Aa),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A5),C) )
     => ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Ba),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,B5),C) )
       => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Aa),Ba)),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A5),B5)),C) ) ) ) ).

% mod_mult_cong
tff(fact_5346_mod__mult__mult2,axiom,
    ! [Aa: nat,C: nat,Ba: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ba),C)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Aa),Ba)),C) ).

% mod_mult_mult2
tff(fact_5347_mod__mult__mult2,axiom,
    ! [Aa: int,C: int,Ba: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,Aa),C)),aa_int_int(aa_int_fun_int_int(times_times_int,Ba),C)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),Ba)),C) ).

% mod_mult_mult2
tff(fact_5348_mod__mult__mult2,axiom,
    ! [Aa: code_integer,C: code_integer,Ba: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Aa),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Ba),C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Aa),Ba)),C) ).

% mod_mult_mult2
tff(fact_5349_mult__mod__right,axiom,
    ! [C: nat,Aa: nat,Ba: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Aa),Ba)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),Aa)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),Ba)) ).

% mult_mod_right
tff(fact_5350_mult__mod__right,axiom,
    ! [C: int,Aa: int,Ba: int] : aa_int_int(aa_int_fun_int_int(times_times_int,C),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),Ba)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),Aa)),aa_int_int(aa_int_fun_int_int(times_times_int,C),Ba)) ).

% mult_mod_right
tff(fact_5351_mult__mod__right,axiom,
    ! [C: code_integer,Aa: code_integer,Ba: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Aa),Ba)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),Aa)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),Ba)) ).

% mult_mod_right
tff(fact_5352_mod__mult__left__eq,axiom,
    ! [Aa: nat,C: nat,Ba: 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,Aa),C)),Ba)),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),Ba)),C) ).

% mod_mult_left_eq
tff(fact_5353_mod__mult__left__eq,axiom,
    ! [Aa: int,C: int,Ba: 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,Aa),C)),Ba)),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,Aa),Ba)),C) ).

% mod_mult_left_eq
tff(fact_5354_mod__mult__left__eq,axiom,
    ! [Aa: code_integer,C: code_integer,Ba: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Aa),C)),Ba)),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Aa),Ba)),C) ).

% mod_mult_left_eq
tff(fact_5355_mod__mult__right__eq,axiom,
    ! [Aa: nat,Ba: 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,Aa),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Ba),C))),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),Ba)),C) ).

% mod_mult_right_eq
tff(fact_5356_mod__mult__right__eq,axiom,
    ! [Aa: int,Ba: 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,Aa),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Ba),C))),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,Aa),Ba)),C) ).

% mod_mult_right_eq
tff(fact_5357_mod__mult__right__eq,axiom,
    ! [Aa: code_integer,Ba: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Aa),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Ba),C))),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Aa),Ba)),C) ).

% mod_mult_right_eq
tff(fact_5358_mod__add__eq,axiom,
    ! [Aa: nat,C: nat,Ba: 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,Aa),C)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Ba),C))),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),Ba)),C) ).

% mod_add_eq
tff(fact_5359_mod__add__eq,axiom,
    ! [Aa: int,C: int,Ba: 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,Aa),C)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Ba),C))),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),Ba)),C) ).

% mod_add_eq
tff(fact_5360_mod__add__eq,axiom,
    ! [Aa: code_integer,C: code_integer,Ba: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Aa),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Ba),C))),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Aa),Ba)),C) ).

% mod_add_eq
tff(fact_5361_mod__add__cong,axiom,
    ! [Aa: nat,C: nat,A5: nat,Ba: nat,B5: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Aa),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A5),C) )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Ba),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B5),C) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),Ba)),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A5),B5)),C) ) ) ) ).

% mod_add_cong
tff(fact_5362_mod__add__cong,axiom,
    ! [Aa: int,C: int,A5: int,Ba: int,B5: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A5),C) )
     => ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Ba),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B5),C) )
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),Ba)),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A5),B5)),C) ) ) ) ).

% mod_add_cong
tff(fact_5363_mod__add__cong,axiom,
    ! [Aa: code_integer,C: code_integer,A5: code_integer,Ba: code_integer,B5: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Aa),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A5),C) )
     => ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Ba),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,B5),C) )
       => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Aa),Ba)),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A5),B5)),C) ) ) ) ).

% mod_add_cong
tff(fact_5364_mod__add__left__eq,axiom,
    ! [Aa: nat,C: nat,Ba: 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,Aa),C)),Ba)),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),Ba)),C) ).

% mod_add_left_eq
tff(fact_5365_mod__add__left__eq,axiom,
    ! [Aa: int,C: int,Ba: 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,Aa),C)),Ba)),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),Ba)),C) ).

% mod_add_left_eq
tff(fact_5366_mod__add__left__eq,axiom,
    ! [Aa: code_integer,C: code_integer,Ba: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Aa),C)),Ba)),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Aa),Ba)),C) ).

% mod_add_left_eq
tff(fact_5367_mod__add__right__eq,axiom,
    ! [Aa: nat,Ba: 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,Aa),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Ba),C))),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),Ba)),C) ).

% mod_add_right_eq
tff(fact_5368_mod__add__right__eq,axiom,
    ! [Aa: int,Ba: 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,Aa),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Ba),C))),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),Ba)),C) ).

% mod_add_right_eq
tff(fact_5369_mod__add__right__eq,axiom,
    ! [Aa: code_integer,Ba: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Aa),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Ba),C))),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Aa),Ba)),C) ).

% mod_add_right_eq
tff(fact_5370_mod__diff__eq,axiom,
    ! [Aa: int,C: int,Ba: 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,Aa),C)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Ba),C))),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Aa),Ba)),C) ).

% mod_diff_eq
tff(fact_5371_mod__diff__eq,axiom,
    ! [Aa: code_integer,C: code_integer,Ba: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Aa),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Ba),C))),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,Aa),Ba)),C) ).

% mod_diff_eq
tff(fact_5372_mod__diff__cong,axiom,
    ! [Aa: int,C: int,A5: int,Ba: int,B5: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A5),C) )
     => ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Ba),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B5),C) )
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Aa),Ba)),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A5),B5)),C) ) ) ) ).

% mod_diff_cong
tff(fact_5373_mod__diff__cong,axiom,
    ! [Aa: code_integer,C: code_integer,A5: code_integer,Ba: code_integer,B5: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Aa),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A5),C) )
     => ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Ba),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,B5),C) )
       => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,Aa),Ba)),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,A5),B5)),C) ) ) ) ).

% mod_diff_cong
tff(fact_5374_mod__diff__left__eq,axiom,
    ! [Aa: int,C: int,Ba: 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,Aa),C)),Ba)),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Aa),Ba)),C) ).

% mod_diff_left_eq
tff(fact_5375_mod__diff__left__eq,axiom,
    ! [Aa: code_integer,C: code_integer,Ba: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Aa),C)),Ba)),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,Aa),Ba)),C) ).

% mod_diff_left_eq
tff(fact_5376_mod__diff__right__eq,axiom,
    ! [Aa: int,Ba: 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,Aa),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Ba),C))),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Aa),Ba)),C) ).

% mod_diff_right_eq
tff(fact_5377_mod__diff__right__eq,axiom,
    ! [Aa: code_integer,Ba: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,Aa),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Ba),C))),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,Aa),Ba)),C) ).

% mod_diff_right_eq
tff(fact_5378_mod__minus__eq,axiom,
    ! [Aa: int,Ba: 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,Aa),Ba))),Ba) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,Aa)),Ba) ).

% mod_minus_eq
tff(fact_5379_mod__minus__eq,axiom,
    ! [Aa: code_integer,Ba: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Aa),Ba))),Ba) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa)),Ba) ).

% mod_minus_eq
tff(fact_5380_mod__minus__cong,axiom,
    ! [Aa: int,Ba: int,A5: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),Ba) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A5),Ba) )
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,Aa)),Ba) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,A5)),Ba) ) ) ).

% mod_minus_cong
tff(fact_5381_mod__minus__cong,axiom,
    ! [Aa: code_integer,Ba: code_integer,A5: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Aa),Ba) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,A5),Ba) )
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa)),Ba) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A5)),Ba) ) ) ).

% mod_minus_cong
tff(fact_5382_mod__minus__right,axiom,
    ! [Aa: int,Ba: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),aa_int_int(uminus_uminus_int,Ba)) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,Aa)),Ba)) ).

% mod_minus_right
tff(fact_5383_mod__minus__right,axiom,
    ! [Aa: code_integer,Ba: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Aa),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Ba)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa)),Ba)) ).

% mod_minus_right
tff(fact_5384_power__mod,axiom,
    ! [Aa: nat,Ba: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Aa),Ba)),Na)),Ba) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Aa),Na)),Ba) ).

% power_mod
tff(fact_5385_power__mod,axiom,
    ! [Aa: int,Ba: int,Na: 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,Aa),Ba)),Na)),Ba) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_nat_int(power_power_int(Aa),Na)),Ba) ).

% power_mod
tff(fact_5386_power__mod,axiom,
    ! [Aa: code_integer,Ba: code_integer,Na: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Aa),Ba)),Na)),Ba) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_nat_Code_integer(power_8256067586552552935nteger(Aa),Na)),Ba) ).

% power_mod
tff(fact_5387_dvd__mod__iff,axiom,
    ! [C: nat,Ba: nat,Aa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),Ba)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Aa),Ba))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),Aa) ) ) ).

% dvd_mod_iff
tff(fact_5388_dvd__mod__iff,axiom,
    ! [C: int,Ba: int,Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),Ba)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),Ba))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),Aa) ) ) ).

% dvd_mod_iff
tff(fact_5389_dvd__mod__iff,axiom,
    ! [C: code_integer,Ba: code_integer,Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),Ba)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Aa),Ba))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),Aa) ) ) ).

% dvd_mod_iff
tff(fact_5390_dvd__mod__imp__dvd,axiom,
    ! [C: nat,Aa: nat,Ba: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Aa),Ba))
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),Ba)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),Aa) ) ) ).

% dvd_mod_imp_dvd
tff(fact_5391_dvd__mod__imp__dvd,axiom,
    ! [C: int,Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),Ba))
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),Ba)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),Aa) ) ) ).

% dvd_mod_imp_dvd
tff(fact_5392_dvd__mod__imp__dvd,axiom,
    ! [C: code_integer,Aa: code_integer,Ba: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Aa),Ba))
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),Ba)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),Aa) ) ) ).

% dvd_mod_imp_dvd
tff(fact_5393_dvd__mod,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),Na)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Na)) ) ) ).

% dvd_mod
tff(fact_5394_dvd__mod,axiom,
    ! [K: int,M: int,Na: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,K),M)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,K),Na)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,K),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,M),Na)) ) ) ).

% dvd_mod
tff(fact_5395_dvd__mod,axiom,
    ! [K: code_integer,M: code_integer,Na: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,K),M)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,K),Na)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,K),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,M),Na)) ) ) ).

% dvd_mod
tff(fact_5396_mod__mod__cancel,axiom,
    ! [C: nat,Ba: nat,Aa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),Ba)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Aa),Ba)),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Aa),C) ) ) ).

% mod_mod_cancel
tff(fact_5397_mod__mod__cancel,axiom,
    ! [C: int,Ba: int,Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),Ba)
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),Ba)),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),C) ) ) ).

% mod_mod_cancel
tff(fact_5398_mod__mod__cancel,axiom,
    ! [C: code_integer,Ba: code_integer,Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),Ba)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Aa),Ba)),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Aa),C) ) ) ).

% mod_mod_cancel
tff(fact_5399_mod__Suc__eq,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Na))),Na) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),Na) ).

% mod_Suc_eq
tff(fact_5400_mod__Suc__Suc__eq,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Na)))),Na) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,M))),Na) ).

% mod_Suc_Suc_eq
tff(fact_5401_mod__less__eq__dividend,axiom,
    ! [M: nat,Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Na)),M) ).

% mod_less_eq_dividend
tff(fact_5402_numeral__code_I2_J,axiom,
    ! [Na: num] :
      aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,Na)) = $let(
        m: rat,
        m:= aa_num_rat(numeral_numeral_rat,Na),
        aa_rat_rat(plus_plus_rat(m),m) ) ).

% numeral_code(2)
tff(fact_5403_numeral__code_I2_J,axiom,
    ! [Na: num] :
      aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Na)) = $let(
        m: nat,
        m:= aa_num_nat(numeral_numeral_nat,Na),
        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,m),m) ) ).

% numeral_code(2)
tff(fact_5404_numeral__code_I2_J,axiom,
    ! [Na: num] :
      aa_num_real(numeral_numeral_real,aa_num_num(bit0,Na)) = $let(
        m: real,
        m:= aa_num_real(numeral_numeral_real,Na),
        aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,m),m) ) ).

% numeral_code(2)
tff(fact_5405_numeral__code_I2_J,axiom,
    ! [Na: num] :
      aa_num_int(numeral_numeral_int,aa_num_num(bit0,Na)) = $let(
        m: int,
        m:= aa_num_int(numeral_numeral_int,Na),
        aa_int_int(aa_int_fun_int_int(plus_plus_int,m),m) ) ).

% numeral_code(2)
tff(fact_5406_numeral__code_I2_J,axiom,
    ! [Na: num] :
      numera1916890842035813515d_enat(aa_num_num(bit0,Na)) = $let(
        m: extended_enat,
        m:= numera1916890842035813515d_enat(Na),
        plus_p3455044024723400733d_enat(m,m) ) ).

% numeral_code(2)
tff(fact_5407_numeral__code_I2_J,axiom,
    ! [Na: num] :
      aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Na)) = $let(
        m: code_integer,
        m:= aa_num_Code_integer(numera6620942414471956472nteger,Na),
        aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,m),m) ) ).

% numeral_code(2)
tff(fact_5408_set__vebt__def,axiom,
    ! [Ta: vEBT_VEBT] : vEBT_set_vebt(Ta) = collect_nat(vEBT_V8194947554948674370ptions(Ta)) ).

% set_vebt_def
tff(fact_5409_finite__int__segment,axiom,
    ! [Aa: real,Ba: real] : finite_finite_real(collect_real(aa_real_fun_real_o(aTP_Lamm_cp(Aa),Ba))) ).

% finite_int_segment
tff(fact_5410_finite__int__segment,axiom,
    ! [Aa: rat,Ba: rat] : finite_finite_rat(collect_rat(aa_rat_fun_rat_o(aTP_Lamm_cq(Aa),Ba))) ).

% finite_int_segment
tff(fact_5411_finite__roots__unity,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Na)
     => finite_finite_real(collect_real(aa_nat_fun_real_o(aTP_Lamm_cr,Na))) ) ).

% finite_roots_unity
tff(fact_5412_finite__roots__unity,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Na)
     => aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_nat_fun_complex_o(aTP_Lamm_cs,Na))) ) ).

% finite_roots_unity
tff(fact_5413_finite__divisors__nat,axiom,
    ! [M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
     => aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_cf,M))) ) ).

% finite_divisors_nat
tff(fact_5414_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [Aa: code_integer,Ba: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Aa)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Aa),Ba)),Aa) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
tff(fact_5415_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [Aa: nat,Ba: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Aa)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Aa),Ba)),Aa) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
tff(fact_5416_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Aa)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),Ba)),Aa) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
tff(fact_5417_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [Ba: nat,Aa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Ba)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Aa),Ba)),Ba) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
tff(fact_5418_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [Ba: int,Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Ba)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),Ba)),Ba) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
tff(fact_5419_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [Ba: code_integer,Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),Ba)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Aa),Ba)),Ba) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
tff(fact_5420_mod__eq__self__iff__div__eq__0,axiom,
    ! [Aa: nat,Ba: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Aa),Ba) = Aa )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Aa),Ba) = zero_zero_nat ) ) ).

% mod_eq_self_iff_div_eq_0
tff(fact_5421_mod__eq__self__iff__div__eq__0,axiom,
    ! [Aa: int,Ba: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),Ba) = Aa )
    <=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),Ba) = zero_zero_int ) ) ).

% mod_eq_self_iff_div_eq_0
tff(fact_5422_mod__eq__self__iff__div__eq__0,axiom,
    ! [Aa: code_integer,Ba: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Aa),Ba) = Aa )
    <=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Aa),Ba) = zero_z3403309356797280102nteger ) ) ).

% mod_eq_self_iff_div_eq_0
tff(fact_5423_mod__eqE,axiom,
    ! [Aa: int,C: int,Ba: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Ba),C) )
     => ~ ! [D4: int] : Ba != aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),aa_int_int(aa_int_fun_int_int(times_times_int,C),D4)) ) ).

% mod_eqE
tff(fact_5424_mod__eqE,axiom,
    ! [Aa: code_integer,C: code_integer,Ba: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Aa),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Ba),C) )
     => ~ ! [D4: code_integer] : Ba != aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Aa),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),D4)) ) ).

% mod_eqE
tff(fact_5425_mod__0__imp__dvd,axiom,
    ! [Aa: nat,Ba: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Aa),Ba) = zero_zero_nat )
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Ba),Aa) ) ).

% mod_0_imp_dvd
tff(fact_5426_mod__0__imp__dvd,axiom,
    ! [Aa: int,Ba: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),Ba) = zero_zero_int )
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Ba),Aa) ) ).

% mod_0_imp_dvd
tff(fact_5427_mod__0__imp__dvd,axiom,
    ! [Aa: code_integer,Ba: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Aa),Ba) = zero_z3403309356797280102nteger )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Ba),Aa) ) ).

% mod_0_imp_dvd
tff(fact_5428_dvd__eq__mod__eq__0,axiom,
    ! [Aa: nat,Ba: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Aa),Ba)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Ba),Aa) = zero_zero_nat ) ) ).

% dvd_eq_mod_eq_0
tff(fact_5429_dvd__eq__mod__eq__0,axiom,
    ! [Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Aa),Ba)
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Ba),Aa) = zero_zero_int ) ) ).

% dvd_eq_mod_eq_0
tff(fact_5430_dvd__eq__mod__eq__0,axiom,
    ! [Aa: code_integer,Ba: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Aa),Ba)
    <=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Ba),Aa) = zero_z3403309356797280102nteger ) ) ).

% dvd_eq_mod_eq_0
tff(fact_5431_mod__eq__0__iff__dvd,axiom,
    ! [Aa: nat,Ba: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Aa),Ba) = zero_zero_nat )
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Ba),Aa) ) ).

% mod_eq_0_iff_dvd
tff(fact_5432_mod__eq__0__iff__dvd,axiom,
    ! [Aa: int,Ba: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),Ba) = zero_zero_int )
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Ba),Aa) ) ).

% mod_eq_0_iff_dvd
tff(fact_5433_mod__eq__0__iff__dvd,axiom,
    ! [Aa: code_integer,Ba: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Aa),Ba) = zero_z3403309356797280102nteger )
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Ba),Aa) ) ).

% mod_eq_0_iff_dvd
tff(fact_5434_div__add1__eq,axiom,
    ! [Aa: nat,Ba: 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,Aa),Ba)),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,Aa),C)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Ba),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,Aa),C)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Ba),C))),C)) ).

% div_add1_eq
tff(fact_5435_div__add1__eq,axiom,
    ! [Aa: int,Ba: 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,Aa),Ba)),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,Aa),C)),aa_int_int(aa_int_fun_int_int(divide_divide_int,Ba),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,Aa),C)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Ba),C))),C)) ).

% div_add1_eq
tff(fact_5436_div__add1__eq,axiom,
    ! [Aa: code_integer,Ba: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Aa),Ba)),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Aa),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Ba),C))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Aa),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Ba),C))),C)) ).

% div_add1_eq
tff(fact_5437_dvd__minus__mod,axiom,
    ! [Ba: nat,Aa: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Ba),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Aa),Ba))) ).

% dvd_minus_mod
tff(fact_5438_dvd__minus__mod,axiom,
    ! [Ba: int,Aa: int] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Ba),aa_int_int(aa_int_fun_int_int(minus_minus_int,Aa),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),Ba))) ).

% dvd_minus_mod
tff(fact_5439_dvd__minus__mod,axiom,
    ! [Ba: code_integer,Aa: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Ba),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,Aa),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Aa),Ba))) ).

% dvd_minus_mod
tff(fact_5440_mod__eq__dvd__iff,axiom,
    ! [Aa: int,C: int,Ba: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Ba),C) )
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),aa_int_int(aa_int_fun_int_int(minus_minus_int,Aa),Ba)) ) ).

% mod_eq_dvd_iff
tff(fact_5441_mod__eq__dvd__iff,axiom,
    ! [Aa: code_integer,C: code_integer,Ba: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Aa),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Ba),C) )
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,Aa),Ba)) ) ).

% mod_eq_dvd_iff
tff(fact_5442_mod__Suc,axiom,
    ! [M: nat,Na: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),Na) = $ite(aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Na)) = Na,zero_zero_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Na))) ).

% mod_Suc
tff(fact_5443_mod__induct,axiom,
    ! [P: fun_nat_o,Na: nat,P3: nat,M: nat] :
      ( aa_nat_o(P,Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),P3)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),P3)
         => ( ! [N: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),P3)
               => ( aa_nat_o(P,N)
                 => aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,N)),P3)) ) )
           => aa_nat_o(P,M) ) ) ) ) ).

% mod_induct
tff(fact_5444_mod__less__divisor,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Na)),Na) ) ).

% mod_less_divisor
tff(fact_5445_gcd__nat__induct,axiom,
    ! [P: fun_nat_fun_nat_o,M: nat,Na: nat] :
      ( ! [M3: nat] : aa_nat_o(aa_nat_fun_nat_o(P,M3),zero_zero_nat)
     => ( ! [M3: 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,M3),N))
             => aa_nat_o(aa_nat_fun_nat_o(P,M3),N) ) )
       => aa_nat_o(aa_nat_fun_nat_o(P,M),Na) ) ) ).

% gcd_nat_induct
tff(fact_5446_mod__Suc__le__divisor,axiom,
    ! [M: nat,Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_nat_nat(suc,Na))),Na) ).

% mod_Suc_le_divisor
tff(fact_5447_mod__eq__0D,axiom,
    ! [M: nat,D: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),D) = zero_zero_nat )
     => ? [Q3: nat] : M = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,D),Q3) ) ).

% mod_eq_0D
tff(fact_5448_mod__geq,axiom,
    ! [M: nat,Na: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Na) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na)),Na) ) ) ).

% mod_geq
tff(fact_5449_mod__if,axiom,
    ! [M: nat,Na: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Na) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na),M,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na)),Na)) ).

% mod_if
tff(fact_5450_le__mod__geq,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Na) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na)),Na) ) ) ).

% le_mod_geq
tff(fact_5451_finite__lists__length__eq,axiom,
    ! [A3: set_complex,Na: nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A3)
     => finite8712137658972009173omplex(collect_list_complex(aa_nat5940423528545770789plex_o(aTP_Lamm_ct(A3),Na))) ) ).

% finite_lists_length_eq
tff(fact_5452_finite__lists__length__eq,axiom,
    ! [A3: set_VEBT_VEBT,Na: nat] :
      ( finite5795047828879050333T_VEBT(A3)
     => finite3004134309566078307T_VEBT(collec5608196760682091941T_VEBT(aa_nat1274389614848952367VEBT_o(aTP_Lamm_cu(A3),Na))) ) ).

% finite_lists_length_eq
tff(fact_5453_finite__lists__length__eq,axiom,
    ! [A3: set_o,Na: nat] :
      ( finite_finite_o(A3)
     => finite_finite_list_o(collect_list_o(aa_nat_fun_list_o_o(aTP_Lamm_cv(A3),Na))) ) ).

% finite_lists_length_eq
tff(fact_5454_finite__lists__length__eq,axiom,
    ! [A3: set_nat,Na: nat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => finite8100373058378681591st_nat(collect_list_nat(aa_nat5883842100386203267_nat_o(aTP_Lamm_cw(A3),Na))) ) ).

% finite_lists_length_eq
tff(fact_5455_finite__lists__length__eq,axiom,
    ! [A3: set_int,Na: nat] :
      ( aa_set_int_o(finite_finite_int,A3)
     => finite3922522038869484883st_int(collect_list_int(aa_nat3638697132715757351_int_o(aTP_Lamm_cx(A3),Na))) ) ).

% finite_lists_length_eq
tff(fact_5456_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [Aa: code_integer,Ba: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Aa)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Aa),Ba)
       => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Aa),Ba) = Aa ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
tff(fact_5457_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [Aa: nat,Ba: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Aa)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),Ba)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Aa),Ba) = Aa ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
tff(fact_5458_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Aa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),Ba)
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),Ba) = Aa ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
tff(fact_5459_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [Ba: code_integer,Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),Ba)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Aa),Ba)) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
tff(fact_5460_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [Ba: nat,Aa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Ba)
     => 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,Aa),Ba)) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
tff(fact_5461_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [Ba: int,Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Ba)
     => 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,Aa),Ba)) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
tff(fact_5462_cong__exp__iff__simps_I2_J,axiom,
    ! [Na: num,Q4: num] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Na))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q4))) = zero_zero_nat )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Na)),aa_num_nat(numeral_numeral_nat,Q4)) = zero_zero_nat ) ) ).

% cong_exp_iff_simps(2)
tff(fact_5463_cong__exp__iff__simps_I2_J,axiom,
    ! [Na: num,Q4: num] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Na))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q4))) = zero_zero_int )
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,Na)),aa_num_int(numeral_numeral_int,Q4)) = zero_zero_int ) ) ).

% cong_exp_iff_simps(2)
tff(fact_5464_cong__exp__iff__simps_I2_J,axiom,
    ! [Na: num,Q4: num] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Na))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Q4))) = zero_z3403309356797280102nteger )
    <=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_num_Code_integer(numera6620942414471956472nteger,Na)),aa_num_Code_integer(numera6620942414471956472nteger,Q4)) = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(2)
tff(fact_5465_cong__exp__iff__simps_I1_J,axiom,
    ! [Na: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Na)),aa_num_nat(numeral_numeral_nat,one)) = zero_zero_nat ).

% cong_exp_iff_simps(1)
tff(fact_5466_cong__exp__iff__simps_I1_J,axiom,
    ! [Na: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,Na)),aa_num_int(numeral_numeral_int,one)) = zero_zero_int ).

% cong_exp_iff_simps(1)
tff(fact_5467_cong__exp__iff__simps_I1_J,axiom,
    ! [Na: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_num_Code_integer(numera6620942414471956472nteger,Na)),aa_num_Code_integer(numera6620942414471956472nteger,one)) = zero_z3403309356797280102nteger ).

% cong_exp_iff_simps(1)
tff(fact_5468_finite__lists__length__le,axiom,
    ! [A3: set_complex,Na: nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A3)
     => finite8712137658972009173omplex(collect_list_complex(aa_nat5940423528545770789plex_o(aTP_Lamm_cy(A3),Na))) ) ).

% finite_lists_length_le
tff(fact_5469_finite__lists__length__le,axiom,
    ! [A3: set_VEBT_VEBT,Na: nat] :
      ( finite5795047828879050333T_VEBT(A3)
     => finite3004134309566078307T_VEBT(collec5608196760682091941T_VEBT(aa_nat1274389614848952367VEBT_o(aTP_Lamm_cz(A3),Na))) ) ).

% finite_lists_length_le
tff(fact_5470_finite__lists__length__le,axiom,
    ! [A3: set_o,Na: nat] :
      ( finite_finite_o(A3)
     => finite_finite_list_o(collect_list_o(aa_nat_fun_list_o_o(aTP_Lamm_da(A3),Na))) ) ).

% finite_lists_length_le
tff(fact_5471_finite__lists__length__le,axiom,
    ! [A3: set_nat,Na: nat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => finite8100373058378681591st_nat(collect_list_nat(aa_nat5883842100386203267_nat_o(aTP_Lamm_db(A3),Na))) ) ).

% finite_lists_length_le
tff(fact_5472_finite__lists__length__le,axiom,
    ! [A3: set_int,Na: nat] :
      ( aa_set_int_o(finite_finite_int,A3)
     => finite3922522038869484883st_int(collect_list_int(aa_nat3638697132715757351_int_o(aTP_Lamm_dc(A3),Na))) ) ).

% finite_lists_length_le
tff(fact_5473_cancel__div__mod__rules_I2_J,axiom,
    ! [Ba: nat,Aa: 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,Ba),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Aa),Ba))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Aa),Ba))),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),C) ).

% cancel_div_mod_rules(2)
tff(fact_5474_cancel__div__mod__rules_I2_J,axiom,
    ! [Ba: int,Aa: 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,Ba),aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),Ba))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),Ba))),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),C) ).

% cancel_div_mod_rules(2)
tff(fact_5475_cancel__div__mod__rules_I2_J,axiom,
    ! [Ba: code_integer,Aa: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Ba),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Aa),Ba))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Aa),Ba))),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Aa),C) ).

% cancel_div_mod_rules(2)
tff(fact_5476_cancel__div__mod__rules_I1_J,axiom,
    ! [Aa: nat,Ba: 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,Aa),Ba)),Ba)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Aa),Ba))),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),C) ).

% cancel_div_mod_rules(1)
tff(fact_5477_cancel__div__mod__rules_I1_J,axiom,
    ! [Aa: int,Ba: 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,Aa),Ba)),Ba)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),Ba))),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),C) ).

% cancel_div_mod_rules(1)
tff(fact_5478_cancel__div__mod__rules_I1_J,axiom,
    ! [Aa: code_integer,Ba: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Aa),Ba)),Ba)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Aa),Ba))),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Aa),C) ).

% cancel_div_mod_rules(1)
tff(fact_5479_mod__div__decomp,axiom,
    ! [Aa: nat,Ba: nat] : Aa = 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,Aa),Ba)),Ba)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Aa),Ba)) ).

% mod_div_decomp
tff(fact_5480_mod__div__decomp,axiom,
    ! [Aa: int,Ba: int] : Aa = 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,Aa),Ba)),Ba)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),Ba)) ).

% mod_div_decomp
tff(fact_5481_mod__div__decomp,axiom,
    ! [Aa: code_integer,Ba: code_integer] : Aa = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Aa),Ba)),Ba)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Aa),Ba)) ).

% mod_div_decomp
tff(fact_5482_div__mult__mod__eq,axiom,
    ! [Aa: nat,Ba: 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,Aa),Ba)),Ba)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Aa),Ba)) = Aa ).

% div_mult_mod_eq
tff(fact_5483_div__mult__mod__eq,axiom,
    ! [Aa: int,Ba: 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,Aa),Ba)),Ba)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),Ba)) = Aa ).

% div_mult_mod_eq
tff(fact_5484_div__mult__mod__eq,axiom,
    ! [Aa: code_integer,Ba: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Aa),Ba)),Ba)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Aa),Ba)) = Aa ).

% div_mult_mod_eq
tff(fact_5485_mod__div__mult__eq,axiom,
    ! [Aa: nat,Ba: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Aa),Ba)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Aa),Ba)),Ba)) = Aa ).

% mod_div_mult_eq
tff(fact_5486_mod__div__mult__eq,axiom,
    ! [Aa: int,Ba: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),Ba)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),Ba)),Ba)) = Aa ).

% mod_div_mult_eq
tff(fact_5487_mod__div__mult__eq,axiom,
    ! [Aa: code_integer,Ba: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Aa),Ba)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Aa),Ba)),Ba)) = Aa ).

% mod_div_mult_eq
tff(fact_5488_mod__mult__div__eq,axiom,
    ! [Aa: nat,Ba: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Aa),Ba)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ba),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Aa),Ba))) = Aa ).

% mod_mult_div_eq
tff(fact_5489_mod__mult__div__eq,axiom,
    ! [Aa: int,Ba: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),Ba)),aa_int_int(aa_int_fun_int_int(times_times_int,Ba),aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),Ba))) = Aa ).

% mod_mult_div_eq
tff(fact_5490_mod__mult__div__eq,axiom,
    ! [Aa: code_integer,Ba: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Aa),Ba)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Ba),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Aa),Ba))) = Aa ).

% mod_mult_div_eq
tff(fact_5491_mult__div__mod__eq,axiom,
    ! [Ba: nat,Aa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ba),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Aa),Ba))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Aa),Ba)) = Aa ).

% mult_div_mod_eq
tff(fact_5492_mult__div__mod__eq,axiom,
    ! [Ba: int,Aa: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Ba),aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),Ba))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),Ba)) = Aa ).

% mult_div_mod_eq
tff(fact_5493_mult__div__mod__eq,axiom,
    ! [Ba: code_integer,Aa: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Ba),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Aa),Ba))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Aa),Ba)) = Aa ).

% mult_div_mod_eq
tff(fact_5494_div__mult1__eq,axiom,
    ! [Aa: nat,Ba: 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,Aa),Ba)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Ba),C))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Ba),C))),C)) ).

% div_mult1_eq
tff(fact_5495_div__mult1__eq,axiom,
    ! [Aa: int,Ba: 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,Aa),Ba)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Aa),aa_int_int(aa_int_fun_int_int(divide_divide_int,Ba),C))),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,Aa),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Ba),C))),C)) ).

% div_mult1_eq
tff(fact_5496_div__mult1__eq,axiom,
    ! [Aa: code_integer,Ba: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Aa),Ba)),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Aa),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Ba),C))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Aa),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Ba),C))),C)) ).

% div_mult1_eq
tff(fact_5497_unit__imp__mod__eq__0,axiom,
    ! [Ba: nat,Aa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Ba),one_one_nat)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Aa),Ba) = zero_zero_nat ) ) ).

% unit_imp_mod_eq_0
tff(fact_5498_unit__imp__mod__eq__0,axiom,
    ! [Ba: int,Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Ba),one_one_int)
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),Ba) = zero_zero_int ) ) ).

% unit_imp_mod_eq_0
tff(fact_5499_unit__imp__mod__eq__0,axiom,
    ! [Ba: code_integer,Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Ba),one_one_Code_integer)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Aa),Ba) = zero_z3403309356797280102nteger ) ) ).

% unit_imp_mod_eq_0
tff(fact_5500_minus__div__mult__eq__mod,axiom,
    ! [Aa: nat,Ba: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Aa),Ba)),Ba)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Aa),Ba) ).

% minus_div_mult_eq_mod
tff(fact_5501_minus__div__mult__eq__mod,axiom,
    ! [Aa: int,Ba: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,Aa),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),Ba)),Ba)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),Ba) ).

% minus_div_mult_eq_mod
tff(fact_5502_minus__div__mult__eq__mod,axiom,
    ! [Aa: code_integer,Ba: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,Aa),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Aa),Ba)),Ba)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Aa),Ba) ).

% minus_div_mult_eq_mod
tff(fact_5503_minus__mod__eq__div__mult,axiom,
    ! [Aa: nat,Ba: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Aa),Ba)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Aa),Ba)),Ba) ).

% minus_mod_eq_div_mult
tff(fact_5504_minus__mod__eq__div__mult,axiom,
    ! [Aa: int,Ba: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,Aa),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),Ba)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),Ba)),Ba) ).

% minus_mod_eq_div_mult
tff(fact_5505_minus__mod__eq__div__mult,axiom,
    ! [Aa: code_integer,Ba: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,Aa),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Aa),Ba)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Aa),Ba)),Ba) ).

% minus_mod_eq_div_mult
tff(fact_5506_minus__mod__eq__mult__div,axiom,
    ! [Aa: nat,Ba: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Aa),Ba)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ba),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Aa),Ba)) ).

% minus_mod_eq_mult_div
tff(fact_5507_minus__mod__eq__mult__div,axiom,
    ! [Aa: int,Ba: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,Aa),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),Ba)) = aa_int_int(aa_int_fun_int_int(times_times_int,Ba),aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),Ba)) ).

% minus_mod_eq_mult_div
tff(fact_5508_minus__mod__eq__mult__div,axiom,
    ! [Aa: code_integer,Ba: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,Aa),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Aa),Ba)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Ba),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Aa),Ba)) ).

% minus_mod_eq_mult_div
tff(fact_5509_minus__mult__div__eq__mod,axiom,
    ! [Aa: nat,Ba: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ba),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Aa),Ba))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Aa),Ba) ).

% minus_mult_div_eq_mod
tff(fact_5510_minus__mult__div__eq__mod,axiom,
    ! [Aa: int,Ba: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,Aa),aa_int_int(aa_int_fun_int_int(times_times_int,Ba),aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),Ba))) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),Ba) ).

% minus_mult_div_eq_mod
tff(fact_5511_minus__mult__div__eq__mod,axiom,
    ! [Aa: code_integer,Ba: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,Aa),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Ba),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Aa),Ba))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Aa),Ba) ).

% minus_mult_div_eq_mod
tff(fact_5512_fact__mod,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_nat_int(semiri1406184849735516958ct_int,Na)),aa_nat_int(semiri1406184849735516958ct_int,M)) = zero_zero_int ) ) ).

% fact_mod
tff(fact_5513_fact__mod,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_nat_Code_integer(semiri3624122377584611663nteger,Na)),aa_nat_Code_integer(semiri3624122377584611663nteger,M)) = zero_z3403309356797280102nteger ) ) ).

% fact_mod
tff(fact_5514_fact__mod,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Na)),aa_nat_nat(semiri1408675320244567234ct_nat,M)) = zero_zero_nat ) ) ).

% fact_mod
tff(fact_5515_mod__le__divisor,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Na)),Na) ) ).

% mod_le_divisor
tff(fact_5516_mod__size__less,axiom,
    ! [Ba: code_integer,Aa: code_integer] :
      ( ( Ba != zero_z3403309356797280102nteger )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_Code_integer_nat(euclid6377331345833325938nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Aa),Ba))),aa_Code_integer_nat(euclid6377331345833325938nteger,Ba)) ) ).

% mod_size_less
tff(fact_5517_mod__size__less,axiom,
    ! [Ba: int,Aa: int] :
      ( ( Ba != zero_zero_int )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(euclid4774559944035922753ze_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),Ba))),aa_int_nat(euclid4774559944035922753ze_int,Ba)) ) ).

% mod_size_less
tff(fact_5518_mod__size__less,axiom,
    ! [Ba: nat,Aa: nat] :
      ( ( Ba != zero_zero_nat )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(euclid4777050414544973029ze_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Aa),Ba))),aa_nat_nat(euclid4777050414544973029ze_nat,Ba)) ) ).

% mod_size_less
tff(fact_5519_div__less__mono,axiom,
    ! [A3: nat,B4: nat,Na: 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),Na)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A3),Na) = zero_zero_nat )
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B4),Na) = zero_zero_nat )
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A3),Na)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B4),Na)) ) ) ) ) ).

% div_less_mono
tff(fact_5520_mod__greater__zero__iff__not__dvd,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Na))
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Na),M) ) ).

% mod_greater_zero_iff_not_dvd
tff(fact_5521_mod__eq__nat1E,axiom,
    ! [M: nat,Q4: nat,Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Q4) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Na),Q4) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
       => ~ ! [S: nat] : M != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Q4),S)) ) ) ).

% mod_eq_nat1E
tff(fact_5522_mod__eq__nat2E,axiom,
    ! [M: nat,Q4: nat,Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Q4) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Na),Q4) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
       => ~ ! [S: nat] : Na != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Q4),S)) ) ) ).

% mod_eq_nat2E
tff(fact_5523_nat__mod__eq__lemma,axiom,
    ! [Xb: nat,Na: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Na) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Y),Na) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Xb)
       => ? [Q3: nat] : Xb = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),Q3)) ) ) ).

% nat_mod_eq_lemma
tff(fact_5524_mod__eq__dvd__iff__nat,axiom,
    ! [Na: nat,M: nat,Q4: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Q4) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Na),Q4) )
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Q4),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na)) ) ) ).

% mod_eq_dvd_iff_nat
tff(fact_5525_mod__mult2__eq,axiom,
    ! [M: nat,Na: nat,Q4: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),Q4)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na)),Q4))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Na)) ).

% mod_mult2_eq
tff(fact_5526_modulo__nat__def,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Na) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na)),Na)) ).

% modulo_nat_def
tff(fact_5527_split__mod,axiom,
    ! [P: fun_nat_o,M: nat,Na: nat] :
      ( aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Na))
    <=> ( ( ( Na = zero_zero_nat )
         => aa_nat_o(P,M) )
        & ( ( Na != zero_zero_nat )
         => ! [I3: nat,J3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),Na)
             => ( ( M = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),I3)),J3) )
               => aa_nat_o(P,J3) ) ) ) ) ) ).

% split_mod
tff(fact_5528_mod__nat__eqI,axiom,
    ! [R2: nat,Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,R2),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,R2),M)
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),R2))
         => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Na) = R2 ) ) ) ) ).

% mod_nat_eqI
tff(fact_5529_vebt__buildup_Osimps_I3_J,axiom,
    ! [Va: nat] :
      vEBT_vebt_buildup(aa_nat_nat(suc,aa_nat_nat(suc,Va))) = $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.simps(3)
tff(fact_5530_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C: code_integer,Aa: code_integer,Ba: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Aa),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Ba),C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Ba),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Aa),Ba)),C))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Aa),Ba)) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_5531_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C: nat,Aa: nat,Ba: 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,Aa),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ba),C)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ba),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Aa),Ba)),C))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Aa),Ba)) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_5532_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C: int,Aa: int,Ba: 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,Aa),aa_int_int(aa_int_fun_int_int(times_times_int,Ba),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Ba),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),Ba)),C))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),Ba)) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_5533_even__iff__mod__2__eq__zero,axiom,
    ! [Aa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Aa)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ) ) ).

% even_iff_mod_2_eq_zero
tff(fact_5534_even__iff__mod__2__eq__zero,axiom,
    ! [Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Aa)
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int ) ) ).

% even_iff_mod_2_eq_zero
tff(fact_5535_even__iff__mod__2__eq__zero,axiom,
    ! [Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Aa)
    <=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Aa),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = zero_z3403309356797280102nteger ) ) ).

% even_iff_mod_2_eq_zero
tff(fact_5536_odd__iff__mod__2__eq__one,axiom,
    ! [Aa: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Aa)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_nat ) ) ).

% odd_iff_mod_2_eq_one
tff(fact_5537_odd__iff__mod__2__eq__one,axiom,
    ! [Aa: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Aa)
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int ) ) ).

% odd_iff_mod_2_eq_one
tff(fact_5538_odd__iff__mod__2__eq__one,axiom,
    ! [Aa: code_integer] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Aa)
    <=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Aa),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = one_one_Code_integer ) ) ).

% odd_iff_mod_2_eq_one
tff(fact_5539_Suc__times__mod__eq,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),M)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na))),M) = one_one_nat ) ) ).

% Suc_times_mod_eq
tff(fact_5540_nth__rotate1,axiom,
    ! [Na: nat,Xs: list_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_size_list_int(Xs))
     => ( aa_nat_int(nth_int(rotate1_int(Xs)),Na) = aa_nat_int(nth_int(Xs),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,Na)),size_size_list_int(Xs))) ) ) ).

% nth_rotate1
tff(fact_5541_nth__rotate1,axiom,
    ! [Na: nat,Xs: list_VEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_s6755466524823107622T_VEBT(Xs))
     => ( nth_VEBT_VEBT(rotate1_VEBT_VEBT(Xs),Na) = nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,Na)),size_s6755466524823107622T_VEBT(Xs))) ) ) ).

% nth_rotate1
tff(fact_5542_nth__rotate1,axiom,
    ! [Na: nat,Xs: list_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_size_list_o(Xs))
     => ( aa_nat_o(nth_o(rotate1_o(Xs)),Na)
      <=> aa_nat_o(nth_o(Xs),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,Na)),size_size_list_o(Xs))) ) ) ).

% nth_rotate1
tff(fact_5543_nth__rotate1,axiom,
    ! [Na: nat,Xs: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_size_list_nat(Xs))
     => ( aa_nat_nat(nth_nat(rotate1_nat(Xs)),Na) = aa_nat_nat(nth_nat(Xs),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,Na)),size_size_list_nat(Xs))) ) ) ).

% nth_rotate1
tff(fact_5544_vebt__buildup_Oelims,axiom,
    ! [Xb: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_buildup(Xb) = Y )
     => ( ( ( Xb = zero_zero_nat )
         => ( Y != vEBT_Leaf($false,$false) ) )
       => ( ( ( Xb = aa_nat_nat(suc,zero_zero_nat) )
           => ( Y != vEBT_Leaf($false,$false) ) )
         => ~ ! [Va2: nat] :
                ( ( Xb = aa_nat_nat(suc,aa_nat_nat(suc,Va2)) )
               => ( Y != $ite(
                      aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,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.elims
tff(fact_5545_divmod__digit__0_I2_J,axiom,
    ! [Ba: nat,Aa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Ba)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Ba))),Ba)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Ba)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Aa),Ba) ) ) ) ).

% divmod_digit_0(2)
tff(fact_5546_divmod__digit__0_I2_J,axiom,
    ! [Ba: int,Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Ba)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Ba))),Ba)
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Ba)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),Ba) ) ) ) ).

% divmod_digit_0(2)
tff(fact_5547_divmod__digit__0_I2_J,axiom,
    ! [Ba: code_integer,Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),Ba)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Aa),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Ba))),Ba)
       => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Aa),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Ba)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Aa),Ba) ) ) ) ).

% divmod_digit_0(2)
tff(fact_5548_bits__stable__imp__add__self,axiom,
    ! [Aa: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = Aa )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = zero_zero_nat ) ) ).

% bits_stable_imp_add_self
tff(fact_5549_bits__stable__imp__add__self,axiom,
    ! [Aa: int] :
      ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = Aa )
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) = zero_zero_int ) ) ).

% bits_stable_imp_add_self
tff(fact_5550_bits__stable__imp__add__self,axiom,
    ! [Aa: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Aa),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = Aa )
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Aa),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Aa),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))) = zero_z3403309356797280102nteger ) ) ).

% bits_stable_imp_add_self
tff(fact_5551_parity__cases,axiom,
    ! [Aa: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Aa)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Aa),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))),Aa)
         => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) != one_one_nat ) ) ) ).

% parity_cases
tff(fact_5552_parity__cases,axiom,
    ! [Aa: int] :
      ( ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Aa)
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),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))),Aa)
         => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) != one_one_int ) ) ) ).

% parity_cases
tff(fact_5553_parity__cases,axiom,
    ! [Aa: code_integer] :
      ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Aa)
       => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Aa),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) != zero_z3403309356797280102nteger ) )
     => ~ ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Aa)
         => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Aa),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) != one_one_Code_integer ) ) ) ).

% parity_cases
tff(fact_5554_mod2__eq__if,axiom,
    ! [Aa: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Aa),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))),Aa),zero_zero_nat,one_one_nat) ).

% mod2_eq_if
tff(fact_5555_mod2__eq__if,axiom,
    ! [Aa: int] :
      aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),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))),Aa),zero_zero_int,one_one_int) ).

% mod2_eq_if
tff(fact_5556_mod2__eq__if,axiom,
    ! [Aa: code_integer] :
      aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Aa),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Aa),zero_z3403309356797280102nteger,one_one_Code_integer) ).

% mod2_eq_if
tff(fact_5557_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
    ! [Uy: option4927543243414619207at_nat,V: nat,TreeList: list_VEBT_VEBT,S3: vEBT_VEBT,Xb: nat] :
      ( aa_nat_o(vEBT_V5719532721284313246member(vEBT_Node(Uy,aa_nat_nat(suc,V),TreeList,S3)),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)))),
          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ).

% VEBT_internal.naive_member.simps(3)
tff(fact_5558_verit__le__mono__div,axiom,
    ! [A3: nat,B4: nat,Na: 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),Na)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,
              aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A3),Na)),
                $ite(aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B4),Na) = zero_zero_nat,one_one_nat,zero_zero_nat))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B4),Na)) ) ) ).

% verit_le_mono_div
tff(fact_5559_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
    ! [V: nat,TreeList: list_VEBT_VEBT,Vd: vEBT_VEBT,Xb: nat] :
      ( aa_nat_o(vEBT_VEBT_membermima(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V),TreeList,Vd)),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)))),
          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ).

% VEBT_internal.membermima.simps(5)
tff(fact_5560_divmod__digit__0_I1_J,axiom,
    ! [Ba: nat,Aa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Ba)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Ba))),Ba)
       => ( 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,Aa),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Ba))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Aa),Ba) ) ) ) ).

% divmod_digit_0(1)
tff(fact_5561_divmod__digit__0_I1_J,axiom,
    ! [Ba: int,Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Ba)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Ba))),Ba)
       => ( 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,Aa),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Ba))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),Ba) ) ) ) ).

% divmod_digit_0(1)
tff(fact_5562_divmod__digit__0_I1_J,axiom,
    ! [Ba: code_integer,Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),Ba)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Aa),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Ba))),Ba)
       => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Aa),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Ba))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Aa),Ba) ) ) ) ).

% divmod_digit_0(1)
tff(fact_5563_vebt__member_Osimps_I5_J,axiom,
    ! [Mi2: nat,Ma2: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Xb: nat] :
      ( aa_nat_o(vEBT_vebt_member(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary)),Xb)
    <=> $ite(
          Xb = Mi2,
          $true,
          $ite(
            Xb = Ma2,
            $true,
            $ite(
              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Mi2),
              $false,
              $ite(
                aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),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,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                  $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(TreeList,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ).

% vebt_member.simps(5)
tff(fact_5564_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
    ! [Mi2: nat,Ma2: nat,V: nat,TreeList: 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(Mi2),Ma2)),aa_nat_nat(suc,V),TreeList,Vc)),Xb)
    <=> ( ( Xb = Mi2 )
        | ( Xb = Ma2 )
        | $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)))),
            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ).

% VEBT_internal.membermima.simps(4)
tff(fact_5565_mult__exp__mod__exp__eq,axiom,
    ! [M: nat,Na: nat,Aa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),M)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) ) ) ).

% mult_exp_mod_exp_eq
tff(fact_5566_mult__exp__mod__exp__eq,axiom,
    ! [M: nat,Na: nat,Aa: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,Aa),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),M)))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M)) ) ) ).

% mult_exp_mod_exp_eq
tff(fact_5567_mult__exp__mod__exp__eq,axiom,
    ! [M: nat,Na: nat,Aa: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Aa),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),M))),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Na)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Aa),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),M)))),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),M)) ) ) ).

% mult_exp_mod_exp_eq
tff(fact_5568_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( aa_nat_o(vEBT_V5719532721284313246member(Xb),Xaa)
      <=> (Y) )
     => ( ! [A2: $o,B2: $o] :
            ( ( Xb = vEBT_Leaf((A2),(B2)) )
           => ( (Y)
            <=> ~ $ite(
                    Xaa = zero_zero_nat,
                    (A2),
                    $ite(Xaa = one_one_nat,(B2),$false) ) ) )
       => ( ( ? [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : Xb = vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2)
           => (Y) )
         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT] :
                ( ? [S: vEBT_VEBT] : Xb = vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList2,S)
               => ( (Y)
                <=> ~ $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)))),
                        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList2,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(1)
tff(fact_5569_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(vEBT_V5719532721284313246member(Xb),Xaa)
     => ( ! [A2: $o,B2: $o] :
            ( ( Xb = vEBT_Leaf((A2),(B2)) )
           => ~ $ite(
                  Xaa = zero_zero_nat,
                  (A2),
                  $ite(Xaa = one_one_nat,(B2),$false) ) )
       => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT] :
              ( ? [S: vEBT_VEBT] : Xb = vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList2,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)))),
                    $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList2,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ).

% VEBT_internal.naive_member.elims(2)
tff(fact_5570_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(vEBT_V5719532721284313246member(Xb),Xaa)
     => ( ! [A2: $o,B2: $o] :
            ( ( Xb = vEBT_Leaf((A2),(B2)) )
           => $ite(
                Xaa = zero_zero_nat,
                (A2),
                $ite(Xaa = one_one_nat,(B2),$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,TreeList2: list_VEBT_VEBT] :
                ( ? [S: vEBT_VEBT] : Xb = vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList2,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)))),
                    $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList2,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(3)
tff(fact_5571_eucl__rel__int__iff,axiom,
    ! [K: int,L: int,Q4: int,R2: int] :
      ( aa_Pro4592068669816069595_int_o(eucl_rel_int(K,L),product_Pair_int_int(Q4,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),Q4)),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) ),
              Q4 = zero_zero_int ) ) ) ) ).

% eucl_rel_int_iff
tff(fact_5572_mod__double__modulus,axiom,
    ! [M: code_integer,Xb: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),M)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Xb)
       => ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Xb),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),M)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Xb),M) )
          | ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Xb),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),M)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Xb),M)),M) ) ) ) ) ).

% mod_double_modulus
tff(fact_5573_mod__double__modulus,axiom,
    ! [M: nat,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),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))),M)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),M) )
          | ( 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))),M)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),M)),M) ) ) ) ) ).

% mod_double_modulus
tff(fact_5574_mod__double__modulus,axiom,
    ! [M: int,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),M)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),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))),M)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Xb),M) )
          | ( 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))),M)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Xb),M)),M) ) ) ) ) ).

% mod_double_modulus
tff(fact_5575_divmod__digit__1_I2_J,axiom,
    ! [Aa: code_integer,Ba: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Aa)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),Ba)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Ba),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Aa),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Ba)))
         => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Aa),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Ba))),Ba) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Aa),Ba) ) ) ) ) ).

% divmod_digit_1(2)
tff(fact_5576_divmod__digit__1_I2_J,axiom,
    ! [Aa: nat,Ba: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Aa)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Ba)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ba),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Ba)))
         => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Ba))),Ba) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Aa),Ba) ) ) ) ) ).

% divmod_digit_1(2)
tff(fact_5577_divmod__digit__1_I2_J,axiom,
    ! [Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Aa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Ba)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ba),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Ba)))
         => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Ba))),Ba) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),Ba) ) ) ) ) ).

% divmod_digit_1(2)
tff(fact_5578_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,TreeList2: 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),TreeList2,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)))),
                      $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList2,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) )
         => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT] :
                ( ? [Vd2: vEBT_VEBT] : Xb = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2,Vd2)
               => ~ $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)))),
                      $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList2,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(2)
tff(fact_5579_unset__bit__Suc,axiom,
    ! [Na: nat,Aa: int] : aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,aa_nat_nat(suc,Na)),Aa) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,Na),aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% unset_bit_Suc
tff(fact_5580_unset__bit__Suc,axiom,
    ! [Na: nat,Aa: code_integer] : aa_Cod1783974652533396728nteger(aa_nat6383147911832638789nteger(bit_se8260200283734997820nteger,aa_nat_nat(suc,Na)),Aa) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Aa),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_nat6383147911832638789nteger(bit_se8260200283734997820nteger,Na),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Aa),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))))) ).

% unset_bit_Suc
tff(fact_5581_unset__bit__Suc,axiom,
    ! [Na: nat,Aa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,aa_nat_nat(suc,Na)),Aa) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% unset_bit_Suc
tff(fact_5582_set__bit__Suc,axiom,
    ! [Na: nat,Aa: int] : aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,aa_nat_nat(suc,Na)),Aa) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,Na),aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% set_bit_Suc
tff(fact_5583_set__bit__Suc,axiom,
    ! [Na: nat,Aa: code_integer] : aa_Cod1783974652533396728nteger(aa_nat6383147911832638789nteger(bit_se2793503036327961859nteger,aa_nat_nat(suc,Na)),Aa) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Aa),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_nat6383147911832638789nteger(bit_se2793503036327961859nteger,Na),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Aa),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))))) ).

% set_bit_Suc
tff(fact_5584_set__bit__Suc,axiom,
    ! [Na: nat,Aa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,aa_nat_nat(suc,Na)),Aa) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% set_bit_Suc
tff(fact_5585_even__mod__4__div__2,axiom,
    ! [Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_nat_nat(suc,zero_zero_nat) )
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(suc,zero_zero_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% even_mod_4_div_2
tff(fact_5586_vebt__member_Oelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(vEBT_vebt_member(Xb),Xaa)
     => ( ! [A2: $o,B2: $o] :
            ( ( Xb = vEBT_Leaf((A2),(B2)) )
           => ~ $ite(
                  Xaa = zero_zero_nat,
                  (A2),
                  $ite(Xaa = one_one_nat,(B2),$false) ) )
       => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList2: list_VEBT_VEBT] :
              ( ? [Summary2: 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,Va2)),TreeList2,Summary2)
             => ~ $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,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(TreeList2,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(2)
tff(fact_5587_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,TreeList2: 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),TreeList2,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)))),
                        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList2,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) )
             => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT] :
                    ( ? [Vd2: vEBT_VEBT] : Xb = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2,Vd2)
                   => $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)))),
                        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList2,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(3)
tff(fact_5588_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( aa_nat_o(vEBT_VEBT_membermima(Xb),Xaa)
      <=> (Y) )
     => ( ( ? [Uu2: $o,Uv2: $o] : Xb = vEBT_Leaf((Uu2),(Uv2))
         => (Y) )
       => ( ( ? [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : Xb = vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2,Uy2)
           => (Y) )
         => ( ! [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)
               => ( (Y)
                <=> ~ ( ( Xaa = Mi )
                      | ( Xaa = Ma ) ) ) )
           => ( ! [Mi: nat,Ma: nat,V2: nat,TreeList2: 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),TreeList2,Vc2)
                 => ( (Y)
                  <=> ~ ( ( 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)))),
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList2,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) )
             => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT] :
                    ( ? [Vd2: vEBT_VEBT] : Xb = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2,Vd2)
                   => ( (Y)
                    <=> ~ $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)))),
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList2,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(1)
tff(fact_5589_vebt__member_Oelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( aa_nat_o(vEBT_vebt_member(Xb),Xaa)
      <=> (Y) )
     => ( ! [A2: $o,B2: $o] :
            ( ( Xb = vEBT_Leaf((A2),(B2)) )
           => ( (Y)
            <=> ~ $ite(
                    Xaa = zero_zero_nat,
                    (A2),
                    $ite(Xaa = one_one_nat,(B2),$false) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] : Xb = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2)
           => (Y) )
         => ( ( ? [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy2,Uz2)
             => (Y) )
           => ( ( ? [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)
               => (Y) )
             => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList2: list_VEBT_VEBT] :
                    ( ? [Summary2: 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,Va2)),TreeList2,Summary2)
                   => ( (Y)
                    <=> ~ $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,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                    $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(TreeList2,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(1)
tff(fact_5590_vebt__member_Oelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(vEBT_vebt_member(Xb),Xaa)
     => ( ! [A2: $o,B2: $o] :
            ( ( Xb = vEBT_Leaf((A2),(B2)) )
           => $ite(
                Xaa = zero_zero_nat,
                (A2),
                $ite(Xaa = one_one_nat,(B2),$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,Va2: nat,TreeList2: list_VEBT_VEBT] :
                    ( ? [Summary2: 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,Va2)),TreeList2,Summary2)
                   => $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,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(TreeList2,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(3)
tff(fact_5591_divmod__digit__1_I1_J,axiom,
    ! [Aa: code_integer,Ba: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Aa)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),Ba)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Ba),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Aa),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Ba)))
         => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Aa),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Ba)))),one_one_Code_integer) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Aa),Ba) ) ) ) ) ).

% divmod_digit_1(1)
tff(fact_5592_divmod__digit__1_I1_J,axiom,
    ! [Aa: nat,Ba: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Aa)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Ba)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ba),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Ba)))
         => ( 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,Aa),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Ba)))),one_one_nat) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Aa),Ba) ) ) ) ) ).

% divmod_digit_1(1)
tff(fact_5593_divmod__digit__1_I1_J,axiom,
    ! [Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Aa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Ba)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ba),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Ba)))
         => ( 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,Aa),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Ba)))),one_one_int) = aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),Ba) ) ) ) ) ).

% divmod_digit_1(1)
tff(fact_5594_pos__eucl__rel__int__mult__2,axiom,
    ! [Ba: int,Aa: int,Q4: int,R2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Ba)
     => ( aa_Pro4592068669816069595_int_o(eucl_rel_int(Aa,Ba),product_Pair_int_int(Q4,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))),Aa)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Ba)),product_Pair_int_int(Q4,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_5595_vebt__pred_Osimps_I7_J,axiom,
    ! [Mi2: nat,Ma2: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Xb: nat] :
      vEBT_vebt_pred(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary),Xb) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),Xb),
        some_nat(Ma2),
        $let(
          l: nat,
          l:= 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,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
          $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,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)),
              $let(
                minlow: option_nat,
                minlow:= vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,h)),
                $ite(
                  ( ( minlow != none_nat )
                  & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_greater,some_nat(l)),minlow) ),
                  aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(h))),vEBT_vebt_pred(nth_VEBT_VEBT(TreeList,h),l)),
                  $let(
                    pr: option_nat,
                    pr:= vEBT_vebt_pred(Summary,h),
                    $ite(
                      pr = none_nat,
                      $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi2),Xb),some_nat(Mi2),none_nat),
                      aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),pr)),vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList,the_nat(pr)))) ) ) ) ),
              none_nat ) ) ) ) ).

% vebt_pred.simps(7)
tff(fact_5596_arcosh__1,axiom,
    aa_real_real(arcosh_real,one_one_real) = zero_zero_real ).

% arcosh_1
tff(fact_5597_finite__nth__roots,axiom,
    ! [Na: nat,C: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_com5808801942499186519plex_o(aTP_Lamm_dd(Na),C))) ) ).

% finite_nth_roots
tff(fact_5598_vebt__pred_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: option_nat] :
      ( ( vEBT_vebt_pred(Xb,Xaa) = Y )
     => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel),produc738532404422230701BT_nat(Xb,Xaa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( Xb = vEBT_Leaf((Uu2),(Uv2)) )
             => ( ( Xaa = zero_zero_nat )
               => ( ( Y = none_nat )
                 => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel),produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(Uv2)),zero_zero_nat)) ) ) )
         => ( ! [A2: $o,Uw2: $o] :
                ( ( Xb = vEBT_Leaf((A2),(Uw2)) )
               => ( ( Xaa = aa_nat_nat(suc,zero_zero_nat) )
                 => ( ( Y = $ite((A2),some_nat(zero_zero_nat),none_nat) )
                   => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel),produc738532404422230701BT_nat(vEBT_Leaf((A2),(Uw2)),aa_nat_nat(suc,zero_zero_nat))) ) ) )
           => ( ! [A2: $o,B2: $o] :
                  ( ( Xb = vEBT_Leaf((A2),(B2)) )
                 => ! [Va2: nat] :
                      ( ( Xaa = aa_nat_nat(suc,aa_nat_nat(suc,Va2)) )
                     => ( ( Y = $ite(
                              (B2),
                              some_nat(one_one_nat),
                              $ite((A2),some_nat(zero_zero_nat),none_nat) ) )
                       => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel),produc738532404422230701BT_nat(vEBT_Leaf((A2),(B2)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)))) ) ) )
             => ( ! [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2,Va3) )
                   => ( ( Y = none_nat )
                     => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel),produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2,Va3),Xaa)) ) )
               => ( ! [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vd2,Ve2) )
                     => ( ( Y = none_nat )
                       => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel),produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vd2,Ve2),Xaa)) ) )
                 => ( ! [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT] :
                        ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vh2,Vi2) )
                       => ( ( Y = none_nat )
                         => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel),produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vh2,Vi2),Xaa)) ) )
                   => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary2: 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,Va2)),TreeList2,Summary2) )
                         => ( ( Y = $ite(
                                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xaa),
                                  some_nat(Ma),
                                  $let(
                                    l: nat,
                                    l:= 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,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                    $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,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(TreeList2)),
                                        $let(
                                          minlow: option_nat,
                                          minlow:= vEBT_vebt_mint(nth_VEBT_VEBT(TreeList2,h)),
                                          $ite(
                                            ( ( minlow != none_nat )
                                            & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_greater,some_nat(l)),minlow) ),
                                            aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(h))),vEBT_vebt_pred(nth_VEBT_VEBT(TreeList2,h),l)),
                                            $let(
                                              pr: option_nat,
                                              pr:= vEBT_vebt_pred(Summary2,h),
                                              $ite(
                                                pr = none_nat,
                                                $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi),Xaa),some_nat(Mi),none_nat),
                                                aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),pr)),vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList2,the_nat(pr)))) ) ) ) ),
                                        none_nat ) ) ) ) )
                           => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_pred_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,Va2)),TreeList2,Summary2),Xaa)) ) ) ) ) ) ) ) ) ) ) ).

% vebt_pred.pelims
tff(fact_5599_product__nth,axiom,
    ! [Na: nat,Xs: list_Code_integer,Ys2: list_Code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s3445333598471063425nteger(Xs)),size_s3445333598471063425nteger(Ys2)))
     => ( nth_Pr2304437835452373666nteger(produc8792966785426426881nteger(Xs,Ys2),Na) = produc1086072967326762835nteger(aa_nat_Code_integer(nth_Code_integer(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),size_s3445333598471063425nteger(Ys2))),aa_nat_Code_integer(nth_Code_integer(Ys2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Na),size_s3445333598471063425nteger(Ys2)))) ) ) ).

% product_nth
tff(fact_5600_product__nth,axiom,
    ! [Na: nat,Xs: list_int,Ys2: list_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_int(Xs)),size_size_list_int(Ys2)))
     => ( nth_Pr4439495888332055232nt_int(product_int_int(Xs,Ys2),Na) = product_Pair_int_int(aa_nat_int(nth_int(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),size_size_list_int(Ys2))),aa_nat_int(nth_int(Ys2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Na),size_size_list_int(Ys2)))) ) ) ).

% product_nth
tff(fact_5601_product__nth,axiom,
    ! [Na: nat,Xs: list_int,Ys2: list_VEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_int(Xs)),size_s6755466524823107622T_VEBT(Ys2)))
     => ( nth_Pr3474266648193625910T_VEBT(produc662631939642741121T_VEBT(Xs,Ys2),Na) = produc3329399203697025711T_VEBT(aa_nat_int(nth_int(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),size_s6755466524823107622T_VEBT(Ys2))),nth_VEBT_VEBT(Ys2,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Na),size_s6755466524823107622T_VEBT(Ys2)))) ) ) ).

% product_nth
tff(fact_5602_product__nth,axiom,
    ! [Na: nat,Xs: list_int,Ys2: list_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_int(Xs)),size_size_list_o(Ys2)))
     => ( nth_Pr7514405829937366042_int_o(product_int_o(Xs,Ys2),Na) = product_Pair_int_o(aa_nat_int(nth_int(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),size_size_list_o(Ys2))),aa_nat_o(nth_o(Ys2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Na),size_size_list_o(Ys2)))) ) ) ).

% product_nth
tff(fact_5603_product__nth,axiom,
    ! [Na: nat,Xs: list_Code_integer,Ys2: list_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s3445333598471063425nteger(Xs)),size_size_list_o(Ys2)))
     => ( nth_Pr8522763379788166057eger_o(produc3607205314601156340eger_o(Xs,Ys2),Na) = produc6677183202524767010eger_o(aa_nat_Code_integer(nth_Code_integer(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),size_size_list_o(Ys2))),aa_nat_o(nth_o(Ys2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Na),size_size_list_o(Ys2)))) ) ) ).

% product_nth
tff(fact_5604_product__nth,axiom,
    ! [Na: nat,Xs: list_int,Ys2: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_int(Xs)),size_size_list_nat(Ys2)))
     => ( nth_Pr8617346907841251940nt_nat(product_int_nat(Xs,Ys2),Na) = product_Pair_int_nat(aa_nat_int(nth_int(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),size_size_list_nat(Ys2))),aa_nat_nat(nth_nat(Ys2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Na),size_size_list_nat(Ys2)))) ) ) ).

% product_nth
tff(fact_5605_product__nth,axiom,
    ! [Na: nat,Xs: list_VEBT_VEBT,Ys2: list_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_size_list_int(Ys2)))
     => ( nth_Pr6837108013167703752BT_int(produc7292646706713671643BT_int(Xs,Ys2),Na) = produc736041933913180425BT_int(nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),size_size_list_int(Ys2))),aa_nat_int(nth_int(Ys2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Na),size_size_list_int(Ys2)))) ) ) ).

% product_nth
tff(fact_5606_product__nth,axiom,
    ! [Na: nat,Xs: list_VEBT_VEBT,Ys2: list_VEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_s6755466524823107622T_VEBT(Ys2)))
     => ( nth_Pr4953567300277697838T_VEBT(produc4743750530478302277T_VEBT(Xs,Ys2),Na) = produc537772716801021591T_VEBT(nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),size_s6755466524823107622T_VEBT(Ys2))),nth_VEBT_VEBT(Ys2,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Na),size_s6755466524823107622T_VEBT(Ys2)))) ) ) ).

% product_nth
tff(fact_5607_product__nth,axiom,
    ! [Na: nat,Xs: list_VEBT_VEBT,Ys2: list_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_size_list_o(Ys2)))
     => ( nth_Pr4606735188037164562VEBT_o(product_VEBT_VEBT_o(Xs,Ys2),Na) = produc8721562602347293563VEBT_o(nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),size_size_list_o(Ys2))),aa_nat_o(nth_o(Ys2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Na),size_size_list_o(Ys2)))) ) ) ).

% product_nth
tff(fact_5608_product__nth,axiom,
    ! [Na: nat,Xs: list_VEBT_VEBT,Ys2: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_size_list_nat(Ys2)))
     => ( nth_Pr1791586995822124652BT_nat(produc7295137177222721919BT_nat(Xs,Ys2),Na) = produc738532404422230701BT_nat(nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),size_size_list_nat(Ys2))),aa_nat_nat(nth_nat(Ys2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Na),size_size_list_nat(Ys2)))) ) ) ).

% product_nth
tff(fact_5609_flip__bit__Suc,axiom,
    ! [Na: nat,Aa: int] : aa_int_int(aa_nat_fun_int_int(bit_se2159334234014336723it_int,aa_nat_nat(suc,Na)),Aa) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),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_se2159334234014336723it_int,Na),aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% flip_bit_Suc
tff(fact_5610_flip__bit__Suc,axiom,
    ! [Na: nat,Aa: code_integer] : aa_Cod1783974652533396728nteger(aa_nat6383147911832638789nteger(bit_se1345352211410354436nteger,aa_nat_nat(suc,Na)),Aa) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Aa),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_nat6383147911832638789nteger(bit_se1345352211410354436nteger,Na),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Aa),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))))) ).

% flip_bit_Suc
tff(fact_5611_flip__bit__Suc,axiom,
    ! [Na: nat,Aa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,aa_nat_nat(suc,Na)),Aa) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% flip_bit_Suc
tff(fact_5612_signed__take__bit__rec,axiom,
    ! [Na: nat,Aa: code_integer] :
      aa_Cod1783974652533396728nteger(bit_ri6519982836138164636nteger(Na),Aa) = $ite(Na = zero_zero_nat,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Aa),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Aa),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(bit_ri6519982836138164636nteger(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Aa),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))))))) ).

% signed_take_bit_rec
tff(fact_5613_signed__take__bit__rec,axiom,
    ! [Na: nat,Aa: int] :
      aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,Na),Aa) = $ite(Na = zero_zero_nat,aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),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,Aa),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_ri631733984087533419it_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat)),aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))))) ).

% signed_take_bit_rec
tff(fact_5614_artanh__0,axiom,
    aa_real_real(artanh_real,zero_zero_real) = zero_zero_real ).

% artanh_0
tff(fact_5615_flip__bit__nonnegative__int__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_nat_fun_int_int(bit_se2159334234014336723it_int,Na),K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K) ) ).

% flip_bit_nonnegative_int_iff
tff(fact_5616_flip__bit__negative__int__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_nat_fun_int_int(bit_se2159334234014336723it_int,Na),K)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int) ) ).

% flip_bit_negative_int_iff
tff(fact_5617_signed__take__bit__of__0,axiom,
    ! [Na: nat] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,Na),zero_zero_int) = zero_zero_int ).

% signed_take_bit_of_0
tff(fact_5618_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_5619_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_5620_signed__take__bit__of__minus__1,axiom,
    ! [Na: nat] : aa_Cod1783974652533396728nteger(bit_ri6519982836138164636nteger(Na),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% signed_take_bit_of_minus_1
tff(fact_5621_signed__take__bit__of__minus__1,axiom,
    ! [Na: nat] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,Na),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,one_one_int) ).

% signed_take_bit_of_minus_1
tff(fact_5622_signed__take__bit__Suc__1,axiom,
    ! [Na: nat] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,aa_nat_nat(suc,Na)),one_one_int) = one_one_int ).

% signed_take_bit_Suc_1
tff(fact_5623_signed__take__bit__numeral__of__1,axiom,
    ! [K: num] : aa_int_int(aa_nat_fun_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_5624_signed__take__bit__Suc__bit0,axiom,
    ! [Na: nat,K: num] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,aa_nat_nat(suc,Na)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,K))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,Na),aa_num_int(numeral_numeral_int,K))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% signed_take_bit_Suc_bit0
tff(fact_5625_signed__take__bit__Suc__minus__bit0,axiom,
    ! [Na: nat,K: num] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,aa_nat_nat(suc,Na)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,K)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,Na),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% signed_take_bit_Suc_minus_bit0
tff(fact_5626_signed__take__bit__0,axiom,
    ! [Aa: code_integer] : aa_Cod1783974652533396728nteger(bit_ri6519982836138164636nteger(zero_zero_nat),Aa) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Aa),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))) ).

% signed_take_bit_0
tff(fact_5627_signed__take__bit__0,axiom,
    ! [Aa: int] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,zero_zero_nat),Aa) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ).

% signed_take_bit_0
tff(fact_5628_zmod__le__nonneg__dividend,axiom,
    ! [M: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),M)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,M),K)),M) ) ).

% zmod_le_nonneg_dividend
tff(fact_5629_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_5630_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_5631_neg__mod__conj,axiom,
    ! [Ba: int,Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Ba),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,Aa),Ba)),zero_zero_int)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,Ba),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),Ba)) ) ) ).

% neg_mod_conj
tff(fact_5632_pos__mod__conj,axiom,
    ! [Ba: int,Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Ba)
     => ( 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,Aa),Ba))
        & aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),Ba)),Ba) ) ) ).

% pos_mod_conj
tff(fact_5633_zmod__trivial__iff,axiom,
    ! [I: int,K: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,I),K) = I )
    <=> ( ( K = zero_zero_int )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),I)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,I),K) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,K),I) ) ) ) ).

% zmod_trivial_iff
tff(fact_5634_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_5635_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_5636_zdiv__mono__strict,axiom,
    ! [A3: int,B4: int,Na: 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),Na)
       => ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A3),Na) = zero_zero_int )
         => ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B4),Na) = zero_zero_int )
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A3),Na)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B4),Na)) ) ) ) ) ).

% zdiv_mono_strict
tff(fact_5637_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_5638_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_5639_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_5640_int__mod__pos__eq,axiom,
    ! [Aa: int,Ba: int,Q4: int,R2: int] :
      ( ( Aa = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Ba),Q4)),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),Ba)
         => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),Ba) = R2 ) ) ) ) ).

% int_mod_pos_eq
tff(fact_5641_int__mod__neg__eq,axiom,
    ! [Aa: int,Ba: int,Q4: int,R2: int] :
      ( ( Aa = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Ba),Q4)),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,Ba),R2)
         => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),Ba) = R2 ) ) ) ) ).

% int_mod_neg_eq
tff(fact_5642_split__zmod,axiom,
    ! [P: fun_int_o,Na: int,K: int] :
      ( aa_int_o(P,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Na),K))
    <=> ( ( ( K = zero_zero_int )
         => aa_int_o(P,Na) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
         => ! [I3: int,J3: int] :
              ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),J3)
                & aa_int_o(aa_int_fun_int_o(ord_less_int,J3),K)
                & ( Na = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I3)),J3) ) )
             => aa_int_o(P,J3) ) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
         => ! [I3: int,J3: int] :
              ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),J3)
                & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J3),zero_zero_int)
                & ( Na = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I3)),J3) ) )
             => aa_int_o(P,J3) ) ) ) ) ).

% split_zmod
tff(fact_5643_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_5644_zmod__minus1,axiom,
    ! [Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Ba)
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,one_one_int)),Ba) = aa_int_int(aa_int_fun_int_int(minus_minus_int,Ba),one_one_int) ) ) ).

% zmod_minus1
tff(fact_5645_signed__take__bit__int__less__exp,axiom,
    ! [Na: nat,K: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,Na),K)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)) ).

% signed_take_bit_int_less_exp
tff(fact_5646_zmod__zmult2__eq,axiom,
    ! [C: int,Aa: int,Ba: 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,Aa),aa_int_int(aa_int_fun_int_int(times_times_int,Ba),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Ba),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),Ba)),C))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),Ba)) ) ) ).

% zmod_zmult2_eq
tff(fact_5647_verit__le__mono__div__int,axiom,
    ! [A3: int,B4: int,Na: 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),Na)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,
              aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A3),Na)),
                $ite(aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B4),Na) = zero_zero_int,one_one_int,zero_zero_int))),aa_int_int(aa_int_fun_int_int(divide_divide_int,B4),Na)) ) ) ).

% verit_le_mono_div_int
tff(fact_5648_signed__take__bit__int__less__self__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,Na),K)),K)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)),K) ) ).

% signed_take_bit_int_less_self_iff
tff(fact_5649_signed__take__bit__int__greater__eq__self__iff,axiom,
    ! [K: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,Na),K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)) ) ).

% signed_take_bit_int_greater_eq_self_iff
tff(fact_5650_split__pos__lemma,axiom,
    ! [K: int,P: fun_int_fun_int_o,Na: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
     => ( aa_int_o(aa_int_fun_int_o(P,aa_int_int(aa_int_fun_int_int(divide_divide_int,Na),K)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Na),K))
      <=> ! [I3: int,J3: int] :
            ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),J3)
              & aa_int_o(aa_int_fun_int_o(ord_less_int,J3),K)
              & ( Na = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I3)),J3) ) )
           => aa_int_o(aa_int_fun_int_o(P,I3),J3) ) ) ) ).

% split_pos_lemma
tff(fact_5651_split__neg__lemma,axiom,
    ! [K: int,P: fun_int_fun_int_o,Na: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(P,aa_int_int(aa_int_fun_int_int(divide_divide_int,Na),K)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Na),K))
      <=> ! [I3: int,J3: int] :
            ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),J3)
              & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J3),zero_zero_int)
              & ( Na = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I3)),J3) ) )
           => aa_int_o(aa_int_fun_int_o(P,I3),J3) ) ) ) ).

% split_neg_lemma
tff(fact_5652_signed__take__bit__int__greater__eq__minus__exp,axiom,
    ! [Na: nat,K: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na))),aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,Na),K)) ).

% signed_take_bit_int_greater_eq_minus_exp
tff(fact_5653_signed__take__bit__int__less__eq__self__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,Na),K)),K)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na))),K) ) ).

% signed_take_bit_int_less_eq_self_iff
tff(fact_5654_signed__take__bit__int__greater__self__iff,axiom,
    ! [K: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,Na),K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na))) ) ).

% signed_take_bit_int_greater_self_iff
tff(fact_5655_signed__take__bit__int__less__eq,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)),K)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,Na),K)),aa_int_int(aa_int_fun_int_int(minus_minus_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,Na)))) ) ).

% signed_take_bit_int_less_eq
tff(fact_5656_signed__take__bit__int__eq__self,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na))),K)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na))
       => ( aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,Na),K) = K ) ) ) ).

% signed_take_bit_int_eq_self
tff(fact_5657_signed__take__bit__int__eq__self__iff,axiom,
    ! [Na: nat,K: int] :
      ( ( aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,Na),K) = K )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na))),K)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)) ) ) ).

% signed_take_bit_int_eq_self_iff
tff(fact_5658_pos__zmod__mult__2,axiom,
    ! [Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Aa)
     => ( 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))),Ba))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Aa)) = 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,Ba),Aa))) ) ) ).

% pos_zmod_mult_2
tff(fact_5659_neg__zmod__mult__2,axiom,
    ! [Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),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))),Ba))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Aa)) = 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,Ba),one_one_int)),Aa))),one_one_int) ) ) ).

% neg_zmod_mult_2
tff(fact_5660_signed__take__bit__int__greater__eq,axiom,
    ! [K: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,Na)))),aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,Na),K)) ) ).

% signed_take_bit_int_greater_eq
tff(fact_5661_even__flip__bit__iff,axiom,
    ! [M: nat,Aa: 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_se2159334234014336723it_int,M),Aa))
    <=> ~ ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Aa)
        <=> ( M = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
tff(fact_5662_even__flip__bit__iff,axiom,
    ! [M: nat,Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_nat6383147911832638789nteger(bit_se1345352211410354436nteger,M),Aa))
    <=> ~ ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Aa)
        <=> ( M = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
tff(fact_5663_even__flip__bit__iff,axiom,
    ! [M: nat,Aa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,M),Aa))
    <=> ~ ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Aa)
        <=> ( M = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
tff(fact_5664_signed__take__bit__Suc,axiom,
    ! [Na: nat,Aa: code_integer] : aa_Cod1783974652533396728nteger(bit_ri6519982836138164636nteger(aa_nat_nat(suc,Na)),Aa) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Aa),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(bit_ri6519982836138164636nteger(Na),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Aa),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))))) ).

% signed_take_bit_Suc
tff(fact_5665_signed__take__bit__Suc,axiom,
    ! [Na: nat,Aa: int] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,aa_nat_nat(suc,Na)),Aa) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),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_ri631733984087533419it_int,Na),aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% signed_take_bit_Suc
tff(fact_5666_vebt__member_Opelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(vEBT_vebt_member(Xb),Xaa)
     => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel),produc738532404422230701BT_nat(Xb,Xaa))
       => ( ! [A2: $o,B2: $o] :
              ( ( Xb = vEBT_Leaf((A2),(B2)) )
             => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel),produc738532404422230701BT_nat(vEBT_Leaf((A2),(B2)),Xaa))
               => $ite(
                    Xaa = zero_zero_nat,
                    (A2),
                    $ite(Xaa = one_one_nat,(B2),$false) ) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2) )
               => ~ aa_Pro632845113475883111_nat_o(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) )
                 => ~ aa_Pro632845113475883111_nat_o(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) )
                   => ~ aa_Pro632845113475883111_nat_o(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,Va2: nat,TreeList2: list_VEBT_VEBT,Summary2: 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,Va2)),TreeList2,Summary2) )
                     => ( aa_Pro632845113475883111_nat_o(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,Va2)),TreeList2,Summary2),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,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                    $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(TreeList2,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(3)
tff(fact_5667_vebt__member_Opelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( aa_nat_o(vEBT_vebt_member(Xb),Xaa)
      <=> (Y) )
     => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel),produc738532404422230701BT_nat(Xb,Xaa))
       => ( ! [A2: $o,B2: $o] :
              ( ( Xb = vEBT_Leaf((A2),(B2)) )
             => ( ( (Y)
                <=> $ite(
                      Xaa = zero_zero_nat,
                      (A2),
                      $ite(Xaa = one_one_nat,(B2),$false) ) )
               => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel),produc738532404422230701BT_nat(vEBT_Leaf((A2),(B2)),Xaa)) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv2,Uw2) )
               => ( ~ (Y)
                 => ~ aa_Pro632845113475883111_nat_o(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) )
                 => ( ~ (Y)
                   => ~ aa_Pro632845113475883111_nat_o(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) )
                   => ( ~ (Y)
                     => ~ aa_Pro632845113475883111_nat_o(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,Va2: nat,TreeList2: list_VEBT_VEBT,Summary2: 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,Va2)),TreeList2,Summary2) )
                     => ( ( (Y)
                        <=> $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,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                      $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(TreeList2,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) )
                       => ~ aa_Pro632845113475883111_nat_o(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,Va2)),TreeList2,Summary2),Xaa)) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(1)
tff(fact_5668_VEBT__internal_Onaive__member_Opelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( aa_nat_o(vEBT_V5719532721284313246member(Xb),Xaa)
      <=> (Y) )
     => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel),produc738532404422230701BT_nat(Xb,Xaa))
       => ( ! [A2: $o,B2: $o] :
              ( ( Xb = vEBT_Leaf((A2),(B2)) )
             => ( ( (Y)
                <=> $ite(
                      Xaa = zero_zero_nat,
                      (A2),
                      $ite(Xaa = one_one_nat,(B2),$false) ) )
               => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel),produc738532404422230701BT_nat(vEBT_Leaf((A2),(B2)),Xaa)) ) )
         => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2) )
               => ( ~ (Y)
                 => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel),produc738532404422230701BT_nat(vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2),Xaa)) ) )
           => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT,S: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList2,S) )
                 => ( ( (Y)
                    <=> $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)))),
                          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList2,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) )
                   => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel),produc738532404422230701BT_nat(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList2,S),Xaa)) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(1)
tff(fact_5669_VEBT__internal_Onaive__member_Opelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(vEBT_V5719532721284313246member(Xb),Xaa)
     => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel),produc738532404422230701BT_nat(Xb,Xaa))
       => ( ! [A2: $o,B2: $o] :
              ( ( Xb = vEBT_Leaf((A2),(B2)) )
             => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel),produc738532404422230701BT_nat(vEBT_Leaf((A2),(B2)),Xaa))
               => ~ $ite(
                      Xaa = zero_zero_nat,
                      (A2),
                      $ite(Xaa = one_one_nat,(B2),$false) ) ) )
         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT,S: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList2,S) )
               => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel),produc738532404422230701BT_nat(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList2,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)))),
                        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList2,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(2)
tff(fact_5670_VEBT__internal_Onaive__member_Opelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(vEBT_V5719532721284313246member(Xb),Xaa)
     => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel),produc738532404422230701BT_nat(Xb,Xaa))
       => ( ! [A2: $o,B2: $o] :
              ( ( Xb = vEBT_Leaf((A2),(B2)) )
             => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel),produc738532404422230701BT_nat(vEBT_Leaf((A2),(B2)),Xaa))
               => $ite(
                    Xaa = zero_zero_nat,
                    (A2),
                    $ite(Xaa = one_one_nat,(B2),$false) ) ) )
         => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2) )
               => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel),produc738532404422230701BT_nat(vEBT_Node(Uu2,zero_zero_nat,Uv2,Uw2),Xaa)) )
           => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT,S: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList2,S) )
                 => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel),produc738532404422230701BT_nat(vEBT_Node(Uy2,aa_nat_nat(suc,V2),TreeList2,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)))),
                        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList2,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(3)
tff(fact_5671_vebt__member_Opelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(vEBT_vebt_member(Xb),Xaa)
     => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel),produc738532404422230701BT_nat(Xb,Xaa))
       => ( ! [A2: $o,B2: $o] :
              ( ( Xb = vEBT_Leaf((A2),(B2)) )
             => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel),produc738532404422230701BT_nat(vEBT_Leaf((A2),(B2)),Xaa))
               => ~ $ite(
                      Xaa = zero_zero_nat,
                      (A2),
                      $ite(Xaa = one_one_nat,(B2),$false) ) ) )
         => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary2: 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,Va2)),TreeList2,Summary2) )
               => ( aa_Pro632845113475883111_nat_o(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,Va2)),TreeList2,Summary2),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,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(TreeList2,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(2)
tff(fact_5672_VEBT__internal_Omembermima_Opelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( aa_nat_o(vEBT_VEBT_membermima(Xb),Xaa)
      <=> (Y) )
     => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel),produc738532404422230701BT_nat(Xb,Xaa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( Xb = vEBT_Leaf((Uu2),(Uv2)) )
             => ( ~ (Y)
               => ~ aa_Pro632845113475883111_nat_o(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) )
               => ( ~ (Y)
                 => ~ aa_Pro632845113475883111_nat_o(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) )
                 => ( ( (Y)
                    <=> ( ( Xaa = Mi )
                        | ( Xaa = Ma ) ) )
                   => ~ aa_Pro632845113475883111_nat_o(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,TreeList2: 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),TreeList2,Vc2) )
                   => ( ( (Y)
                      <=> ( ( 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)))),
                              $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList2,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) )
                     => ~ aa_Pro632845113475883111_nat_o(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),TreeList2,Vc2),Xaa)) ) )
               => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2,Vd2) )
                     => ( ( (Y)
                        <=> $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)))),
                              $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList2,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) )
                       => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel),produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2,Vd2),Xaa)) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(1)
tff(fact_5673_VEBT__internal_Omembermima_Opelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(vEBT_VEBT_membermima(Xb),Xaa)
     => ( aa_Pro632845113475883111_nat_o(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) )
             => ( aa_Pro632845113475883111_nat_o(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,TreeList2: 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),TreeList2,Vc2) )
               => ( aa_Pro632845113475883111_nat_o(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),TreeList2,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)))),
                          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList2,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) )
           => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2,Vd2) )
                 => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel),produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2,Vd2),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)))),
                          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList2,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(2)
tff(fact_5674_VEBT__internal_Omembermima_Opelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(vEBT_VEBT_membermima(Xb),Xaa)
     => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel),produc738532404422230701BT_nat(Xb,Xaa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( Xb = vEBT_Leaf((Uu2),(Uv2)) )
             => ~ aa_Pro632845113475883111_nat_o(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) )
               => ~ aa_Pro632845113475883111_nat_o(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) )
                 => ( aa_Pro632845113475883111_nat_o(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,TreeList2: 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),TreeList2,Vc2) )
                   => ( aa_Pro632845113475883111_nat_o(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),TreeList2,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)))),
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList2,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) )
               => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2,Vd2) )
                     => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel),produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2,Vd2),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)))),
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList2,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(3)
tff(fact_5675_artanh__def,axiom,
    ! [Xb: real] : aa_real_real(artanh_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xb)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),Xb)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% artanh_def
tff(fact_5676_arsinh__0,axiom,
    aa_real_real(arsinh_real,zero_zero_real) = zero_zero_real ).

% arsinh_0
tff(fact_5677_signed__take__bit__Suc__minus__bit1,axiom,
    ! [Na: nat,K: num] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,aa_nat_nat(suc,Na)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,Na),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))),one_one_int))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).

% signed_take_bit_Suc_minus_bit1
tff(fact_5678_accp__subset,axiom,
    ! [R1: fun_VE2962686792283380846VEBT_o,R22: fun_VE2962686792283380846VEBT_o] :
      ( ord_le1270047517488690190VEBT_o(R1,R22)
     => ord_le88303844413636359VEBT_o(accp_VEBT_VEBT(R22),accp_VEBT_VEBT(R1)) ) ).

% accp_subset
tff(fact_5679_accp__subset,axiom,
    ! [R1: fun_Pr8877776288780692068_nat_o,R22: fun_Pr8877776288780692068_nat_o] :
      ( ord_le4038806025055212036_nat_o(R1,R22)
     => ord_le1769015658490769310_nat_o(accp_P2887432264394892906BT_nat(R22),accp_P2887432264394892906BT_nat(R1)) ) ).

% accp_subset
tff(fact_5680_accp__subset,axiom,
    ! [R1: fun_nat_fun_nat_o,R22: fun_nat_fun_nat_o] :
      ( ord_le768339454706049738_nat_o(R1,R22)
     => ord_le7240017405068453697_nat_o(accp_nat(R22),accp_nat(R1)) ) ).

% accp_subset
tff(fact_5681_accp__subset,axiom,
    ! [R1: fun_Pr2028404973800837934_nat_o,R22: fun_Pr2028404973800837934_nat_o] :
      ( ord_le3650406611588909774_nat_o(R1,R22)
     => ord_le1572679713340831354_nat_o(accp_P4275260045618599050at_nat(R22),accp_P4275260045618599050at_nat(R1)) ) ).

% accp_subset
tff(fact_5682_accp__subset,axiom,
    ! [R1: fun_Pr9018117161326028078_int_o,R22: fun_Pr9018117161326028078_int_o] :
      ( ord_le1416746762259324110_int_o(R1,R22)
     => ord_le14110778710936002_int_o(accp_P1096762738010456898nt_int(R22),accp_P1096762738010456898nt_int(R1)) ) ).

% accp_subset
tff(fact_5683_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_real,Xb: fun_real_complex,Y: fun_real_complex] :
      ( finite_finite_real(collect_real(aa_fun5218151893725926128real_o(aTP_Lamm_de(I5),Xb)))
     => ( finite_finite_real(collect_real(aa_fun5218151893725926128real_o(aTP_Lamm_de(I5),Y)))
       => finite_finite_real(collect_real(aa_fun5218151893725926128real_o(aa_fun1467772539989157170real_o(aTP_Lamm_df(I5),Xb),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_5684_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,Xb: fun_nat_complex,Y: fun_nat_complex] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun3019658233935452328_nat_o(aTP_Lamm_dg(I5),Xb)))
     => ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun3019658233935452328_nat_o(aTP_Lamm_dg(I5),Y)))
       => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun3019658233935452328_nat_o(aa_fun894001447060918982_nat_o(aTP_Lamm_dh(I5),Xb),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_5685_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_int,Xb: fun_int_complex,Y: fun_int_complex] :
      ( aa_set_int_o(finite_finite_int,collect_int(aa_fun5261878201068379632_int_o(aTP_Lamm_di(I5),Xb)))
     => ( aa_set_int_o(finite_finite_int,collect_int(aa_fun5261878201068379632_int_o(aTP_Lamm_di(I5),Y)))
       => aa_set_int_o(finite_finite_int,collect_int(aa_fun5261878201068379632_int_o(aa_fun2392861216585087154_int_o(aTP_Lamm_dj(I5),Xb),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_5686_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_complex,Xb: fun_complex_complex,Y: fun_complex_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun573566726631320556plex_o(aTP_Lamm_dk(I5),Xb)))
     => ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun573566726631320556plex_o(aTP_Lamm_dk(I5),Y)))
       => aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun573566726631320556plex_o(aa_fun4378733901284544684plex_o(aTP_Lamm_dl(I5),Xb),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_5687_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_real,Xb: fun_real_real,Y: fun_real_real] :
      ( finite_finite_real(collect_real(aa_fun2895569901277098738real_o(aTP_Lamm_dm(I5),Xb)))
     => ( finite_finite_real(collect_real(aa_fun2895569901277098738real_o(aTP_Lamm_dm(I5),Y)))
       => finite_finite_real(collect_real(aa_fun2895569901277098738real_o(aa_fun3693225987673163702real_o(aTP_Lamm_dn(I5),Xb),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_5688_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,Xb: fun_nat_real,Y: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun6456149469358400426_nat_o(aTP_Lamm_do(I5),Xb)))
     => ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun6456149469358400426_nat_o(aTP_Lamm_do(I5),Y)))
       => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun6456149469358400426_nat_o(aa_fun5464864432877286986_nat_o(aTP_Lamm_dp(I5),Xb),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_5689_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_int,Xb: fun_int_real,Y: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,collect_int(aa_fun4444401208693448434_int_o(aTP_Lamm_dq(I5),Xb)))
     => ( aa_set_int_o(finite_finite_int,collect_int(aa_fun4444401208693448434_int_o(aTP_Lamm_dq(I5),Y)))
       => aa_set_int_o(finite_finite_int,collect_int(aa_fun4444401208693448434_int_o(aa_fun135101165924023862_int_o(aTP_Lamm_dr(I5),Xb),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_5690_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_complex,Xb: fun_complex_real,Y: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun6195079535267300846plex_o(aTP_Lamm_ds(I5),Xb)))
     => ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun6195079535267300846plex_o(aTP_Lamm_ds(I5),Y)))
       => aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun6195079535267300846plex_o(aa_fun805843229300662576plex_o(aTP_Lamm_dt(I5),Xb),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_5691_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_real,Xb: fun_real_rat,Y: fun_real_rat] :
      ( finite_finite_real(collect_real(aa_fun3209485225100452678real_o(aTP_Lamm_du(I5),Xb)))
     => ( finite_finite_real(collect_real(aa_fun3209485225100452678real_o(aTP_Lamm_du(I5),Y)))
       => finite_finite_real(collect_real(aa_fun3209485225100452678real_o(aa_fun8459995508950063326real_o(aTP_Lamm_dv(I5),Xb),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_5692_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,Xb: fun_nat_rat,Y: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun1118977351124560894_nat_o(aTP_Lamm_dw(I5),Xb)))
     => ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun1118977351124560894_nat_o(aTP_Lamm_dw(I5),Y)))
       => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun1118977351124560894_nat_o(aa_fun8314661095172591474_nat_o(aTP_Lamm_dx(I5),Xb),Y))) ) ) ).

% prod.finite_Collect_op
tff(fact_5693_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_real,Xb: fun_real_real,Y: fun_real_real] :
      ( finite_finite_real(collect_real(aa_fun2895569901277098738real_o(aTP_Lamm_dy(I5),Xb)))
     => ( finite_finite_real(collect_real(aa_fun2895569901277098738real_o(aTP_Lamm_dy(I5),Y)))
       => finite_finite_real(collect_real(aa_fun2895569901277098738real_o(aa_fun3693225987673163702real_o(aTP_Lamm_dz(I5),Xb),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_5694_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,Xb: fun_nat_real,Y: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun6456149469358400426_nat_o(aTP_Lamm_ea(I5),Xb)))
     => ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun6456149469358400426_nat_o(aTP_Lamm_ea(I5),Y)))
       => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun6456149469358400426_nat_o(aa_fun5464864432877286986_nat_o(aTP_Lamm_eb(I5),Xb),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_5695_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_int,Xb: fun_int_real,Y: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,collect_int(aa_fun4444401208693448434_int_o(aTP_Lamm_ec(I5),Xb)))
     => ( aa_set_int_o(finite_finite_int,collect_int(aa_fun4444401208693448434_int_o(aTP_Lamm_ec(I5),Y)))
       => aa_set_int_o(finite_finite_int,collect_int(aa_fun4444401208693448434_int_o(aa_fun135101165924023862_int_o(aTP_Lamm_ed(I5),Xb),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_5696_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_complex,Xb: fun_complex_real,Y: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun6195079535267300846plex_o(aTP_Lamm_ee(I5),Xb)))
     => ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun6195079535267300846plex_o(aTP_Lamm_ee(I5),Y)))
       => aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun6195079535267300846plex_o(aa_fun805843229300662576plex_o(aTP_Lamm_ef(I5),Xb),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_5697_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_real,Xb: fun_real_rat,Y: fun_real_rat] :
      ( finite_finite_real(collect_real(aa_fun3209485225100452678real_o(aTP_Lamm_eg(I5),Xb)))
     => ( finite_finite_real(collect_real(aa_fun3209485225100452678real_o(aTP_Lamm_eg(I5),Y)))
       => finite_finite_real(collect_real(aa_fun3209485225100452678real_o(aa_fun8459995508950063326real_o(aTP_Lamm_eh(I5),Xb),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_5698_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,Xb: fun_nat_rat,Y: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun1118977351124560894_nat_o(aTP_Lamm_ei(I5),Xb)))
     => ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun1118977351124560894_nat_o(aTP_Lamm_ei(I5),Y)))
       => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun1118977351124560894_nat_o(aa_fun8314661095172591474_nat_o(aTP_Lamm_ej(I5),Xb),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_5699_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_int,Xb: fun_int_rat,Y: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,collect_int(aa_fun7094158214180239686_int_o(aTP_Lamm_ek(I5),Xb)))
     => ( aa_set_int_o(finite_finite_int,collect_int(aa_fun7094158214180239686_int_o(aTP_Lamm_ek(I5),Y)))
       => aa_set_int_o(finite_finite_int,collect_int(aa_fun7094158214180239686_int_o(aa_fun1445893531507182942_int_o(aTP_Lamm_el(I5),Xb),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_5700_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_complex,Xb: fun_complex_rat,Y: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun5099116602557847362plex_o(aTP_Lamm_em(I5),Xb)))
     => ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun5099116602557847362plex_o(aTP_Lamm_em(I5),Y)))
       => aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun5099116602557847362plex_o(aa_fun4586712283274990936plex_o(aTP_Lamm_en(I5),Xb),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_5701_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_real,Xb: fun_real_nat,Y: fun_real_nat] :
      ( finite_finite_real(collect_real(aa_fun7775777430665565006real_o(aTP_Lamm_eo(I5),Xb)))
     => ( finite_finite_real(collect_real(aa_fun7775777430665565006real_o(aTP_Lamm_eo(I5),Y)))
       => finite_finite_real(collect_real(aa_fun7775777430665565006real_o(aa_fun1158036852828689646real_o(aTP_Lamm_ep(I5),Xb),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_5702_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,Xb: fun_nat_nat,Y: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun8673969331427483654_nat_o(aTP_Lamm_eq(I5),Xb)))
     => ( aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun8673969331427483654_nat_o(aTP_Lamm_eq(I5),Y)))
       => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun8673969331427483654_nat_o(aa_fun6348600740168882050_nat_o(aTP_Lamm_er(I5),Xb),Y))) ) ) ).

% sum.finite_Collect_op
tff(fact_5703_insert__simp__excp,axiom,
    ! [Mi2: nat,Deg: nat,TreeList: list_VEBT_VEBT,Xb: nat,Ma2: nat,Summary: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mi2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Mi2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)
         => ( ( Xb != Ma2 )
           => ( vEBT_vebt_insert(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Deg,TreeList,Summary),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,Mi2),Ma2))),Deg,list_u1324408373059187874T_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mi2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),vEBT_vebt_insert(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mi2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Mi2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),
                  $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mi2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_insert(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mi2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),Summary)) ) ) ) ) ) ).

% insert_simp_excp
tff(fact_5704_ln__inj__iff,axiom,
    ! [Xb: real,Y: 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),Y)
       => ( ( aa_real_real(ln_ln_real,Xb) = aa_real_real(ln_ln_real,Y) )
        <=> ( Xb = Y ) ) ) ) ).

% ln_inj_iff
tff(fact_5705_ln__less__cancel__iff,axiom,
    ! [Xb: real,Y: 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),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(ln_ln_real,Xb)),aa_real_real(ln_ln_real,Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y) ) ) ) ).

% ln_less_cancel_iff
tff(fact_5706_max__bot,axiom,
    ! [Xb: filter_nat] : ord_max_filter_nat(bot_bot_filter_nat,Xb) = Xb ).

% max_bot
tff(fact_5707_max__bot,axiom,
    ! [Xb: set_real] : ord_max_set_real(bot_bot_set_real,Xb) = Xb ).

% max_bot
tff(fact_5708_max__bot,axiom,
    ! [Xb: set_nat] : ord_max_set_nat(bot_bot_set_nat,Xb) = Xb ).

% max_bot
tff(fact_5709_max__bot,axiom,
    ! [Xb: set_int] : ord_max_set_int(bot_bot_set_int,Xb) = Xb ).

% max_bot
tff(fact_5710_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_5711_max__bot2,axiom,
    ! [Xb: filter_nat] : ord_max_filter_nat(Xb,bot_bot_filter_nat) = Xb ).

% max_bot2
tff(fact_5712_max__bot2,axiom,
    ! [Xb: set_real] : ord_max_set_real(Xb,bot_bot_set_real) = Xb ).

% max_bot2
tff(fact_5713_max__bot2,axiom,
    ! [Xb: set_nat] : ord_max_set_nat(Xb,bot_bot_set_nat) = Xb ).

% max_bot2
tff(fact_5714_max__bot2,axiom,
    ! [Xb: set_int] : ord_max_set_int(Xb,bot_bot_set_int) = Xb ).

% max_bot2
tff(fact_5715_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_5716_max__Suc__Suc,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,Na)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,M),Na)) ).

% max_Suc_Suc
tff(fact_5717_max__nat_Oeq__neutr__iff,axiom,
    ! [Aa: nat,Ba: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Aa),Ba) = zero_zero_nat )
    <=> ( ( Aa = zero_zero_nat )
        & ( Ba = zero_zero_nat ) ) ) ).

% max_nat.eq_neutr_iff
tff(fact_5718_max__nat_Oleft__neutral,axiom,
    ! [Aa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,zero_zero_nat),Aa) = Aa ).

% max_nat.left_neutral
tff(fact_5719_max__nat_Oneutr__eq__iff,axiom,
    ! [Aa: nat,Ba: nat] :
      ( ( zero_zero_nat = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Aa),Ba) )
    <=> ( ( Aa = zero_zero_nat )
        & ( Ba = zero_zero_nat ) ) ) ).

% max_nat.neutr_eq_iff
tff(fact_5720_max__nat_Oright__neutral,axiom,
    ! [Aa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Aa),zero_zero_nat) = Aa ).

% max_nat.right_neutral
tff(fact_5721_max__0L,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,zero_zero_nat),Na) = Na ).

% max_0L
tff(fact_5722_max__0R,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Na),zero_zero_nat) = Na ).

% max_0R
tff(fact_5723_semiring__norm_I73_J,axiom,
    ! [M: num,Na: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(bit1,M)),aa_num_num(bit1,Na))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,M),Na) ) ).

% semiring_norm(73)
tff(fact_5724_semiring__norm_I80_J,axiom,
    ! [M: num,Na: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(bit1,M)),aa_num_num(bit1,Na))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,M),Na) ) ).

% semiring_norm(80)
tff(fact_5725_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      aa_real_real(aa_rea7340605660300375871l_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_5726_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      ord_ma741700101516333627d_enat(numera1916890842035813515d_enat(U),numera1916890842035813515d_enat(V)) = $ite(aa_Extended_enat_o(ord_le2932123472753598470d_enat(numera1916890842035813515d_enat(U)),numera1916890842035813515d_enat(V)),numera1916890842035813515d_enat(V),numera1916890842035813515d_enat(U)) ).

% max_number_of(1)
tff(fact_5727_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      ord_max_Code_integer(aa_num_Code_integer(numera6620942414471956472nteger,U),aa_num_Code_integer(numera6620942414471956472nteger,V)) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_num_Code_integer(numera6620942414471956472nteger,U)),aa_num_Code_integer(numera6620942414471956472nteger,V)),aa_num_Code_integer(numera6620942414471956472nteger,V),aa_num_Code_integer(numera6620942414471956472nteger,U)) ).

% max_number_of(1)
tff(fact_5728_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_5729_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_5730_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      aa_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_5731_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_5732_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_5733_max__0__1_I3_J,axiom,
    ! [Xb: num] : aa_real_real(aa_rea7340605660300375871l_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_5734_max__0__1_I3_J,axiom,
    ! [Xb: num] : aa_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_5735_max__0__1_I3_J,axiom,
    ! [Xb: num] : ord_ma741700101516333627d_enat(zero_z5237406670263579293d_enat,numera1916890842035813515d_enat(Xb)) = numera1916890842035813515d_enat(Xb) ).

% max_0_1(3)
tff(fact_5736_max__0__1_I3_J,axiom,
    ! [Xb: num] : ord_max_Code_integer(zero_z3403309356797280102nteger,aa_num_Code_integer(numera6620942414471956472nteger,Xb)) = aa_num_Code_integer(numera6620942414471956472nteger,Xb) ).

% max_0_1(3)
tff(fact_5737_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_5738_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_5739_max__0__1_I4_J,axiom,
    ! [Xb: num] : aa_real_real(aa_rea7340605660300375871l_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_5740_max__0__1_I4_J,axiom,
    ! [Xb: num] : aa_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_5741_max__0__1_I4_J,axiom,
    ! [Xb: num] : ord_ma741700101516333627d_enat(numera1916890842035813515d_enat(Xb),zero_z5237406670263579293d_enat) = numera1916890842035813515d_enat(Xb) ).

% max_0_1(4)
tff(fact_5742_max__0__1_I4_J,axiom,
    ! [Xb: num] : ord_max_Code_integer(aa_num_Code_integer(numera6620942414471956472nteger,Xb),zero_z3403309356797280102nteger) = aa_num_Code_integer(numera6620942414471956472nteger,Xb) ).

% max_0_1(4)
tff(fact_5743_max__0__1_I1_J,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,zero_zero_real),one_one_real) = one_one_real ).

% max_0_1(1)
tff(fact_5744_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_5745_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_5746_max__0__1_I1_J,axiom,
    ord_max_Code_integer(zero_z3403309356797280102nteger,one_one_Code_integer) = one_one_Code_integer ).

% max_0_1(1)
tff(fact_5747_max__0__1_I1_J,axiom,
    aa_int_int(ord_max_int(zero_zero_int),one_one_int) = one_one_int ).

% max_0_1(1)
tff(fact_5748_max__0__1_I2_J,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,one_one_real),zero_zero_real) = one_one_real ).

% max_0_1(2)
tff(fact_5749_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_5750_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_5751_max__0__1_I2_J,axiom,
    ord_max_Code_integer(one_one_Code_integer,zero_z3403309356797280102nteger) = one_one_Code_integer ).

% max_0_1(2)
tff(fact_5752_max__0__1_I2_J,axiom,
    aa_int_int(ord_max_int(one_one_int),zero_zero_int) = one_one_int ).

% max_0_1(2)
tff(fact_5753_ln__le__cancel__iff,axiom,
    ! [Xb: real,Y: 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),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,Xb)),aa_real_real(ln_ln_real,Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y) ) ) ) ).

% ln_le_cancel_iff
tff(fact_5754_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_5755_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_5756_max__0__1_I6_J,axiom,
    ! [Xb: num] : aa_real_real(aa_rea7340605660300375871l_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_5757_max__0__1_I6_J,axiom,
    ! [Xb: num] : aa_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_5758_max__0__1_I6_J,axiom,
    ! [Xb: num] : ord_ma741700101516333627d_enat(numera1916890842035813515d_enat(Xb),one_on7984719198319812577d_enat) = numera1916890842035813515d_enat(Xb) ).

% max_0_1(6)
tff(fact_5759_max__0__1_I6_J,axiom,
    ! [Xb: num] : ord_max_Code_integer(aa_num_Code_integer(numera6620942414471956472nteger,Xb),one_one_Code_integer) = aa_num_Code_integer(numera6620942414471956472nteger,Xb) ).

% max_0_1(6)
tff(fact_5760_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_5761_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_5762_max__0__1_I5_J,axiom,
    ! [Xb: num] : aa_real_real(aa_rea7340605660300375871l_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_5763_max__0__1_I5_J,axiom,
    ! [Xb: num] : aa_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_5764_max__0__1_I5_J,axiom,
    ! [Xb: num] : ord_ma741700101516333627d_enat(one_on7984719198319812577d_enat,numera1916890842035813515d_enat(Xb)) = numera1916890842035813515d_enat(Xb) ).

% max_0_1(5)
tff(fact_5765_max__0__1_I5_J,axiom,
    ! [Xb: num] : ord_max_Code_integer(one_one_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,Xb)) = aa_num_Code_integer(numera6620942414471956472nteger,Xb) ).

% max_0_1(5)
tff(fact_5766_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_5767_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_5768_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_5769_ln__one,axiom,
    aa_real_real(ln_ln_real,one_one_real) = zero_zero_real ).

% ln_one
tff(fact_5770_list__update__beyond,axiom,
    ! [Xs: list_VEBT_VEBT,I: nat,Xb: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_s6755466524823107622T_VEBT(Xs)),I)
     => ( list_u1324408373059187874T_VEBT(Xs,I,Xb) = Xs ) ) ).

% list_update_beyond
tff(fact_5771_list__update__beyond,axiom,
    ! [Xs: list_o,I: nat,Xb: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_o(Xs)),I)
     => ( list_update_o(Xs,I,(Xb)) = Xs ) ) ).

% list_update_beyond
tff(fact_5772_list__update__beyond,axiom,
    ! [Xs: list_nat,I: nat,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_nat(Xs)),I)
     => ( list_update_nat(Xs,I,Xb) = Xs ) ) ).

% list_update_beyond
tff(fact_5773_semiring__norm_I72_J,axiom,
    ! [M: num,Na: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(bit0,M)),aa_num_num(bit1,Na))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,M),Na) ) ).

% semiring_norm(72)
tff(fact_5774_semiring__norm_I81_J,axiom,
    ! [M: num,Na: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(bit1,M)),aa_num_num(bit0,Na))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,M),Na) ) ).

% semiring_norm(81)
tff(fact_5775_semiring__norm_I70_J,axiom,
    ! [M: num] : ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(bit1,M)),one) ).

% semiring_norm(70)
tff(fact_5776_semiring__norm_I77_J,axiom,
    ! [Na: num] : aa_num_o(aa_num_fun_num_o(ord_less_num,one),aa_num_num(bit1,Na)) ).

% semiring_norm(77)
tff(fact_5777_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_5778_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_5779_dbl__inc__simps_I5_J,axiom,
    ! [K: num] : neg_nu5831290666863070958nteger(aa_num_Code_integer(numera6620942414471956472nteger,K)) = aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,K)) ).

% dbl_inc_simps(5)
tff(fact_5780_max__number__of_I4_J,axiom,
    ! [U: num,V: num] :
      aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,U))),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))) = $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_5781_max__number__of_I4_J,axiom,
    ! [U: num,V: num] :
      ord_max_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,U)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,V))) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,U))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,V))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,V)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,U))) ).

% max_number_of(4)
tff(fact_5782_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_5783_max__number__of_I4_J,axiom,
    ! [U: num,V: num] :
      aa_int_int(ord_max_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))) = $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_5784_max__number__of_I3_J,axiom,
    ! [U: num,V: num] :
      aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,U))),aa_num_real(numeral_numeral_real,V)) = $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_5785_max__number__of_I3_J,axiom,
    ! [U: num,V: num] :
      ord_max_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,U)),aa_num_Code_integer(numera6620942414471956472nteger,V)) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,U))),aa_num_Code_integer(numera6620942414471956472nteger,V)),aa_num_Code_integer(numera6620942414471956472nteger,V),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,U))) ).

% max_number_of(3)
tff(fact_5786_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_5787_max__number__of_I3_J,axiom,
    ! [U: num,V: num] :
      aa_int_int(ord_max_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U))),aa_num_int(numeral_numeral_int,V)) = $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_5788_max__number__of_I2_J,axiom,
    ! [U: num,V: num] :
      aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_num_real(numeral_numeral_real,U)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))) = $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_5789_max__number__of_I2_J,axiom,
    ! [U: num,V: num] :
      ord_max_Code_integer(aa_num_Code_integer(numera6620942414471956472nteger,U),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,V))) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_num_Code_integer(numera6620942414471956472nteger,U)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,V))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,V)),aa_num_Code_integer(numera6620942414471956472nteger,U)) ).

% max_number_of(2)
tff(fact_5790_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_5791_max__number__of_I2_J,axiom,
    ! [U: num,V: num] :
      aa_int_int(ord_max_int(aa_num_int(numeral_numeral_int,U)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))) = $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_5792_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_5793_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_5794_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_int,Xb: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_int(Xs))
     => ( aa_nat_int(nth_int(list_update_int(Xs,I,Xb)),I) = Xb ) ) ).

% nth_list_update_eq
tff(fact_5795_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,Xb: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs))
     => ( nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(Xs,I,Xb),I) = Xb ) ) ).

% nth_list_update_eq
tff(fact_5796_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_o,Xb: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_o(Xs))
     => ( aa_nat_o(nth_o(list_update_o(Xs,I,(Xb))),I)
      <=> (Xb) ) ) ).

% nth_list_update_eq
tff(fact_5797_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_nat,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_nat(Xs))
     => ( aa_nat_nat(nth_nat(list_update_nat(Xs,I,Xb)),I) = Xb ) ) ).

% nth_list_update_eq
tff(fact_5798_semiring__norm_I74_J,axiom,
    ! [M: num,Na: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(bit1,M)),aa_num_num(bit0,Na))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,M),Na) ) ).

% semiring_norm(74)
tff(fact_5799_semiring__norm_I79_J,axiom,
    ! [M: num,Na: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(bit0,M)),aa_num_num(bit1,Na))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,M),Na) ) ).

% semiring_norm(79)
tff(fact_5800_insert__simp__norm,axiom,
    ! [Xb: nat,Deg: nat,TreeList: list_VEBT_VEBT,Mi2: nat,Ma2: nat,Summary: 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,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeList))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi2),Xb)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)
         => ( ( Xb != Ma2 )
           => ( vEBT_vebt_insert(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),Deg,TreeList,Summary),Xb) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xb),Ma2))),Deg,list_u1324408373059187874T_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),vEBT_vebt_insert(nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),
                  $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_insert(Summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),Summary)) ) ) ) ) ) ).

% insert_simp_norm
tff(fact_5801_set__swap,axiom,
    ! [I: nat,Xs: list_int,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),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,I,aa_nat_int(nth_int(Xs),J)),J,aa_nat_int(nth_int(Xs),I))) = set_int2(Xs) ) ) ) ).

% set_swap
tff(fact_5802_set__swap,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),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,I,nth_VEBT_VEBT(Xs,J)),J,nth_VEBT_VEBT(Xs,I))) = set_VEBT_VEBT2(Xs) ) ) ) ).

% set_swap
tff(fact_5803_set__swap,axiom,
    ! [I: nat,Xs: list_o,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),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,I,aa_nat_o(nth_o(Xs),J)),J,aa_nat_o(nth_o(Xs),I))) = set_o2(Xs) ) ) ) ).

% set_swap
tff(fact_5804_set__swap,axiom,
    ! [I: nat,Xs: list_nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),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,I,aa_nat_nat(nth_nat(Xs),J)),J,aa_nat_nat(nth_nat(Xs),I))) = set_nat2(Xs) ) ) ) ).

% set_swap
tff(fact_5805_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_5806_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_5807_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_5808_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_5809_dbl__inc__simps_I3_J,axiom,
    neg_nu5831290666863070958nteger(one_one_Code_integer) = aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,one)) ).

% dbl_inc_simps(3)
tff(fact_5810_div__Suc__eq__div__add3,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,Na)))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),Na)) ).

% div_Suc_eq_div_add3
tff(fact_5811_Suc__div__eq__add3__div__numeral,axiom,
    ! [M: nat,V: num] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,M)))),aa_num_nat(numeral_numeral_nat,V)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),M)),aa_num_nat(numeral_numeral_nat,V)) ).

% Suc_div_eq_add3_div_numeral
tff(fact_5812_mod__Suc__eq__mod__add3,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,Na)))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),Na)) ).

% mod_Suc_eq_mod_add3
tff(fact_5813_Suc__mod__eq__add3__mod__numeral,axiom,
    ! [M: nat,V: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,M)))),aa_num_nat(numeral_numeral_nat,V)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),M)),aa_num_nat(numeral_numeral_nat,V)) ).

% Suc_mod_eq_add3_mod_numeral
tff(fact_5814_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_5815_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_5816_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_5817_dbl__dec__simps_I4_J,axiom,
    neg_nu7757733837767384882nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,one))) ).

% dbl_dec_simps(4)
tff(fact_5818_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_5819_signed__take__bit__Suc__bit1,axiom,
    ! [Na: nat,K: num] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,aa_nat_nat(suc,Na)),aa_num_int(numeral_numeral_int,aa_num_num(bit1,K))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,Na),aa_num_int(numeral_numeral_int,K))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).

% signed_take_bit_Suc_bit1
tff(fact_5820_max__def,axiom,
    ! [Aa: code_integer,Ba: code_integer] :
      ord_max_Code_integer(Aa,Ba) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Aa),Ba),Ba,Aa) ).

% max_def
tff(fact_5821_max__def,axiom,
    ! [Aa: set_int,Ba: set_int] :
      ord_max_set_int(Aa,Ba) = $ite(aa_set_int_o(ord_less_eq_set_int(Aa),Ba),Ba,Aa) ).

% max_def
tff(fact_5822_max__def,axiom,
    ! [Aa: rat,Ba: rat] :
      aa_rat_rat(ord_max_rat(Aa),Ba) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),Ba),Ba,Aa) ).

% max_def
tff(fact_5823_max__def,axiom,
    ! [Aa: num,Ba: num] :
      aa_num_num(aa_num_fun_num_num(ord_max_num,Aa),Ba) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Aa),Ba),Ba,Aa) ).

% max_def
tff(fact_5824_max__def,axiom,
    ! [Aa: nat,Ba: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Aa),Ba) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),Ba),Ba,Aa) ).

% max_def
tff(fact_5825_max__def,axiom,
    ! [Aa: int,Ba: int] :
      aa_int_int(ord_max_int(Aa),Ba) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),Ba),Ba,Aa) ).

% max_def
tff(fact_5826_max__absorb1,axiom,
    ! [Y: code_integer,Xb: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Y),Xb)
     => ( ord_max_Code_integer(Xb,Y) = Xb ) ) ).

% max_absorb1
tff(fact_5827_max__absorb1,axiom,
    ! [Y: set_int,Xb: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(Y),Xb)
     => ( ord_max_set_int(Xb,Y) = Xb ) ) ).

% max_absorb1
tff(fact_5828_max__absorb1,axiom,
    ! [Y: rat,Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Xb)
     => ( aa_rat_rat(ord_max_rat(Xb),Y) = Xb ) ) ).

% max_absorb1
tff(fact_5829_max__absorb1,axiom,
    ! [Y: num,Xb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),Xb)
     => ( aa_num_num(aa_num_fun_num_num(ord_max_num,Xb),Y) = Xb ) ) ).

% max_absorb1
tff(fact_5830_max__absorb1,axiom,
    ! [Y: nat,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Xb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xb),Y) = Xb ) ) ).

% max_absorb1
tff(fact_5831_max__absorb1,axiom,
    ! [Y: int,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Xb)
     => ( aa_int_int(ord_max_int(Xb),Y) = Xb ) ) ).

% max_absorb1
tff(fact_5832_max__absorb2,axiom,
    ! [Xb: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Xb),Y)
     => ( ord_max_Code_integer(Xb,Y) = Y ) ) ).

% max_absorb2
tff(fact_5833_max__absorb2,axiom,
    ! [Xb: set_int,Y: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(Xb),Y)
     => ( ord_max_set_int(Xb,Y) = Y ) ) ).

% max_absorb2
tff(fact_5834_max__absorb2,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y)
     => ( aa_rat_rat(ord_max_rat(Xb),Y) = Y ) ) ).

% max_absorb2
tff(fact_5835_max__absorb2,axiom,
    ! [Xb: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Y)
     => ( aa_num_num(aa_num_fun_num_num(ord_max_num,Xb),Y) = Y ) ) ).

% max_absorb2
tff(fact_5836_max__absorb2,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xb),Y) = Y ) ) ).

% max_absorb2
tff(fact_5837_max__absorb2,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y)
     => ( aa_int_int(ord_max_int(Xb),Y) = Y ) ) ).

% max_absorb2
tff(fact_5838_max__add__distrib__right,axiom,
    ! [Xb: real,Y: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,Y),Z)) = aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Z)) ).

% max_add_distrib_right
tff(fact_5839_max__add__distrib__right,axiom,
    ! [Xb: rat,Y: rat,Z: rat] : aa_rat_rat(plus_plus_rat(Xb),aa_rat_rat(ord_max_rat(Y),Z)) = aa_rat_rat(ord_max_rat(aa_rat_rat(plus_plus_rat(Xb),Y)),aa_rat_rat(plus_plus_rat(Xb),Z)) ).

% max_add_distrib_right
tff(fact_5840_max__add__distrib__right,axiom,
    ! [Xb: nat,Y: 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,Y),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xb),Y)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xb),Z)) ).

% max_add_distrib_right
tff(fact_5841_max__add__distrib__right,axiom,
    ! [Xb: code_integer,Y: code_integer,Z: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Xb),ord_max_Code_integer(Y,Z)) = ord_max_Code_integer(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Xb),Y),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Xb),Z)) ).

% max_add_distrib_right
tff(fact_5842_max__add__distrib__right,axiom,
    ! [Xb: int,Y: int,Z: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),aa_int_int(ord_max_int(Y),Z)) = aa_int_int(ord_max_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Y)),aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Z)) ).

% max_add_distrib_right
tff(fact_5843_max__add__distrib__left,axiom,
    ! [Xb: real,Y: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,Xb),Y)),Z) = aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Z)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Y),Z)) ).

% max_add_distrib_left
tff(fact_5844_max__add__distrib__left,axiom,
    ! [Xb: rat,Y: rat,Z: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(ord_max_rat(Xb),Y)),Z) = aa_rat_rat(ord_max_rat(aa_rat_rat(plus_plus_rat(Xb),Z)),aa_rat_rat(plus_plus_rat(Y),Z)) ).

% max_add_distrib_left
tff(fact_5845_max__add__distrib__left,axiom,
    ! [Xb: nat,Y: nat,Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xb),Y)),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,Y),Z)) ).

% max_add_distrib_left
tff(fact_5846_max__add__distrib__left,axiom,
    ! [Xb: code_integer,Y: code_integer,Z: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,ord_max_Code_integer(Xb,Y)),Z) = ord_max_Code_integer(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Xb),Z),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Y),Z)) ).

% max_add_distrib_left
tff(fact_5847_max__add__distrib__left,axiom,
    ! [Xb: int,Y: int,Z: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(ord_max_int(Xb),Y)),Z) = aa_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,Y),Z)) ).

% max_add_distrib_left
tff(fact_5848_max__diff__distrib__left,axiom,
    ! [Xb: code_integer,Y: code_integer,Z: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,ord_max_Code_integer(Xb,Y)),Z) = ord_max_Code_integer(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,Xb),Z),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,Y),Z)) ).

% max_diff_distrib_left
tff(fact_5849_max__diff__distrib__left,axiom,
    ! [Xb: real,Y: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,Xb),Y)),Z) = aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Z)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y),Z)) ).

% max_diff_distrib_left
tff(fact_5850_max__diff__distrib__left,axiom,
    ! [Xb: rat,Y: rat,Z: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(ord_max_rat(Xb),Y)),Z) = aa_rat_rat(ord_max_rat(aa_rat_rat(minus_minus_rat(Xb),Z)),aa_rat_rat(minus_minus_rat(Y),Z)) ).

% max_diff_distrib_left
tff(fact_5851_max__diff__distrib__left,axiom,
    ! [Xb: int,Y: int,Z: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(ord_max_int(Xb),Y)),Z) = aa_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,Y),Z)) ).

% max_diff_distrib_left
tff(fact_5852_nat__add__max__right,axiom,
    ! [M: nat,Na: nat,Q4: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Na),Q4)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Q4)) ).

% nat_add_max_right
tff(fact_5853_nat__add__max__left,axiom,
    ! [M: nat,Na: nat,Q4: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,M),Na)),Q4) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Q4)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),Q4)) ).

% nat_add_max_left
tff(fact_5854_nat__mult__max__right,axiom,
    ! [M: nat,Na: nat,Q4: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Na),Q4)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Q4)) ).

% nat_mult_max_right
tff(fact_5855_nat__mult__max__left,axiom,
    ! [M: nat,Na: nat,Q4: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,M),Na)),Q4) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Q4)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),Q4)) ).

% nat_mult_max_left
tff(fact_5856_max__def__raw,axiom,
    ! [X3: code_integer,Xa: code_integer] :
      ord_max_Code_integer(X3,Xa) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X3),Xa),Xa,X3) ).

% max_def_raw
tff(fact_5857_max__def__raw,axiom,
    ! [X3: set_int,Xa: set_int] :
      ord_max_set_int(X3,Xa) = $ite(aa_set_int_o(ord_less_eq_set_int(X3),Xa),Xa,X3) ).

% max_def_raw
tff(fact_5858_max__def__raw,axiom,
    ! [X3: rat,Xa: rat] :
      aa_rat_rat(ord_max_rat(X3),Xa) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Xa),Xa,X3) ).

% max_def_raw
tff(fact_5859_max__def__raw,axiom,
    ! [X3: num,Xa: num] :
      aa_num_num(aa_num_fun_num_num(ord_max_num,X3),Xa) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Xa),Xa,X3) ).

% max_def_raw
tff(fact_5860_max__def__raw,axiom,
    ! [X3: nat,Xa: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X3),Xa) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Xa),Xa,X3) ).

% max_def_raw
tff(fact_5861_max__def__raw,axiom,
    ! [X3: int,Xa: int] :
      aa_int_int(ord_max_int(X3),Xa) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),Xa),Xa,X3) ).

% max_def_raw
tff(fact_5862_ln__less__self,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)),Xb) ) ).

% ln_less_self
tff(fact_5863_nat__minus__add__max,axiom,
    ! [Na: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),M)),M) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Na),M) ).

% nat_minus_add_max
tff(fact_5864_num_Oexhaust,axiom,
    ! [Y: num] :
      ( ( Y != one )
     => ( ! [X23: num] : Y != aa_num_num(bit0,X23)
       => ~ ! [X32: num] : Y != aa_num_num(bit1,X32) ) ) ).

% num.exhaust
tff(fact_5865_list__update__code_I3_J,axiom,
    ! [Xb: int,Xs: list_int,I: nat,Y: int] : list_update_int(cons_int(Xb,Xs),aa_nat_nat(suc,I),Y) = cons_int(Xb,list_update_int(Xs,I,Y)) ).

% list_update_code(3)
tff(fact_5866_list__update__code_I3_J,axiom,
    ! [Xb: nat,Xs: list_nat,I: nat,Y: nat] : list_update_nat(cons_nat(Xb,Xs),aa_nat_nat(suc,I),Y) = cons_nat(Xb,list_update_nat(Xs,I,Y)) ).

% list_update_code(3)
tff(fact_5867_list__update__code_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xs: list_VEBT_VEBT,I: nat,Y: vEBT_VEBT] : list_u1324408373059187874T_VEBT(cons_VEBT_VEBT(Xb,Xs),aa_nat_nat(suc,I),Y) = cons_VEBT_VEBT(Xb,list_u1324408373059187874T_VEBT(Xs,I,Y)) ).

% list_update_code(3)
tff(fact_5868_list__update__code_I2_J,axiom,
    ! [Xb: int,Xs: list_int,Y: int] : list_update_int(cons_int(Xb,Xs),zero_zero_nat,Y) = cons_int(Y,Xs) ).

% list_update_code(2)
tff(fact_5869_list__update__code_I2_J,axiom,
    ! [Xb: nat,Xs: list_nat,Y: nat] : list_update_nat(cons_nat(Xb,Xs),zero_zero_nat,Y) = cons_nat(Y,Xs) ).

% list_update_code(2)
tff(fact_5870_list__update__code_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xs: list_VEBT_VEBT,Y: vEBT_VEBT] : list_u1324408373059187874T_VEBT(cons_VEBT_VEBT(Xb,Xs),zero_zero_nat,Y) = cons_VEBT_VEBT(Y,Xs) ).

% list_update_code(2)
tff(fact_5871_set__update__subsetI,axiom,
    ! [Xs: list_P6011104703257516679at_nat,A3: set_Pr1261947904930325089at_nat,Xb: product_prod_nat_nat,I: nat] :
      ( ord_le3146513528884898305at_nat(set_Pr5648618587558075414at_nat(Xs),A3)
     => ( member8440522571783428010at_nat(Xb,A3)
       => ord_le3146513528884898305at_nat(set_Pr5648618587558075414at_nat(list_u6180841689913720943at_nat(Xs,I,Xb)),A3) ) ) ).

% set_update_subsetI
tff(fact_5872_set__update__subsetI,axiom,
    ! [Xs: list_real,A3: set_real,Xb: real,I: 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,I,Xb)),A3) ) ) ).

% set_update_subsetI
tff(fact_5873_set__update__subsetI,axiom,
    ! [Xs: list_set_nat,A3: set_set_nat,Xb: set_nat,I: 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,I,Xb)),A3) ) ) ).

% set_update_subsetI
tff(fact_5874_set__update__subsetI,axiom,
    ! [Xs: list_nat,A3: set_nat,Xb: nat,I: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,set_nat2(Xs)),A3)
     => ( aa_set_nat_o(member_nat(Xb),A3)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,set_nat2(list_update_nat(Xs,I,Xb))),A3) ) ) ).

% set_update_subsetI
tff(fact_5875_set__update__subsetI,axiom,
    ! [Xs: list_VEBT_VEBT,A3: set_VEBT_VEBT,Xb: vEBT_VEBT,I: nat] :
      ( ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(Xs),A3)
     => ( member_VEBT_VEBT(Xb,A3)
       => ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(Xs,I,Xb)),A3) ) ) ).

% set_update_subsetI
tff(fact_5876_set__update__subsetI,axiom,
    ! [Xs: list_int,A3: set_int,Xb: int,I: nat] :
      ( aa_set_int_o(ord_less_eq_set_int(set_int2(Xs)),A3)
     => ( aa_set_int_o(member_int(Xb),A3)
       => aa_set_int_o(ord_less_eq_set_int(set_int2(list_update_int(Xs,I,Xb))),A3) ) ) ).

% set_update_subsetI
tff(fact_5877_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_5878_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_5879_ln__bound,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)),Xb) ) ).

% ln_bound
tff(fact_5880_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(aa_rea7340605660300375871l_real(minus_minus_real,Xb),one_one_real) )
       => ( Xb = one_one_real ) ) ) ).

% ln_eq_minus_one
tff(fact_5881_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_5882_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_5883_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_5884_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_5885_ln__div,axiom,
    ! [Xb: real,Y: 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),Y)
       => ( aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(ln_ln_real,Xb)),aa_real_real(ln_ln_real,Y)) ) ) ) ).

% ln_div
tff(fact_5886_numeral__Bit1,axiom,
    ! [Na: num] : numera6690914467698888265omplex(aa_num_num(bit1,Na)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(Na)),numera6690914467698888265omplex(Na))),one_one_complex) ).

% numeral_Bit1
tff(fact_5887_numeral__Bit1,axiom,
    ! [Na: num] : aa_num_rat(numeral_numeral_rat,aa_num_num(bit1,Na)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,Na)),aa_num_rat(numeral_numeral_rat,Na))),one_one_rat) ).

% numeral_Bit1
tff(fact_5888_numeral__Bit1,axiom,
    ! [Na: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Na)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,Na)),aa_num_nat(numeral_numeral_nat,Na))),one_one_nat) ).

% numeral_Bit1
tff(fact_5889_numeral__Bit1,axiom,
    ! [Na: num] : aa_num_real(numeral_numeral_real,aa_num_num(bit1,Na)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,Na)),aa_num_real(numeral_numeral_real,Na))),one_one_real) ).

% numeral_Bit1
tff(fact_5890_numeral__Bit1,axiom,
    ! [Na: num] : aa_num_int(numeral_numeral_int,aa_num_num(bit1,Na)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,Na)),aa_num_int(numeral_numeral_int,Na))),one_one_int) ).

% numeral_Bit1
tff(fact_5891_numeral__Bit1,axiom,
    ! [Na: num] : numera1916890842035813515d_enat(aa_num_num(bit1,Na)) = plus_p3455044024723400733d_enat(plus_p3455044024723400733d_enat(numera1916890842035813515d_enat(Na),numera1916890842035813515d_enat(Na)),one_on7984719198319812577d_enat) ).

% numeral_Bit1
tff(fact_5892_numeral__Bit1,axiom,
    ! [Na: num] : aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,Na)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_num_Code_integer(numera6620942414471956472nteger,Na)),aa_num_Code_integer(numera6620942414471956472nteger,Na))),one_one_Code_integer) ).

% numeral_Bit1
tff(fact_5893_eval__nat__numeral_I3_J,axiom,
    ! [Na: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Na)) = aa_nat_nat(suc,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Na))) ).

% eval_nat_numeral(3)
tff(fact_5894_set__update__memI,axiom,
    ! [Na: nat,Xs: list_P6011104703257516679at_nat,Xb: product_prod_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_s5460976970255530739at_nat(Xs))
     => member8440522571783428010at_nat(Xb,set_Pr5648618587558075414at_nat(list_u6180841689913720943at_nat(Xs,Na,Xb))) ) ).

% set_update_memI
tff(fact_5895_set__update__memI,axiom,
    ! [Na: nat,Xs: list_real,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_size_list_real(Xs))
     => member_real(Xb,set_real2(list_update_real(Xs,Na,Xb))) ) ).

% set_update_memI
tff(fact_5896_set__update__memI,axiom,
    ! [Na: nat,Xs: list_set_nat,Xb: set_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_s3254054031482475050et_nat(Xs))
     => member_set_nat(Xb,set_set_nat2(list_update_set_nat(Xs,Na,Xb))) ) ).

% set_update_memI
tff(fact_5897_set__update__memI,axiom,
    ! [Na: nat,Xs: list_int,Xb: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_size_list_int(Xs))
     => aa_set_int_o(member_int(Xb),set_int2(list_update_int(Xs,Na,Xb))) ) ).

% set_update_memI
tff(fact_5898_set__update__memI,axiom,
    ! [Na: nat,Xs: list_VEBT_VEBT,Xb: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_s6755466524823107622T_VEBT(Xs))
     => member_VEBT_VEBT(Xb,set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(Xs,Na,Xb))) ) ).

% set_update_memI
tff(fact_5899_set__update__memI,axiom,
    ! [Na: nat,Xs: list_o,Xb: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_size_list_o(Xs))
     => member_o((Xb),set_o2(list_update_o(Xs,Na,(Xb)))) ) ).

% set_update_memI
tff(fact_5900_set__update__memI,axiom,
    ! [Na: nat,Xs: list_nat,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),size_size_list_nat(Xs))
     => aa_set_nat_o(member_nat(Xb),set_nat2(list_update_nat(Xs,Na,Xb))) ) ).

% set_update_memI
tff(fact_5901_nth__list__update,axiom,
    ! [I: nat,Xs: list_int,Xb: int,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_int(Xs))
     => ( aa_nat_int(nth_int(list_update_int(Xs,I,Xb)),J) = $ite(I = J,Xb,aa_nat_int(nth_int(Xs),J)) ) ) ).

% nth_list_update
tff(fact_5902_nth__list__update,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,Xb: vEBT_VEBT,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs))
     => ( nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(Xs,I,Xb),J) = $ite(I = J,Xb,nth_VEBT_VEBT(Xs,J)) ) ) ).

% nth_list_update
tff(fact_5903_nth__list__update,axiom,
    ! [I: nat,Xs: list_o,Xb: $o,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_o(Xs))
     => ( aa_nat_o(nth_o(list_update_o(Xs,I,(Xb))),J)
      <=> $ite(I = J,(Xb),aa_nat_o(nth_o(Xs),J)) ) ) ).

% nth_list_update
tff(fact_5904_nth__list__update,axiom,
    ! [I: nat,Xs: list_nat,Xb: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_nat(Xs))
     => ( aa_nat_nat(nth_nat(list_update_nat(Xs,I,Xb)),J) = $ite(I = J,Xb,aa_nat_nat(nth_nat(Xs),J)) ) ) ).

% nth_list_update
tff(fact_5905_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_int,Xb: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_int(Xs))
     => ( ( list_update_int(Xs,I,Xb) = Xs )
      <=> ( aa_nat_int(nth_int(Xs),I) = Xb ) ) ) ).

% list_update_same_conv
tff(fact_5906_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,Xb: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs))
     => ( ( list_u1324408373059187874T_VEBT(Xs,I,Xb) = Xs )
      <=> ( nth_VEBT_VEBT(Xs,I) = Xb ) ) ) ).

% list_update_same_conv
tff(fact_5907_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_o,Xb: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_o(Xs))
     => ( ( list_update_o(Xs,I,(Xb)) = Xs )
      <=> ( aa_nat_o(nth_o(Xs),I)
        <=> (Xb) ) ) ) ).

% list_update_same_conv
tff(fact_5908_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_nat,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_nat(Xs))
     => ( ( list_update_nat(Xs,I,Xb) = Xs )
      <=> ( aa_nat_nat(nth_nat(Xs),I) = Xb ) ) ) ).

% list_update_same_conv
tff(fact_5909_numeral__code_I3_J,axiom,
    ! [Na: num] :
      numera6690914467698888265omplex(aa_num_num(bit1,Na)) = $let(
        m: complex,
        m:= numera6690914467698888265omplex(Na),
        aa_complex_complex(plus_plus_complex(aa_complex_complex(plus_plus_complex(m),m)),one_one_complex) ) ).

% numeral_code(3)
tff(fact_5910_numeral__code_I3_J,axiom,
    ! [Na: num] :
      aa_num_rat(numeral_numeral_rat,aa_num_num(bit1,Na)) = $let(
        m: rat,
        m:= aa_num_rat(numeral_numeral_rat,Na),
        aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(m),m)),one_one_rat) ) ).

% numeral_code(3)
tff(fact_5911_numeral__code_I3_J,axiom,
    ! [Na: num] :
      aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Na)) = $let(
        m: nat,
        m:= aa_num_nat(numeral_numeral_nat,Na),
        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,m),m)),one_one_nat) ) ).

% numeral_code(3)
tff(fact_5912_numeral__code_I3_J,axiom,
    ! [Na: num] :
      aa_num_real(numeral_numeral_real,aa_num_num(bit1,Na)) = $let(
        m: real,
        m:= aa_num_real(numeral_numeral_real,Na),
        aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,m),m)),one_one_real) ) ).

% numeral_code(3)
tff(fact_5913_numeral__code_I3_J,axiom,
    ! [Na: num] :
      aa_num_int(numeral_numeral_int,aa_num_num(bit1,Na)) = $let(
        m: int,
        m:= aa_num_int(numeral_numeral_int,Na),
        aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,m),m)),one_one_int) ) ).

% numeral_code(3)
tff(fact_5914_numeral__code_I3_J,axiom,
    ! [Na: num] :
      numera1916890842035813515d_enat(aa_num_num(bit1,Na)) = $let(
        m: extended_enat,
        m:= numera1916890842035813515d_enat(Na),
        plus_p3455044024723400733d_enat(plus_p3455044024723400733d_enat(m,m),one_on7984719198319812577d_enat) ) ).

% numeral_code(3)
tff(fact_5915_numeral__code_I3_J,axiom,
    ! [Na: num] :
      aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,Na)) = $let(
        m: code_integer,
        m:= aa_num_Code_integer(numera6620942414471956472nteger,Na),
        aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,m),m)),one_one_Code_integer) ) ).

% numeral_code(3)
tff(fact_5916_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_5917_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(aa_rea7340605660300375871l_real(minus_minus_real,Xb),one_one_real)) ) ).

% ln_le_minus_one
tff(fact_5918_ln__diff__le,axiom,
    ! [Xb: real,Y: 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),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(ln_ln_real,Xb)),aa_real_real(ln_ln_real,Y))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Y)),Y)) ) ) ).

% ln_diff_le
tff(fact_5919_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(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xb))),Xb) ) ).

% ln_add_one_self_le_self
tff(fact_5920_ln__mult,axiom,
    ! [Xb: real,Y: 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),Y)
       => ( aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(ln_ln_real,Xb)),aa_real_real(ln_ln_real,Y)) ) ) ) ).

% ln_mult
tff(fact_5921_numeral__Bit1__div__2,axiom,
    ! [Na: num] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Na))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_num_nat(numeral_numeral_nat,Na) ).

% numeral_Bit1_div_2
tff(fact_5922_numeral__Bit1__div__2,axiom,
    ! [Na: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Na))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_num_int(numeral_numeral_int,Na) ).

% numeral_Bit1_div_2
tff(fact_5923_numeral__Bit1__div__2,axiom,
    ! [Na: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,Na))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = aa_num_Code_integer(numera6620942414471956472nteger,Na) ).

% numeral_Bit1_div_2
tff(fact_5924_cong__exp__iff__simps_I3_J,axiom,
    ! [Na: num,Q4: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Na))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q4))) != zero_zero_nat ).

% cong_exp_iff_simps(3)
tff(fact_5925_cong__exp__iff__simps_I3_J,axiom,
    ! [Na: num,Q4: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Na))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q4))) != zero_zero_int ).

% cong_exp_iff_simps(3)
tff(fact_5926_cong__exp__iff__simps_I3_J,axiom,
    ! [Na: num,Q4: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,Na))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Q4))) != zero_z3403309356797280102nteger ).

% cong_exp_iff_simps(3)
tff(fact_5927_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_5928_Suc3__eq__add__3,axiom,
    ! [Na: nat] : aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,Na))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),Na) ).

% Suc3_eq_add_3
tff(fact_5929_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_5930_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(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),Xb))),aa_real_real(uminus_uminus_real,Xb)) ) ) ).

% ln_one_minus_pos_upper_bound
tff(fact_5931_num_Osize__gen_I3_J,axiom,
    ! [X33: num] : aa_num_nat(size_num,aa_num_num(bit1,X33)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(size_num,X33)),aa_nat_nat(suc,zero_zero_nat)) ).

% num.size_gen(3)
tff(fact_5932_num_Osize_I6_J,axiom,
    ! [X33: num] : aa_num_nat(size_size_num,aa_num_num(bit1,X33)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(size_size_num,X33)),aa_nat_nat(suc,zero_zero_nat)) ).

% num.size(6)
tff(fact_5933_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(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xb))),Xb) ) ).

% ln_add_one_self_le_self2
tff(fact_5934_cong__exp__iff__simps_I11_J,axiom,
    ! [M: num,Q4: num] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,M))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q4))) = 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,Q4))) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,Q4)) = zero_zero_nat ) ) ).

% cong_exp_iff_simps(11)
tff(fact_5935_cong__exp__iff__simps_I11_J,axiom,
    ! [M: num,Q4: num] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q4))) = 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,Q4))) )
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,Q4)) = zero_zero_int ) ) ).

% cong_exp_iff_simps(11)
tff(fact_5936_cong__exp__iff__simps_I11_J,axiom,
    ! [M: num,Q4: num] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,M))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Q4))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_num_Code_integer(numera6620942414471956472nteger,one)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Q4))) )
    <=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)),aa_num_Code_integer(numera6620942414471956472nteger,Q4)) = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(11)
tff(fact_5937_cong__exp__iff__simps_I7_J,axiom,
    ! [Q4: num,Na: num] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,one)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q4))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Na))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q4))) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Na)),aa_num_nat(numeral_numeral_nat,Q4)) = zero_zero_nat ) ) ).

% cong_exp_iff_simps(7)
tff(fact_5938_cong__exp__iff__simps_I7_J,axiom,
    ! [Q4: num,Na: 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,Q4))) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Na))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q4))) )
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,Na)),aa_num_int(numeral_numeral_int,Q4)) = zero_zero_int ) ) ).

% cong_exp_iff_simps(7)
tff(fact_5939_cong__exp__iff__simps_I7_J,axiom,
    ! [Q4: num,Na: num] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_num_Code_integer(numera6620942414471956472nteger,one)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Q4))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,Na))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,Q4))) )
    <=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_num_Code_integer(numera6620942414471956472nteger,Na)),aa_num_Code_integer(numera6620942414471956472nteger,Q4)) = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(7)
tff(fact_5940_Suc__div__eq__add3__div,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,M)))),Na) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),M)),Na) ).

% Suc_div_eq_add3_div
tff(fact_5941_Suc__mod__eq__add3__mod,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,M)))),Na) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),M)),Na) ).

% Suc_mod_eq_add3_mod
tff(fact_5942_mod__exhaust__less__4,axiom,
    ! [M: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = zero_zero_nat )
      | ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = one_one_nat )
      | ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) )
      | ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) ) ) ).

% mod_exhaust_less_4
tff(fact_5943_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(aa_rea7340605660300375871l_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(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xb))) ) ) ).

% ln_one_plus_pos_lower_bound
tff(fact_5944_vebt__insert_Osimps_I5_J,axiom,
    ! [Mi2: nat,Ma2: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Xb: nat] :
      vEBT_vebt_insert(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary),Xb) = $let(
        xn: nat,
        xn:= 
          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Mi2),Mi2,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,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
          $ite(
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList))
            & ~ ( ( Xb = Mi2 )
                | ( Xb = Ma2 ) ) ),
            vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat($ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Mi2),Xb,Mi2)),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,xn),Ma2))),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(Mi2),Ma2)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary) ) ) ) ).

% vebt_insert.simps(5)
tff(fact_5945_accp__subset__induct,axiom,
    ! [D6: fun_VEBT_VEBT_o,R: fun_VE2962686792283380846VEBT_o,Xb: vEBT_VEBT,P: fun_VEBT_VEBT_o] :
      ( ord_le88303844413636359VEBT_o(D6,accp_VEBT_VEBT(R))
     => ( ! [X: vEBT_VEBT,Z3: vEBT_VEBT] :
            ( aa_VEBT_VEBT_o(D6,X)
           => ( aa_VEBT_VEBT_o(aa_VEB1521802747257091359VEBT_o(R,Z3),X)
             => aa_VEBT_VEBT_o(D6,Z3) ) )
       => ( aa_VEBT_VEBT_o(D6,Xb)
         => ( ! [X: vEBT_VEBT] :
                ( aa_VEBT_VEBT_o(D6,X)
               => ( ! [Z4: vEBT_VEBT] :
                      ( aa_VEBT_VEBT_o(aa_VEB1521802747257091359VEBT_o(R,Z4),X)
                     => aa_VEBT_VEBT_o(P,Z4) )
                 => aa_VEBT_VEBT_o(P,X) ) )
           => aa_VEBT_VEBT_o(P,Xb) ) ) ) ) ).

% accp_subset_induct
tff(fact_5946_accp__subset__induct,axiom,
    ! [D6: fun_Pr7546709192626087934_nat_o,R: fun_Pr8877776288780692068_nat_o,Xb: produc9072475918466114483BT_nat,P: fun_Pr7546709192626087934_nat_o] :
      ( ord_le1769015658490769310_nat_o(D6,accp_P2887432264394892906BT_nat(R))
     => ( ! [X: produc9072475918466114483BT_nat,Z3: produc9072475918466114483BT_nat] :
            ( aa_Pro632845113475883111_nat_o(D6,X)
           => ( aa_Pro632845113475883111_nat_o(aa_Pro1578556086376436173_nat_o(R,Z3),X)
             => aa_Pro632845113475883111_nat_o(D6,Z3) ) )
       => ( aa_Pro632845113475883111_nat_o(D6,Xb)
         => ( ! [X: produc9072475918466114483BT_nat] :
                ( aa_Pro632845113475883111_nat_o(D6,X)
               => ( ! [Z4: produc9072475918466114483BT_nat] :
                      ( aa_Pro632845113475883111_nat_o(aa_Pro1578556086376436173_nat_o(R,Z4),X)
                     => aa_Pro632845113475883111_nat_o(P,Z4) )
                 => aa_Pro632845113475883111_nat_o(P,X) ) )
           => aa_Pro632845113475883111_nat_o(P,Xb) ) ) ) ) ).

% accp_subset_induct
tff(fact_5947_accp__subset__induct,axiom,
    ! [D6: fun_nat_o,R: fun_nat_fun_nat_o,Xb: nat,P: fun_nat_o] :
      ( ord_le7240017405068453697_nat_o(D6,accp_nat(R))
     => ( ! [X: nat,Z3: nat] :
            ( aa_nat_o(D6,X)
           => ( aa_nat_o(aa_nat_fun_nat_o(R,Z3),X)
             => aa_nat_o(D6,Z3) ) )
       => ( aa_nat_o(D6,Xb)
         => ( ! [X: nat] :
                ( aa_nat_o(D6,X)
               => ( ! [Z4: nat] :
                      ( aa_nat_o(aa_nat_fun_nat_o(R,Z4),X)
                     => aa_nat_o(P,Z4) )
                 => aa_nat_o(P,X) ) )
           => aa_nat_o(P,Xb) ) ) ) ) ).

% accp_subset_induct
tff(fact_5948_accp__subset__induct,axiom,
    ! [D6: fun_Pr4796943543331071514_nat_o,R: fun_Pr2028404973800837934_nat_o,Xb: product_prod_nat_nat,P: fun_Pr4796943543331071514_nat_o] :
      ( ord_le1572679713340831354_nat_o(D6,accp_P4275260045618599050at_nat(R))
     => ( ! [X: product_prod_nat_nat,Z3: product_prod_nat_nat] :
            ( aa_Pro6150637604445964947_nat_o(D6,X)
           => ( aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(R,Z3),X)
             => aa_Pro6150637604445964947_nat_o(D6,Z3) ) )
       => ( aa_Pro6150637604445964947_nat_o(D6,Xb)
         => ( ! [X: product_prod_nat_nat] :
                ( aa_Pro6150637604445964947_nat_o(D6,X)
               => ( ! [Z4: product_prod_nat_nat] :
                      ( aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(R,Z4),X)
                     => aa_Pro6150637604445964947_nat_o(P,Z4) )
                 => aa_Pro6150637604445964947_nat_o(P,X) ) )
           => aa_Pro6150637604445964947_nat_o(P,Xb) ) ) ) ) ).

% accp_subset_induct
tff(fact_5949_accp__subset__induct,axiom,
    ! [D6: fun_Pr3238374608701176162_int_o,R: fun_Pr9018117161326028078_int_o,Xb: product_prod_int_int,P: fun_Pr3238374608701176162_int_o] :
      ( ord_le14110778710936002_int_o(D6,accp_P1096762738010456898nt_int(R))
     => ( ! [X: product_prod_int_int,Z3: product_prod_int_int] :
            ( aa_Pro4592068669816069595_int_o(D6,X)
           => ( aa_Pro4592068669816069595_int_o(aa_Pro3952998155787061599_int_o(R,Z3),X)
             => aa_Pro4592068669816069595_int_o(D6,Z3) ) )
       => ( aa_Pro4592068669816069595_int_o(D6,Xb)
         => ( ! [X: product_prod_int_int] :
                ( aa_Pro4592068669816069595_int_o(D6,X)
               => ( ! [Z4: product_prod_int_int] :
                      ( aa_Pro4592068669816069595_int_o(aa_Pro3952998155787061599_int_o(R,Z4),X)
                     => aa_Pro4592068669816069595_int_o(P,Z4) )
                 => aa_Pro4592068669816069595_int_o(P,X) ) )
           => aa_Pro4592068669816069595_int_o(P,Xb) ) ) ) ) ).

% accp_subset_induct
tff(fact_5950_odd__mod__4__div__2,axiom,
    ! [Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) )
     => ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(suc,zero_zero_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% odd_mod_4_div_2
tff(fact_5951_vebt__insert_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_insert(Xb,Xaa) = Y )
     => ( ! [A2: $o,B2: $o] :
            ( ( Xb = vEBT_Leaf((A2),(B2)) )
           => ( Y != $ite(
                  Xaa = zero_zero_nat,
                  vEBT_Leaf($true,(B2)),
                  $ite(Xaa = one_one_nat,vEBT_Leaf((A2),$true),vEBT_Leaf((A2),(B2))) ) ) )
       => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT] :
              ( ( Xb = vEBT_Node(Info2,zero_zero_nat,Ts,S) )
             => ( Y != vEBT_Node(Info2,zero_zero_nat,Ts,S) ) )
         => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts,S) )
               => ( Y != vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts,S) ) )
           => ( ! [V2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2,Summary2) )
                 => ( Y != vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xaa),Xaa)),aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2,Summary2) ) )
             => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary2: 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,Va2)),TreeList2,Summary2) )
                   => ( Y != $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,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                            $ite(
                              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2))
                              & ~ ( ( Xaa = Mi )
                                  | ( Xaa = Ma ) ) ),
                              vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat($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,Va2)),list_u1324408373059187874T_VEBT(TreeList2,h,vEBT_vebt_insert(nth_VEBT_VEBT(TreeList2,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(TreeList2,h)),vEBT_vebt_insert(Summary2,h),Summary2)),
                              vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary2) ) ) ) ) ) ) ) ) ) ) ).

% vebt_insert.elims
tff(fact_5952_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(aa_rea7340605660300375871l_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(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,Xb)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),Xb))) ) ) ).

% ln_one_minus_pos_lower_bound
tff(fact_5953_vebt__insert_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_insert(Xb,Xaa) = Y )
     => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel),produc738532404422230701BT_nat(Xb,Xaa))
       => ( ! [A2: $o,B2: $o] :
              ( ( Xb = vEBT_Leaf((A2),(B2)) )
             => ( ( Y = $ite(
                      Xaa = zero_zero_nat,
                      vEBT_Leaf($true,(B2)),
                      $ite(Xaa = one_one_nat,vEBT_Leaf((A2),$true),vEBT_Leaf((A2),(B2))) ) )
               => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel),produc738532404422230701BT_nat(vEBT_Leaf((A2),(B2)),Xaa)) ) )
         => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Info2,zero_zero_nat,Ts,S) )
               => ( ( Y = vEBT_Node(Info2,zero_zero_nat,Ts,S) )
                 => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel),produc738532404422230701BT_nat(vEBT_Node(Info2,zero_zero_nat,Ts,S),Xaa)) ) )
           => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts,S) )
                 => ( ( Y = vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts,S) )
                   => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel),produc738532404422230701BT_nat(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts,S),Xaa)) ) )
             => ( ! [V2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2,Summary2) )
                   => ( ( Y = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xaa),Xaa)),aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2,Summary2) )
                     => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel),produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2,Summary2),Xaa)) ) )
               => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary2: 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,Va2)),TreeList2,Summary2) )
                     => ( ( Y = $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,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                $ite(
                                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2))
                                  & ~ ( ( Xaa = Mi )
                                      | ( Xaa = Ma ) ) ),
                                  vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat($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,Va2)),list_u1324408373059187874T_VEBT(TreeList2,h,vEBT_vebt_insert(nth_VEBT_VEBT(TreeList2,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(TreeList2,h)),vEBT_vebt_insert(Summary2,h),Summary2)),
                                  vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2,Summary2) ) ) ) )
                       => ~ aa_Pro632845113475883111_nat_o(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,Va2)),TreeList2,Summary2),Xaa)) ) ) ) ) ) ) ) ) ).

% vebt_insert.pelims
tff(fact_5954_vebt__buildup_Opelims,axiom,
    ! [Xb: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_buildup(Xb) = Y )
     => ( aa_nat_o(accp_nat(vEBT_v4011308405150292612up_rel),Xb)
       => ( ( ( Xb = zero_zero_nat )
           => ( ( Y = vEBT_Leaf($false,$false) )
             => ~ aa_nat_o(accp_nat(vEBT_v4011308405150292612up_rel),zero_zero_nat) ) )
         => ( ( ( Xb = aa_nat_nat(suc,zero_zero_nat) )
             => ( ( Y = vEBT_Leaf($false,$false) )
               => ~ aa_nat_o(accp_nat(vEBT_v4011308405150292612up_rel),aa_nat_nat(suc,zero_zero_nat)) ) )
           => ~ ! [Va2: nat] :
                  ( ( Xb = aa_nat_nat(suc,aa_nat_nat(suc,Va2)) )
                 => ( ( Y = $ite(
                          aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,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))) ) ) )
                   => ~ aa_nat_o(accp_nat(vEBT_v4011308405150292612up_rel),aa_nat_nat(suc,aa_nat_nat(suc,Va2))) ) ) ) ) ) ) ).

% vebt_buildup.pelims
tff(fact_5955_prod__decode__aux_Opelims,axiom,
    ! [Xb: nat,Xaa: nat,Y: product_prod_nat_nat] :
      ( ( nat_prod_decode_aux(Xb,Xaa) = Y )
     => ( aa_Pro6150637604445964947_nat_o(accp_P4275260045618599050at_nat(nat_pr5047031295181774490ux_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xb),Xaa))
       => ~ ( ( Y = $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_5956_max_Oabsorb3,axiom,
    ! [Ba: code_integer,Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Ba),Aa)
     => ( ord_max_Code_integer(Aa,Ba) = Aa ) ) ).

% max.absorb3
tff(fact_5957_max_Oabsorb3,axiom,
    ! [Ba: real,Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Ba),Aa)
     => ( aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,Aa),Ba) = Aa ) ) ).

% max.absorb3
tff(fact_5958_max_Oabsorb3,axiom,
    ! [Ba: rat,Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ba),Aa)
     => ( aa_rat_rat(ord_max_rat(Aa),Ba) = Aa ) ) ).

% max.absorb3
tff(fact_5959_max_Oabsorb3,axiom,
    ! [Ba: num,Aa: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Ba),Aa)
     => ( aa_num_num(aa_num_fun_num_num(ord_max_num,Aa),Ba) = Aa ) ) ).

% max.absorb3
tff(fact_5960_max_Oabsorb3,axiom,
    ! [Ba: nat,Aa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ba),Aa)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Aa),Ba) = Aa ) ) ).

% max.absorb3
tff(fact_5961_max_Oabsorb3,axiom,
    ! [Ba: int,Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Ba),Aa)
     => ( aa_int_int(ord_max_int(Aa),Ba) = Aa ) ) ).

% max.absorb3
tff(fact_5962_max_Oabsorb4,axiom,
    ! [Aa: code_integer,Ba: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Aa),Ba)
     => ( ord_max_Code_integer(Aa,Ba) = Ba ) ) ).

% max.absorb4
tff(fact_5963_max_Oabsorb4,axiom,
    ! [Aa: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),Ba)
     => ( aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,Aa),Ba) = Ba ) ) ).

% max.absorb4
tff(fact_5964_max_Oabsorb4,axiom,
    ! [Aa: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),Ba)
     => ( aa_rat_rat(ord_max_rat(Aa),Ba) = Ba ) ) ).

% max.absorb4
tff(fact_5965_max_Oabsorb4,axiom,
    ! [Aa: num,Ba: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Aa),Ba)
     => ( aa_num_num(aa_num_fun_num_num(ord_max_num,Aa),Ba) = Ba ) ) ).

% max.absorb4
tff(fact_5966_max_Oabsorb4,axiom,
    ! [Aa: nat,Ba: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),Ba)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Aa),Ba) = Ba ) ) ).

% max.absorb4
tff(fact_5967_max_Oabsorb4,axiom,
    ! [Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),Ba)
     => ( aa_int_int(ord_max_int(Aa),Ba) = Ba ) ) ).

% max.absorb4
tff(fact_5968_max__less__iff__conj,axiom,
    ! [Xb: code_integer,Y: code_integer,Z: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,ord_max_Code_integer(Xb,Y)),Z)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Xb),Z)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Y),Z) ) ) ).

% max_less_iff_conj
tff(fact_5969_max__less__iff__conj,axiom,
    ! [Xb: real,Y: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,Xb),Y)),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,Y),Z) ) ) ).

% max_less_iff_conj
tff(fact_5970_max__less__iff__conj,axiom,
    ! [Xb: rat,Y: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(ord_max_rat(Xb),Y)),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,Y),Z) ) ) ).

% max_less_iff_conj
tff(fact_5971_max__less__iff__conj,axiom,
    ! [Xb: num,Y: num,Z: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(aa_num_fun_num_num(ord_max_num,Xb),Y)),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,Y),Z) ) ) ).

% max_less_iff_conj
tff(fact_5972_max__less__iff__conj,axiom,
    ! [Xb: nat,Y: 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),Y)),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,Y),Z) ) ) ).

% max_less_iff_conj
tff(fact_5973_max__less__iff__conj,axiom,
    ! [Xb: int,Y: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(ord_max_int(Xb),Y)),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,Y),Z) ) ) ).

% max_less_iff_conj
tff(fact_5974_max_Obounded__iff,axiom,
    ! [Ba: code_integer,C: code_integer,Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,ord_max_Code_integer(Ba,C)),Aa)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Ba),Aa)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),Aa) ) ) ).

% max.bounded_iff
tff(fact_5975_max_Obounded__iff,axiom,
    ! [Ba: rat,C: rat,Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(ord_max_rat(Ba),C)),Aa)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ba),Aa)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),Aa) ) ) ).

% max.bounded_iff
tff(fact_5976_max_Obounded__iff,axiom,
    ! [Ba: num,C: num,Aa: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(aa_num_fun_num_num(ord_max_num,Ba),C)),Aa)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ba),Aa)
        & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),Aa) ) ) ).

% max.bounded_iff
tff(fact_5977_max_Obounded__iff,axiom,
    ! [Ba: nat,C: nat,Aa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Ba),C)),Aa)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ba),Aa)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),Aa) ) ) ).

% max.bounded_iff
tff(fact_5978_max_Obounded__iff,axiom,
    ! [Ba: int,C: int,Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ord_max_int(Ba),C)),Aa)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ba),Aa)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),Aa) ) ) ).

% max.bounded_iff
tff(fact_5979_max_Oabsorb1,axiom,
    ! [Ba: code_integer,Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Ba),Aa)
     => ( ord_max_Code_integer(Aa,Ba) = Aa ) ) ).

% max.absorb1
tff(fact_5980_max_Oabsorb1,axiom,
    ! [Ba: rat,Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ba),Aa)
     => ( aa_rat_rat(ord_max_rat(Aa),Ba) = Aa ) ) ).

% max.absorb1
tff(fact_5981_max_Oabsorb1,axiom,
    ! [Ba: num,Aa: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ba),Aa)
     => ( aa_num_num(aa_num_fun_num_num(ord_max_num,Aa),Ba) = Aa ) ) ).

% max.absorb1
tff(fact_5982_max_Oabsorb1,axiom,
    ! [Ba: nat,Aa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ba),Aa)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Aa),Ba) = Aa ) ) ).

% max.absorb1
tff(fact_5983_max_Oabsorb1,axiom,
    ! [Ba: int,Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ba),Aa)
     => ( aa_int_int(ord_max_int(Aa),Ba) = Aa ) ) ).

% max.absorb1
tff(fact_5984_max_Oabsorb2,axiom,
    ! [Aa: code_integer,Ba: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Aa),Ba)
     => ( ord_max_Code_integer(Aa,Ba) = Ba ) ) ).

% max.absorb2
tff(fact_5985_max_Oabsorb2,axiom,
    ! [Aa: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),Ba)
     => ( aa_rat_rat(ord_max_rat(Aa),Ba) = Ba ) ) ).

% max.absorb2
tff(fact_5986_max_Oabsorb2,axiom,
    ! [Aa: num,Ba: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Aa),Ba)
     => ( aa_num_num(aa_num_fun_num_num(ord_max_num,Aa),Ba) = Ba ) ) ).

% max.absorb2
tff(fact_5987_max_Oabsorb2,axiom,
    ! [Aa: nat,Ba: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),Ba)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Aa),Ba) = Ba ) ) ).

% max.absorb2
tff(fact_5988_max_Oabsorb2,axiom,
    ! [Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),Ba)
     => ( aa_int_int(ord_max_int(Aa),Ba) = Ba ) ) ).

% max.absorb2
tff(fact_5989_max_Omono,axiom,
    ! [C: code_integer,Aa: code_integer,D: code_integer,Ba: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),Aa)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,D),Ba)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,ord_max_Code_integer(C,D)),ord_max_Code_integer(Aa,Ba)) ) ) ).

% max.mono
tff(fact_5990_max_Omono,axiom,
    ! [C: rat,Aa: rat,D: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),Aa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,D),Ba)
       => 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(Aa),Ba)) ) ) ).

% max.mono
tff(fact_5991_max_Omono,axiom,
    ! [C: num,Aa: num,D: num,Ba: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),Aa)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,D),Ba)
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(aa_num_fun_num_num(ord_max_num,C),D)),aa_num_num(aa_num_fun_num_num(ord_max_num,Aa),Ba)) ) ) ).

% max.mono
tff(fact_5992_max_Omono,axiom,
    ! [C: nat,Aa: nat,D: nat,Ba: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),Aa)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,D),Ba)
       => 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,Aa),Ba)) ) ) ).

% max.mono
tff(fact_5993_max_Omono,axiom,
    ! [C: int,Aa: int,D: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),Aa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,D),Ba)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ord_max_int(C),D)),aa_int_int(ord_max_int(Aa),Ba)) ) ) ).

% max.mono
tff(fact_5994_max_OorderE,axiom,
    ! [Ba: code_integer,Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Ba),Aa)
     => ( Aa = ord_max_Code_integer(Aa,Ba) ) ) ).

% max.orderE
tff(fact_5995_max_OorderE,axiom,
    ! [Ba: rat,Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ba),Aa)
     => ( Aa = aa_rat_rat(ord_max_rat(Aa),Ba) ) ) ).

% max.orderE
tff(fact_5996_max_OorderE,axiom,
    ! [Ba: num,Aa: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ba),Aa)
     => ( Aa = aa_num_num(aa_num_fun_num_num(ord_max_num,Aa),Ba) ) ) ).

% max.orderE
tff(fact_5997_max_OorderE,axiom,
    ! [Ba: nat,Aa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ba),Aa)
     => ( Aa = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Aa),Ba) ) ) ).

% max.orderE
tff(fact_5998_max_OorderE,axiom,
    ! [Ba: int,Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ba),Aa)
     => ( Aa = aa_int_int(ord_max_int(Aa),Ba) ) ) ).

% max.orderE
tff(fact_5999_max_OorderI,axiom,
    ! [Aa: code_integer,Ba: code_integer] :
      ( ( Aa = ord_max_Code_integer(Aa,Ba) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Ba),Aa) ) ).

% max.orderI
tff(fact_6000_max_OorderI,axiom,
    ! [Aa: rat,Ba: rat] :
      ( ( Aa = aa_rat_rat(ord_max_rat(Aa),Ba) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ba),Aa) ) ).

% max.orderI
tff(fact_6001_max_OorderI,axiom,
    ! [Aa: num,Ba: num] :
      ( ( Aa = aa_num_num(aa_num_fun_num_num(ord_max_num,Aa),Ba) )
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ba),Aa) ) ).

% max.orderI
tff(fact_6002_max_OorderI,axiom,
    ! [Aa: nat,Ba: nat] :
      ( ( Aa = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Aa),Ba) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ba),Aa) ) ).

% max.orderI
tff(fact_6003_max_OorderI,axiom,
    ! [Aa: int,Ba: int] :
      ( ( Aa = aa_int_int(ord_max_int(Aa),Ba) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ba),Aa) ) ).

% max.orderI
tff(fact_6004_max_OboundedE,axiom,
    ! [Ba: code_integer,C: code_integer,Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,ord_max_Code_integer(Ba,C)),Aa)
     => ~ ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Ba),Aa)
         => ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),Aa) ) ) ).

% max.boundedE
tff(fact_6005_max_OboundedE,axiom,
    ! [Ba: rat,C: rat,Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(ord_max_rat(Ba),C)),Aa)
     => ~ ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ba),Aa)
         => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),Aa) ) ) ).

% max.boundedE
tff(fact_6006_max_OboundedE,axiom,
    ! [Ba: num,C: num,Aa: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(aa_num_fun_num_num(ord_max_num,Ba),C)),Aa)
     => ~ ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ba),Aa)
         => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),Aa) ) ) ).

% max.boundedE
tff(fact_6007_max_OboundedE,axiom,
    ! [Ba: nat,C: nat,Aa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Ba),C)),Aa)
     => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ba),Aa)
         => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),Aa) ) ) ).

% max.boundedE
tff(fact_6008_max_OboundedE,axiom,
    ! [Ba: int,C: int,Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ord_max_int(Ba),C)),Aa)
     => ~ ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ba),Aa)
         => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),Aa) ) ) ).

% max.boundedE
tff(fact_6009_max_OboundedI,axiom,
    ! [Ba: code_integer,Aa: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Ba),Aa)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),Aa)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,ord_max_Code_integer(Ba,C)),Aa) ) ) ).

% max.boundedI
tff(fact_6010_max_OboundedI,axiom,
    ! [Ba: rat,Aa: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ba),Aa)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),Aa)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(ord_max_rat(Ba),C)),Aa) ) ) ).

% max.boundedI
tff(fact_6011_max_OboundedI,axiom,
    ! [Ba: num,Aa: num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ba),Aa)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),Aa)
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(aa_num_fun_num_num(ord_max_num,Ba),C)),Aa) ) ) ).

% max.boundedI
tff(fact_6012_max_OboundedI,axiom,
    ! [Ba: nat,Aa: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ba),Aa)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),Aa)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Ba),C)),Aa) ) ) ).

% max.boundedI
tff(fact_6013_max_OboundedI,axiom,
    ! [Ba: int,Aa: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ba),Aa)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),Aa)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ord_max_int(Ba),C)),Aa) ) ) ).

% max.boundedI
tff(fact_6014_max_Oorder__iff,axiom,
    ! [Ba: code_integer,Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Ba),Aa)
    <=> ( Aa = ord_max_Code_integer(Aa,Ba) ) ) ).

% max.order_iff
tff(fact_6015_max_Oorder__iff,axiom,
    ! [Ba: rat,Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ba),Aa)
    <=> ( Aa = aa_rat_rat(ord_max_rat(Aa),Ba) ) ) ).

% max.order_iff
tff(fact_6016_max_Oorder__iff,axiom,
    ! [Ba: num,Aa: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ba),Aa)
    <=> ( Aa = aa_num_num(aa_num_fun_num_num(ord_max_num,Aa),Ba) ) ) ).

% max.order_iff
tff(fact_6017_max_Oorder__iff,axiom,
    ! [Ba: nat,Aa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ba),Aa)
    <=> ( Aa = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Aa),Ba) ) ) ).

% max.order_iff
tff(fact_6018_max_Oorder__iff,axiom,
    ! [Ba: int,Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ba),Aa)
    <=> ( Aa = aa_int_int(ord_max_int(Aa),Ba) ) ) ).

% max.order_iff
tff(fact_6019_max_Ocobounded1,axiom,
    ! [Aa: code_integer,Ba: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Aa),ord_max_Code_integer(Aa,Ba)) ).

% max.cobounded1
tff(fact_6020_max_Ocobounded1,axiom,
    ! [Aa: rat,Ba: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),aa_rat_rat(ord_max_rat(Aa),Ba)) ).

% max.cobounded1
tff(fact_6021_max_Ocobounded1,axiom,
    ! [Aa: num,Ba: num] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Aa),aa_num_num(aa_num_fun_num_num(ord_max_num,Aa),Ba)) ).

% max.cobounded1
tff(fact_6022_max_Ocobounded1,axiom,
    ! [Aa: nat,Ba: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Aa),Ba)) ).

% max.cobounded1
tff(fact_6023_max_Ocobounded1,axiom,
    ! [Aa: int,Ba: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),aa_int_int(ord_max_int(Aa),Ba)) ).

% max.cobounded1
tff(fact_6024_max_Ocobounded2,axiom,
    ! [Ba: code_integer,Aa: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Ba),ord_max_Code_integer(Aa,Ba)) ).

% max.cobounded2
tff(fact_6025_max_Ocobounded2,axiom,
    ! [Ba: rat,Aa: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ba),aa_rat_rat(ord_max_rat(Aa),Ba)) ).

% max.cobounded2
tff(fact_6026_max_Ocobounded2,axiom,
    ! [Ba: num,Aa: num] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ba),aa_num_num(aa_num_fun_num_num(ord_max_num,Aa),Ba)) ).

% max.cobounded2
tff(fact_6027_max_Ocobounded2,axiom,
    ! [Ba: nat,Aa: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ba),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Aa),Ba)) ).

% max.cobounded2
tff(fact_6028_max_Ocobounded2,axiom,
    ! [Ba: int,Aa: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ba),aa_int_int(ord_max_int(Aa),Ba)) ).

% max.cobounded2
tff(fact_6029_le__max__iff__disj,axiom,
    ! [Z: code_integer,Xb: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Z),ord_max_Code_integer(Xb,Y))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Z),Xb)
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Z),Y) ) ) ).

% le_max_iff_disj
tff(fact_6030_le__max__iff__disj,axiom,
    ! [Z: rat,Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z),aa_rat_rat(ord_max_rat(Xb),Y))
    <=> ( 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),Y) ) ) ).

% le_max_iff_disj
tff(fact_6031_le__max__iff__disj,axiom,
    ! [Z: num,Xb: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Z),aa_num_num(aa_num_fun_num_num(ord_max_num,Xb),Y))
    <=> ( 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),Y) ) ) ).

% le_max_iff_disj
tff(fact_6032_le__max__iff__disj,axiom,
    ! [Z: nat,Xb: nat,Y: 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),Y))
    <=> ( 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),Y) ) ) ).

% le_max_iff_disj
tff(fact_6033_le__max__iff__disj,axiom,
    ! [Z: int,Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),aa_int_int(ord_max_int(Xb),Y))
    <=> ( 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),Y) ) ) ).

% le_max_iff_disj
tff(fact_6034_max_Oabsorb__iff1,axiom,
    ! [Ba: code_integer,Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Ba),Aa)
    <=> ( ord_max_Code_integer(Aa,Ba) = Aa ) ) ).

% max.absorb_iff1
tff(fact_6035_max_Oabsorb__iff1,axiom,
    ! [Ba: rat,Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ba),Aa)
    <=> ( aa_rat_rat(ord_max_rat(Aa),Ba) = Aa ) ) ).

% max.absorb_iff1
tff(fact_6036_max_Oabsorb__iff1,axiom,
    ! [Ba: num,Aa: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ba),Aa)
    <=> ( aa_num_num(aa_num_fun_num_num(ord_max_num,Aa),Ba) = Aa ) ) ).

% max.absorb_iff1
tff(fact_6037_max_Oabsorb__iff1,axiom,
    ! [Ba: nat,Aa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ba),Aa)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Aa),Ba) = Aa ) ) ).

% max.absorb_iff1
tff(fact_6038_max_Oabsorb__iff1,axiom,
    ! [Ba: int,Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ba),Aa)
    <=> ( aa_int_int(ord_max_int(Aa),Ba) = Aa ) ) ).

% max.absorb_iff1
tff(fact_6039_max_Oabsorb__iff2,axiom,
    ! [Aa: code_integer,Ba: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Aa),Ba)
    <=> ( ord_max_Code_integer(Aa,Ba) = Ba ) ) ).

% max.absorb_iff2
tff(fact_6040_max_Oabsorb__iff2,axiom,
    ! [Aa: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),Ba)
    <=> ( aa_rat_rat(ord_max_rat(Aa),Ba) = Ba ) ) ).

% max.absorb_iff2
tff(fact_6041_max_Oabsorb__iff2,axiom,
    ! [Aa: num,Ba: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Aa),Ba)
    <=> ( aa_num_num(aa_num_fun_num_num(ord_max_num,Aa),Ba) = Ba ) ) ).

% max.absorb_iff2
tff(fact_6042_max_Oabsorb__iff2,axiom,
    ! [Aa: nat,Ba: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),Ba)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Aa),Ba) = Ba ) ) ).

% max.absorb_iff2
tff(fact_6043_max_Oabsorb__iff2,axiom,
    ! [Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),Ba)
    <=> ( aa_int_int(ord_max_int(Aa),Ba) = Ba ) ) ).

% max.absorb_iff2
tff(fact_6044_max_OcoboundedI1,axiom,
    ! [C: code_integer,Aa: code_integer,Ba: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),Aa)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),ord_max_Code_integer(Aa,Ba)) ) ).

% max.coboundedI1
tff(fact_6045_max_OcoboundedI1,axiom,
    ! [C: rat,Aa: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),Aa)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),aa_rat_rat(ord_max_rat(Aa),Ba)) ) ).

% max.coboundedI1
tff(fact_6046_max_OcoboundedI1,axiom,
    ! [C: num,Aa: num,Ba: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),Aa)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),aa_num_num(aa_num_fun_num_num(ord_max_num,Aa),Ba)) ) ).

% max.coboundedI1
tff(fact_6047_max_OcoboundedI1,axiom,
    ! [C: nat,Aa: nat,Ba: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),Aa)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Aa),Ba)) ) ).

% max.coboundedI1
tff(fact_6048_max_OcoboundedI1,axiom,
    ! [C: int,Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),Aa)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),aa_int_int(ord_max_int(Aa),Ba)) ) ).

% max.coboundedI1
tff(fact_6049_max_OcoboundedI2,axiom,
    ! [C: code_integer,Ba: code_integer,Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),Ba)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C),ord_max_Code_integer(Aa,Ba)) ) ).

% max.coboundedI2
tff(fact_6050_max_OcoboundedI2,axiom,
    ! [C: rat,Ba: rat,Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),Ba)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),aa_rat_rat(ord_max_rat(Aa),Ba)) ) ).

% max.coboundedI2
tff(fact_6051_max_OcoboundedI2,axiom,
    ! [C: num,Ba: num,Aa: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),Ba)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),aa_num_num(aa_num_fun_num_num(ord_max_num,Aa),Ba)) ) ).

% max.coboundedI2
tff(fact_6052_max_OcoboundedI2,axiom,
    ! [C: nat,Ba: nat,Aa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),Ba)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Aa),Ba)) ) ).

% max.coboundedI2
tff(fact_6053_max_OcoboundedI2,axiom,
    ! [C: int,Ba: int,Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),Ba)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),aa_int_int(ord_max_int(Aa),Ba)) ) ).

% max.coboundedI2
tff(fact_6054_less__max__iff__disj,axiom,
    ! [Z: code_integer,Xb: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Z),ord_max_Code_integer(Xb,Y))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Z),Xb)
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Z),Y) ) ) ).

% less_max_iff_disj
tff(fact_6055_less__max__iff__disj,axiom,
    ! [Z: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z),aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,Xb),Y))
    <=> ( 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),Y) ) ) ).

% less_max_iff_disj
tff(fact_6056_less__max__iff__disj,axiom,
    ! [Z: rat,Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),aa_rat_rat(ord_max_rat(Xb),Y))
    <=> ( 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),Y) ) ) ).

% less_max_iff_disj
tff(fact_6057_less__max__iff__disj,axiom,
    ! [Z: num,Xb: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z),aa_num_num(aa_num_fun_num_num(ord_max_num,Xb),Y))
    <=> ( 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),Y) ) ) ).

% less_max_iff_disj
tff(fact_6058_less__max__iff__disj,axiom,
    ! [Z: nat,Xb: nat,Y: 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),Y))
    <=> ( 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),Y) ) ) ).

% less_max_iff_disj
tff(fact_6059_less__max__iff__disj,axiom,
    ! [Z: int,Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z),aa_int_int(ord_max_int(Xb),Y))
    <=> ( 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),Y) ) ) ).

% less_max_iff_disj
tff(fact_6060_max_Ostrict__boundedE,axiom,
    ! [Ba: code_integer,C: code_integer,Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,ord_max_Code_integer(Ba,C)),Aa)
     => ~ ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Ba),Aa)
         => ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),Aa) ) ) ).

% max.strict_boundedE
tff(fact_6061_max_Ostrict__boundedE,axiom,
    ! [Ba: real,C: real,Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,Ba),C)),Aa)
     => ~ ( aa_real_o(aa_real_fun_real_o(ord_less_real,Ba),Aa)
         => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,C),Aa) ) ) ).

% max.strict_boundedE
tff(fact_6062_max_Ostrict__boundedE,axiom,
    ! [Ba: rat,C: rat,Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(ord_max_rat(Ba),C)),Aa)
     => ~ ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ba),Aa)
         => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),Aa) ) ) ).

% max.strict_boundedE
tff(fact_6063_max_Ostrict__boundedE,axiom,
    ! [Ba: num,C: num,Aa: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(aa_num_fun_num_num(ord_max_num,Ba),C)),Aa)
     => ~ ( aa_num_o(aa_num_fun_num_o(ord_less_num,Ba),Aa)
         => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,C),Aa) ) ) ).

% max.strict_boundedE
tff(fact_6064_max_Ostrict__boundedE,axiom,
    ! [Ba: nat,C: nat,Aa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Ba),C)),Aa)
     => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ba),Aa)
         => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),Aa) ) ) ).

% max.strict_boundedE
tff(fact_6065_max_Ostrict__boundedE,axiom,
    ! [Ba: int,C: int,Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(ord_max_int(Ba),C)),Aa)
     => ~ ( aa_int_o(aa_int_fun_int_o(ord_less_int,Ba),Aa)
         => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,C),Aa) ) ) ).

% max.strict_boundedE
tff(fact_6066_max_Ostrict__order__iff,axiom,
    ! [Ba: code_integer,Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Ba),Aa)
    <=> ( ( Aa = ord_max_Code_integer(Aa,Ba) )
        & ( Aa != Ba ) ) ) ).

% max.strict_order_iff
tff(fact_6067_max_Ostrict__order__iff,axiom,
    ! [Ba: real,Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Ba),Aa)
    <=> ( ( Aa = aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,Aa),Ba) )
        & ( Aa != Ba ) ) ) ).

% max.strict_order_iff
tff(fact_6068_max_Ostrict__order__iff,axiom,
    ! [Ba: rat,Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ba),Aa)
    <=> ( ( Aa = aa_rat_rat(ord_max_rat(Aa),Ba) )
        & ( Aa != Ba ) ) ) ).

% max.strict_order_iff
tff(fact_6069_max_Ostrict__order__iff,axiom,
    ! [Ba: num,Aa: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Ba),Aa)
    <=> ( ( Aa = aa_num_num(aa_num_fun_num_num(ord_max_num,Aa),Ba) )
        & ( Aa != Ba ) ) ) ).

% max.strict_order_iff
tff(fact_6070_max_Ostrict__order__iff,axiom,
    ! [Ba: nat,Aa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ba),Aa)
    <=> ( ( Aa = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Aa),Ba) )
        & ( Aa != Ba ) ) ) ).

% max.strict_order_iff
tff(fact_6071_max_Ostrict__order__iff,axiom,
    ! [Ba: int,Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Ba),Aa)
    <=> ( ( Aa = aa_int_int(ord_max_int(Aa),Ba) )
        & ( Aa != Ba ) ) ) ).

% max.strict_order_iff
tff(fact_6072_max_Ostrict__coboundedI1,axiom,
    ! [C: code_integer,Aa: code_integer,Ba: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),Aa)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),ord_max_Code_integer(Aa,Ba)) ) ).

% max.strict_coboundedI1
tff(fact_6073_max_Ostrict__coboundedI1,axiom,
    ! [C: real,Aa: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),Aa)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,C),aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,Aa),Ba)) ) ).

% max.strict_coboundedI1
tff(fact_6074_max_Ostrict__coboundedI1,axiom,
    ! [C: rat,Aa: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),Aa)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),aa_rat_rat(ord_max_rat(Aa),Ba)) ) ).

% max.strict_coboundedI1
tff(fact_6075_max_Ostrict__coboundedI1,axiom,
    ! [C: num,Aa: num,Ba: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,C),Aa)
     => aa_num_o(aa_num_fun_num_o(ord_less_num,C),aa_num_num(aa_num_fun_num_num(ord_max_num,Aa),Ba)) ) ).

% max.strict_coboundedI1
tff(fact_6076_max_Ostrict__coboundedI1,axiom,
    ! [C: nat,Aa: nat,Ba: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),Aa)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Aa),Ba)) ) ).

% max.strict_coboundedI1
tff(fact_6077_max_Ostrict__coboundedI1,axiom,
    ! [C: int,Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),Aa)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,C),aa_int_int(ord_max_int(Aa),Ba)) ) ).

% max.strict_coboundedI1
tff(fact_6078_max_Ostrict__coboundedI2,axiom,
    ! [C: code_integer,Ba: code_integer,Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),Ba)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C),ord_max_Code_integer(Aa,Ba)) ) ).

% max.strict_coboundedI2
tff(fact_6079_max_Ostrict__coboundedI2,axiom,
    ! [C: real,Ba: real,Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),Ba)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,C),aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,Aa),Ba)) ) ).

% max.strict_coboundedI2
tff(fact_6080_max_Ostrict__coboundedI2,axiom,
    ! [C: rat,Ba: rat,Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),Ba)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),aa_rat_rat(ord_max_rat(Aa),Ba)) ) ).

% max.strict_coboundedI2
tff(fact_6081_max_Ostrict__coboundedI2,axiom,
    ! [C: num,Ba: num,Aa: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,C),Ba)
     => aa_num_o(aa_num_fun_num_o(ord_less_num,C),aa_num_num(aa_num_fun_num_num(ord_max_num,Aa),Ba)) ) ).

% max.strict_coboundedI2
tff(fact_6082_max_Ostrict__coboundedI2,axiom,
    ! [C: nat,Ba: nat,Aa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),Ba)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Aa),Ba)) ) ).

% max.strict_coboundedI2
tff(fact_6083_max_Ostrict__coboundedI2,axiom,
    ! [C: int,Ba: int,Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),Ba)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,C),aa_int_int(ord_max_int(Aa),Ba)) ) ).

% max.strict_coboundedI2
tff(fact_6084_fold__atLeastAtMost__nat_Opinduct,axiom,
    ! [A0: fun_nat_fun_nat_nat,A1: nat,A22: nat,A32: nat,P: fun_fu2571895429041709858_nat_o] :
      ( accp_P4694664897847628410at_nat(set_fo3699595496184130361el_nat,produc2818902965508115497at_nat(A0,produc487386426758144856at_nat(A1,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A22),A32))))
     => ( ! [F2: fun_nat_fun_nat_nat,A2: nat,B2: nat,Acc: nat] :
            ( accp_P4694664897847628410at_nat(set_fo3699595496184130361el_nat,produc2818902965508115497at_nat(F2,produc487386426758144856at_nat(A2,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B2),Acc))))
           => ( ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),A2)
               => aa_nat_o(aa_nat_fun_nat_o(aa_nat8314798964848609180_nat_o(aa_fun3751150903526668555_nat_o(P,F2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),one_one_nat)),B2),aa_nat_nat(aa_nat_fun_nat_nat(F2,A2),Acc)) )
             => aa_nat_o(aa_nat_fun_nat_o(aa_nat8314798964848609180_nat_o(aa_fun3751150903526668555_nat_o(P,F2),A2),B2),Acc) ) )
       => aa_nat_o(aa_nat_fun_nat_o(aa_nat8314798964848609180_nat_o(aa_fun3751150903526668555_nat_o(P,A0),A1),A22),A32) ) ) ).

% fold_atLeastAtMost_nat.pinduct
tff(fact_6085_VEBT__internal_Ooption__shift_Opelims,axiom,
    ! [Xb: fun_Pr2477239776880934807at_nat,Xaa: option4927543243414619207at_nat,Xba: option4927543243414619207at_nat,Y: option4927543243414619207at_nat] :
      ( ( vEBT_V1502963449132264192at_nat(Xb,Xaa,Xba) = Y )
     => ( accp_P4760198183577772864at_nat(vEBT_V7235779383477046023at_nat,produc7814092408141220007at_nat(Xb,produc488173922507101015at_nat(Xaa,Xba)))
       => ( ( ( Xaa = none_P5556105721700978146at_nat )
           => ( ( Y = none_P5556105721700978146at_nat )
             => ~ accp_P4760198183577772864at_nat(vEBT_V7235779383477046023at_nat,produc7814092408141220007at_nat(Xb,produc488173922507101015at_nat(none_P5556105721700978146at_nat,Xba))) ) )
         => ( ! [V2: product_prod_nat_nat] :
                ( ( Xaa = some_P7363390416028606310at_nat(V2) )
               => ( ( Xba = none_P5556105721700978146at_nat )
                 => ( ( Y = none_P5556105721700978146at_nat )
                   => ~ accp_P4760198183577772864at_nat(vEBT_V7235779383477046023at_nat,produc7814092408141220007at_nat(Xb,produc488173922507101015at_nat(some_P7363390416028606310at_nat(V2),none_P5556105721700978146at_nat))) ) ) )
           => ~ ! [A2: product_prod_nat_nat] :
                  ( ( Xaa = some_P7363390416028606310at_nat(A2) )
                 => ! [B2: product_prod_nat_nat] :
                      ( ( Xba = some_P7363390416028606310at_nat(B2) )
                     => ( ( Y = some_P7363390416028606310at_nat(aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(Xb,A2),B2)) )
                       => ~ accp_P4760198183577772864at_nat(vEBT_V7235779383477046023at_nat,produc7814092408141220007at_nat(Xb,produc488173922507101015at_nat(some_P7363390416028606310at_nat(A2),some_P7363390416028606310at_nat(B2)))) ) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.pelims
tff(fact_6086_VEBT__internal_Ooption__shift_Opelims,axiom,
    ! [Xb: fun_num_fun_num_num,Xaa: option_num,Xba: option_num,Y: option_num] :
      ( ( vEBT_V819420779217536731ft_num(Xb,Xaa,Xba) = Y )
     => ( accp_P1225658569124232473on_num(vEBT_V452583751252753300el_num,produc6360833703471310868on_num(Xb,produc8585076106096196333on_num(Xaa,Xba)))
       => ( ( ( Xaa = none_num )
           => ( ( Y = none_num )
             => ~ accp_P1225658569124232473on_num(vEBT_V452583751252753300el_num,produc6360833703471310868on_num(Xb,produc8585076106096196333on_num(none_num,Xba))) ) )
         => ( ! [V2: num] :
                ( ( Xaa = some_num(V2) )
               => ( ( Xba = none_num )
                 => ( ( Y = none_num )
                   => ~ accp_P1225658569124232473on_num(vEBT_V452583751252753300el_num,produc6360833703471310868on_num(Xb,produc8585076106096196333on_num(some_num(V2),none_num))) ) ) )
           => ~ ! [A2: num] :
                  ( ( Xaa = some_num(A2) )
                 => ! [B2: num] :
                      ( ( Xba = some_num(B2) )
                     => ( ( Y = some_num(aa_num_num(aa_num_fun_num_num(Xb,A2),B2)) )
                       => ~ accp_P1225658569124232473on_num(vEBT_V452583751252753300el_num,produc6360833703471310868on_num(Xb,produc8585076106096196333on_num(some_num(A2),some_num(B2)))) ) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.pelims
tff(fact_6087_VEBT__internal_Ooption__shift_Opelims,axiom,
    ! [Xb: fun_nat_fun_nat_nat,Xaa: option_nat,Xba: option_nat,Y: option_nat] :
      ( ( aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_V4262088993061758097ft_nat(Xb),Xaa),Xba) = Y )
     => ( accp_P5526603486790338531on_nat(vEBT_V3895251965096974666el_nat,produc6699827698519632094on_nat(Xb,produc5098337634421038937on_nat(Xaa,Xba)))
       => ( ( ( Xaa = none_nat )
           => ( ( Y = none_nat )
             => ~ accp_P5526603486790338531on_nat(vEBT_V3895251965096974666el_nat,produc6699827698519632094on_nat(Xb,produc5098337634421038937on_nat(none_nat,Xba))) ) )
         => ( ! [V2: nat] :
                ( ( Xaa = some_nat(V2) )
               => ( ( Xba = none_nat )
                 => ( ( Y = none_nat )
                   => ~ accp_P5526603486790338531on_nat(vEBT_V3895251965096974666el_nat,produc6699827698519632094on_nat(Xb,produc5098337634421038937on_nat(some_nat(V2),none_nat))) ) ) )
           => ~ ! [A2: nat] :
                  ( ( Xaa = some_nat(A2) )
                 => ! [B2: nat] :
                      ( ( Xba = some_nat(B2) )
                     => ( ( Y = some_nat(aa_nat_nat(aa_nat_fun_nat_nat(Xb,A2),B2)) )
                       => ~ accp_P5526603486790338531on_nat(vEBT_V3895251965096974666el_nat,produc6699827698519632094on_nat(Xb,produc5098337634421038937on_nat(some_nat(A2),some_nat(B2)))) ) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.pelims
tff(fact_6088_Bolzano,axiom,
    ! [Aa: real,Ba: real,P: fun_real_fun_real_o] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),Ba)
     => ( ! [A2: real,B2: real,C3: real] :
            ( aa_real_o(aa_real_fun_real_o(P,A2),B2)
           => ( aa_real_o(aa_real_fun_real_o(P,B2),C3)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B2)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),C3)
                 => aa_real_o(aa_real_fun_real_o(P,A2),C3) ) ) ) )
       => ( ! [X: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),X)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Ba)
               => ? [D3: real] :
                    ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D3)
                    & ! [A2: real,B2: real] :
                        ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),X)
                          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),B2)
                          & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B2),A2)),D3) )
                       => aa_real_o(aa_real_fun_real_o(P,A2),B2) ) ) ) )
         => aa_real_o(aa_real_fun_real_o(P,Aa),Ba) ) ) ) ).

% Bolzano
tff(fact_6089_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(aa_rea7340605660300375871l_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(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xb))),Xb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ).

% abs_ln_one_plus_x_minus_x_bound_nonpos
tff(fact_6090_divmod__algorithm__code_I8_J,axiom,
    ! [M: num,Na: num] :
      unique3479559517661332726nteger(aa_num_num(bit1,M),aa_num_num(bit1,Na)) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_num,M),Na),produc1086072967326762835nteger(zero_z3403309356797280102nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,M))),unique4921790084139445826nteger(aa_num_num(bit1,Na),unique3479559517661332726nteger(aa_num_num(bit1,M),aa_num_num(bit0,aa_num_num(bit1,Na))))) ).

% divmod_algorithm_code(8)
tff(fact_6091_divmod__algorithm__code_I8_J,axiom,
    ! [M: num,Na: num] :
      unique5055182867167087721od_nat(aa_num_num(bit1,M),aa_num_num(bit1,Na)) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_num,M),Na),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,M))),unique5026877609467782581ep_nat(aa_num_num(bit1,Na),unique5055182867167087721od_nat(aa_num_num(bit1,M),aa_num_num(bit0,aa_num_num(bit1,Na))))) ).

% divmod_algorithm_code(8)
tff(fact_6092_divmod__algorithm__code_I8_J,axiom,
    ! [M: num,Na: num] :
      unique5052692396658037445od_int(aa_num_num(bit1,M),aa_num_num(bit1,Na)) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_num,M),Na),product_Pair_int_int(zero_zero_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))),unique5024387138958732305ep_int(aa_num_num(bit1,Na),unique5052692396658037445od_int(aa_num_num(bit1,M),aa_num_num(bit0,aa_num_num(bit1,Na))))) ).

% divmod_algorithm_code(8)
tff(fact_6093_divmod__algorithm__code_I7_J,axiom,
    ! [M: num,Na: num] :
      unique3479559517661332726nteger(aa_num_num(bit0,M),aa_num_num(bit1,Na)) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_eq_num,M),Na),produc1086072967326762835nteger(zero_z3403309356797280102nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,M))),unique4921790084139445826nteger(aa_num_num(bit1,Na),unique3479559517661332726nteger(aa_num_num(bit0,M),aa_num_num(bit0,aa_num_num(bit1,Na))))) ).

% divmod_algorithm_code(7)
tff(fact_6094_divmod__algorithm__code_I7_J,axiom,
    ! [M: num,Na: num] :
      unique5055182867167087721od_nat(aa_num_num(bit0,M),aa_num_num(bit1,Na)) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_eq_num,M),Na),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,M))),unique5026877609467782581ep_nat(aa_num_num(bit1,Na),unique5055182867167087721od_nat(aa_num_num(bit0,M),aa_num_num(bit0,aa_num_num(bit1,Na))))) ).

% divmod_algorithm_code(7)
tff(fact_6095_divmod__algorithm__code_I7_J,axiom,
    ! [M: num,Na: num] :
      unique5052692396658037445od_int(aa_num_num(bit0,M),aa_num_num(bit1,Na)) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_eq_num,M),Na),product_Pair_int_int(zero_zero_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,M))),unique5024387138958732305ep_int(aa_num_num(bit1,Na),unique5052692396658037445od_int(aa_num_num(bit0,M),aa_num_num(bit0,aa_num_num(bit1,Na))))) ).

% divmod_algorithm_code(7)
tff(fact_6096_abs__idempotent,axiom,
    ! [Aa: real] : aa_real_real(abs_abs_real,aa_real_real(abs_abs_real,Aa)) = aa_real_real(abs_abs_real,Aa) ).

% abs_idempotent
tff(fact_6097_abs__idempotent,axiom,
    ! [Aa: int] : aa_int_int(abs_abs_int,aa_int_int(abs_abs_int,Aa)) = aa_int_int(abs_abs_int,Aa) ).

% abs_idempotent
tff(fact_6098_abs__idempotent,axiom,
    ! [Aa: code_integer] : aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,Aa)) = aa_Cod1783974652533396728nteger(abs_abs_Code_integer,Aa) ).

% abs_idempotent
tff(fact_6099_abs__idempotent,axiom,
    ! [Aa: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(abs_abs_rat,Aa)) = aa_rat_rat(abs_abs_rat,Aa) ).

% abs_idempotent
tff(fact_6100_abs__abs,axiom,
    ! [Aa: real] : aa_real_real(abs_abs_real,aa_real_real(abs_abs_real,Aa)) = aa_real_real(abs_abs_real,Aa) ).

% abs_abs
tff(fact_6101_abs__abs,axiom,
    ! [Aa: int] : aa_int_int(abs_abs_int,aa_int_int(abs_abs_int,Aa)) = aa_int_int(abs_abs_int,Aa) ).

% abs_abs
tff(fact_6102_abs__abs,axiom,
    ! [Aa: code_integer] : aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,Aa)) = aa_Cod1783974652533396728nteger(abs_abs_Code_integer,Aa) ).

% abs_abs
tff(fact_6103_abs__abs,axiom,
    ! [Aa: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(abs_abs_rat,Aa)) = aa_rat_rat(abs_abs_rat,Aa) ).

% abs_abs
tff(fact_6104_abs__0,axiom,
    aa_Cod1783974652533396728nteger(abs_abs_Code_integer,zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% abs_0
tff(fact_6105_abs__0,axiom,
    aa_real_real(abs_abs_real,zero_zero_real) = zero_zero_real ).

% abs_0
tff(fact_6106_abs__0,axiom,
    aa_rat_rat(abs_abs_rat,zero_zero_rat) = zero_zero_rat ).

% abs_0
tff(fact_6107_abs__0,axiom,
    aa_int_int(abs_abs_int,zero_zero_int) = zero_zero_int ).

% abs_0
tff(fact_6108_abs__zero,axiom,
    aa_Cod1783974652533396728nteger(abs_abs_Code_integer,zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% abs_zero
tff(fact_6109_abs__zero,axiom,
    aa_real_real(abs_abs_real,zero_zero_real) = zero_zero_real ).

% abs_zero
tff(fact_6110_abs__zero,axiom,
    aa_rat_rat(abs_abs_rat,zero_zero_rat) = zero_zero_rat ).

% abs_zero
tff(fact_6111_abs__zero,axiom,
    aa_int_int(abs_abs_int,zero_zero_int) = zero_zero_int ).

% abs_zero
tff(fact_6112_abs__eq__0,axiom,
    ! [Aa: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(abs_abs_Code_integer,Aa) = zero_z3403309356797280102nteger )
    <=> ( Aa = zero_z3403309356797280102nteger ) ) ).

% abs_eq_0
tff(fact_6113_abs__eq__0,axiom,
    ! [Aa: real] :
      ( ( aa_real_real(abs_abs_real,Aa) = zero_zero_real )
    <=> ( Aa = zero_zero_real ) ) ).

% abs_eq_0
tff(fact_6114_abs__eq__0,axiom,
    ! [Aa: rat] :
      ( ( aa_rat_rat(abs_abs_rat,Aa) = zero_zero_rat )
    <=> ( Aa = zero_zero_rat ) ) ).

% abs_eq_0
tff(fact_6115_abs__eq__0,axiom,
    ! [Aa: int] :
      ( ( aa_int_int(abs_abs_int,Aa) = zero_zero_int )
    <=> ( Aa = zero_zero_int ) ) ).

% abs_eq_0
tff(fact_6116_abs__0__eq,axiom,
    ! [Aa: code_integer] :
      ( ( zero_z3403309356797280102nteger = aa_Cod1783974652533396728nteger(abs_abs_Code_integer,Aa) )
    <=> ( Aa = zero_z3403309356797280102nteger ) ) ).

% abs_0_eq
tff(fact_6117_abs__0__eq,axiom,
    ! [Aa: real] :
      ( ( zero_zero_real = aa_real_real(abs_abs_real,Aa) )
    <=> ( Aa = zero_zero_real ) ) ).

% abs_0_eq
tff(fact_6118_abs__0__eq,axiom,
    ! [Aa: rat] :
      ( ( zero_zero_rat = aa_rat_rat(abs_abs_rat,Aa) )
    <=> ( Aa = zero_zero_rat ) ) ).

% abs_0_eq
tff(fact_6119_abs__0__eq,axiom,
    ! [Aa: int] :
      ( ( zero_zero_int = aa_int_int(abs_abs_int,Aa) )
    <=> ( Aa = zero_zero_int ) ) ).

% abs_0_eq
tff(fact_6120_abs__numeral,axiom,
    ! [Na: num] : aa_rat_rat(abs_abs_rat,aa_num_rat(numeral_numeral_rat,Na)) = aa_num_rat(numeral_numeral_rat,Na) ).

% abs_numeral
tff(fact_6121_abs__numeral,axiom,
    ! [Na: num] : aa_real_real(abs_abs_real,aa_num_real(numeral_numeral_real,Na)) = aa_num_real(numeral_numeral_real,Na) ).

% abs_numeral
tff(fact_6122_abs__numeral,axiom,
    ! [Na: num] : aa_int_int(abs_abs_int,aa_num_int(numeral_numeral_int,Na)) = aa_num_int(numeral_numeral_int,Na) ).

% abs_numeral
tff(fact_6123_abs__numeral,axiom,
    ! [Na: num] : aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,Na)) = aa_num_Code_integer(numera6620942414471956472nteger,Na) ).

% abs_numeral
tff(fact_6124_abs__mult__self__eq,axiom,
    ! [Aa: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,Aa)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,Aa)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Aa),Aa) ).

% abs_mult_self_eq
tff(fact_6125_abs__mult__self__eq,axiom,
    ! [Aa: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(abs_abs_real,Aa)),aa_real_real(abs_abs_real,Aa)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),Aa) ).

% abs_mult_self_eq
tff(fact_6126_abs__mult__self__eq,axiom,
    ! [Aa: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(abs_abs_rat,Aa)),aa_rat_rat(abs_abs_rat,Aa)) = aa_rat_rat(times_times_rat(Aa),Aa) ).

% abs_mult_self_eq
tff(fact_6127_abs__mult__self__eq,axiom,
    ! [Aa: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,Aa)),aa_int_int(abs_abs_int,Aa)) = aa_int_int(aa_int_fun_int_int(times_times_int,Aa),Aa) ).

% abs_mult_self_eq
tff(fact_6128_abs__1,axiom,
    aa_Cod1783974652533396728nteger(abs_abs_Code_integer,one_one_Code_integer) = one_one_Code_integer ).

% abs_1
tff(fact_6129_abs__1,axiom,
    aa_complex_complex(abs_abs_complex,one_one_complex) = one_one_complex ).

% abs_1
tff(fact_6130_abs__1,axiom,
    aa_real_real(abs_abs_real,one_one_real) = one_one_real ).

% abs_1
tff(fact_6131_abs__1,axiom,
    aa_rat_rat(abs_abs_rat,one_one_rat) = one_one_rat ).

% abs_1
tff(fact_6132_abs__1,axiom,
    aa_int_int(abs_abs_int,one_one_int) = one_one_int ).

% abs_1
tff(fact_6133_abs__add__abs,axiom,
    ! [Aa: code_integer,Ba: code_integer] : aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,Aa)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,Ba))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,Aa)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,Ba)) ).

% abs_add_abs
tff(fact_6134_abs__add__abs,axiom,
    ! [Aa: real,Ba: real] : aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,Aa)),aa_real_real(abs_abs_real,Ba))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,Aa)),aa_real_real(abs_abs_real,Ba)) ).

% abs_add_abs
tff(fact_6135_abs__add__abs,axiom,
    ! [Aa: rat,Ba: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(abs_abs_rat,Aa)),aa_rat_rat(abs_abs_rat,Ba))) = aa_rat_rat(plus_plus_rat(aa_rat_rat(abs_abs_rat,Aa)),aa_rat_rat(abs_abs_rat,Ba)) ).

% abs_add_abs
tff(fact_6136_abs__add__abs,axiom,
    ! [Aa: int,Ba: int] : aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,Aa)),aa_int_int(abs_abs_int,Ba))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,Aa)),aa_int_int(abs_abs_int,Ba)) ).

% abs_add_abs
tff(fact_6137_abs__minus__cancel,axiom,
    ! [Aa: real] : aa_real_real(abs_abs_real,aa_real_real(uminus_uminus_real,Aa)) = aa_real_real(abs_abs_real,Aa) ).

% abs_minus_cancel
tff(fact_6138_abs__minus__cancel,axiom,
    ! [Aa: int] : aa_int_int(abs_abs_int,aa_int_int(uminus_uminus_int,Aa)) = aa_int_int(abs_abs_int,Aa) ).

% abs_minus_cancel
tff(fact_6139_abs__minus__cancel,axiom,
    ! [Aa: code_integer] : aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa)) = aa_Cod1783974652533396728nteger(abs_abs_Code_integer,Aa) ).

% abs_minus_cancel
tff(fact_6140_abs__minus__cancel,axiom,
    ! [Aa: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(uminus_uminus_rat,Aa)) = aa_rat_rat(abs_abs_rat,Aa) ).

% abs_minus_cancel
tff(fact_6141_abs__minus,axiom,
    ! [Aa: real] : aa_real_real(abs_abs_real,aa_real_real(uminus_uminus_real,Aa)) = aa_real_real(abs_abs_real,Aa) ).

% abs_minus
tff(fact_6142_abs__minus,axiom,
    ! [Aa: int] : aa_int_int(abs_abs_int,aa_int_int(uminus_uminus_int,Aa)) = aa_int_int(abs_abs_int,Aa) ).

% abs_minus
tff(fact_6143_abs__minus,axiom,
    ! [Aa: code_integer] : aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa)) = aa_Cod1783974652533396728nteger(abs_abs_Code_integer,Aa) ).

% abs_minus
tff(fact_6144_abs__minus,axiom,
    ! [Aa: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(uminus_uminus_rat,Aa)) = aa_rat_rat(abs_abs_rat,Aa) ).

% abs_minus
tff(fact_6145_dvd__abs__iff,axiom,
    ! [M: real,K: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,M),aa_real_real(abs_abs_real,K))
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,M),K) ) ).

% dvd_abs_iff
tff(fact_6146_dvd__abs__iff,axiom,
    ! [M: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,M),aa_int_int(abs_abs_int,K))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,M),K) ) ).

% dvd_abs_iff
tff(fact_6147_dvd__abs__iff,axiom,
    ! [M: code_integer,K: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,M),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,K))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,M),K) ) ).

% dvd_abs_iff
tff(fact_6148_dvd__abs__iff,axiom,
    ! [M: rat,K: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,M),aa_rat_rat(abs_abs_rat,K))
    <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,M),K) ) ).

% dvd_abs_iff
tff(fact_6149_abs__dvd__iff,axiom,
    ! [M: real,K: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_real_real(abs_abs_real,M)),K)
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,M),K) ) ).

% abs_dvd_iff
tff(fact_6150_abs__dvd__iff,axiom,
    ! [M: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(abs_abs_int,M)),K)
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,M),K) ) ).

% abs_dvd_iff
tff(fact_6151_abs__dvd__iff,axiom,
    ! [M: code_integer,K: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,M)),K)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,M),K) ) ).

% abs_dvd_iff
tff(fact_6152_abs__dvd__iff,axiom,
    ! [M: rat,K: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(abs_abs_rat,M)),K)
    <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,M),K) ) ).

% abs_dvd_iff
tff(fact_6153_abs__le__zero__iff,axiom,
    ! [Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,Aa)),zero_z3403309356797280102nteger)
    <=> ( Aa = zero_z3403309356797280102nteger ) ) ).

% abs_le_zero_iff
tff(fact_6154_abs__le__zero__iff,axiom,
    ! [Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Aa)),zero_zero_real)
    <=> ( Aa = zero_zero_real ) ) ).

% abs_le_zero_iff
tff(fact_6155_abs__le__zero__iff,axiom,
    ! [Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,Aa)),zero_zero_rat)
    <=> ( Aa = zero_zero_rat ) ) ).

% abs_le_zero_iff
tff(fact_6156_abs__le__zero__iff,axiom,
    ! [Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,Aa)),zero_zero_int)
    <=> ( Aa = zero_zero_int ) ) ).

% abs_le_zero_iff
tff(fact_6157_abs__le__self__iff,axiom,
    ! [Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,Aa)),Aa)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Aa) ) ).

% abs_le_self_iff
tff(fact_6158_abs__le__self__iff,axiom,
    ! [Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Aa)),Aa)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Aa) ) ).

% abs_le_self_iff
tff(fact_6159_abs__le__self__iff,axiom,
    ! [Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,Aa)),Aa)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Aa) ) ).

% abs_le_self_iff
tff(fact_6160_abs__le__self__iff,axiom,
    ! [Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,Aa)),Aa)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Aa) ) ).

% abs_le_self_iff
tff(fact_6161_abs__of__nonneg,axiom,
    ! [Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Aa)
     => ( aa_Cod1783974652533396728nteger(abs_abs_Code_integer,Aa) = Aa ) ) ).

% abs_of_nonneg
tff(fact_6162_abs__of__nonneg,axiom,
    ! [Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Aa)
     => ( aa_real_real(abs_abs_real,Aa) = Aa ) ) ).

% abs_of_nonneg
tff(fact_6163_abs__of__nonneg,axiom,
    ! [Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Aa)
     => ( aa_rat_rat(abs_abs_rat,Aa) = Aa ) ) ).

% abs_of_nonneg
tff(fact_6164_abs__of__nonneg,axiom,
    ! [Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Aa)
     => ( aa_int_int(abs_abs_int,Aa) = Aa ) ) ).

% abs_of_nonneg
tff(fact_6165_zero__less__abs__iff,axiom,
    ! [Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,Aa))
    <=> ( Aa != zero_z3403309356797280102nteger ) ) ).

% zero_less_abs_iff
tff(fact_6166_zero__less__abs__iff,axiom,
    ! [Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(abs_abs_real,Aa))
    <=> ( Aa != zero_zero_real ) ) ).

% zero_less_abs_iff
tff(fact_6167_zero__less__abs__iff,axiom,
    ! [Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(abs_abs_rat,Aa))
    <=> ( Aa != zero_zero_rat ) ) ).

% zero_less_abs_iff
tff(fact_6168_zero__less__abs__iff,axiom,
    ! [Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(abs_abs_int,Aa))
    <=> ( Aa != zero_zero_int ) ) ).

% zero_less_abs_iff
tff(fact_6169_abs__neg__numeral,axiom,
    ! [Na: num] : aa_real_real(abs_abs_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Na))) = aa_num_real(numeral_numeral_real,Na) ).

% abs_neg_numeral
tff(fact_6170_abs__neg__numeral,axiom,
    ! [Na: num] : aa_int_int(abs_abs_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Na))) = aa_num_int(numeral_numeral_int,Na) ).

% abs_neg_numeral
tff(fact_6171_abs__neg__numeral,axiom,
    ! [Na: num] : aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Na))) = aa_num_Code_integer(numera6620942414471956472nteger,Na) ).

% abs_neg_numeral
tff(fact_6172_abs__neg__numeral,axiom,
    ! [Na: num] : aa_rat_rat(abs_abs_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Na))) = aa_num_rat(numeral_numeral_rat,Na) ).

% abs_neg_numeral
tff(fact_6173_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_6174_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_6175_abs__neg__one,axiom,
    aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = one_one_Code_integer ).

% abs_neg_one
tff(fact_6176_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_6177_zero__le__divide__abs__iff,axiom,
    ! [Aa: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Aa),aa_real_real(abs_abs_real,Ba)))
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Aa)
        | ( Ba = zero_zero_real ) ) ) ).

% zero_le_divide_abs_iff
tff(fact_6178_zero__le__divide__abs__iff,axiom,
    ! [Aa: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(Aa),aa_rat_rat(abs_abs_rat,Ba)))
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Aa)
        | ( Ba = zero_zero_rat ) ) ) ).

% zero_le_divide_abs_iff
tff(fact_6179_divide__le__0__abs__iff,axiom,
    ! [Aa: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Aa),aa_real_real(abs_abs_real,Ba))),zero_zero_real)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),zero_zero_real)
        | ( Ba = zero_zero_real ) ) ) ).

% divide_le_0_abs_iff
tff(fact_6180_divide__le__0__abs__iff,axiom,
    ! [Aa: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(Aa),aa_rat_rat(abs_abs_rat,Ba))),zero_zero_rat)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),zero_zero_rat)
        | ( Ba = zero_zero_rat ) ) ) ).

% divide_le_0_abs_iff
tff(fact_6181_abs__of__nonpos,axiom,
    ! [Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),zero_zero_real)
     => ( aa_real_real(abs_abs_real,Aa) = aa_real_real(uminus_uminus_real,Aa) ) ) ).

% abs_of_nonpos
tff(fact_6182_abs__of__nonpos,axiom,
    ! [Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Aa),zero_z3403309356797280102nteger)
     => ( aa_Cod1783974652533396728nteger(abs_abs_Code_integer,Aa) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa) ) ) ).

% abs_of_nonpos
tff(fact_6183_abs__of__nonpos,axiom,
    ! [Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),zero_zero_rat)
     => ( aa_rat_rat(abs_abs_rat,Aa) = aa_rat_rat(uminus_uminus_rat,Aa) ) ) ).

% abs_of_nonpos
tff(fact_6184_abs__of__nonpos,axiom,
    ! [Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),zero_zero_int)
     => ( aa_int_int(abs_abs_int,Aa) = aa_int_int(uminus_uminus_int,Aa) ) ) ).

% abs_of_nonpos
tff(fact_6185_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_6186_zero__less__power__abs__iff,axiom,
    ! [Aa: code_integer,Na: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(abs_abs_Code_integer,Aa)),Na))
    <=> ( ( Aa != zero_z3403309356797280102nteger )
        | ( Na = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
tff(fact_6187_zero__less__power__abs__iff,axiom,
    ! [Aa: real,Na: 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,Aa)),Na))
    <=> ( ( Aa != zero_zero_real )
        | ( Na = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
tff(fact_6188_zero__less__power__abs__iff,axiom,
    ! [Aa: rat,Na: 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,Aa)),Na))
    <=> ( ( Aa != zero_zero_rat )
        | ( Na = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
tff(fact_6189_zero__less__power__abs__iff,axiom,
    ! [Aa: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(aa_int_int(abs_abs_int,Aa)),Na))
    <=> ( ( Aa != zero_zero_int )
        | ( Na = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
tff(fact_6190_divmod__algorithm__code_I2_J,axiom,
    ! [M: num] : unique5052692396658037445od_int(M,one) = product_Pair_int_int(aa_num_int(numeral_numeral_int,M),zero_zero_int) ).

% divmod_algorithm_code(2)
tff(fact_6191_divmod__algorithm__code_I2_J,axiom,
    ! [M: num] : unique3479559517661332726nteger(M,one) = produc1086072967326762835nteger(aa_num_Code_integer(numera6620942414471956472nteger,M),zero_z3403309356797280102nteger) ).

% divmod_algorithm_code(2)
tff(fact_6192_divmod__algorithm__code_I2_J,axiom,
    ! [M: num] : unique5055182867167087721od_nat(M,one) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_num_nat(numeral_numeral_nat,M)),zero_zero_nat) ).

% divmod_algorithm_code(2)
tff(fact_6193_divmod__algorithm__code_I3_J,axiom,
    ! [Na: num] : unique5052692396658037445od_int(one,aa_num_num(bit0,Na)) = product_Pair_int_int(zero_zero_int,aa_num_int(numeral_numeral_int,one)) ).

% divmod_algorithm_code(3)
tff(fact_6194_divmod__algorithm__code_I3_J,axiom,
    ! [Na: num] : unique3479559517661332726nteger(one,aa_num_num(bit0,Na)) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,aa_num_Code_integer(numera6620942414471956472nteger,one)) ).

% divmod_algorithm_code(3)
tff(fact_6195_divmod__algorithm__code_I3_J,axiom,
    ! [Na: num] : unique5055182867167087721od_nat(one,aa_num_num(bit0,Na)) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,one)) ).

% divmod_algorithm_code(3)
tff(fact_6196_divmod__algorithm__code_I4_J,axiom,
    ! [Na: num] : unique5052692396658037445od_int(one,aa_num_num(bit1,Na)) = product_Pair_int_int(zero_zero_int,aa_num_int(numeral_numeral_int,one)) ).

% divmod_algorithm_code(4)
tff(fact_6197_divmod__algorithm__code_I4_J,axiom,
    ! [Na: num] : unique3479559517661332726nteger(one,aa_num_num(bit1,Na)) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,aa_num_Code_integer(numera6620942414471956472nteger,one)) ).

% divmod_algorithm_code(4)
tff(fact_6198_divmod__algorithm__code_I4_J,axiom,
    ! [Na: num] : unique5055182867167087721od_nat(one,aa_num_num(bit1,Na)) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,one)) ).

% divmod_algorithm_code(4)
tff(fact_6199_abs__ge__self,axiom,
    ! [Aa: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),aa_real_real(abs_abs_real,Aa)) ).

% abs_ge_self
tff(fact_6200_abs__ge__self,axiom,
    ! [Aa: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Aa),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,Aa)) ).

% abs_ge_self
tff(fact_6201_abs__ge__self,axiom,
    ! [Aa: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),aa_rat_rat(abs_abs_rat,Aa)) ).

% abs_ge_self
tff(fact_6202_abs__ge__self,axiom,
    ! [Aa: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),aa_int_int(abs_abs_int,Aa)) ).

% abs_ge_self
tff(fact_6203_abs__le__D1,axiom,
    ! [Aa: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Aa)),Ba)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),Ba) ) ).

% abs_le_D1
tff(fact_6204_abs__le__D1,axiom,
    ! [Aa: code_integer,Ba: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,Aa)),Ba)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Aa),Ba) ) ).

% abs_le_D1
tff(fact_6205_abs__le__D1,axiom,
    ! [Aa: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,Aa)),Ba)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),Ba) ) ).

% abs_le_D1
tff(fact_6206_abs__le__D1,axiom,
    ! [Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,Aa)),Ba)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),Ba) ) ).

% abs_le_D1
tff(fact_6207_abs__eq__0__iff,axiom,
    ! [Aa: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(abs_abs_Code_integer,Aa) = zero_z3403309356797280102nteger )
    <=> ( Aa = zero_z3403309356797280102nteger ) ) ).

% abs_eq_0_iff
tff(fact_6208_abs__eq__0__iff,axiom,
    ! [Aa: real] :
      ( ( aa_real_real(abs_abs_real,Aa) = zero_zero_real )
    <=> ( Aa = zero_zero_real ) ) ).

% abs_eq_0_iff
tff(fact_6209_abs__eq__0__iff,axiom,
    ! [Aa: rat] :
      ( ( aa_rat_rat(abs_abs_rat,Aa) = zero_zero_rat )
    <=> ( Aa = zero_zero_rat ) ) ).

% abs_eq_0_iff
tff(fact_6210_abs__eq__0__iff,axiom,
    ! [Aa: int] :
      ( ( aa_int_int(abs_abs_int,Aa) = zero_zero_int )
    <=> ( Aa = zero_zero_int ) ) ).

% abs_eq_0_iff
tff(fact_6211_abs__mult,axiom,
    ! [Aa: code_integer,Ba: code_integer] : aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Aa),Ba)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,Aa)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,Ba)) ).

% abs_mult
tff(fact_6212_abs__mult,axiom,
    ! [Aa: real,Ba: real] : aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),Ba)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(abs_abs_real,Aa)),aa_real_real(abs_abs_real,Ba)) ).

% abs_mult
tff(fact_6213_abs__mult,axiom,
    ! [Aa: rat,Ba: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(times_times_rat(Aa),Ba)) = aa_rat_rat(times_times_rat(aa_rat_rat(abs_abs_rat,Aa)),aa_rat_rat(abs_abs_rat,Ba)) ).

% abs_mult
tff(fact_6214_abs__mult,axiom,
    ! [Aa: int,Ba: int] : aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(times_times_int,Aa),Ba)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,Aa)),aa_int_int(abs_abs_int,Ba)) ).

% abs_mult
tff(fact_6215_abs__one,axiom,
    aa_Cod1783974652533396728nteger(abs_abs_Code_integer,one_one_Code_integer) = one_one_Code_integer ).

% abs_one
tff(fact_6216_abs__one,axiom,
    aa_real_real(abs_abs_real,one_one_real) = one_one_real ).

% abs_one
tff(fact_6217_abs__one,axiom,
    aa_rat_rat(abs_abs_rat,one_one_rat) = one_one_rat ).

% abs_one
tff(fact_6218_abs__one,axiom,
    aa_int_int(abs_abs_int,one_one_int) = one_one_int ).

% abs_one
tff(fact_6219_abs__minus__commute,axiom,
    ! [Aa: code_integer,Ba: code_integer] : aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,Aa),Ba)) = aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,Ba),Aa)) ).

% abs_minus_commute
tff(fact_6220_abs__minus__commute,axiom,
    ! [Aa: real,Ba: real] : aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Aa),Ba)) = aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Ba),Aa)) ).

% abs_minus_commute
tff(fact_6221_abs__minus__commute,axiom,
    ! [Aa: rat,Ba: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(Aa),Ba)) = aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(Ba),Aa)) ).

% abs_minus_commute
tff(fact_6222_abs__minus__commute,axiom,
    ! [Aa: int,Ba: int] : aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Aa),Ba)) = aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Ba),Aa)) ).

% abs_minus_commute
tff(fact_6223_abs__eq__iff,axiom,
    ! [Xb: real,Y: real] :
      ( ( aa_real_real(abs_abs_real,Xb) = aa_real_real(abs_abs_real,Y) )
    <=> ( ( Xb = Y )
        | ( Xb = aa_real_real(uminus_uminus_real,Y) ) ) ) ).

% abs_eq_iff
tff(fact_6224_abs__eq__iff,axiom,
    ! [Xb: int,Y: int] :
      ( ( aa_int_int(abs_abs_int,Xb) = aa_int_int(abs_abs_int,Y) )
    <=> ( ( Xb = Y )
        | ( Xb = aa_int_int(uminus_uminus_int,Y) ) ) ) ).

% abs_eq_iff
tff(fact_6225_abs__eq__iff,axiom,
    ! [Xb: code_integer,Y: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(abs_abs_Code_integer,Xb) = aa_Cod1783974652533396728nteger(abs_abs_Code_integer,Y) )
    <=> ( ( Xb = Y )
        | ( Xb = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Y) ) ) ) ).

% abs_eq_iff
tff(fact_6226_abs__eq__iff,axiom,
    ! [Xb: rat,Y: rat] :
      ( ( aa_rat_rat(abs_abs_rat,Xb) = aa_rat_rat(abs_abs_rat,Y) )
    <=> ( ( Xb = Y )
        | ( Xb = aa_rat_rat(uminus_uminus_rat,Y) ) ) ) ).

% abs_eq_iff
tff(fact_6227_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_6228_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_6229_dvd__if__abs__eq,axiom,
    ! [L: code_integer,K: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(abs_abs_Code_integer,L) = aa_Cod1783974652533396728nteger(abs_abs_Code_integer,K) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,L),K) ) ).

% dvd_if_abs_eq
tff(fact_6230_dvd__if__abs__eq,axiom,
    ! [L: rat,K: rat] :
      ( ( aa_rat_rat(abs_abs_rat,L) = aa_rat_rat(abs_abs_rat,K) )
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,L),K) ) ).

% dvd_if_abs_eq
tff(fact_6231_abs__ge__zero,axiom,
    ! [Aa: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,Aa)) ).

% abs_ge_zero
tff(fact_6232_abs__ge__zero,axiom,
    ! [Aa: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(abs_abs_real,Aa)) ).

% abs_ge_zero
tff(fact_6233_abs__ge__zero,axiom,
    ! [Aa: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(abs_abs_rat,Aa)) ).

% abs_ge_zero
tff(fact_6234_abs__ge__zero,axiom,
    ! [Aa: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(abs_abs_int,Aa)) ).

% abs_ge_zero
tff(fact_6235_abs__not__less__zero,axiom,
    ! [Aa: code_integer] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,Aa)),zero_z3403309356797280102nteger) ).

% abs_not_less_zero
tff(fact_6236_abs__not__less__zero,axiom,
    ! [Aa: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Aa)),zero_zero_real) ).

% abs_not_less_zero
tff(fact_6237_abs__not__less__zero,axiom,
    ! [Aa: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,Aa)),zero_zero_rat) ).

% abs_not_less_zero
tff(fact_6238_abs__not__less__zero,axiom,
    ! [Aa: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,Aa)),zero_zero_int) ).

% abs_not_less_zero
tff(fact_6239_abs__of__pos,axiom,
    ! [Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),Aa)
     => ( aa_Cod1783974652533396728nteger(abs_abs_Code_integer,Aa) = Aa ) ) ).

% abs_of_pos
tff(fact_6240_abs__of__pos,axiom,
    ! [Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Aa)
     => ( aa_real_real(abs_abs_real,Aa) = Aa ) ) ).

% abs_of_pos
tff(fact_6241_abs__of__pos,axiom,
    ! [Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Aa)
     => ( aa_rat_rat(abs_abs_rat,Aa) = Aa ) ) ).

% abs_of_pos
tff(fact_6242_abs__of__pos,axiom,
    ! [Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Aa)
     => ( aa_int_int(abs_abs_int,Aa) = Aa ) ) ).

% abs_of_pos
tff(fact_6243_abs__triangle__ineq,axiom,
    ! [Aa: code_integer,Ba: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Aa),Ba))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,Aa)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,Ba))) ).

% abs_triangle_ineq
tff(fact_6244_abs__triangle__ineq,axiom,
    ! [Aa: real,Ba: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),Ba))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,Aa)),aa_real_real(abs_abs_real,Ba))) ).

% abs_triangle_ineq
tff(fact_6245_abs__triangle__ineq,axiom,
    ! [Aa: rat,Ba: 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(Aa),Ba))),aa_rat_rat(plus_plus_rat(aa_rat_rat(abs_abs_rat,Aa)),aa_rat_rat(abs_abs_rat,Ba))) ).

% abs_triangle_ineq
tff(fact_6246_abs__triangle__ineq,axiom,
    ! [Aa: int,Ba: 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,Aa),Ba))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,Aa)),aa_int_int(abs_abs_int,Ba))) ).

% abs_triangle_ineq
tff(fact_6247_abs__mult__less,axiom,
    ! [Aa: code_integer,C: code_integer,Ba: code_integer,D: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,Aa)),C)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,Ba)),D)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,Aa)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,Ba))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),D)) ) ) ).

% abs_mult_less
tff(fact_6248_abs__mult__less,axiom,
    ! [Aa: real,C: real,Ba: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Aa)),C)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Ba)),D)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(abs_abs_real,Aa)),aa_real_real(abs_abs_real,Ba))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),D)) ) ) ).

% abs_mult_less
tff(fact_6249_abs__mult__less,axiom,
    ! [Aa: rat,C: rat,Ba: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,Aa)),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,Ba)),D)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(aa_rat_rat(abs_abs_rat,Aa)),aa_rat_rat(abs_abs_rat,Ba))),aa_rat_rat(times_times_rat(C),D)) ) ) ).

% abs_mult_less
tff(fact_6250_abs__mult__less,axiom,
    ! [Aa: int,C: int,Ba: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,Aa)),C)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,Ba)),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,Aa)),aa_int_int(abs_abs_int,Ba))),aa_int_int(aa_int_fun_int_int(times_times_int,C),D)) ) ) ).

% abs_mult_less
tff(fact_6251_abs__triangle__ineq2__sym,axiom,
    ! [Aa: code_integer,Ba: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,Aa)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,Ba))),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,Ba),Aa))) ).

% abs_triangle_ineq2_sym
tff(fact_6252_abs__triangle__ineq2__sym,axiom,
    ! [Aa: real,Ba: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(abs_abs_real,Aa)),aa_real_real(abs_abs_real,Ba))),aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Ba),Aa))) ).

% abs_triangle_ineq2_sym
tff(fact_6253_abs__triangle__ineq2__sym,axiom,
    ! [Aa: rat,Ba: 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,Aa)),aa_rat_rat(abs_abs_rat,Ba))),aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(Ba),Aa))) ).

% abs_triangle_ineq2_sym
tff(fact_6254_abs__triangle__ineq2__sym,axiom,
    ! [Aa: int,Ba: 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,Aa)),aa_int_int(abs_abs_int,Ba))),aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Ba),Aa))) ).

% abs_triangle_ineq2_sym
tff(fact_6255_abs__triangle__ineq3,axiom,
    ! [Aa: code_integer,Ba: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,Aa)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,Ba)))),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,Aa),Ba))) ).

% abs_triangle_ineq3
tff(fact_6256_abs__triangle__ineq3,axiom,
    ! [Aa: real,Ba: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(abs_abs_real,Aa)),aa_real_real(abs_abs_real,Ba)))),aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Aa),Ba))) ).

% abs_triangle_ineq3
tff(fact_6257_abs__triangle__ineq3,axiom,
    ! [Aa: rat,Ba: 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,Aa)),aa_rat_rat(abs_abs_rat,Ba)))),aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(Aa),Ba))) ).

% abs_triangle_ineq3
tff(fact_6258_abs__triangle__ineq3,axiom,
    ! [Aa: int,Ba: 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,Aa)),aa_int_int(abs_abs_int,Ba)))),aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Aa),Ba))) ).

% abs_triangle_ineq3
tff(fact_6259_abs__triangle__ineq2,axiom,
    ! [Aa: code_integer,Ba: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,Aa)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,Ba))),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,Aa),Ba))) ).

% abs_triangle_ineq2
tff(fact_6260_abs__triangle__ineq2,axiom,
    ! [Aa: real,Ba: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(abs_abs_real,Aa)),aa_real_real(abs_abs_real,Ba))),aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Aa),Ba))) ).

% abs_triangle_ineq2
tff(fact_6261_abs__triangle__ineq2,axiom,
    ! [Aa: rat,Ba: 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,Aa)),aa_rat_rat(abs_abs_rat,Ba))),aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(Aa),Ba))) ).

% abs_triangle_ineq2
tff(fact_6262_abs__triangle__ineq2,axiom,
    ! [Aa: int,Ba: 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,Aa)),aa_int_int(abs_abs_int,Ba))),aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Aa),Ba))) ).

% abs_triangle_ineq2
tff(fact_6263_nonzero__abs__divide,axiom,
    ! [Ba: rat,Aa: rat] :
      ( ( Ba != zero_zero_rat )
     => ( aa_rat_rat(abs_abs_rat,aa_rat_rat(divide_divide_rat(Aa),Ba)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(abs_abs_rat,Aa)),aa_rat_rat(abs_abs_rat,Ba)) ) ) ).

% nonzero_abs_divide
tff(fact_6264_nonzero__abs__divide,axiom,
    ! [Ba: real,Aa: real] :
      ( ( Ba != zero_zero_real )
     => ( aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Aa),Ba)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(abs_abs_real,Aa)),aa_real_real(abs_abs_real,Ba)) ) ) ).

% nonzero_abs_divide
tff(fact_6265_abs__leI,axiom,
    ! [Aa: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),Ba)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,Aa)),Ba)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Aa)),Ba) ) ) ).

% abs_leI
tff(fact_6266_abs__leI,axiom,
    ! [Aa: code_integer,Ba: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Aa),Ba)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa)),Ba)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,Aa)),Ba) ) ) ).

% abs_leI
tff(fact_6267_abs__leI,axiom,
    ! [Aa: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),Ba)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,Aa)),Ba)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,Aa)),Ba) ) ) ).

% abs_leI
tff(fact_6268_abs__leI,axiom,
    ! [Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),Ba)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,Aa)),Ba)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,Aa)),Ba) ) ) ).

% abs_leI
tff(fact_6269_abs__le__D2,axiom,
    ! [Aa: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Aa)),Ba)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,Aa)),Ba) ) ).

% abs_le_D2
tff(fact_6270_abs__le__D2,axiom,
    ! [Aa: code_integer,Ba: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,Aa)),Ba)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa)),Ba) ) ).

% abs_le_D2
tff(fact_6271_abs__le__D2,axiom,
    ! [Aa: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,Aa)),Ba)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,Aa)),Ba) ) ).

% abs_le_D2
tff(fact_6272_abs__le__D2,axiom,
    ! [Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,Aa)),Ba)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,Aa)),Ba) ) ).

% abs_le_D2
tff(fact_6273_abs__le__iff,axiom,
    ! [Aa: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Aa)),Ba)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),Ba)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,Aa)),Ba) ) ) ).

% abs_le_iff
tff(fact_6274_abs__le__iff,axiom,
    ! [Aa: code_integer,Ba: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,Aa)),Ba)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Aa),Ba)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa)),Ba) ) ) ).

% abs_le_iff
tff(fact_6275_abs__le__iff,axiom,
    ! [Aa: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,Aa)),Ba)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),Ba)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,Aa)),Ba) ) ) ).

% abs_le_iff
tff(fact_6276_abs__le__iff,axiom,
    ! [Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,Aa)),Ba)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),Ba)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,Aa)),Ba) ) ) ).

% abs_le_iff
tff(fact_6277_abs__ge__minus__self,axiom,
    ! [Aa: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,Aa)),aa_real_real(abs_abs_real,Aa)) ).

% abs_ge_minus_self
tff(fact_6278_abs__ge__minus__self,axiom,
    ! [Aa: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,Aa)) ).

% abs_ge_minus_self
tff(fact_6279_abs__ge__minus__self,axiom,
    ! [Aa: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,Aa)),aa_rat_rat(abs_abs_rat,Aa)) ).

% abs_ge_minus_self
tff(fact_6280_abs__ge__minus__self,axiom,
    ! [Aa: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,Aa)),aa_int_int(abs_abs_int,Aa)) ).

% abs_ge_minus_self
tff(fact_6281_abs__less__iff,axiom,
    ! [Aa: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Aa)),Ba)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),Ba)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,Aa)),Ba) ) ) ).

% abs_less_iff
tff(fact_6282_abs__less__iff,axiom,
    ! [Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,Aa)),Ba)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),Ba)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,Aa)),Ba) ) ) ).

% abs_less_iff
tff(fact_6283_abs__less__iff,axiom,
    ! [Aa: code_integer,Ba: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,Aa)),Ba)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Aa),Ba)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa)),Ba) ) ) ).

% abs_less_iff
tff(fact_6284_abs__less__iff,axiom,
    ! [Aa: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,Aa)),Ba)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),Ba)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,Aa)),Ba) ) ) ).

% abs_less_iff
tff(fact_6285_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_6286_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_6287_abs__eq__mult,axiom,
    ! [Aa: code_integer,Ba: code_integer] :
      ( ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Aa)
          | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Aa),zero_z3403309356797280102nteger) )
        & ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Ba)
          | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Ba),zero_z3403309356797280102nteger) ) )
     => ( aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Aa),Ba)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,Aa)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,Ba)) ) ) ).

% abs_eq_mult
tff(fact_6288_abs__eq__mult,axiom,
    ! [Aa: real,Ba: real] :
      ( ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Aa)
          | aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),zero_zero_real) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Ba)
          | aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ba),zero_zero_real) ) )
     => ( aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),Ba)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(abs_abs_real,Aa)),aa_real_real(abs_abs_real,Ba)) ) ) ).

% abs_eq_mult
tff(fact_6289_abs__eq__mult,axiom,
    ! [Aa: rat,Ba: rat] :
      ( ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Aa)
          | aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Aa),zero_zero_rat) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Ba)
          | aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ba),zero_zero_rat) ) )
     => ( aa_rat_rat(abs_abs_rat,aa_rat_rat(times_times_rat(Aa),Ba)) = aa_rat_rat(times_times_rat(aa_rat_rat(abs_abs_rat,Aa)),aa_rat_rat(abs_abs_rat,Ba)) ) ) ).

% abs_eq_mult
tff(fact_6290_abs__eq__mult,axiom,
    ! [Aa: int,Ba: int] :
      ( ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Aa)
          | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),zero_zero_int) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Ba)
          | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ba),zero_zero_int) ) )
     => ( aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(times_times_int,Aa),Ba)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,Aa)),aa_int_int(abs_abs_int,Ba)) ) ) ).

% abs_eq_mult
tff(fact_6291_abs__mult__pos,axiom,
    ! [Xb: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Xb)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,Y)),Xb) = aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Y),Xb)) ) ) ).

% abs_mult_pos
tff(fact_6292_abs__mult__pos,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(abs_abs_real,Y)),Xb) = aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Xb)) ) ) ).

% abs_mult_pos
tff(fact_6293_abs__mult__pos,axiom,
    ! [Xb: rat,Y: 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,Y)),Xb) = aa_rat_rat(abs_abs_rat,aa_rat_rat(times_times_rat(Y),Xb)) ) ) ).

% abs_mult_pos
tff(fact_6294_abs__mult__pos,axiom,
    ! [Xb: int,Y: 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,Y)),Xb) = aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(times_times_int,Y),Xb)) ) ) ).

% abs_mult_pos
tff(fact_6295_abs__minus__le__zero,axiom,
    ! [Aa: 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,Aa))),zero_zero_real) ).

% abs_minus_le_zero
tff(fact_6296_abs__minus__le__zero,axiom,
    ! [Aa: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,Aa))),zero_z3403309356797280102nteger) ).

% abs_minus_le_zero
tff(fact_6297_abs__minus__le__zero,axiom,
    ! [Aa: 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,Aa))),zero_zero_rat) ).

% abs_minus_le_zero
tff(fact_6298_abs__minus__le__zero,axiom,
    ! [Aa: 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,Aa))),zero_zero_int) ).

% abs_minus_le_zero
tff(fact_6299_eq__abs__iff_H,axiom,
    ! [Aa: real,Ba: real] :
      ( ( Aa = aa_real_real(abs_abs_real,Ba) )
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Aa)
        & ( ( Ba = Aa )
          | ( Ba = aa_real_real(uminus_uminus_real,Aa) ) ) ) ) ).

% eq_abs_iff'
tff(fact_6300_eq__abs__iff_H,axiom,
    ! [Aa: code_integer,Ba: code_integer] :
      ( ( Aa = aa_Cod1783974652533396728nteger(abs_abs_Code_integer,Ba) )
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Aa)
        & ( ( Ba = Aa )
          | ( Ba = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa) ) ) ) ) ).

% eq_abs_iff'
tff(fact_6301_eq__abs__iff_H,axiom,
    ! [Aa: rat,Ba: rat] :
      ( ( Aa = aa_rat_rat(abs_abs_rat,Ba) )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Aa)
        & ( ( Ba = Aa )
          | ( Ba = aa_rat_rat(uminus_uminus_rat,Aa) ) ) ) ) ).

% eq_abs_iff'
tff(fact_6302_eq__abs__iff_H,axiom,
    ! [Aa: int,Ba: int] :
      ( ( Aa = aa_int_int(abs_abs_int,Ba) )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Aa)
        & ( ( Ba = Aa )
          | ( Ba = aa_int_int(uminus_uminus_int,Aa) ) ) ) ) ).

% eq_abs_iff'
tff(fact_6303_abs__eq__iff_H,axiom,
    ! [Aa: real,Ba: real] :
      ( ( aa_real_real(abs_abs_real,Aa) = Ba )
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Ba)
        & ( ( Aa = Ba )
          | ( Aa = aa_real_real(uminus_uminus_real,Ba) ) ) ) ) ).

% abs_eq_iff'
tff(fact_6304_abs__eq__iff_H,axiom,
    ! [Aa: code_integer,Ba: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(abs_abs_Code_integer,Aa) = Ba )
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Ba)
        & ( ( Aa = Ba )
          | ( Aa = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Ba) ) ) ) ) ).

% abs_eq_iff'
tff(fact_6305_abs__eq__iff_H,axiom,
    ! [Aa: rat,Ba: rat] :
      ( ( aa_rat_rat(abs_abs_rat,Aa) = Ba )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Ba)
        & ( ( Aa = Ba )
          | ( Aa = aa_rat_rat(uminus_uminus_rat,Ba) ) ) ) ) ).

% abs_eq_iff'
tff(fact_6306_abs__eq__iff_H,axiom,
    ! [Aa: int,Ba: int] :
      ( ( aa_int_int(abs_abs_int,Aa) = Ba )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Ba)
        & ( ( Aa = Ba )
          | ( Aa = aa_int_int(uminus_uminus_int,Ba) ) ) ) ) ).

% abs_eq_iff'
tff(fact_6307_abs__div__pos,axiom,
    ! [Y: rat,Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y)
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(abs_abs_rat,Xb)),Y) = aa_rat_rat(abs_abs_rat,aa_rat_rat(divide_divide_rat(Xb),Y)) ) ) ).

% abs_div_pos
tff(fact_6308_abs__div__pos,axiom,
    ! [Y: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(abs_abs_real,Xb)),Y) = aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)) ) ) ).

% abs_div_pos
tff(fact_6309_zero__le__power__abs,axiom,
    ! [Aa: code_integer,Na: nat] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(abs_abs_Code_integer,Aa)),Na)) ).

% zero_le_power_abs
tff(fact_6310_zero__le__power__abs,axiom,
    ! [Aa: real,Na: 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,Aa)),Na)) ).

% zero_le_power_abs
tff(fact_6311_zero__le__power__abs,axiom,
    ! [Aa: rat,Na: 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,Aa)),Na)) ).

% zero_le_power_abs
tff(fact_6312_zero__le__power__abs,axiom,
    ! [Aa: int,Na: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(aa_int_int(abs_abs_int,Aa)),Na)) ).

% zero_le_power_abs
tff(fact_6313_abs__if__raw,axiom,
    ! [X3: real] :
      aa_real_real(abs_abs_real,X3) = $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,X3),zero_zero_real),aa_real_real(uminus_uminus_real,X3),X3) ).

% abs_if_raw
tff(fact_6314_abs__if__raw,axiom,
    ! [X3: int] :
      aa_int_int(abs_abs_int,X3) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,X3),zero_zero_int),aa_int_int(uminus_uminus_int,X3),X3) ).

% abs_if_raw
tff(fact_6315_abs__if__raw,axiom,
    ! [X3: code_integer] :
      aa_Cod1783974652533396728nteger(abs_abs_Code_integer,X3) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X3),zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,X3),X3) ).

% abs_if_raw
tff(fact_6316_abs__if__raw,axiom,
    ! [X3: rat] :
      aa_rat_rat(abs_abs_rat,X3) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),zero_zero_rat),aa_rat_rat(uminus_uminus_rat,X3),X3) ).

% abs_if_raw
tff(fact_6317_abs__of__neg,axiom,
    ! [Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),zero_zero_real)
     => ( aa_real_real(abs_abs_real,Aa) = aa_real_real(uminus_uminus_real,Aa) ) ) ).

% abs_of_neg
tff(fact_6318_abs__of__neg,axiom,
    ! [Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),zero_zero_int)
     => ( aa_int_int(abs_abs_int,Aa) = aa_int_int(uminus_uminus_int,Aa) ) ) ).

% abs_of_neg
tff(fact_6319_abs__of__neg,axiom,
    ! [Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Aa),zero_z3403309356797280102nteger)
     => ( aa_Cod1783974652533396728nteger(abs_abs_Code_integer,Aa) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa) ) ) ).

% abs_of_neg
tff(fact_6320_abs__of__neg,axiom,
    ! [Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),zero_zero_rat)
     => ( aa_rat_rat(abs_abs_rat,Aa) = aa_rat_rat(uminus_uminus_rat,Aa) ) ) ).

% abs_of_neg
tff(fact_6321_abs__if,axiom,
    ! [Aa: real] :
      aa_real_real(abs_abs_real,Aa) = $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),zero_zero_real),aa_real_real(uminus_uminus_real,Aa),Aa) ).

% abs_if
tff(fact_6322_abs__if,axiom,
    ! [Aa: int] :
      aa_int_int(abs_abs_int,Aa) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),zero_zero_int),aa_int_int(uminus_uminus_int,Aa),Aa) ).

% abs_if
tff(fact_6323_abs__if,axiom,
    ! [Aa: code_integer] :
      aa_Cod1783974652533396728nteger(abs_abs_Code_integer,Aa) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Aa),zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Aa),Aa) ).

% abs_if
tff(fact_6324_abs__if,axiom,
    ! [Aa: rat] :
      aa_rat_rat(abs_abs_rat,Aa) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),zero_zero_rat),aa_rat_rat(uminus_uminus_rat,Aa),Aa) ).

% abs_if
tff(fact_6325_abs__triangle__ineq4,axiom,
    ! [Aa: code_integer,Ba: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,Aa),Ba))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,Aa)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,Ba))) ).

% abs_triangle_ineq4
tff(fact_6326_abs__triangle__ineq4,axiom,
    ! [Aa: real,Ba: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Aa),Ba))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,Aa)),aa_real_real(abs_abs_real,Ba))) ).

% abs_triangle_ineq4
tff(fact_6327_abs__triangle__ineq4,axiom,
    ! [Aa: rat,Ba: 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),Ba))),aa_rat_rat(plus_plus_rat(aa_rat_rat(abs_abs_rat,Aa)),aa_rat_rat(abs_abs_rat,Ba))) ).

% abs_triangle_ineq4
tff(fact_6328_abs__triangle__ineq4,axiom,
    ! [Aa: int,Ba: 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),Ba))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,Aa)),aa_int_int(abs_abs_int,Ba))) ).

% abs_triangle_ineq4
tff(fact_6329_abs__diff__triangle__ineq,axiom,
    ! [Aa: code_integer,Ba: code_integer,C: code_integer,D: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Aa),Ba)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),D)))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,Aa),C))),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,Ba),D)))) ).

% abs_diff_triangle_ineq
tff(fact_6330_abs__diff__triangle__ineq,axiom,
    ! [Aa: real,Ba: real,C: real,D: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),Ba)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),D)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Aa),C))),aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Ba),D)))) ).

% abs_diff_triangle_ineq
tff(fact_6331_abs__diff__triangle__ineq,axiom,
    ! [Aa: rat,Ba: 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(Aa),Ba)),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(Aa),C))),aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(Ba),D)))) ).

% abs_diff_triangle_ineq
tff(fact_6332_abs__diff__triangle__ineq,axiom,
    ! [Aa: int,Ba: 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,Aa),Ba)),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,Aa),C))),aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Ba),D)))) ).

% abs_diff_triangle_ineq
tff(fact_6333_abs__diff__le__iff,axiom,
    ! [Xb: code_integer,Aa: code_integer,R2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,Xb),Aa))),R2)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,Aa),R2)),Xb)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Xb),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Aa),R2)) ) ) ).

% abs_diff_le_iff
tff(fact_6334_abs__diff__le__iff,axiom,
    ! [Xb: real,Aa: real,R2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Aa))),R2)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Aa),R2)),Xb)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),R2)) ) ) ).

% abs_diff_le_iff
tff(fact_6335_abs__diff__le__iff,axiom,
    ! [Xb: rat,Aa: 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),Aa))),R2)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(Aa),R2)),Xb)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),aa_rat_rat(plus_plus_rat(Aa),R2)) ) ) ).

% abs_diff_le_iff
tff(fact_6336_abs__diff__le__iff,axiom,
    ! [Xb: int,Aa: 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),Aa))),R2)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Aa),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,Aa),R2)) ) ) ).

% abs_diff_le_iff
tff(fact_6337_abs__diff__less__iff,axiom,
    ! [Xb: code_integer,Aa: code_integer,R2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,Xb),Aa))),R2)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,Aa),R2)),Xb)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Xb),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Aa),R2)) ) ) ).

% abs_diff_less_iff
tff(fact_6338_abs__diff__less__iff,axiom,
    ! [Xb: real,Aa: real,R2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Aa))),R2)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Aa),R2)),Xb)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),R2)) ) ) ).

% abs_diff_less_iff
tff(fact_6339_abs__diff__less__iff,axiom,
    ! [Xb: rat,Aa: 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),Aa))),R2)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(Aa),R2)),Xb)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),aa_rat_rat(plus_plus_rat(Aa),R2)) ) ) ).

% abs_diff_less_iff
tff(fact_6340_abs__diff__less__iff,axiom,
    ! [Xb: int,Aa: 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),Aa))),R2)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Aa),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,Aa),R2)) ) ) ).

% abs_diff_less_iff
tff(fact_6341_abs__real__def,axiom,
    ! [Aa: real] :
      aa_real_real(abs_abs_real,Aa) = $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),zero_zero_real),aa_real_real(uminus_uminus_real,Aa),Aa) ).

% abs_real_def
tff(fact_6342_sin__bound__lemma,axiom,
    ! [Xb: real,Y: real,U: real,V: real] :
      ( ( Xb = Y )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,U)),V)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),U)),Y))),V) ) ) ).

% sin_bound_lemma
tff(fact_6343_finite__abs__int__segment,axiom,
    ! [Aa: real] : finite_finite_real(collect_real(aa_real_fun_real_o(aTP_Lamm_es,Aa))) ).

% finite_abs_int_segment
tff(fact_6344_finite__abs__int__segment,axiom,
    ! [Aa: rat] : finite_finite_rat(collect_rat(aa_rat_fun_rat_o(aTP_Lamm_et,Aa))) ).

% finite_abs_int_segment
tff(fact_6345_abs__add__one__gt__zero,axiom,
    ! [Xb: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,Xb))) ).

% abs_add_one_gt_zero
tff(fact_6346_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(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(abs_abs_real,Xb))) ).

% abs_add_one_gt_zero
tff(fact_6347_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_6348_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_6349_Ints__nonzero__abs__ge1,axiom,
    ! [Xb: code_integer] :
      ( member_Code_integer(Xb,ring_11222124179247155820nteger)
     => ( ( Xb != zero_z3403309356797280102nteger )
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,Xb)) ) ) ).

% Ints_nonzero_abs_ge1
tff(fact_6350_Ints__nonzero__abs__ge1,axiom,
    ! [Xb: real] :
      ( member_real(Xb,ring_1_Ints_real)
     => ( ( Xb != zero_zero_real )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(abs_abs_real,Xb)) ) ) ).

% Ints_nonzero_abs_ge1
tff(fact_6351_Ints__nonzero__abs__ge1,axiom,
    ! [Xb: rat] :
      ( member_rat(Xb,ring_1_Ints_rat)
     => ( ( Xb != zero_zero_rat )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_rat_rat(abs_abs_rat,Xb)) ) ) ).

% Ints_nonzero_abs_ge1
tff(fact_6352_Ints__nonzero__abs__ge1,axiom,
    ! [Xb: int] :
      ( aa_set_int_o(member_int(Xb),ring_1_Ints_int)
     => ( ( Xb != zero_zero_int )
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_int_int(abs_abs_int,Xb)) ) ) ).

% Ints_nonzero_abs_ge1
tff(fact_6353_Ints__nonzero__abs__less1,axiom,
    ! [Xb: code_integer] :
      ( member_Code_integer(Xb,ring_11222124179247155820nteger)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,Xb)),one_one_Code_integer)
       => ( Xb = zero_z3403309356797280102nteger ) ) ) ).

% Ints_nonzero_abs_less1
tff(fact_6354_Ints__nonzero__abs__less1,axiom,
    ! [Xb: real] :
      ( member_real(Xb,ring_1_Ints_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xb)),one_one_real)
       => ( Xb = zero_zero_real ) ) ) ).

% Ints_nonzero_abs_less1
tff(fact_6355_Ints__nonzero__abs__less1,axiom,
    ! [Xb: rat] :
      ( member_rat(Xb,ring_1_Ints_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,Xb)),one_one_rat)
       => ( Xb = zero_zero_rat ) ) ) ).

% Ints_nonzero_abs_less1
tff(fact_6356_Ints__nonzero__abs__less1,axiom,
    ! [Xb: int] :
      ( aa_set_int_o(member_int(Xb),ring_1_Ints_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,Xb)),one_one_int)
       => ( Xb = zero_zero_int ) ) ) ).

% Ints_nonzero_abs_less1
tff(fact_6357_Ints__eq__abs__less1,axiom,
    ! [Xb: code_integer,Y: code_integer] :
      ( member_Code_integer(Xb,ring_11222124179247155820nteger)
     => ( member_Code_integer(Y,ring_11222124179247155820nteger)
       => ( ( Xb = Y )
        <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,Xb),Y))),one_one_Code_integer) ) ) ) ).

% Ints_eq_abs_less1
tff(fact_6358_Ints__eq__abs__less1,axiom,
    ! [Xb: real,Y: real] :
      ( member_real(Xb,ring_1_Ints_real)
     => ( member_real(Y,ring_1_Ints_real)
       => ( ( Xb = Y )
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Y))),one_one_real) ) ) ) ).

% Ints_eq_abs_less1
tff(fact_6359_Ints__eq__abs__less1,axiom,
    ! [Xb: rat,Y: rat] :
      ( member_rat(Xb,ring_1_Ints_rat)
     => ( member_rat(Y,ring_1_Ints_rat)
       => ( ( Xb = Y )
        <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(Xb),Y))),one_one_rat) ) ) ) ).

% Ints_eq_abs_less1
tff(fact_6360_Ints__eq__abs__less1,axiom,
    ! [Xb: int,Y: int] :
      ( aa_set_int_o(member_int(Xb),ring_1_Ints_int)
     => ( aa_set_int_o(member_int(Y),ring_1_Ints_int)
       => ( ( Xb = Y )
        <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),Y))),one_one_int) ) ) ) ).

% Ints_eq_abs_less1
tff(fact_6361_abs__le__square__iff,axiom,
    ! [Xb: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,Xb)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,Y))
    <=> 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(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% abs_le_square_iff
tff(fact_6362_abs__le__square__iff,axiom,
    ! [Xb: real,Y: 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,Y))
    <=> 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,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% abs_le_square_iff
tff(fact_6363_abs__le__square__iff,axiom,
    ! [Xb: rat,Y: 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,Y))
    <=> 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(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% abs_le_square_iff
tff(fact_6364_abs__le__square__iff,axiom,
    ! [Xb: int,Y: 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,Y))
    <=> 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(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% abs_le_square_iff
tff(fact_6365_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 )
    <=> ( aa_Cod1783974652533396728nteger(abs_abs_Code_integer,Xb) = one_one_Code_integer ) ) ).

% abs_square_eq_1
tff(fact_6366_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_6367_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_6368_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_6369_abs__sqrt__wlog,axiom,
    ! [P: fun_Co2081214437782444048eger_o,Xb: code_integer] :
      ( ! [X: code_integer] :
          ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),X)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(P,X),aa_nat_Code_integer(power_8256067586552552935nteger(X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(P,aa_Cod1783974652533396728nteger(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_6370_abs__sqrt__wlog,axiom,
    ! [P: fun_real_fun_real_o,Xb: real] :
      ( ! [X: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
         => aa_real_o(aa_real_fun_real_o(P,X),aa_nat_real(aa_real_fun_nat_real(power_power_real,X),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) )
     => aa_real_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_6371_abs__sqrt__wlog,axiom,
    ! [P: fun_rat_fun_rat_o,Xb: rat] :
      ( ! [X: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),X)
         => aa_rat_o(aa_rat_fun_rat_o(P,X),aa_nat_rat(power_power_rat(X),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_6372_abs__sqrt__wlog,axiom,
    ! [P: fun_int_fun_int_o,Xb: int] :
      ( ! [X: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
         => aa_int_o(aa_int_fun_int_o(P,X),aa_nat_int(power_power_int(X),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_6373_power2__le__iff__abs__le,axiom,
    ! [Y: code_integer,Xb: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Y)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_Code_integer(power_8256067586552552935nteger(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,Xb)),Y) ) ) ).

% power2_le_iff_abs_le
tff(fact_6374_power2__le__iff__abs__le,axiom,
    ! [Y: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(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,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),Y) ) ) ).

% power2_le_iff_abs_le
tff(fact_6375_power2__le__iff__abs__le,axiom,
    ! [Y: rat,Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,Xb)),Y) ) ) ).

% power2_le_iff_abs_le
tff(fact_6376_power2__le__iff__abs__le,axiom,
    ! [Y: int,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,Xb)),Y) ) ) ).

% power2_le_iff_abs_le
tff(fact_6377_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,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,Xb)),one_one_Code_integer) ) ).

% abs_square_le_1
tff(fact_6378_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_6379_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_6380_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_6381_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,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,Xb)),one_one_Code_integer) ) ).

% abs_square_less_1
tff(fact_6382_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_6383_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_6384_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_6385_power__mono__even,axiom,
    ! [Na: nat,Aa: code_integer,Ba: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,Aa)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,Ba))
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(Aa),Na)),aa_nat_Code_integer(power_8256067586552552935nteger(Ba),Na)) ) ) ).

% power_mono_even
tff(fact_6386_power__mono__even,axiom,
    ! [Na: nat,Aa: real,Ba: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Aa)),aa_real_real(abs_abs_real,Ba))
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),Na)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ba),Na)) ) ) ).

% power_mono_even
tff(fact_6387_power__mono__even,axiom,
    ! [Na: nat,Aa: rat,Ba: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,Aa)),aa_rat_rat(abs_abs_rat,Ba))
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(Aa),Na)),aa_nat_rat(power_power_rat(Ba),Na)) ) ) ).

% power_mono_even
tff(fact_6388_power__mono__even,axiom,
    ! [Na: nat,Aa: int,Ba: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,Aa)),aa_int_int(abs_abs_int,Ba))
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(Aa),Na)),aa_nat_int(power_power_int(Ba),Na)) ) ) ).

% power_mono_even
tff(fact_6389_divmod__divmod__step,axiom,
    ! [M: num,Na: num] :
      unique3479559517661332726nteger(M,Na) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_num,M),Na),produc1086072967326762835nteger(zero_z3403309356797280102nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)),unique4921790084139445826nteger(Na,unique3479559517661332726nteger(M,aa_num_num(bit0,Na)))) ).

% divmod_divmod_step
tff(fact_6390_divmod__divmod__step,axiom,
    ! [M: num,Na: num] :
      unique5055182867167087721od_nat(M,Na) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_num,M),Na),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,M)),unique5026877609467782581ep_nat(Na,unique5055182867167087721od_nat(M,aa_num_num(bit0,Na)))) ).

% divmod_divmod_step
tff(fact_6391_divmod__divmod__step,axiom,
    ! [M: num,Na: num] :
      unique5052692396658037445od_int(M,Na) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_num,M),Na),product_Pair_int_int(zero_zero_int,aa_num_int(numeral_numeral_int,M)),unique5024387138958732305ep_int(Na,unique5052692396658037445od_int(M,aa_num_num(bit0,Na)))) ).

% divmod_divmod_step
tff(fact_6392_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(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_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_6393_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(aa_rea7340605660300375871l_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(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xb))),Xb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% abs_ln_one_plus_x_minus_x_bound
tff(fact_6394_lemma__interval,axiom,
    ! [Aa: real,Xb: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Ba)
       => ? [D4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D4)
            & ! [Y4: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Y4))),D4)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),Y4)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y4),Ba) ) ) ) ) ) ).

% lemma_interval
tff(fact_6395_fold__atLeastAtMost__nat_Opsimps,axiom,
    ! [F: fun_nat_fun_nat_nat,Aa: nat,Ba: nat,Acc2: nat] :
      ( accp_P4694664897847628410at_nat(set_fo3699595496184130361el_nat,produc2818902965508115497at_nat(F,produc487386426758144856at_nat(Aa,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Ba),Acc2))))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(F,Aa),Ba),Acc2) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ba),Aa),Acc2,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),one_one_nat)),Ba),aa_nat_nat(aa_nat_fun_nat_nat(F,Aa),Acc2))) ) ) ).

% fold_atLeastAtMost_nat.psimps
tff(fact_6396_fold__atLeastAtMost__nat_Opelims,axiom,
    ! [Xb: fun_nat_fun_nat_nat,Xaa: nat,Xba: nat,Xc: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(Xb,Xaa),Xba),Xc) = Y )
     => ( accp_P4694664897847628410at_nat(set_fo3699595496184130361el_nat,produc2818902965508115497at_nat(Xb,produc487386426758144856at_nat(Xaa,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xba),Xc))))
       => ~ ( ( Y = $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))) )
           => ~ accp_P4694664897847628410at_nat(set_fo3699595496184130361el_nat,produc2818902965508115497at_nat(Xb,produc487386426758144856at_nat(Xaa,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xba),Xc)))) ) ) ) ).

% fold_atLeastAtMost_nat.pelims
tff(fact_6397_lemma__interval__lt,axiom,
    ! [Aa: real,Xb: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Ba)
       => ? [D4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D4)
            & ! [Y4: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Y4))),D4)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),Y4)
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,Y4),Ba) ) ) ) ) ) ).

% lemma_interval_lt
tff(fact_6398_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(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)) ) ) ).

% tanh_ln_real
tff(fact_6399_in__measure,axiom,
    ! [Xb: code_integer,Y: code_integer,F: fun_Code_integer_nat] :
      ( member157494554546826820nteger(produc1086072967326762835nteger(Xb,Y),measure_Code_integer(F))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_Code_integer_nat(F,Xb)),aa_Code_integer_nat(F,Y)) ) ).

% in_measure
tff(fact_6400_in__measure,axiom,
    ! [Xb: product_prod_nat_nat,Y: product_prod_nat_nat,F: fun_Pr2602117396471556196at_nat] :
      ( member8206827879206165904at_nat(produc6161850002892822231at_nat(Xb,Y),measur8038558561449204169at_nat(F))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_Pro4243173770044397973at_nat(F,Xb)),aa_Pro4243173770044397973at_nat(F,Y)) ) ).

% in_measure
tff(fact_6401_in__measure,axiom,
    ! [Xb: nat,Y: nat,F: fun_nat_nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xb),Y),measure_nat(F))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,Xb)),aa_nat_nat(F,Y)) ) ).

% in_measure
tff(fact_6402_in__measure,axiom,
    ! [Xb: int,Y: int,F: fun_int_nat] :
      ( member5262025264175285858nt_int(product_Pair_int_int(Xb,Y),measure_int(F))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(F,Xb)),aa_int_nat(F,Y)) ) ).

% in_measure
tff(fact_6403_tanh__0,axiom,
    aa_real_real(tanh_real,zero_zero_real) = zero_zero_real ).

% tanh_0
tff(fact_6404_tanh__real__less__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(tanh_real,Xb)),aa_real_real(tanh_real,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y) ) ).

% tanh_real_less_iff
tff(fact_6405_tanh__real__le__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(tanh_real,Xb)),aa_real_real(tanh_real,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y) ) ).

% tanh_real_le_iff
tff(fact_6406_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_6407_tanh__real__pos__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(tanh_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb) ) ).

% tanh_real_pos_iff
tff(fact_6408_tanh__real__neg__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(tanh_real,Xb)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real) ) ).

% tanh_real_neg_iff
tff(fact_6409_tanh__real__nonneg__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(tanh_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb) ) ).

% tanh_real_nonneg_iff
tff(fact_6410_tanh__real__nonpos__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(tanh_real,Xb)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real) ) ).

% tanh_real_nonpos_iff
tff(fact_6411_infinite__int__iff__unbounded__le,axiom,
    ! [S2: set_int] :
      ( ~ aa_set_int_o(finite_finite_int,S2)
    <=> ! [M6: int] :
        ? [N4: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,M6),aa_int_int(abs_abs_int,N4))
          & aa_set_int_o(member_int(N4),S2) ) ) ).

% infinite_int_iff_unbounded_le
tff(fact_6412_infinite__int__iff__unbounded,axiom,
    ! [S2: set_int] :
      ( ~ aa_set_int_o(finite_finite_int,S2)
    <=> ! [M6: int] :
        ? [N4: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,M6),aa_int_int(abs_abs_int,N4))
          & aa_set_int_o(member_int(N4),S2) ) ) ).

% infinite_int_iff_unbounded
tff(fact_6413_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_6414_zabs__def,axiom,
    ! [I: int] :
      aa_int_int(abs_abs_int,I) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,I),zero_zero_int),aa_int_int(uminus_uminus_int,I),I) ).

% zabs_def
tff(fact_6415_dvd__imp__le__int,axiom,
    ! [I: int,D: int] :
      ( ( I != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),I)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,D)),aa_int_int(abs_abs_int,I)) ) ) ).

% dvd_imp_le_int
tff(fact_6416_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_6417_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_6418_fold__atLeastAtMost__nat_Oelims,axiom,
    ! [Xb: fun_nat_fun_nat_nat,Xaa: nat,Xba: nat,Xc: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(Xb,Xaa),Xba),Xc) = Y )
     => ( Y = $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_6419_fold__atLeastAtMost__nat_Osimps,axiom,
    ! [F: fun_nat_fun_nat_nat,Aa: nat,Ba: nat,Acc2: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(F,Aa),Ba),Acc2) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ba),Aa),Acc2,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),one_one_nat)),Ba),aa_nat_nat(aa_nat_fun_nat_nat(F,Aa),Acc2))) ).

% fold_atLeastAtMost_nat.simps
tff(fact_6420_nat__intermed__int__val,axiom,
    ! [M: nat,Na: nat,F: fun_nat_int,K: int] :
      ( ! [I2: nat] :
          ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),I2)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),Na) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,aa_nat_nat(suc,I2))),aa_nat_int(F,I2)))),one_one_int) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,M)),K)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),aa_nat_int(F,Na))
           => ? [I2: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),I2)
                & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),Na)
                & ( aa_nat_int(F,I2) = K ) ) ) ) ) ) ).

% nat_intermed_int_val
tff(fact_6421_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_6422_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_6423_nat__ivt__aux,axiom,
    ! [Na: nat,F: fun_nat_int,K: int] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),Na)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,aa_nat_nat(suc,I2))),aa_nat_int(F,I2)))),one_one_int) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,zero_zero_nat)),K)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),aa_nat_int(F,Na))
         => ? [I2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),Na)
              & ( aa_nat_int(F,I2) = K ) ) ) ) ) ).

% nat_ivt_aux
tff(fact_6424_nat0__intermed__int__val,axiom,
    ! [Na: nat,F: fun_nat_int,K: int] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),Na)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),one_one_nat))),aa_nat_int(F,I2)))),one_one_int) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,zero_zero_nat)),K)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),aa_nat_int(F,Na))
         => ? [I2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),Na)
              & ( aa_nat_int(F,I2) = K ) ) ) ) ) ).

% nat0_intermed_int_val
tff(fact_6425_in__finite__psubset,axiom,
    ! [A3: set_nat,B4: set_nat] :
      ( member8277197624267554838et_nat(produc4532415448927165861et_nat(A3,B4),finite_psubset_nat)
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A3),B4)
        & aa_set_nat_o(finite_finite_nat,B4) ) ) ).

% in_finite_psubset
tff(fact_6426_in__finite__psubset,axiom,
    ! [A3: set_int,B4: set_int] :
      ( member2572552093476627150et_int(produc6363374080413544029et_int(A3,B4),finite_psubset_int)
    <=> ( aa_set_int_o(ord_less_set_int(A3),B4)
        & aa_set_int_o(finite_finite_int,B4) ) ) ).

% in_finite_psubset
tff(fact_6427_in__finite__psubset,axiom,
    ! [A3: set_complex,B4: set_complex] :
      ( member351165363924911826omplex(produc3790773574474814305omplex(A3,B4),finite8643634255014194347omplex)
    <=> ( aa_set_complex_o(ord_less_set_complex(A3),B4)
        & aa_set_complex_o(finite3207457112153483333omplex,B4) ) ) ).

% in_finite_psubset
tff(fact_6428_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(aa_rea7340605660300375871l_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(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Xb)),aa_real_real(aa_rea7340605660300375871l_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_6429_concat__bit__Suc,axiom,
    ! [Na: nat,K: int,L: int] : aa_int_int(bit_concat_bit(aa_nat_nat(suc,Na),K),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(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(bit_concat_bit(Na,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),L))) ).

% concat_bit_Suc
tff(fact_6430_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_6431_of__int__code__if,axiom,
    ! [K: int] :
      aa_int_real(ring_1_of_int_real,K) = $ite(
        K = zero_zero_int,
        zero_zero_real,
        $ite(
          aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int),
          aa_real_real(uminus_uminus_real,aa_int_real(ring_1_of_int_real,aa_int_int(uminus_uminus_int,K))),
          $let(
            l: real,
            l:= aa_real_real(aa_rea7340605660300375871l_real(times_times_real,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(aa_rea7340605660300375871l_real(plus_plus_real,l),one_one_real)) ) ) ) ).

% of_int_code_if
tff(fact_6432_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_6433_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,aa_num_Code_integer(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,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,l),one_one_Code_integer)) ) ) ) ).

% of_int_code_if
tff(fact_6434_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_6435_divmod__algorithm__code_I6_J,axiom,
    ! [M: num,Na: num] : unique5052692396658037445od_int(aa_num_num(bit1,M),aa_num_num(bit0,Na)) = produc4245557441103728435nt_int(aTP_Lamm_eu,unique5052692396658037445od_int(M,Na)) ).

% divmod_algorithm_code(6)
tff(fact_6436_divmod__algorithm__code_I6_J,axiom,
    ! [M: num,Na: num] : unique3479559517661332726nteger(aa_num_num(bit1,M),aa_num_num(bit0,Na)) = produc6916734918728496179nteger(aTP_Lamm_ev,unique3479559517661332726nteger(M,Na)) ).

% divmod_algorithm_code(6)
tff(fact_6437_divmod__algorithm__code_I6_J,axiom,
    ! [M: num,Na: num] : unique5055182867167087721od_nat(aa_num_num(bit1,M),aa_num_num(bit0,Na)) = produc2626176000494625587at_nat(aTP_Lamm_ew,unique5055182867167087721od_nat(M,Na)) ).

% divmod_algorithm_code(6)
tff(fact_6438_flip__bit__0,axiom,
    ! [Aa: int] : aa_int_int(aa_nat_fun_int_int(bit_se2159334234014336723it_int,zero_zero_nat),Aa) = 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))),Aa))),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,Aa),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))) ).

% flip_bit_0
tff(fact_6439_flip__bit__0,axiom,
    ! [Aa: code_integer] : aa_Cod1783974652533396728nteger(aa_nat6383147911832638789nteger(bit_se1345352211410354436nteger,zero_zero_nat),Aa) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,zero_n356916108424825756nteger(aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Aa))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Aa),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))))) ).

% flip_bit_0
tff(fact_6440_flip__bit__0,axiom,
    ! [Aa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,zero_zero_nat),Aa) = 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))),Aa))),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,Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% flip_bit_0
tff(fact_6441_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_6442_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_6443_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_6444_of__bool__eq_I1_J,axiom,
    zero_n3304061248610475627l_real($false) = zero_zero_real ).

% of_bool_eq(1)
tff(fact_6445_of__bool__eq_I1_J,axiom,
    zero_n2052037380579107095ol_rat($false) = zero_zero_rat ).

% of_bool_eq(1)
tff(fact_6446_of__bool__eq_I1_J,axiom,
    zero_n2687167440665602831ol_nat($false) = zero_zero_nat ).

% of_bool_eq(1)
tff(fact_6447_of__bool__eq_I1_J,axiom,
    aa_o_int(zero_n2684676970156552555ol_int,$false) = zero_zero_int ).

% of_bool_eq(1)
tff(fact_6448_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( zero_n3304061248610475627l_real((P)) = zero_zero_real )
    <=> ~ (P) ) ).

% of_bool_eq_0_iff
tff(fact_6449_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( zero_n2052037380579107095ol_rat((P)) = zero_zero_rat )
    <=> ~ (P) ) ).

% of_bool_eq_0_iff
tff(fact_6450_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( zero_n2687167440665602831ol_nat((P)) = zero_zero_nat )
    <=> ~ (P) ) ).

% of_bool_eq_0_iff
tff(fact_6451_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_6452_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_6453_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_6454_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_6455_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_6456_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( zero_n1201886186963655149omplex((P)) = one_one_complex )
    <=> (P) ) ).

% of_bool_eq_1_iff
tff(fact_6457_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( zero_n3304061248610475627l_real((P)) = one_one_real )
    <=> (P) ) ).

% of_bool_eq_1_iff
tff(fact_6458_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( zero_n2052037380579107095ol_rat((P)) = one_one_rat )
    <=> (P) ) ).

% of_bool_eq_1_iff
tff(fact_6459_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( zero_n2687167440665602831ol_nat((P)) = one_one_nat )
    <=> (P) ) ).

% of_bool_eq_1_iff
tff(fact_6460_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_6461_of__bool__eq_I2_J,axiom,
    zero_n1201886186963655149omplex($true) = one_one_complex ).

% of_bool_eq(2)
tff(fact_6462_of__bool__eq_I2_J,axiom,
    zero_n3304061248610475627l_real($true) = one_one_real ).

% of_bool_eq(2)
tff(fact_6463_of__bool__eq_I2_J,axiom,
    zero_n2052037380579107095ol_rat($true) = one_one_rat ).

% of_bool_eq(2)
tff(fact_6464_of__bool__eq_I2_J,axiom,
    zero_n2687167440665602831ol_nat($true) = one_one_nat ).

% of_bool_eq(2)
tff(fact_6465_of__bool__eq_I2_J,axiom,
    aa_o_int(zero_n2684676970156552555ol_int,$true) = one_one_int ).

% of_bool_eq(2)
tff(fact_6466_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_6467_abs__bool__eq,axiom,
    ! [P: $o] : aa_Cod1783974652533396728nteger(abs_abs_Code_integer,zero_n356916108424825756nteger((P))) = zero_n356916108424825756nteger((P)) ).

% abs_bool_eq
tff(fact_6468_abs__bool__eq,axiom,
    ! [P: $o] : aa_rat_rat(abs_abs_rat,zero_n2052037380579107095ol_rat((P))) = zero_n2052037380579107095ol_rat((P)) ).

% abs_bool_eq
tff(fact_6469_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_6470_of__int__fact,axiom,
    ! [Na: nat] : aa_int_Code_integer(ring_18347121197199848620nteger,aa_nat_int(semiri1406184849735516958ct_int,Na)) = aa_nat_Code_integer(semiri3624122377584611663nteger,Na) ).

% of_int_fact
tff(fact_6471_of__int__fact,axiom,
    ! [Na: nat] : aa_int_rat(ring_1_of_int_rat,aa_nat_int(semiri1406184849735516958ct_int,Na)) = aa_nat_rat(semiri773545260158071498ct_rat,Na) ).

% of_int_fact
tff(fact_6472_of__int__fact,axiom,
    ! [Na: nat] : aa_int_real(ring_1_of_int_real,aa_nat_int(semiri1406184849735516958ct_int,Na)) = aa_nat_real(semiri2265585572941072030t_real,Na) ).

% of_int_fact
tff(fact_6473_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_6474_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_6475_of__bool__or__iff,axiom,
    ! [P: $o,Q: $o] :
      aa_o_int(zero_n2684676970156552555ol_int,
        ( (P)
        | (Q) )) = aa_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_6476_concat__bit__0,axiom,
    ! [K: int,L: int] : aa_int_int(bit_concat_bit(zero_zero_nat,K),L) = L ).

% concat_bit_0
tff(fact_6477_of__int__0,axiom,
    aa_int_int(ring_1_of_int_int,zero_zero_int) = zero_zero_int ).

% of_int_0
tff(fact_6478_of__int__0,axiom,
    aa_int_real(ring_1_of_int_real,zero_zero_int) = zero_zero_real ).

% of_int_0
tff(fact_6479_of__int__0,axiom,
    aa_int_Code_integer(ring_18347121197199848620nteger,zero_zero_int) = zero_z3403309356797280102nteger ).

% of_int_0
tff(fact_6480_of__int__0,axiom,
    aa_int_rat(ring_1_of_int_rat,zero_zero_int) = zero_zero_rat ).

% of_int_0
tff(fact_6481_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_6482_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_6483_of__int__0__eq__iff,axiom,
    ! [Z: int] :
      ( ( zero_z3403309356797280102nteger = aa_int_Code_integer(ring_18347121197199848620nteger,Z) )
    <=> ( Z = zero_zero_int ) ) ).

% of_int_0_eq_iff
tff(fact_6484_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_6485_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_6486_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_6487_of__int__eq__0__iff,axiom,
    ! [Z: int] :
      ( ( aa_int_Code_integer(ring_18347121197199848620nteger,Z) = zero_z3403309356797280102nteger )
    <=> ( Z = zero_zero_int ) ) ).

% of_int_eq_0_iff
tff(fact_6488_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_6489_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_6490_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_6491_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_6492_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_6493_of__int__le__iff,axiom,
    ! [W2: int,Z: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,W2)),aa_int_real(ring_1_of_int_real,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W2),Z) ) ).

% of_int_le_iff
tff(fact_6494_of__int__le__iff,axiom,
    ! [W2: int,Z: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_int_Code_integer(ring_18347121197199848620nteger,W2)),aa_int_Code_integer(ring_18347121197199848620nteger,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W2),Z) ) ).

% of_int_le_iff
tff(fact_6495_of__int__le__iff,axiom,
    ! [W2: int,Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,W2)),aa_int_rat(ring_1_of_int_rat,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W2),Z) ) ).

% of_int_le_iff
tff(fact_6496_of__int__le__iff,axiom,
    ! [W2: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ring_1_of_int_int,W2)),aa_int_int(ring_1_of_int_int,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W2),Z) ) ).

% of_int_le_iff
tff(fact_6497_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_6498_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_6499_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_6500_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_6501_of__int__less__iff,axiom,
    ! [W2: int,Z: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_int_Code_integer(ring_18347121197199848620nteger,W2)),aa_int_Code_integer(ring_18347121197199848620nteger,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W2),Z) ) ).

% of_int_less_iff
tff(fact_6502_of__int__less__iff,axiom,
    ! [W2: int,Z: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,W2)),aa_int_real(ring_1_of_int_real,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W2),Z) ) ).

% of_int_less_iff
tff(fact_6503_of__int__less__iff,axiom,
    ! [W2: int,Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,W2)),aa_int_rat(ring_1_of_int_rat,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W2),Z) ) ).

% of_int_less_iff
tff(fact_6504_of__int__less__iff,axiom,
    ! [W2: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(ring_1_of_int_int,W2)),aa_int_int(ring_1_of_int_int,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W2),Z) ) ).

% of_int_less_iff
tff(fact_6505_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_6506_of__bool__not__iff,axiom,
    ! [P: $o] : zero_n3304061248610475627l_real(~ (P)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),zero_n3304061248610475627l_real((P))) ).

% of_bool_not_iff
tff(fact_6507_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_6508_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_6509_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_6510_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_6511_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_6512_of__int__eq__1__iff,axiom,
    ! [Z: int] :
      ( ( aa_int_Code_integer(ring_18347121197199848620nteger,Z) = one_one_Code_integer )
    <=> ( Z = one_one_int ) ) ).

% of_int_eq_1_iff
tff(fact_6513_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_6514_of__int__1,axiom,
    aa_int_complex(ring_17405671764205052669omplex,one_one_int) = one_one_complex ).

% of_int_1
tff(fact_6515_of__int__1,axiom,
    aa_int_int(ring_1_of_int_int,one_one_int) = one_one_int ).

% of_int_1
tff(fact_6516_of__int__1,axiom,
    aa_int_real(ring_1_of_int_real,one_one_int) = one_one_real ).

% of_int_1
tff(fact_6517_of__int__1,axiom,
    aa_int_Code_integer(ring_18347121197199848620nteger,one_one_int) = one_one_Code_integer ).

% of_int_1
tff(fact_6518_of__int__1,axiom,
    aa_int_rat(ring_1_of_int_rat,one_one_int) = one_one_rat ).

% of_int_1
tff(fact_6519_Suc__0__mod__eq,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,zero_zero_nat)),Na) = zero_n2687167440665602831ol_nat(Na != aa_nat_nat(suc,zero_zero_nat)) ).

% Suc_0_mod_eq
tff(fact_6520_arctan__less__zero__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arctan,Xb)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real) ) ).

% arctan_less_zero_iff
tff(fact_6521_zero__less__arctan__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(arctan,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb) ) ).

% zero_less_arctan_iff
tff(fact_6522_zero__le__arctan__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(arctan,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb) ) ).

% zero_le_arctan_iff
tff(fact_6523_arctan__le__zero__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arctan,Xb)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real) ) ).

% arctan_le_zero_iff
tff(fact_6524_frac__of__int,axiom,
    ! [Z: int] : aa_real_real(archim2898591450579166408c_real,aa_int_real(ring_1_of_int_real,Z)) = zero_zero_real ).

% frac_of_int
tff(fact_6525_frac__of__int,axiom,
    ! [Z: int] : aa_rat_rat(archimedean_frac_rat,aa_int_rat(ring_1_of_int_rat,Z)) = zero_zero_rat ).

% frac_of_int
tff(fact_6526_concat__bit__nonnegative__iff,axiom,
    ! [Na: nat,K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(bit_concat_bit(Na,K),L))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),L) ) ).

% concat_bit_nonnegative_iff
tff(fact_6527_concat__bit__negative__iff,axiom,
    ! [Na: nat,K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_concat_bit(Na,K),L)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int) ) ).

% concat_bit_negative_iff
tff(fact_6528_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_6529_of__int__le__0__iff,axiom,
    ! [Z: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_int_Code_integer(ring_18347121197199848620nteger,Z)),zero_z3403309356797280102nteger)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),zero_zero_int) ) ).

% of_int_le_0_iff
tff(fact_6530_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_6531_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_6532_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_6533_of__int__0__le__iff,axiom,
    ! [Z: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(ring_18347121197199848620nteger,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z) ) ).

% of_int_0_le_iff
tff(fact_6534_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_6535_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_6536_of__int__less__0__iff,axiom,
    ! [Z: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_int_Code_integer(ring_18347121197199848620nteger,Z)),zero_z3403309356797280102nteger)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),zero_zero_int) ) ).

% of_int_less_0_iff
tff(fact_6537_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_6538_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_6539_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_6540_of__int__0__less__iff,axiom,
    ! [Z: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(ring_18347121197199848620nteger,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z) ) ).

% of_int_0_less_iff
tff(fact_6541_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_6542_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_6543_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_6544_of__int__le__numeral__iff,axiom,
    ! [Z: int,Na: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z)),aa_num_real(numeral_numeral_real,Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),aa_num_int(numeral_numeral_int,Na)) ) ).

% of_int_le_numeral_iff
tff(fact_6545_of__int__le__numeral__iff,axiom,
    ! [Z: int,Na: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_int_Code_integer(ring_18347121197199848620nteger,Z)),aa_num_Code_integer(numera6620942414471956472nteger,Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),aa_num_int(numeral_numeral_int,Na)) ) ).

% of_int_le_numeral_iff
tff(fact_6546_of__int__le__numeral__iff,axiom,
    ! [Z: int,Na: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z)),aa_num_rat(numeral_numeral_rat,Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),aa_num_int(numeral_numeral_int,Na)) ) ).

% of_int_le_numeral_iff
tff(fact_6547_of__int__le__numeral__iff,axiom,
    ! [Z: int,Na: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ring_1_of_int_int,Z)),aa_num_int(numeral_numeral_int,Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),aa_num_int(numeral_numeral_int,Na)) ) ).

% of_int_le_numeral_iff
tff(fact_6548_of__int__numeral__le__iff,axiom,
    ! [Na: num,Z: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,Na)),aa_int_real(ring_1_of_int_real,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Na)),Z) ) ).

% of_int_numeral_le_iff
tff(fact_6549_of__int__numeral__le__iff,axiom,
    ! [Na: num,Z: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_num_Code_integer(numera6620942414471956472nteger,Na)),aa_int_Code_integer(ring_18347121197199848620nteger,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Na)),Z) ) ).

% of_int_numeral_le_iff
tff(fact_6550_of__int__numeral__le__iff,axiom,
    ! [Na: num,Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,Na)),aa_int_rat(ring_1_of_int_rat,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Na)),Z) ) ).

% of_int_numeral_le_iff
tff(fact_6551_of__int__numeral__le__iff,axiom,
    ! [Na: num,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Na)),aa_int_int(ring_1_of_int_int,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Na)),Z) ) ).

% of_int_numeral_le_iff
tff(fact_6552_of__int__numeral__less__iff,axiom,
    ! [Na: num,Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,Na)),aa_int_rat(ring_1_of_int_rat,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Na)),Z) ) ).

% of_int_numeral_less_iff
tff(fact_6553_of__int__numeral__less__iff,axiom,
    ! [Na: num,Z: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,Na)),aa_int_real(ring_1_of_int_real,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Na)),Z) ) ).

% of_int_numeral_less_iff
tff(fact_6554_of__int__numeral__less__iff,axiom,
    ! [Na: num,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Na)),aa_int_int(ring_1_of_int_int,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Na)),Z) ) ).

% of_int_numeral_less_iff
tff(fact_6555_of__int__numeral__less__iff,axiom,
    ! [Na: num,Z: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_num_Code_integer(numera6620942414471956472nteger,Na)),aa_int_Code_integer(ring_18347121197199848620nteger,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Na)),Z) ) ).

% of_int_numeral_less_iff
tff(fact_6556_of__int__less__numeral__iff,axiom,
    ! [Z: int,Na: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Z)),aa_num_rat(numeral_numeral_rat,Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),aa_num_int(numeral_numeral_int,Na)) ) ).

% of_int_less_numeral_iff
tff(fact_6557_of__int__less__numeral__iff,axiom,
    ! [Z: int,Na: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Z)),aa_num_real(numeral_numeral_real,Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),aa_num_int(numeral_numeral_int,Na)) ) ).

% of_int_less_numeral_iff
tff(fact_6558_of__int__less__numeral__iff,axiom,
    ! [Z: int,Na: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(ring_1_of_int_int,Z)),aa_num_int(numeral_numeral_int,Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),aa_num_int(numeral_numeral_int,Na)) ) ).

% of_int_less_numeral_iff
tff(fact_6559_of__int__less__numeral__iff,axiom,
    ! [Z: int,Na: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_int_Code_integer(ring_18347121197199848620nteger,Z)),aa_num_Code_integer(numera6620942414471956472nteger,Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),aa_num_int(numeral_numeral_int,Na)) ) ).

% of_int_less_numeral_iff
tff(fact_6560_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_6561_of__int__le__1__iff,axiom,
    ! [Z: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_int_Code_integer(ring_18347121197199848620nteger,Z)),one_one_Code_integer)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),one_one_int) ) ).

% of_int_le_1_iff
tff(fact_6562_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_6563_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_6564_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_6565_of__int__1__le__iff,axiom,
    ! [Z: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),aa_int_Code_integer(ring_18347121197199848620nteger,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),Z) ) ).

% of_int_1_le_iff
tff(fact_6566_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_6567_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_6568_of__int__less__1__iff,axiom,
    ! [Z: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_int_Code_integer(ring_18347121197199848620nteger,Z)),one_one_Code_integer)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),one_one_int) ) ).

% of_int_less_1_iff
tff(fact_6569_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_6570_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_6571_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_6572_of__int__1__less__iff,axiom,
    ! [Z: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),aa_int_Code_integer(ring_18347121197199848620nteger,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),Z) ) ).

% of_int_1_less_iff
tff(fact_6573_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_6574_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_6575_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_6576_of__int__le__of__int__power__cancel__iff,axiom,
    ! [Ba: int,W2: nat,Xb: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_int_Code_integer(ring_18347121197199848620nteger,Ba)),W2)),aa_int_Code_integer(ring_18347121197199848620nteger,Xb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(Ba),W2)),Xb) ) ).

% of_int_le_of_int_power_cancel_iff
tff(fact_6577_of__int__le__of__int__power__cancel__iff,axiom,
    ! [Ba: int,W2: 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,Ba)),W2)),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(Ba),W2)),Xb) ) ).

% of_int_le_of_int_power_cancel_iff
tff(fact_6578_of__int__le__of__int__power__cancel__iff,axiom,
    ! [Ba: int,W2: 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,Ba)),W2)),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(Ba),W2)),Xb) ) ).

% of_int_le_of_int_power_cancel_iff
tff(fact_6579_of__int__le__of__int__power__cancel__iff,axiom,
    ! [Ba: int,W2: 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,Ba)),W2)),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(Ba),W2)),Xb) ) ).

% of_int_le_of_int_power_cancel_iff
tff(fact_6580_of__int__power__le__of__int__cancel__iff,axiom,
    ! [Xb: int,Ba: int,W2: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_int_Code_integer(ring_18347121197199848620nteger,Xb)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_int_Code_integer(ring_18347121197199848620nteger,Ba)),W2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),aa_nat_int(power_power_int(Ba),W2)) ) ).

% of_int_power_le_of_int_cancel_iff
tff(fact_6581_of__int__power__le__of__int__cancel__iff,axiom,
    ! [Xb: int,Ba: int,W2: 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,Ba)),W2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),aa_nat_int(power_power_int(Ba),W2)) ) ).

% of_int_power_le_of_int_cancel_iff
tff(fact_6582_of__int__power__le__of__int__cancel__iff,axiom,
    ! [Xb: int,Ba: int,W2: 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,Ba)),W2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),aa_nat_int(power_power_int(Ba),W2)) ) ).

% of_int_power_le_of_int_cancel_iff
tff(fact_6583_of__int__power__le__of__int__cancel__iff,axiom,
    ! [Xb: int,Ba: int,W2: 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,Ba)),W2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),aa_nat_int(power_power_int(Ba),W2)) ) ).

% of_int_power_le_of_int_cancel_iff
tff(fact_6584_of__int__power__less__of__int__cancel__iff,axiom,
    ! [Xb: int,Ba: int,W2: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_int_Code_integer(ring_18347121197199848620nteger,Xb)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_int_Code_integer(ring_18347121197199848620nteger,Ba)),W2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),aa_nat_int(power_power_int(Ba),W2)) ) ).

% of_int_power_less_of_int_cancel_iff
tff(fact_6585_of__int__power__less__of__int__cancel__iff,axiom,
    ! [Xb: int,Ba: int,W2: 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,Ba)),W2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),aa_nat_int(power_power_int(Ba),W2)) ) ).

% of_int_power_less_of_int_cancel_iff
tff(fact_6586_of__int__power__less__of__int__cancel__iff,axiom,
    ! [Xb: int,Ba: int,W2: 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,Ba)),W2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),aa_nat_int(power_power_int(Ba),W2)) ) ).

% of_int_power_less_of_int_cancel_iff
tff(fact_6587_of__int__power__less__of__int__cancel__iff,axiom,
    ! [Xb: int,Ba: int,W2: 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,Ba)),W2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),aa_nat_int(power_power_int(Ba),W2)) ) ).

% of_int_power_less_of_int_cancel_iff
tff(fact_6588_of__int__less__of__int__power__cancel__iff,axiom,
    ! [Ba: int,W2: nat,Xb: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_int_Code_integer(ring_18347121197199848620nteger,Ba)),W2)),aa_int_Code_integer(ring_18347121197199848620nteger,Xb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(Ba),W2)),Xb) ) ).

% of_int_less_of_int_power_cancel_iff
tff(fact_6589_of__int__less__of__int__power__cancel__iff,axiom,
    ! [Ba: int,W2: 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,Ba)),W2)),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(Ba),W2)),Xb) ) ).

% of_int_less_of_int_power_cancel_iff
tff(fact_6590_of__int__less__of__int__power__cancel__iff,axiom,
    ! [Ba: int,W2: 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,Ba)),W2)),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(Ba),W2)),Xb) ) ).

% of_int_less_of_int_power_cancel_iff
tff(fact_6591_of__int__less__of__int__power__cancel__iff,axiom,
    ! [Ba: int,W2: 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,Ba)),W2)),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(Ba),W2)),Xb) ) ).

% of_int_less_of_int_power_cancel_iff
tff(fact_6592_of__bool__half__eq__0,axiom,
    ! [Ba: $o] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,zero_n356916108424825756nteger((Ba))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = zero_z3403309356797280102nteger ).

% of_bool_half_eq_0
tff(fact_6593_of__bool__half__eq__0,axiom,
    ! [Ba: $o] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,zero_n2687167440665602831ol_nat((Ba))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ).

% of_bool_half_eq_0
tff(fact_6594_of__bool__half__eq__0,axiom,
    ! [Ba: $o] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_o_int(zero_n2684676970156552555ol_int,(Ba))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int ).

% of_bool_half_eq_0
tff(fact_6595_numeral__power__le__of__int__cancel__iff,axiom,
    ! [Xb: num,Na: nat,Aa: 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)),Na)),aa_int_real(ring_1_of_int_real,Aa))
    <=> 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)),Na)),Aa) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_6596_numeral__power__le__of__int__cancel__iff,axiom,
    ! [Xb: num,Na: nat,Aa: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,Xb)),Na)),aa_int_Code_integer(ring_18347121197199848620nteger,Aa))
    <=> 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)),Na)),Aa) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_6597_numeral__power__le__of__int__cancel__iff,axiom,
    ! [Xb: num,Na: nat,Aa: 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)),Na)),aa_int_rat(ring_1_of_int_rat,Aa))
    <=> 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)),Na)),Aa) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_6598_numeral__power__le__of__int__cancel__iff,axiom,
    ! [Xb: num,Na: nat,Aa: 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)),Na)),aa_int_int(ring_1_of_int_int,Aa))
    <=> 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)),Na)),Aa) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_6599_of__int__le__numeral__power__cancel__iff,axiom,
    ! [Aa: int,Xb: num,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Aa)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,Xb)),Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Na)) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_6600_of__int__le__numeral__power__cancel__iff,axiom,
    ! [Aa: int,Xb: num,Na: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_int_Code_integer(ring_18347121197199848620nteger,Aa)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,Xb)),Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Na)) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_6601_of__int__le__numeral__power__cancel__iff,axiom,
    ! [Aa: int,Xb: num,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Aa)),aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,Xb)),Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Na)) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_6602_of__int__le__numeral__power__cancel__iff,axiom,
    ! [Aa: int,Xb: num,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ring_1_of_int_int,Aa)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Na)) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_6603_of__int__less__numeral__power__cancel__iff,axiom,
    ! [Aa: int,Xb: num,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Aa)),aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,Xb)),Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Na)) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_6604_of__int__less__numeral__power__cancel__iff,axiom,
    ! [Aa: int,Xb: num,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Aa)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,Xb)),Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Na)) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_6605_of__int__less__numeral__power__cancel__iff,axiom,
    ! [Aa: int,Xb: num,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(ring_1_of_int_int,Aa)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Na)) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_6606_of__int__less__numeral__power__cancel__iff,axiom,
    ! [Aa: int,Xb: num,Na: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_int_Code_integer(ring_18347121197199848620nteger,Aa)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,Xb)),Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Na)) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_6607_numeral__power__less__of__int__cancel__iff,axiom,
    ! [Xb: num,Na: nat,Aa: 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)),Na)),aa_int_rat(ring_1_of_int_rat,Aa))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Na)),Aa) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_6608_numeral__power__less__of__int__cancel__iff,axiom,
    ! [Xb: num,Na: nat,Aa: 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)),Na)),aa_int_real(ring_1_of_int_real,Aa))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Na)),Aa) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_6609_numeral__power__less__of__int__cancel__iff,axiom,
    ! [Xb: num,Na: nat,Aa: 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)),Na)),aa_int_int(ring_1_of_int_int,Aa))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Na)),Aa) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_6610_numeral__power__less__of__int__cancel__iff,axiom,
    ! [Xb: num,Na: nat,Aa: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,Xb)),Na)),aa_int_Code_integer(ring_18347121197199848620nteger,Aa))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Na)),Aa) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_6611_one__div__2__pow__eq,axiom,
    ! [Na: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,one_one_Code_integer),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Na)) = zero_n356916108424825756nteger(Na = zero_zero_nat) ).

% one_div_2_pow_eq
tff(fact_6612_one__div__2__pow__eq,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) = zero_n2687167440665602831ol_nat(Na = zero_zero_nat) ).

% one_div_2_pow_eq
tff(fact_6613_one__div__2__pow__eq,axiom,
    ! [Na: nat] : aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)) = aa_o_int(zero_n2684676970156552555ol_int,Na = zero_zero_nat) ).

% one_div_2_pow_eq
tff(fact_6614_bits__1__div__exp,axiom,
    ! [Na: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,one_one_Code_integer),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Na)) = zero_n356916108424825756nteger(Na = zero_zero_nat) ).

% bits_1_div_exp
tff(fact_6615_bits__1__div__exp,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) = zero_n2687167440665602831ol_nat(Na = zero_zero_nat) ).

% bits_1_div_exp
tff(fact_6616_bits__1__div__exp,axiom,
    ! [Na: nat] : aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)) = aa_o_int(zero_n2684676970156552555ol_int,Na = zero_zero_nat) ).

% bits_1_div_exp
tff(fact_6617_one__mod__2__pow__eq,axiom,
    ! [Na: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,one_one_Code_integer),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Na)) = zero_n356916108424825756nteger(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)) ).

% one_mod_2_pow_eq
tff(fact_6618_one__mod__2__pow__eq,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) = zero_n2687167440665602831ol_nat(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)) ).

% one_mod_2_pow_eq
tff(fact_6619_one__mod__2__pow__eq,axiom,
    ! [Na: 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))),Na)) = aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)) ).

% one_mod_2_pow_eq
tff(fact_6620_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [Xb: num,Na: nat,Aa: 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))),Na)),aa_int_real(ring_1_of_int_real,Aa))
    <=> 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))),Na)),Aa) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_6621_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [Xb: num,Na: nat,Aa: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Xb))),Na)),aa_int_Code_integer(ring_18347121197199848620nteger,Aa))
    <=> 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))),Na)),Aa) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_6622_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [Xb: num,Na: nat,Aa: 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))),Na)),aa_int_rat(ring_1_of_int_rat,Aa))
    <=> 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))),Na)),Aa) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_6623_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [Xb: num,Na: nat,Aa: 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))),Na)),aa_int_int(ring_1_of_int_int,Aa))
    <=> 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))),Na)),Aa) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_6624_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [Aa: int,Xb: num,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Aa)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Xb))),Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Na)) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_6625_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [Aa: int,Xb: num,Na: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_int_Code_integer(ring_18347121197199848620nteger,Aa)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Xb))),Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Na)) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_6626_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [Aa: int,Xb: num,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Aa)),aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Xb))),Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Na)) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_6627_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [Aa: int,Xb: num,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ring_1_of_int_int,Aa)),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Na)) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_6628_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [Aa: int,Xb: num,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Aa)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Xb))),Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Na)) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_6629_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [Aa: int,Xb: num,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(ring_1_of_int_int,Aa)),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Na)) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_6630_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [Aa: int,Xb: num,Na: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_int_Code_integer(ring_18347121197199848620nteger,Aa)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Xb))),Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Na)) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_6631_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [Aa: int,Xb: num,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Aa)),aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Xb))),Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Na)) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_6632_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [Xb: num,Na: nat,Aa: 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))),Na)),aa_int_real(ring_1_of_int_real,Aa))
    <=> 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))),Na)),Aa) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_6633_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [Xb: num,Na: nat,Aa: 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))),Na)),aa_int_int(ring_1_of_int_int,Aa))
    <=> 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))),Na)),Aa) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_6634_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [Xb: num,Na: nat,Aa: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_num_Code_integer(numera6620942414471956472nteger,Xb))),Na)),aa_int_Code_integer(ring_18347121197199848620nteger,Aa))
    <=> 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))),Na)),Aa) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_6635_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [Xb: num,Na: nat,Aa: 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))),Na)),aa_int_rat(ring_1_of_int_rat,Aa))
    <=> 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))),Na)),Aa) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_6636_of__bool__eq__iff,axiom,
    ! [P3: $o,Q4: $o] :
      ( ( zero_n2687167440665602831ol_nat((P3)) = zero_n2687167440665602831ol_nat((Q4)) )
    <=> ( (P3)
      <=> (Q4) ) ) ).

% of_bool_eq_iff
tff(fact_6637_of__bool__eq__iff,axiom,
    ! [P3: $o,Q4: $o] :
      ( ( aa_o_int(zero_n2684676970156552555ol_int,(P3)) = aa_o_int(zero_n2684676970156552555ol_int,(Q4)) )
    <=> ( (P3)
      <=> (Q4) ) ) ).

% of_bool_eq_iff
tff(fact_6638_ex__le__of__int,axiom,
    ! [Xb: real] :
    ? [Z3: int] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_int_real(ring_1_of_int_real,Z3)) ).

% ex_le_of_int
tff(fact_6639_ex__le__of__int,axiom,
    ! [Xb: rat] :
    ? [Z3: int] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),aa_int_rat(ring_1_of_int_rat,Z3)) ).

% ex_le_of_int
tff(fact_6640_ex__less__of__int,axiom,
    ! [Xb: real] :
    ? [Z3: int] : aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_int_real(ring_1_of_int_real,Z3)) ).

% ex_less_of_int
tff(fact_6641_ex__less__of__int,axiom,
    ! [Xb: rat] :
    ? [Z3: int] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),aa_int_rat(ring_1_of_int_rat,Z3)) ).

% ex_less_of_int
tff(fact_6642_ex__of__int__less,axiom,
    ! [Xb: real] :
    ? [Z3: int] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Z3)),Xb) ).

% ex_of_int_less
tff(fact_6643_ex__of__int__less,axiom,
    ! [Xb: rat] :
    ? [Z3: int] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Z3)),Xb) ).

% ex_of_int_less
tff(fact_6644_of__bool__conj,axiom,
    ! [P: $o,Q: $o] :
      zero_n3304061248610475627l_real(( (P)
        & (Q) )) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,zero_n3304061248610475627l_real((P))),zero_n3304061248610475627l_real((Q))) ).

% of_bool_conj
tff(fact_6645_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_6646_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_6647_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_6648_arctan__monotone,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arctan,Xb)),aa_real_real(arctan,Y)) ) ).

% arctan_monotone
tff(fact_6649_arctan__less__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arctan,Xb)),aa_real_real(arctan,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y) ) ).

% arctan_less_iff
tff(fact_6650_arctan__monotone_H,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arctan,Xb)),aa_real_real(arctan,Y)) ) ).

% arctan_monotone'
tff(fact_6651_arctan__le__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arctan,Xb)),aa_real_real(arctan,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y) ) ).

% arctan_le_iff
tff(fact_6652_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_6653_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_6654_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_6655_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_6656_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_6657_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_6658_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_6659_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_6660_split__of__bool__asm,axiom,
    ! [P: fun_complex_o,P3: $o] :
      ( aa_complex_o(P,zero_n1201886186963655149omplex((P3)))
    <=> ~ ( ( (P3)
            & ~ aa_complex_o(P,one_one_complex) )
          | ( ~ (P3)
            & ~ aa_complex_o(P,zero_zero_complex) ) ) ) ).

% split_of_bool_asm
tff(fact_6661_split__of__bool__asm,axiom,
    ! [P: fun_real_o,P3: $o] :
      ( aa_real_o(P,zero_n3304061248610475627l_real((P3)))
    <=> ~ ( ( (P3)
            & ~ aa_real_o(P,one_one_real) )
          | ( ~ (P3)
            & ~ aa_real_o(P,zero_zero_real) ) ) ) ).

% split_of_bool_asm
tff(fact_6662_split__of__bool__asm,axiom,
    ! [P: fun_rat_o,P3: $o] :
      ( aa_rat_o(P,zero_n2052037380579107095ol_rat((P3)))
    <=> ~ ( ( (P3)
            & ~ aa_rat_o(P,one_one_rat) )
          | ( ~ (P3)
            & ~ aa_rat_o(P,zero_zero_rat) ) ) ) ).

% split_of_bool_asm
tff(fact_6663_split__of__bool__asm,axiom,
    ! [P: fun_nat_o,P3: $o] :
      ( aa_nat_o(P,zero_n2687167440665602831ol_nat((P3)))
    <=> ~ ( ( (P3)
            & ~ aa_nat_o(P,one_one_nat) )
          | ( ~ (P3)
            & ~ aa_nat_o(P,zero_zero_nat) ) ) ) ).

% split_of_bool_asm
tff(fact_6664_split__of__bool__asm,axiom,
    ! [P: fun_int_o,P3: $o] :
      ( aa_int_o(P,aa_o_int(zero_n2684676970156552555ol_int,(P3)))
    <=> ~ ( ( (P3)
            & ~ aa_int_o(P,one_one_int) )
          | ( ~ (P3)
            & ~ aa_int_o(P,zero_zero_int) ) ) ) ).

% split_of_bool_asm
tff(fact_6665_split__of__bool,axiom,
    ! [P: fun_complex_o,P3: $o] :
      ( aa_complex_o(P,zero_n1201886186963655149omplex((P3)))
    <=> ( ( (P3)
         => aa_complex_o(P,one_one_complex) )
        & ( ~ (P3)
         => aa_complex_o(P,zero_zero_complex) ) ) ) ).

% split_of_bool
tff(fact_6666_split__of__bool,axiom,
    ! [P: fun_real_o,P3: $o] :
      ( aa_real_o(P,zero_n3304061248610475627l_real((P3)))
    <=> ( ( (P3)
         => aa_real_o(P,one_one_real) )
        & ( ~ (P3)
         => aa_real_o(P,zero_zero_real) ) ) ) ).

% split_of_bool
tff(fact_6667_split__of__bool,axiom,
    ! [P: fun_rat_o,P3: $o] :
      ( aa_rat_o(P,zero_n2052037380579107095ol_rat((P3)))
    <=> ( ( (P3)
         => aa_rat_o(P,one_one_rat) )
        & ( ~ (P3)
         => aa_rat_o(P,zero_zero_rat) ) ) ) ).

% split_of_bool
tff(fact_6668_split__of__bool,axiom,
    ! [P: fun_nat_o,P3: $o] :
      ( aa_nat_o(P,zero_n2687167440665602831ol_nat((P3)))
    <=> ( ( (P3)
         => aa_nat_o(P,one_one_nat) )
        & ( ~ (P3)
         => aa_nat_o(P,zero_zero_nat) ) ) ) ).

% split_of_bool
tff(fact_6669_split__of__bool,axiom,
    ! [P: fun_int_o,P3: $o] :
      ( aa_int_o(P,aa_o_int(zero_n2684676970156552555ol_int,(P3)))
    <=> ( ( (P3)
         => aa_int_o(P,one_one_int) )
        & ( ~ (P3)
         => aa_int_o(P,zero_zero_int) ) ) ) ).

% split_of_bool
tff(fact_6670_of__bool__def,axiom,
    ! [P3: $o] :
      zero_n1201886186963655149omplex((P3)) = $ite((P3),one_one_complex,zero_zero_complex) ).

% of_bool_def
tff(fact_6671_of__bool__def,axiom,
    ! [P3: $o] :
      zero_n3304061248610475627l_real((P3)) = $ite((P3),one_one_real,zero_zero_real) ).

% of_bool_def
tff(fact_6672_of__bool__def,axiom,
    ! [P3: $o] :
      zero_n2052037380579107095ol_rat((P3)) = $ite((P3),one_one_rat,zero_zero_rat) ).

% of_bool_def
tff(fact_6673_of__bool__def,axiom,
    ! [P3: $o] :
      zero_n2687167440665602831ol_nat((P3)) = $ite((P3),one_one_nat,zero_zero_nat) ).

% of_bool_def
tff(fact_6674_of__bool__def,axiom,
    ! [P3: $o] :
      aa_o_int(zero_n2684676970156552555ol_int,(P3)) = $ite((P3),one_one_int,zero_zero_int) ).

% of_bool_def
tff(fact_6675_le__of__int__ceiling,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,Xb))) ).

% le_of_int_ceiling
tff(fact_6676_le__of__int__ceiling,axiom,
    ! [Xb: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,Xb))) ).

% le_of_int_ceiling
tff(fact_6677_divmod__step__integer__def,axiom,
    ! [L: num,Qr: produc8923325533196201883nteger] : unique4921790084139445826nteger(L,Qr) = produc6916734918728496179nteger(aTP_Lamm_ex(L),Qr) ).

% divmod_step_integer_def
tff(fact_6678_ceiling__le__iff,axiom,
    ! [Xb: real,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,Xb)),Z)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_int_real(ring_1_of_int_real,Z)) ) ).

% ceiling_le_iff
tff(fact_6679_ceiling__le__iff,axiom,
    ! [Xb: rat,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,Xb)),Z)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),aa_int_rat(ring_1_of_int_rat,Z)) ) ).

% ceiling_le_iff
tff(fact_6680_ceiling__le,axiom,
    ! [Xb: real,Aa: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_int_real(ring_1_of_int_real,Aa))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,Xb)),Aa) ) ).

% ceiling_le
tff(fact_6681_ceiling__le,axiom,
    ! [Xb: rat,Aa: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),aa_int_rat(ring_1_of_int_rat,Aa))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,Xb)),Aa) ) ).

% ceiling_le
tff(fact_6682_less__ceiling__iff,axiom,
    ! [Z: int,Xb: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z),aa_rat_int(archim2889992004027027881ng_rat,Xb))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Z)),Xb) ) ).

% less_ceiling_iff
tff(fact_6683_less__ceiling__iff,axiom,
    ! [Z: int,Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z),aa_real_int(archim7802044766580827645g_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Z)),Xb) ) ).

% less_ceiling_iff
tff(fact_6684_real__of__int__div4,axiom,
    ! [Na: 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,Na),Xb))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,Na)),aa_int_real(ring_1_of_int_real,Xb))) ).

% real_of_int_div4
tff(fact_6685_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_6686_of__int__nonneg,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(ring_18347121197199848620nteger,Z)) ) ).

% of_int_nonneg
tff(fact_6687_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_6688_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_6689_of__int__leD,axiom,
    ! [Na: 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,Na))),Xb)
     => ( ( Na = zero_zero_int )
        | aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xb) ) ) ).

% of_int_leD
tff(fact_6690_of__int__leD,axiom,
    ! [Na: int,Xb: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_int_Code_integer(ring_18347121197199848620nteger,Na))),Xb)
     => ( ( Na = zero_zero_int )
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),Xb) ) ) ).

% of_int_leD
tff(fact_6691_of__int__leD,axiom,
    ! [Na: 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,Na))),Xb)
     => ( ( Na = zero_zero_int )
        | aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),Xb) ) ) ).

% of_int_leD
tff(fact_6692_of__int__leD,axiom,
    ! [Na: 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,Na))),Xb)
     => ( ( Na = zero_zero_int )
        | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),Xb) ) ) ).

% of_int_leD
tff(fact_6693_of__int__pos,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(ring_18347121197199848620nteger,Z)) ) ).

% of_int_pos
tff(fact_6694_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_6695_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_6696_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_6697_of__int__lessD,axiom,
    ! [Na: int,Xb: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_int_Code_integer(ring_18347121197199848620nteger,Na))),Xb)
     => ( ( Na = zero_zero_int )
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),Xb) ) ) ).

% of_int_lessD
tff(fact_6698_of__int__lessD,axiom,
    ! [Na: 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,Na))),Xb)
     => ( ( Na = zero_zero_int )
        | aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb) ) ) ).

% of_int_lessD
tff(fact_6699_of__int__lessD,axiom,
    ! [Na: 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,Na))),Xb)
     => ( ( Na = zero_zero_int )
        | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),Xb) ) ) ).

% of_int_lessD
tff(fact_6700_of__int__lessD,axiom,
    ! [Na: 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,Na))),Xb)
     => ( ( Na = zero_zero_int )
        | aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),Xb) ) ) ).

% of_int_lessD
tff(fact_6701_floor__exists1,axiom,
    ! [Xb: real] :
    ? [X: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,X)),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,X),one_one_int)))
      & ! [Y4: int] :
          ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Y4)),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,Y4),one_one_int))) )
         => ( Y4 = X ) ) ) ).

% floor_exists1
tff(fact_6702_floor__exists1,axiom,
    ! [Xb: rat] :
    ? [X: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,X)),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,X),one_one_int)))
      & ! [Y4: int] :
          ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Y4)),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,Y4),one_one_int))) )
         => ( Y4 = X ) ) ) ).

% floor_exists1
tff(fact_6703_floor__exists,axiom,
    ! [Xb: real] :
    ? [Z3: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z3)),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,Z3),one_one_int))) ) ).

% floor_exists
tff(fact_6704_floor__exists,axiom,
    ! [Xb: rat] :
    ? [Z3: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z3)),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,Z3),one_one_int))) ) ).

% floor_exists
tff(fact_6705_of__int__ceiling__le__add__one,axiom,
    ! [R2: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,R2))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,R2),one_one_real)) ).

% of_int_ceiling_le_add_one
tff(fact_6706_of__int__ceiling__le__add__one,axiom,
    ! [R2: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,R2))),aa_rat_rat(plus_plus_rat(R2),one_one_rat)) ).

% of_int_ceiling_le_add_one
tff(fact_6707_of__int__ceiling__diff__one__le,axiom,
    ! [R2: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,R2))),one_one_real)),R2) ).

% of_int_ceiling_diff_one_le
tff(fact_6708_of__int__ceiling__diff__one__le,axiom,
    ! [R2: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,R2))),one_one_rat)),R2) ).

% of_int_ceiling_diff_one_le
tff(fact_6709_int__le__real__less,axiom,
    ! [Na: int,M: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Na),M)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Na)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,M)),one_one_real)) ) ).

% int_le_real_less
tff(fact_6710_int__less__real__le,axiom,
    ! [Na: int,M: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Na),M)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,Na)),one_one_real)),aa_int_real(ring_1_of_int_real,M)) ) ).

% int_less_real_le
tff(fact_6711_ceiling__correct,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,Xb))),one_one_real)),Xb)
      & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,Xb))) ) ).

% ceiling_correct
tff(fact_6712_ceiling__correct,axiom,
    ! [Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,Xb))),one_one_rat)),Xb)
      & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,Xb))) ) ).

% ceiling_correct
tff(fact_6713_ceiling__unique,axiom,
    ! [Z: int,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(ring_1_of_int_real,Z)),one_one_real)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_int_real(ring_1_of_int_real,Z))
       => ( aa_real_int(archim7802044766580827645g_real,Xb) = Z ) ) ) ).

% ceiling_unique
tff(fact_6714_ceiling__unique,axiom,
    ! [Z: int,Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat)),Xb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),aa_int_rat(ring_1_of_int_rat,Z))
       => ( aa_rat_int(archim2889992004027027881ng_rat,Xb) = Z ) ) ) ).

% ceiling_unique
tff(fact_6715_ceiling__eq__iff,axiom,
    ! [Xb: real,Aa: int] :
      ( ( aa_real_int(archim7802044766580827645g_real,Xb) = Aa )
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(ring_1_of_int_real,Aa)),one_one_real)),Xb)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_int_real(ring_1_of_int_real,Aa)) ) ) ).

% ceiling_eq_iff
tff(fact_6716_ceiling__eq__iff,axiom,
    ! [Xb: rat,Aa: int] :
      ( ( aa_rat_int(archim2889992004027027881ng_rat,Xb) = Aa )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,Aa)),one_one_rat)),Xb)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),aa_int_rat(ring_1_of_int_rat,Aa)) ) ) ).

% ceiling_eq_iff
tff(fact_6717_ceiling__split,axiom,
    ! [P: fun_int_o,Ta: real] :
      ( aa_int_o(P,aa_real_int(archim7802044766580827645g_real,Ta))
    <=> ! [I3: int] :
          ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(ring_1_of_int_real,I3)),one_one_real)),Ta)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ta),aa_int_real(ring_1_of_int_real,I3)) )
         => aa_int_o(P,I3) ) ) ).

% ceiling_split
tff(fact_6718_ceiling__split,axiom,
    ! [P: fun_int_o,Ta: rat] :
      ( aa_int_o(P,aa_rat_int(archim2889992004027027881ng_rat,Ta))
    <=> ! [I3: int] :
          ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,I3)),one_one_rat)),Ta)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ta),aa_int_rat(ring_1_of_int_rat,I3)) )
         => aa_int_o(P,I3) ) ) ).

% ceiling_split
tff(fact_6719_ceiling__less__iff,axiom,
    ! [Xb: real,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim7802044766580827645g_real,Xb)),Z)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(ring_1_of_int_real,Z)),one_one_real)) ) ).

% ceiling_less_iff
tff(fact_6720_ceiling__less__iff,axiom,
    ! [Xb: rat,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(archim2889992004027027881ng_rat,Xb)),Z)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),aa_rat_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat)) ) ).

% ceiling_less_iff
tff(fact_6721_le__ceiling__iff,axiom,
    ! [Z: int,Xb: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),aa_rat_int(archim2889992004027027881ng_rat,Xb))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat)),Xb) ) ).

% le_ceiling_iff
tff(fact_6722_le__ceiling__iff,axiom,
    ! [Z: int,Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),aa_real_int(archim7802044766580827645g_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(ring_1_of_int_real,Z)),one_one_real)),Xb) ) ).

% le_ceiling_iff
tff(fact_6723_real__of__int__div2,axiom,
    ! [Na: int,Xb: int] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,Na)),aa_int_real(ring_1_of_int_real,Xb))),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(divide_divide_int,Na),Xb)))) ).

% real_of_int_div2
tff(fact_6724_real__of__int__div3,axiom,
    ! [Na: int,Xb: int] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,Na)),aa_int_real(ring_1_of_int_real,Xb))),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(divide_divide_int,Na),Xb)))),one_one_real) ).

% real_of_int_div3
tff(fact_6725_ceiling__divide__upper,axiom,
    ! [Q4: real,P3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Q4)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,P3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,P3),Q4)))),Q4)) ) ).

% ceiling_divide_upper
tff(fact_6726_ceiling__divide__upper,axiom,
    ! [Q4: rat,P3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Q4)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,P3),aa_rat_rat(times_times_rat(aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(divide_divide_rat(P3),Q4)))),Q4)) ) ).

% ceiling_divide_upper
tff(fact_6727_mult__ceiling__le__Ints,axiom,
    ! [Aa: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Aa)
     => ( member_real(Aa,ring_1_Ints_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),Ba)))),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(times_times_int,aa_real_int(archim7802044766580827645g_real,Aa)),aa_real_int(archim7802044766580827645g_real,Ba)))) ) ) ).

% mult_ceiling_le_Ints
tff(fact_6728_mult__ceiling__le__Ints,axiom,
    ! [Aa: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Aa)
     => ( member_real(Aa,ring_1_Ints_real)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_int_Code_integer(ring_18347121197199848620nteger,aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),Ba)))),aa_int_Code_integer(ring_18347121197199848620nteger,aa_int_int(aa_int_fun_int_int(times_times_int,aa_real_int(archim7802044766580827645g_real,Aa)),aa_real_int(archim7802044766580827645g_real,Ba)))) ) ) ).

% mult_ceiling_le_Ints
tff(fact_6729_mult__ceiling__le__Ints,axiom,
    ! [Aa: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Aa)
     => ( member_real(Aa,ring_1_Ints_real)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),Ba)))),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(times_times_int,aa_real_int(archim7802044766580827645g_real,Aa)),aa_real_int(archim7802044766580827645g_real,Ba)))) ) ) ).

% mult_ceiling_le_Ints
tff(fact_6730_mult__ceiling__le__Ints,axiom,
    ! [Aa: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Aa)
     => ( member_real(Aa,ring_1_Ints_real)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ring_1_of_int_int,aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),Ba)))),aa_int_int(ring_1_of_int_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_real_int(archim7802044766580827645g_real,Aa)),aa_real_int(archim7802044766580827645g_real,Ba)))) ) ) ).

% mult_ceiling_le_Ints
tff(fact_6731_mult__ceiling__le__Ints,axiom,
    ! [Aa: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Aa)
     => ( member_rat(Aa,ring_1_Ints_rat)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(times_times_rat(Aa),Ba)))),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(times_times_int,aa_rat_int(archim2889992004027027881ng_rat,Aa)),aa_rat_int(archim2889992004027027881ng_rat,Ba)))) ) ) ).

% mult_ceiling_le_Ints
tff(fact_6732_mult__ceiling__le__Ints,axiom,
    ! [Aa: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Aa)
     => ( member_rat(Aa,ring_1_Ints_rat)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_int_Code_integer(ring_18347121197199848620nteger,aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(times_times_rat(Aa),Ba)))),aa_int_Code_integer(ring_18347121197199848620nteger,aa_int_int(aa_int_fun_int_int(times_times_int,aa_rat_int(archim2889992004027027881ng_rat,Aa)),aa_rat_int(archim2889992004027027881ng_rat,Ba)))) ) ) ).

% mult_ceiling_le_Ints
tff(fact_6733_mult__ceiling__le__Ints,axiom,
    ! [Aa: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Aa)
     => ( member_rat(Aa,ring_1_Ints_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(times_times_rat(Aa),Ba)))),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(times_times_int,aa_rat_int(archim2889992004027027881ng_rat,Aa)),aa_rat_int(archim2889992004027027881ng_rat,Ba)))) ) ) ).

% mult_ceiling_le_Ints
tff(fact_6734_mult__ceiling__le__Ints,axiom,
    ! [Aa: rat,Ba: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Aa)
     => ( member_rat(Aa,ring_1_Ints_rat)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ring_1_of_int_int,aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(times_times_rat(Aa),Ba)))),aa_int_int(ring_1_of_int_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_rat_int(archim2889992004027027881ng_rat,Aa)),aa_rat_int(archim2889992004027027881ng_rat,Ba)))) ) ) ).

% mult_ceiling_le_Ints
tff(fact_6735_ceiling__divide__lower,axiom,
    ! [Q4: rat,P3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Q4)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(divide_divide_rat(P3),Q4)))),one_one_rat)),Q4)),P3) ) ).

% ceiling_divide_lower
tff(fact_6736_ceiling__divide__lower,axiom,
    ! [Q4: real,P3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Q4)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,P3),Q4)))),one_one_real)),Q4)),P3) ) ).

% ceiling_divide_lower
tff(fact_6737_exp__mod__exp,axiom,
    ! [M: nat,Na: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),M)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Na)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,zero_n356916108424825756nteger(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na))),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),M)) ).

% exp_mod_exp
tff(fact_6738_exp__mod__exp,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,zero_n2687167440665602831ol_nat(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) ).

% exp_mod_exp
tff(fact_6739_exp__mod__exp,axiom,
    ! [M: nat,Na: 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))),M)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M)) ).

% exp_mod_exp
tff(fact_6740_ceiling__eq,axiom,
    ! [Na: int,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Na)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,Na)),one_one_real))
       => ( aa_real_int(archim7802044766580827645g_real,Xb) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Na),one_one_int) ) ) ) ).

% ceiling_eq
tff(fact_6741_ceiling__eq,axiom,
    ! [Na: int,Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Na)),Xb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),aa_rat_rat(plus_plus_rat(aa_int_rat(ring_1_of_int_rat,Na)),one_one_rat))
       => ( aa_rat_int(archim2889992004027027881ng_rat,Xb) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Na),one_one_int) ) ) ) ).

% ceiling_eq
tff(fact_6742_divmod__step__nat__def,axiom,
    ! [L: num,Qr: product_prod_nat_nat] : unique5026877609467782581ep_nat(L,Qr) = produc2626176000494625587at_nat(aTP_Lamm_ey(L),Qr) ).

% divmod_step_nat_def
tff(fact_6743_exp__div__exp__eq,axiom,
    ! [M: nat,Na: nat] :
      aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),M)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Na)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,
          zero_n356916108424825756nteger(( ( aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),M) != zero_z3403309356797280102nteger )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M) ))),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na))) ).

% exp_div_exp_eq
tff(fact_6744_exp__div__exp__eq,axiom,
    ! [M: nat,Na: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,
          zero_n2687167440665602831ol_nat(( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M) != zero_zero_nat )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M) ))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na))) ).

% exp_div_exp_eq
tff(fact_6745_exp__div__exp__eq,axiom,
    ! [M: nat,Na: nat] :
      aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)) = aa_int_int(aa_int_fun_int_int(times_times_int,
          aa_o_int(zero_n2684676970156552555ol_int,
            ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M) != zero_zero_int )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M) ))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na))) ).

% exp_div_exp_eq
tff(fact_6746_arctan__add,axiom,
    ! [Xb: real,Y: 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,Y)),one_one_real)
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(arctan,Xb)),aa_real_real(arctan,Y)) = aa_real_real(arctan,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Y)))) ) ) ) ).

% arctan_add
tff(fact_6747_divmod__step__int__def,axiom,
    ! [L: num,Qr: product_prod_int_int] : unique5024387138958732305ep_int(L,Qr) = produc4245557441103728435nt_int(aTP_Lamm_ez(L),Qr) ).

% divmod_step_int_def
tff(fact_6748_divmod__step__def,axiom,
    ! [L: num,Qr: produc8923325533196201883nteger] : unique4921790084139445826nteger(L,Qr) = produc6916734918728496179nteger(aTP_Lamm_ex(L),Qr) ).

% divmod_step_def
tff(fact_6749_divmod__step__def,axiom,
    ! [L: num,Qr: product_prod_nat_nat] : unique5026877609467782581ep_nat(L,Qr) = produc2626176000494625587at_nat(aTP_Lamm_ey(L),Qr) ).

% divmod_step_def
tff(fact_6750_divmod__step__def,axiom,
    ! [L: num,Qr: product_prod_int_int] : unique5024387138958732305ep_int(L,Qr) = produc4245557441103728435nt_int(aTP_Lamm_ez(L),Qr) ).

% divmod_step_def
tff(fact_6751_round__unique,axiom,
    ! [Xb: real,Y: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_int_real(ring_1_of_int_real,Y))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Y)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => ( aa_real_int(archim8280529875227126926d_real,Xb) = Y ) ) ) ).

% round_unique
tff(fact_6752_round__unique,axiom,
    ! [Xb: rat,Y: 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,Y))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Y)),aa_rat_rat(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) = Y ) ) ) ).

% round_unique
tff(fact_6753_round__unique_H,axiom,
    ! [Xb: rat,Na: 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,Na)))),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) = Na ) ) ).

% round_unique'
tff(fact_6754_round__unique_H,axiom,
    ! [Xb: real,Na: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),aa_int_real(ring_1_of_int_real,Na)))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
     => ( aa_real_int(archim8280529875227126926d_real,Xb) = Na ) ) ).

% round_unique'
tff(fact_6755_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(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim8280529875227126926d_real,Xb))),Xb))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ).

% of_int_round_abs_le
tff(fact_6756_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_6757_divmod__nat__if,axiom,
    ! [M: nat,Na: nat] :
      divmod_nat(M,Na) = $ite(
        ( ( Na = zero_zero_nat )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ),
        aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),M),
        produc2626176000494625587at_nat(aTP_Lamm_fa,divmod_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na),Na)) ) ).

% divmod_nat_if
tff(fact_6758_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_6759_of__int__round__gt,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_int_real(ring_1_of_int_real,aa_real_int(archim8280529875227126926d_real,Xb))) ).

% of_int_round_gt
tff(fact_6760_of__int__round__ge,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_int_real(ring_1_of_int_real,aa_real_int(archim8280529875227126926d_real,Xb))) ).

% of_int_round_ge
tff(fact_6761_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_6762_round__0,axiom,
    aa_real_int(archim8280529875227126926d_real,zero_zero_real) = zero_zero_int ).

% round_0
tff(fact_6763_round__0,axiom,
    aa_rat_int(archim7778729529865785530nd_rat,zero_zero_rat) = zero_zero_int ).

% round_0
tff(fact_6764_round__1,axiom,
    aa_real_int(archim8280529875227126926d_real,one_one_real) = one_one_int ).

% round_1
tff(fact_6765_round__1,axiom,
    aa_rat_int(archim7778729529865785530nd_rat,one_one_rat) = one_one_int ).

% round_1
tff(fact_6766_divmod__integer_H__def,axiom,
    ! [M: num,Na: num] : unique3479559517661332726nteger(M,Na) = produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)),aa_num_Code_integer(numera6620942414471956472nteger,Na)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_num_Code_integer(numera6620942414471956472nteger,M)),aa_num_Code_integer(numera6620942414471956472nteger,Na))) ).

% divmod_integer'_def
tff(fact_6767_less__eq__integer__code_I1_J,axiom,
    aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),zero_z3403309356797280102nteger) ).

% less_eq_integer_code(1)
tff(fact_6768_plus__integer__code_I1_J,axiom,
    ! [K: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,K),zero_z3403309356797280102nteger) = K ).

% plus_integer_code(1)
tff(fact_6769_plus__integer__code_I2_J,axiom,
    ! [L: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,zero_z3403309356797280102nteger),L) = L ).

% plus_integer_code(2)
tff(fact_6770_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_6771_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_6772_minus__integer__code_I1_J,axiom,
    ! [K: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,K),zero_z3403309356797280102nteger) = K ).

% minus_integer_code(1)
tff(fact_6773_minus__integer__code_I2_J,axiom,
    ! [L: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,zero_z3403309356797280102nteger),L) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,L) ).

% minus_integer_code(2)
tff(fact_6774_round__mono,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim7778729529865785530nd_rat,Xb)),aa_rat_int(archim7778729529865785530nd_rat,Y)) ) ).

% round_mono
tff(fact_6775_ceiling__ge__round,axiom,
    ! [Xb: real] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim8280529875227126926d_real,Xb)),aa_real_int(archim7802044766580827645g_real,Xb)) ).

% ceiling_ge_round
tff(fact_6776_round__diff__minimal,axiom,
    ! [Z: real,M: int] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Z),aa_int_real(ring_1_of_int_real,aa_real_int(archim8280529875227126926d_real,Z))))),aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Z),aa_int_real(ring_1_of_int_real,M)))) ).

% round_diff_minimal
tff(fact_6777_round__diff__minimal,axiom,
    ! [Z: rat,M: 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,M)))) ).

% round_diff_minimal
tff(fact_6778_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(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% of_int_round_le
tff(fact_6779_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_6780_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,aa_num_Code_integer(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,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,l),one_one_Code_integer)) ) ) ) ).

% integer_of_int_code
tff(fact_6781_upto_Opinduct,axiom,
    ! [A0: int,A1: int,P: fun_int_fun_int_o] :
      ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(upto_rel),product_Pair_int_int(A0,A1))
     => ( ! [I2: int,J2: int] :
            ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(upto_rel),product_Pair_int_int(I2,J2))
           => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),J2)
               => aa_int_o(aa_int_fun_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),one_one_int)),J2) )
             => aa_int_o(aa_int_fun_int_o(P,I2),J2) ) )
       => aa_int_o(aa_int_fun_int_o(P,A0),A1) ) ) ).

% upto.pinduct
tff(fact_6782_binomial__code,axiom,
    ! [Na: nat,K: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),K),
        zero_zero_nat,
        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K)),one_one_nat)),Na),one_one_nat)),aa_nat_nat(semiri1408675320244567234ct_nat,K))) ) ).

% binomial_code
tff(fact_6783_fact__double,axiom,
    ! [Na: nat] : aa_nat_complex(semiri5044797733671781792omplex,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(divide1717551699836669952omplex(one_one_complex),numera6690914467698888265omplex(aa_num_num(bit0,one)))),Na))),aa_nat_complex(semiri5044797733671781792omplex,Na)) ).

% fact_double
tff(fact_6784_fact__double,axiom,
    ! [Na: nat] : aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) = aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(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))),Na))),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)))),Na))),aa_nat_rat(semiri773545260158071498ct_rat,Na)) ).

% fact_double
tff(fact_6785_fact__double,axiom,
    ! [Na: nat] : aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(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))),Na))),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),Na))),aa_nat_real(semiri2265585572941072030t_real,Na)) ).

% fact_double
tff(fact_6786_Sum__Icc__int,axiom,
    ! [M: int,Na: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,M),Na)
     => ( groups4538972089207619220nt_int(aTP_Lamm_cd,set_or1266510415728281911st_int(M,Na)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Na),aa_int_int(aa_int_fun_int_int(plus_plus_int,Na),one_one_int))),aa_int_int(aa_int_fun_int_int(times_times_int,M),aa_int_int(aa_int_fun_int_int(minus_minus_int,M),one_one_int)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ) ) ).

% Sum_Icc_int
tff(fact_6787_take__bit__rec,axiom,
    ! [Na: nat,Aa: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),Aa) = $ite(Na = zero_zero_nat,zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Aa),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,Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% take_bit_rec
tff(fact_6788_take__bit__rec,axiom,
    ! [Na: nat,Aa: code_integer] :
      aa_Cod1783974652533396728nteger(aa_nat6383147911832638789nteger(bit_se1745604003318907178nteger,Na),Aa) = $ite(Na = zero_zero_nat,zero_z3403309356797280102nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_nat6383147911832638789nteger(bit_se1745604003318907178nteger,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Aa),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Aa),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))))) ).

% take_bit_rec
tff(fact_6789_take__bit__rec,axiom,
    ! [Na: nat,Aa: int] :
      aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Na),Aa) = $ite(Na = zero_zero_nat,zero_zero_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat)),aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),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,Aa),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))) ).

% take_bit_rec
tff(fact_6790_integer__of__int__eq__of__int,axiom,
    code_integer_of_int = ring_18347121197199848620nteger ).

% integer_of_int_eq_of_int
tff(fact_6791_take__bit__of__0,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),zero_zero_nat) = zero_zero_nat ).

% take_bit_of_0
tff(fact_6792_take__bit__of__0,axiom,
    ! [Na: nat] : aa_Cod1783974652533396728nteger(aa_nat6383147911832638789nteger(bit_se1745604003318907178nteger,Na),zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% take_bit_of_0
tff(fact_6793_take__bit__of__0,axiom,
    ! [Na: nat] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Na),zero_zero_int) = zero_zero_int ).

% take_bit_of_0
tff(fact_6794_binomial__Suc__n,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,Na)),Na) = aa_nat_nat(suc,Na) ).

% binomial_Suc_n
tff(fact_6795_binomial__n__n,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),Na) = one_one_nat ).

% binomial_n_n
tff(fact_6796_sum_Oneutral__const,axiom,
    ! [A3: set_int] : groups4538972089207619220nt_int(aTP_Lamm_by,A3) = zero_zero_int ).

% sum.neutral_const
tff(fact_6797_sum_Oneutral__const,axiom,
    ! [A3: set_complex] : groups7754918857620584856omplex(aTP_Lamm_fb,A3) = zero_zero_complex ).

% sum.neutral_const
tff(fact_6798_sum_Oneutral__const,axiom,
    ! [A3: set_nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_bx),A3) = zero_zero_nat ).

% sum.neutral_const
tff(fact_6799_sum_Oneutral__const,axiom,
    ! [A3: set_nat] : aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_fc),A3) = zero_zero_real ).

% sum.neutral_const
tff(fact_6800_sum_Oempty,axiom,
    ! [G: fun_real_real] : groups8097168146408367636l_real(G,bot_bot_set_real) = zero_zero_real ).

% sum.empty
tff(fact_6801_sum_Oempty,axiom,
    ! [G: fun_real_rat] : groups1300246762558778688al_rat(G,bot_bot_set_real) = zero_zero_rat ).

% sum.empty
tff(fact_6802_sum_Oempty,axiom,
    ! [G: fun_real_nat] : groups1935376822645274424al_nat(G,bot_bot_set_real) = zero_zero_nat ).

% sum.empty
tff(fact_6803_sum_Oempty,axiom,
    ! [G: fun_real_int] : groups1932886352136224148al_int(G,bot_bot_set_real) = zero_zero_int ).

% sum.empty
tff(fact_6804_sum_Oempty,axiom,
    ! [G: fun_nat_rat] : groups2906978787729119204at_rat(G,bot_bot_set_nat) = zero_zero_rat ).

% sum.empty
tff(fact_6805_sum_Oempty,axiom,
    ! [G: fun_nat_int] : groups3539618377306564664at_int(G,bot_bot_set_nat) = zero_zero_int ).

% sum.empty
tff(fact_6806_sum_Oempty,axiom,
    ! [G: fun_int_real] : groups8778361861064173332t_real(G,bot_bot_set_int) = zero_zero_real ).

% sum.empty
tff(fact_6807_sum_Oempty,axiom,
    ! [G: fun_int_rat] : groups3906332499630173760nt_rat(G,bot_bot_set_int) = zero_zero_rat ).

% sum.empty
tff(fact_6808_sum_Oempty,axiom,
    ! [G: fun_int_nat] : groups4541462559716669496nt_nat(G,bot_bot_set_int) = zero_zero_nat ).

% sum.empty
tff(fact_6809_sum_Oempty,axiom,
    ! [G: fun_int_int] : groups4538972089207619220nt_int(G,bot_bot_set_int) = zero_zero_int ).

% sum.empty
tff(fact_6810_sum_Oinfinite,axiom,
    ! [A3: set_int,G: fun_int_real] :
      ( ~ aa_set_int_o(finite_finite_int,A3)
     => ( groups8778361861064173332t_real(G,A3) = zero_zero_real ) ) ).

% sum.infinite
tff(fact_6811_sum_Oinfinite,axiom,
    ! [A3: set_complex,G: fun_complex_real] :
      ( ~ aa_set_complex_o(finite3207457112153483333omplex,A3)
     => ( groups5808333547571424918x_real(G,A3) = zero_zero_real ) ) ).

% sum.infinite
tff(fact_6812_sum_Oinfinite,axiom,
    ! [A3: set_nat,G: fun_nat_rat] :
      ( ~ aa_set_nat_o(finite_finite_nat,A3)
     => ( groups2906978787729119204at_rat(G,A3) = zero_zero_rat ) ) ).

% sum.infinite
tff(fact_6813_sum_Oinfinite,axiom,
    ! [A3: set_int,G: fun_int_rat] :
      ( ~ aa_set_int_o(finite_finite_int,A3)
     => ( groups3906332499630173760nt_rat(G,A3) = zero_zero_rat ) ) ).

% sum.infinite
tff(fact_6814_sum_Oinfinite,axiom,
    ! [A3: set_complex,G: fun_complex_rat] :
      ( ~ aa_set_complex_o(finite3207457112153483333omplex,A3)
     => ( groups5058264527183730370ex_rat(G,A3) = zero_zero_rat ) ) ).

% sum.infinite
tff(fact_6815_sum_Oinfinite,axiom,
    ! [A3: set_int,G: fun_int_nat] :
      ( ~ aa_set_int_o(finite_finite_int,A3)
     => ( groups4541462559716669496nt_nat(G,A3) = zero_zero_nat ) ) ).

% sum.infinite
tff(fact_6816_sum_Oinfinite,axiom,
    ! [A3: set_complex,G: fun_complex_nat] :
      ( ~ aa_set_complex_o(finite3207457112153483333omplex,A3)
     => ( groups5693394587270226106ex_nat(G,A3) = zero_zero_nat ) ) ).

% sum.infinite
tff(fact_6817_sum_Oinfinite,axiom,
    ! [A3: set_nat,G: fun_nat_int] :
      ( ~ aa_set_nat_o(finite_finite_nat,A3)
     => ( groups3539618377306564664at_int(G,A3) = zero_zero_int ) ) ).

% sum.infinite
tff(fact_6818_sum_Oinfinite,axiom,
    ! [A3: set_complex,G: fun_complex_int] :
      ( ~ aa_set_complex_o(finite3207457112153483333omplex,A3)
     => ( groups5690904116761175830ex_int(G,A3) = zero_zero_int ) ) ).

% sum.infinite
tff(fact_6819_sum_Oinfinite,axiom,
    ! [A3: set_int,G: fun_int_int] :
      ( ~ aa_set_int_o(finite_finite_int,A3)
     => ( groups4538972089207619220nt_int(G,A3) = zero_zero_int ) ) ).

% sum.infinite
tff(fact_6820_sum__eq__0__iff,axiom,
    ! [F3: set_int,F: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,F3)
     => ( ( groups4541462559716669496nt_nat(F,F3) = zero_zero_nat )
      <=> ! [X4: int] :
            ( aa_set_int_o(member_int(X4),F3)
           => ( aa_int_nat(F,X4) = zero_zero_nat ) ) ) ) ).

% sum_eq_0_iff
tff(fact_6821_sum__eq__0__iff,axiom,
    ! [F3: set_complex,F: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,F3)
     => ( ( groups5693394587270226106ex_nat(F,F3) = zero_zero_nat )
      <=> ! [X4: complex] :
            ( aa_set_complex_o(member_complex(X4),F3)
           => ( aa_complex_nat(F,X4) = zero_zero_nat ) ) ) ) ).

% sum_eq_0_iff
tff(fact_6822_sum__eq__0__iff,axiom,
    ! [F3: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,F3)
     => ( ( aa_set_nat_nat(groups3542108847815614940at_nat(F),F3) = zero_zero_nat )
      <=> ! [X4: nat] :
            ( aa_set_nat_o(member_nat(X4),F3)
           => ( aa_nat_nat(F,X4) = zero_zero_nat ) ) ) ) ).

% sum_eq_0_iff
tff(fact_6823_take__bit__0,axiom,
    ! [Aa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,zero_zero_nat),Aa) = zero_zero_nat ).

% take_bit_0
tff(fact_6824_take__bit__0,axiom,
    ! [Aa: code_integer] : aa_Cod1783974652533396728nteger(aa_nat6383147911832638789nteger(bit_se1745604003318907178nteger,zero_zero_nat),Aa) = zero_z3403309356797280102nteger ).

% take_bit_0
tff(fact_6825_take__bit__0,axiom,
    ! [Aa: int] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,zero_zero_nat),Aa) = zero_zero_int ).

% take_bit_0
tff(fact_6826_take__bit__Suc__1,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_nat_nat(suc,Na)),one_one_nat) = one_one_nat ).

% take_bit_Suc_1
tff(fact_6827_take__bit__Suc__1,axiom,
    ! [Na: nat] : aa_Cod1783974652533396728nteger(aa_nat6383147911832638789nteger(bit_se1745604003318907178nteger,aa_nat_nat(suc,Na)),one_one_Code_integer) = one_one_Code_integer ).

% take_bit_Suc_1
tff(fact_6828_take__bit__Suc__1,axiom,
    ! [Na: nat] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_nat_nat(suc,Na)),one_one_int) = one_one_int ).

% take_bit_Suc_1
tff(fact_6829_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_6830_take__bit__numeral__1,axiom,
    ! [L: num] : aa_Cod1783974652533396728nteger(aa_nat6383147911832638789nteger(bit_se1745604003318907178nteger,aa_num_nat(numeral_numeral_nat,L)),one_one_Code_integer) = one_one_Code_integer ).

% take_bit_numeral_1
tff(fact_6831_take__bit__numeral__1,axiom,
    ! [L: num] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_num_nat(numeral_numeral_nat,L)),one_one_int) = one_one_int ).

% take_bit_numeral_1
tff(fact_6832_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_6833_binomial__1,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),aa_nat_nat(suc,zero_zero_nat)) = Na ).

% binomial_1
tff(fact_6834_binomial__eq__0__iff,axiom,
    ! [Na: nat,K: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K) = zero_zero_nat )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),K) ) ).

% binomial_eq_0_iff
tff(fact_6835_binomial__Suc__Suc,axiom,
    ! [Na: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,Na)),aa_nat_nat(suc,K)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),aa_nat_nat(suc,K))) ).

% binomial_Suc_Suc
tff(fact_6836_pochhammer__0,axiom,
    ! [Aa: complex] : aa_nat_complex(comm_s2602460028002588243omplex(Aa),zero_zero_nat) = one_one_complex ).

% pochhammer_0
tff(fact_6837_pochhammer__0,axiom,
    ! [Aa: real] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Aa),zero_zero_nat) = one_one_real ).

% pochhammer_0
tff(fact_6838_pochhammer__0,axiom,
    ! [Aa: rat] : aa_nat_rat(comm_s4028243227959126397er_rat(Aa),zero_zero_nat) = one_one_rat ).

% pochhammer_0
tff(fact_6839_pochhammer__0,axiom,
    ! [Aa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Aa),zero_zero_nat) = one_one_nat ).

% pochhammer_0
tff(fact_6840_pochhammer__0,axiom,
    ! [Aa: int] : aa_nat_int(comm_s4660882817536571857er_int(Aa),zero_zero_nat) = one_one_int ).

% pochhammer_0
tff(fact_6841_binomial__n__0,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),zero_zero_nat) = one_one_nat ).

% binomial_n_0
tff(fact_6842_sum_Odelta,axiom,
    ! [S2: set_real,Aa: real,Ba: fun_real_real] :
      ( finite_finite_real(S2)
     => ( groups8097168146408367636l_real(aa_fun6946558556880649704l_real(aTP_Lamm_fd(Aa),Ba),S2) = $ite(member_real(Aa,S2),aa_real_real(Ba,Aa),zero_zero_real) ) ) ).

% sum.delta
tff(fact_6843_sum_Odelta,axiom,
    ! [S2: set_int,Aa: int,Ba: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,S2)
     => ( groups8778361861064173332t_real(aa_fun3252798400008884712t_real(aTP_Lamm_fe(Aa),Ba),S2) = $ite(aa_set_int_o(member_int(Aa),S2),aa_int_real(Ba,Aa),zero_zero_real) ) ) ).

% sum.delta
tff(fact_6844_sum_Odelta,axiom,
    ! [S2: set_complex,Aa: complex,Ba: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S2)
     => ( groups5808333547571424918x_real(aa_fun8933909628229591020x_real(aTP_Lamm_ff(Aa),Ba),S2) = $ite(aa_set_complex_o(member_complex(Aa),S2),aa_complex_real(Ba,Aa),zero_zero_real) ) ) ).

% sum.delta
tff(fact_6845_sum_Odelta,axiom,
    ! [S2: set_real,Aa: real,Ba: fun_real_rat] :
      ( finite_finite_real(S2)
     => ( groups1300246762558778688al_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_fg(Aa),Ba),S2) = $ite(member_real(Aa,S2),aa_real_rat(Ba,Aa),zero_zero_rat) ) ) ).

% sum.delta
tff(fact_6846_sum_Odelta,axiom,
    ! [S2: set_nat,Aa: nat,Ba: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_fh(Aa),Ba),S2) = $ite(aa_set_nat_o(member_nat(Aa),S2),aa_nat_rat(Ba,Aa),zero_zero_rat) ) ) ).

% sum.delta
tff(fact_6847_sum_Odelta,axiom,
    ! [S2: set_int,Aa: int,Ba: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,S2)
     => ( groups3906332499630173760nt_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_fi(Aa),Ba),S2) = $ite(aa_set_int_o(member_int(Aa),S2),aa_int_rat(Ba,Aa),zero_zero_rat) ) ) ).

% sum.delta
tff(fact_6848_sum_Odelta,axiom,
    ! [S2: set_complex,Aa: complex,Ba: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S2)
     => ( groups5058264527183730370ex_rat(aa_fun7622890401989837636ex_rat(aTP_Lamm_fj(Aa),Ba),S2) = $ite(aa_set_complex_o(member_complex(Aa),S2),aa_complex_rat(Ba,Aa),zero_zero_rat) ) ) ).

% sum.delta
tff(fact_6849_sum_Odelta,axiom,
    ! [S2: set_real,Aa: real,Ba: fun_real_nat] :
      ( finite_finite_real(S2)
     => ( groups1935376822645274424al_nat(aa_fun211602392931233840al_nat(aTP_Lamm_fk(Aa),Ba),S2) = $ite(member_real(Aa,S2),aa_real_nat(Ba,Aa),zero_zero_nat) ) ) ).

% sum.delta
tff(fact_6850_sum_Odelta,axiom,
    ! [S2: set_int,Aa: int,Ba: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,S2)
     => ( groups4541462559716669496nt_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_fl(Aa),Ba),S2) = $ite(aa_set_int_o(member_int(Aa),S2),aa_int_nat(Ba,Aa),zero_zero_nat) ) ) ).

% sum.delta
tff(fact_6851_sum_Odelta,axiom,
    ! [S2: set_complex,Aa: complex,Ba: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S2)
     => ( groups5693394587270226106ex_nat(aa_fun2465681967280144180ex_nat(aTP_Lamm_fm(Aa),Ba),S2) = $ite(aa_set_complex_o(member_complex(Aa),S2),aa_complex_nat(Ba,Aa),zero_zero_nat) ) ) ).

% sum.delta
tff(fact_6852_sum_Odelta_H,axiom,
    ! [S2: set_real,Aa: real,Ba: fun_real_real] :
      ( finite_finite_real(S2)
     => ( groups8097168146408367636l_real(aa_fun6946558556880649704l_real(aTP_Lamm_fn(Aa),Ba),S2) = $ite(member_real(Aa,S2),aa_real_real(Ba,Aa),zero_zero_real) ) ) ).

% sum.delta'
tff(fact_6853_sum_Odelta_H,axiom,
    ! [S2: set_int,Aa: int,Ba: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,S2)
     => ( groups8778361861064173332t_real(aa_fun3252798400008884712t_real(aTP_Lamm_fo(Aa),Ba),S2) = $ite(aa_set_int_o(member_int(Aa),S2),aa_int_real(Ba,Aa),zero_zero_real) ) ) ).

% sum.delta'
tff(fact_6854_sum_Odelta_H,axiom,
    ! [S2: set_complex,Aa: complex,Ba: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S2)
     => ( groups5808333547571424918x_real(aa_fun8933909628229591020x_real(aTP_Lamm_fp(Aa),Ba),S2) = $ite(aa_set_complex_o(member_complex(Aa),S2),aa_complex_real(Ba,Aa),zero_zero_real) ) ) ).

% sum.delta'
tff(fact_6855_sum_Odelta_H,axiom,
    ! [S2: set_real,Aa: real,Ba: fun_real_rat] :
      ( finite_finite_real(S2)
     => ( groups1300246762558778688al_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_fq(Aa),Ba),S2) = $ite(member_real(Aa,S2),aa_real_rat(Ba,Aa),zero_zero_rat) ) ) ).

% sum.delta'
tff(fact_6856_sum_Odelta_H,axiom,
    ! [S2: set_nat,Aa: nat,Ba: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_fr(Aa),Ba),S2) = $ite(aa_set_nat_o(member_nat(Aa),S2),aa_nat_rat(Ba,Aa),zero_zero_rat) ) ) ).

% sum.delta'
tff(fact_6857_sum_Odelta_H,axiom,
    ! [S2: set_int,Aa: int,Ba: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,S2)
     => ( groups3906332499630173760nt_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_fs(Aa),Ba),S2) = $ite(aa_set_int_o(member_int(Aa),S2),aa_int_rat(Ba,Aa),zero_zero_rat) ) ) ).

% sum.delta'
tff(fact_6858_sum_Odelta_H,axiom,
    ! [S2: set_complex,Aa: complex,Ba: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S2)
     => ( groups5058264527183730370ex_rat(aa_fun7622890401989837636ex_rat(aTP_Lamm_ft(Aa),Ba),S2) = $ite(aa_set_complex_o(member_complex(Aa),S2),aa_complex_rat(Ba,Aa),zero_zero_rat) ) ) ).

% sum.delta'
tff(fact_6859_sum_Odelta_H,axiom,
    ! [S2: set_real,Aa: real,Ba: fun_real_nat] :
      ( finite_finite_real(S2)
     => ( groups1935376822645274424al_nat(aa_fun211602392931233840al_nat(aTP_Lamm_fu(Aa),Ba),S2) = $ite(member_real(Aa,S2),aa_real_nat(Ba,Aa),zero_zero_nat) ) ) ).

% sum.delta'
tff(fact_6860_sum_Odelta_H,axiom,
    ! [S2: set_int,Aa: int,Ba: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,S2)
     => ( groups4541462559716669496nt_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_fv(Aa),Ba),S2) = $ite(aa_set_int_o(member_int(Aa),S2),aa_int_nat(Ba,Aa),zero_zero_nat) ) ) ).

% sum.delta'
tff(fact_6861_sum_Odelta_H,axiom,
    ! [S2: set_complex,Aa: complex,Ba: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S2)
     => ( groups5693394587270226106ex_nat(aa_fun2465681967280144180ex_nat(aTP_Lamm_fw(Aa),Ba),S2) = $ite(aa_set_complex_o(member_complex(Aa),S2),aa_complex_nat(Ba,Aa),zero_zero_nat) ) ) ).

% sum.delta'
tff(fact_6862_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_fx(F),A3)) ).

% sum_abs
tff(fact_6863_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_fy(F)),A3)) ).

% sum_abs
tff(fact_6864_take__bit__of__1__eq__0__iff,axiom,
    ! [Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),one_one_nat) = zero_zero_nat )
    <=> ( Na = zero_zero_nat ) ) ).

% take_bit_of_1_eq_0_iff
tff(fact_6865_take__bit__of__1__eq__0__iff,axiom,
    ! [Na: nat] :
      ( ( aa_Cod1783974652533396728nteger(aa_nat6383147911832638789nteger(bit_se1745604003318907178nteger,Na),one_one_Code_integer) = zero_z3403309356797280102nteger )
    <=> ( Na = zero_zero_nat ) ) ).

% take_bit_of_1_eq_0_iff
tff(fact_6866_take__bit__of__1__eq__0__iff,axiom,
    ! [Na: nat] :
      ( ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Na),one_one_int) = zero_zero_int )
    <=> ( Na = zero_zero_nat ) ) ).

% take_bit_of_1_eq_0_iff
tff(fact_6867_zero__less__binomial__iff,axiom,
    ! [Na: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na) ) ).

% zero_less_binomial_iff
tff(fact_6868_take__bit__of__Suc__0,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),aa_nat_nat(suc,zero_zero_nat)) = zero_n2687167440665602831ol_nat(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)) ).

% take_bit_of_Suc_0
tff(fact_6869_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_fx(F),A3)) ).

% sum_abs_ge_zero
tff(fact_6870_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_fy(F)),A3)) ).

% sum_abs_ge_zero
tff(fact_6871_take__bit__of__1,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),one_one_nat) = zero_n2687167440665602831ol_nat(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)) ).

% take_bit_of_1
tff(fact_6872_take__bit__of__1,axiom,
    ! [Na: nat] : aa_Cod1783974652533396728nteger(aa_nat6383147911832638789nteger(bit_se1745604003318907178nteger,Na),one_one_Code_integer) = zero_n356916108424825756nteger(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)) ).

% take_bit_of_1
tff(fact_6873_take__bit__of__1,axiom,
    ! [Na: nat] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Na),one_one_int) = aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)) ).

% take_bit_of_1
tff(fact_6874_even__take__bit__eq,axiom,
    ! [Na: nat,Aa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),Aa))
    <=> ( ( Na = zero_zero_nat )
        | aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Aa) ) ) ).

% even_take_bit_eq
tff(fact_6875_even__take__bit__eq,axiom,
    ! [Na: nat,Aa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_Cod1783974652533396728nteger(aa_nat6383147911832638789nteger(bit_se1745604003318907178nteger,Na),Aa))
    <=> ( ( Na = zero_zero_nat )
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Aa) ) ) ).

% even_take_bit_eq
tff(fact_6876_even__take__bit__eq,axiom,
    ! [Na: nat,Aa: 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_se2923211474154528505it_int,Na),Aa))
    <=> ( ( Na = zero_zero_nat )
        | aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Aa) ) ) ).

% even_take_bit_eq
tff(fact_6877_take__bit__Suc__0,axiom,
    ! [Aa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_nat_nat(suc,zero_zero_nat)),Aa) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% take_bit_Suc_0
tff(fact_6878_take__bit__Suc__0,axiom,
    ! [Aa: code_integer] : aa_Cod1783974652533396728nteger(aa_nat6383147911832638789nteger(bit_se1745604003318907178nteger,aa_nat_nat(suc,zero_zero_nat)),Aa) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Aa),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) ).

% take_bit_Suc_0
tff(fact_6879_take__bit__Suc__0,axiom,
    ! [Aa: int] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_nat_nat(suc,zero_zero_nat)),Aa) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% take_bit_Suc_0
tff(fact_6880_take__bit__of__exp,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,zero_n2687167440665602831ol_nat(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) ).

% take_bit_of_exp
tff(fact_6881_take__bit__of__exp,axiom,
    ! [M: nat,Na: nat] : aa_Cod1783974652533396728nteger(aa_nat6383147911832638789nteger(bit_se1745604003318907178nteger,M),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Na)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,zero_n356916108424825756nteger(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M))),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Na)) ).

% take_bit_of_exp
tff(fact_6882_take__bit__of__exp,axiom,
    ! [M: nat,Na: nat] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,M),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)) ).

% take_bit_of_exp
tff(fact_6883_take__bit__of__2,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,zero_n2687167440665602831ol_nat(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% take_bit_of_2
tff(fact_6884_take__bit__of__2,axiom,
    ! [Na: nat] : aa_Cod1783974652533396728nteger(aa_nat6383147911832638789nteger(bit_se1745604003318907178nteger,Na),aa_num_Code_integer(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))),Na))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) ).

% take_bit_of_2
tff(fact_6885_take__bit__of__2,axiom,
    ! [Na: nat] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Na),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% take_bit_of_2
tff(fact_6886_pochhammer__of__int,axiom,
    ! [Xb: int,Na: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_int_real(ring_1_of_int_real,Xb)),Na) = aa_int_real(ring_1_of_int_real,aa_nat_int(comm_s4660882817536571857er_int(Xb),Na)) ).

% pochhammer_of_int
tff(fact_6887_pochhammer__of__int,axiom,
    ! [Xb: int,Na: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_int_Code_integer(ring_18347121197199848620nteger,Xb)),Na) = aa_int_Code_integer(ring_18347121197199848620nteger,aa_nat_int(comm_s4660882817536571857er_int(Xb),Na)) ).

% pochhammer_of_int
tff(fact_6888_pochhammer__of__int,axiom,
    ! [Xb: int,Na: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(aa_int_rat(ring_1_of_int_rat,Xb)),Na) = aa_int_rat(ring_1_of_int_rat,aa_nat_int(comm_s4660882817536571857er_int(Xb),Na)) ).

% pochhammer_of_int
tff(fact_6889_euclidean__size__integer_Oabs__eq,axiom,
    ! [Xb: int] : aa_Code_integer_nat(euclid6377331345833325938nteger,aa_int_Code_integer(code_integer_of_int,Xb)) = aa_int_nat(euclid4774559944035922753ze_int,Xb) ).

% euclidean_size_integer.abs_eq
tff(fact_6890_take__bit__integer_Oabs__eq,axiom,
    ! [Xaa: nat,Xb: int] : aa_Cod1783974652533396728nteger(aa_nat6383147911832638789nteger(bit_se1745604003318907178nteger,Xaa),aa_int_Code_integer(code_integer_of_int,Xb)) = aa_int_Code_integer(code_integer_of_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Xaa),Xb)) ).

% take_bit_integer.abs_eq
tff(fact_6891_uminus__integer__code_I1_J,axiom,
    aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% uminus_integer_code(1)
tff(fact_6892_less__integer__code_I1_J,axiom,
    ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),zero_z3403309356797280102nteger) ).

% less_integer_code(1)
tff(fact_6893_abs__integer__code,axiom,
    ! [K: code_integer] :
      aa_Cod1783974652533396728nteger(abs_abs_Code_integer,K) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,K),zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,K),K) ).

% abs_integer_code
tff(fact_6894_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_real_real,A3: set_real] :
      ( ( groups8097168146408367636l_real(G,A3) != zero_zero_real )
     => ~ ! [A2: real] :
            ( member_real(A2,A3)
           => ( aa_real_real(G,A2) = zero_zero_real ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_6895_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_int_real,A3: set_int] :
      ( ( groups8778361861064173332t_real(G,A3) != zero_zero_real )
     => ~ ! [A2: int] :
            ( aa_set_int_o(member_int(A2),A3)
           => ( aa_int_real(G,A2) = zero_zero_real ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_6896_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_real_rat,A3: set_real] :
      ( ( groups1300246762558778688al_rat(G,A3) != zero_zero_rat )
     => ~ ! [A2: real] :
            ( member_real(A2,A3)
           => ( aa_real_rat(G,A2) = zero_zero_rat ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_6897_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_nat_rat,A3: set_nat] :
      ( ( groups2906978787729119204at_rat(G,A3) != zero_zero_rat )
     => ~ ! [A2: nat] :
            ( aa_set_nat_o(member_nat(A2),A3)
           => ( aa_nat_rat(G,A2) = zero_zero_rat ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_6898_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_int_rat,A3: set_int] :
      ( ( groups3906332499630173760nt_rat(G,A3) != zero_zero_rat )
     => ~ ! [A2: int] :
            ( aa_set_int_o(member_int(A2),A3)
           => ( aa_int_rat(G,A2) = zero_zero_rat ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_6899_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_real_nat,A3: set_real] :
      ( ( groups1935376822645274424al_nat(G,A3) != zero_zero_nat )
     => ~ ! [A2: real] :
            ( member_real(A2,A3)
           => ( aa_real_nat(G,A2) = zero_zero_nat ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_6900_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_int_nat,A3: set_int] :
      ( ( groups4541462559716669496nt_nat(G,A3) != zero_zero_nat )
     => ~ ! [A2: int] :
            ( aa_set_int_o(member_int(A2),A3)
           => ( aa_int_nat(G,A2) = zero_zero_nat ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_6901_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_real_int,A3: set_real] :
      ( ( groups1932886352136224148al_int(G,A3) != zero_zero_int )
     => ~ ! [A2: real] :
            ( member_real(A2,A3)
           => ( aa_real_int(G,A2) = zero_zero_int ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_6902_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_nat_int,A3: set_nat] :
      ( ( groups3539618377306564664at_int(G,A3) != zero_zero_int )
     => ~ ! [A2: nat] :
            ( aa_set_nat_o(member_nat(A2),A3)
           => ( aa_nat_int(G,A2) = zero_zero_int ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_6903_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_int_int,A3: set_int] :
      ( ( groups4538972089207619220nt_int(G,A3) != zero_zero_int )
     => ~ ! [A2: int] :
            ( aa_set_int_o(member_int(A2),A3)
           => ( aa_int_int(G,A2) = zero_zero_int ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_6904_sum_Oneutral,axiom,
    ! [A3: set_int,G: fun_int_int] :
      ( ! [X: int] :
          ( aa_set_int_o(member_int(X),A3)
         => ( aa_int_int(G,X) = zero_zero_int ) )
     => ( groups4538972089207619220nt_int(G,A3) = zero_zero_int ) ) ).

% sum.neutral
tff(fact_6905_sum_Oneutral,axiom,
    ! [A3: set_complex,G: fun_complex_complex] :
      ( ! [X: complex] :
          ( aa_set_complex_o(member_complex(X),A3)
         => ( aa_complex_complex(G,X) = zero_zero_complex ) )
     => ( groups7754918857620584856omplex(G,A3) = zero_zero_complex ) ) ).

% sum.neutral
tff(fact_6906_sum_Oneutral,axiom,
    ! [A3: set_nat,G: fun_nat_nat] :
      ( ! [X: nat] :
          ( aa_set_nat_o(member_nat(X),A3)
         => ( aa_nat_nat(G,X) = zero_zero_nat ) )
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),A3) = zero_zero_nat ) ) ).

% sum.neutral
tff(fact_6907_sum_Oneutral,axiom,
    ! [A3: set_nat,G: fun_nat_real] :
      ( ! [X: nat] :
          ( aa_set_nat_o(member_nat(X),A3)
         => ( aa_nat_real(G,X) = zero_zero_real ) )
     => ( aa_set_nat_real(groups6591440286371151544t_real(G),A3) = zero_zero_real ) ) ).

% sum.neutral
tff(fact_6908_take__bit__tightened,axiom,
    ! [Na: nat,Aa: nat,Ba: nat,M: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),Aa) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),Ba) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,M),Aa) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,M),Ba) ) ) ) ).

% take_bit_tightened
tff(fact_6909_take__bit__tightened,axiom,
    ! [Na: nat,Aa: code_integer,Ba: code_integer,M: nat] :
      ( ( aa_Cod1783974652533396728nteger(aa_nat6383147911832638789nteger(bit_se1745604003318907178nteger,Na),Aa) = aa_Cod1783974652533396728nteger(aa_nat6383147911832638789nteger(bit_se1745604003318907178nteger,Na),Ba) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
       => ( aa_Cod1783974652533396728nteger(aa_nat6383147911832638789nteger(bit_se1745604003318907178nteger,M),Aa) = aa_Cod1783974652533396728nteger(aa_nat6383147911832638789nteger(bit_se1745604003318907178nteger,M),Ba) ) ) ) ).

% take_bit_tightened
tff(fact_6910_take__bit__tightened,axiom,
    ! [Na: nat,Aa: int,Ba: int,M: nat] :
      ( ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Na),Aa) = aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Na),Ba) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
       => ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,M),Aa) = aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,M),Ba) ) ) ) ).

% take_bit_tightened
tff(fact_6911_take__bit__tightened__less__eq__nat,axiom,
    ! [M: nat,Na: nat,Q4: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,M),Q4)),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),Q4)) ) ).

% take_bit_tightened_less_eq_nat
tff(fact_6912_take__bit__nat__less__eq__self,axiom,
    ! [Na: nat,M: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),M)),M) ).

% take_bit_nat_less_eq_self
tff(fact_6913_zero__integer__def,axiom,
    zero_z3403309356797280102nteger = aa_int_Code_integer(code_integer_of_int,zero_zero_int) ).

% zero_integer_def
tff(fact_6914_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_6915_choose__one,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),one_one_nat) = Na ).

% choose_one
tff(fact_6916_uminus__integer_Oabs__eq,axiom,
    ! [Xb: int] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_int_Code_integer(code_integer_of_int,Xb)) = aa_int_Code_integer(code_integer_of_int,aa_int_int(uminus_uminus_int,Xb)) ).

% uminus_integer.abs_eq
tff(fact_6917_divide__integer_Oabs__eq,axiom,
    ! [Xaa: int,Xb: int] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(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_6918_abs__integer_Oabs__eq,axiom,
    ! [Xb: int] : aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_int_Code_integer(code_integer_of_int,Xb)) = aa_int_Code_integer(code_integer_of_int,aa_int_int(abs_abs_int,Xb)) ).

% abs_integer.abs_eq
tff(fact_6919_modulo__integer_Oabs__eq,axiom,
    ! [Xaa: int,Xb: int] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,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(modulo_modulo_int,Xaa),Xb)) ).

% modulo_integer.abs_eq
tff(fact_6920_sum__mono,axiom,
    ! [K4: set_real,F: fun_real_rat,G: fun_real_rat] :
      ( ! [I2: real] :
          ( member_real(I2,K4)
         => 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,K4)),groups1300246762558778688al_rat(G,K4)) ) ).

% sum_mono
tff(fact_6921_sum__mono,axiom,
    ! [K4: set_nat,F: fun_nat_rat,G: fun_nat_rat] :
      ( ! [I2: nat] :
          ( aa_set_nat_o(member_nat(I2),K4)
         => 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,K4)),groups2906978787729119204at_rat(G,K4)) ) ).

% sum_mono
tff(fact_6922_sum__mono,axiom,
    ! [K4: set_int,F: fun_int_rat,G: fun_int_rat] :
      ( ! [I2: int] :
          ( aa_set_int_o(member_int(I2),K4)
         => 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,K4)),groups3906332499630173760nt_rat(G,K4)) ) ).

% sum_mono
tff(fact_6923_sum__mono,axiom,
    ! [K4: set_real,F: fun_real_nat,G: fun_real_nat] :
      ( ! [I2: real] :
          ( member_real(I2,K4)
         => 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,K4)),groups1935376822645274424al_nat(G,K4)) ) ).

% sum_mono
tff(fact_6924_sum__mono,axiom,
    ! [K4: set_int,F: fun_int_nat,G: fun_int_nat] :
      ( ! [I2: int] :
          ( aa_set_int_o(member_int(I2),K4)
         => 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,K4)),groups4541462559716669496nt_nat(G,K4)) ) ).

% sum_mono
tff(fact_6925_sum__mono,axiom,
    ! [K4: set_real,F: fun_real_int,G: fun_real_int] :
      ( ! [I2: real] :
          ( member_real(I2,K4)
         => 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,K4)),groups1932886352136224148al_int(G,K4)) ) ).

% sum_mono
tff(fact_6926_sum__mono,axiom,
    ! [K4: set_nat,F: fun_nat_int,G: fun_nat_int] :
      ( ! [I2: nat] :
          ( aa_set_nat_o(member_nat(I2),K4)
         => 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,K4)),groups3539618377306564664at_int(G,K4)) ) ).

% sum_mono
tff(fact_6927_sum__mono,axiom,
    ! [K4: set_int,F: fun_int_int,G: fun_int_int] :
      ( ! [I2: int] :
          ( aa_set_int_o(member_int(I2),K4)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(F,I2)),aa_int_int(G,I2)) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups4538972089207619220nt_int(F,K4)),groups4538972089207619220nt_int(G,K4)) ) ).

% sum_mono
tff(fact_6928_sum__mono,axiom,
    ! [K4: set_nat,F: fun_nat_nat,G: fun_nat_nat] :
      ( ! [I2: nat] :
          ( aa_set_nat_o(member_nat(I2),K4)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,I2)),aa_nat_nat(G,I2)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),K4)),aa_set_nat_nat(groups3542108847815614940at_nat(G),K4)) ) ).

% sum_mono
tff(fact_6929_sum__mono,axiom,
    ! [K4: set_nat,F: fun_nat_real,G: fun_nat_real] :
      ( ! [I2: nat] :
          ( aa_set_nat_o(member_nat(I2),K4)
         => 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(groups6591440286371151544t_real(F),K4)),aa_set_nat_real(groups6591440286371151544t_real(G),K4)) ) ).

% sum_mono
tff(fact_6930_unset__bit__integer_Oabs__eq,axiom,
    ! [Xaa: nat,Xb: int] : aa_Cod1783974652533396728nteger(aa_nat6383147911832638789nteger(bit_se8260200283734997820nteger,Xaa),aa_int_Code_integer(code_integer_of_int,Xb)) = aa_int_Code_integer(code_integer_of_int,aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,Xaa),Xb)) ).

% unset_bit_integer.abs_eq
tff(fact_6931_sum_Oswap__restrict,axiom,
    ! [A3: set_real,B4: set_int,G: fun_real_fun_int_int,R: fun_real_fun_int_o] :
      ( finite_finite_real(A3)
     => ( aa_set_int_o(finite_finite_int,B4)
       => ( groups1932886352136224148al_int(aa_fun7555737461897725707al_int(aa_fun1679075639499870878al_int(aTP_Lamm_ga(B4),G),R),A3) = groups4538972089207619220nt_int(aa_fun3996843141751676939nt_int(aa_fun129049606478405534nt_int(aTP_Lamm_gc(A3),G),R),B4) ) ) ) ).

% sum.swap_restrict
tff(fact_6932_sum_Oswap__restrict,axiom,
    ! [A3: set_nat,B4: set_int,G: fun_nat_fun_int_int,R: fun_nat_fun_int_o] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_int_o(finite_finite_int,B4)
       => ( groups3539618377306564664at_int(aa_fun4489122510365962891at_int(aa_fun1158358564283266810at_int(aTP_Lamm_ge(B4),G),R),A3) = groups4538972089207619220nt_int(aa_fun5488476222267017447nt_int(aa_fun3345280138976288598nt_int(aTP_Lamm_gg(A3),G),R),B4) ) ) ) ).

% sum.swap_restrict
tff(fact_6933_sum_Oswap__restrict,axiom,
    ! [A3: set_complex,B4: set_int,G: fun_co333780937270390912nt_int,R: fun_co3431063785546946928_int_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A3)
     => ( aa_set_int_o(finite_finite_int,B4)
       => ( groups5690904116761175830ex_int(aa_fun5055871773053899019ex_int(aa_fun7361901416012365340ex_int(aTP_Lamm_gi(B4),G),R),A3) = groups4538972089207619220nt_int(aa_fun2158622962795119753nt_int(aa_fun607866698265288346nt_int(aTP_Lamm_gl(A3),G),R),B4) ) ) ) ).

% sum.swap_restrict
tff(fact_6934_sum_Oswap__restrict,axiom,
    ! [A3: set_real,B4: set_complex,G: fun_re7208826350674946562omplex,R: fun_re7319081385389557872plex_o] :
      ( finite_finite_real(A3)
     => ( aa_set_complex_o(finite3207457112153483333omplex,B4)
       => ( groups5754745047067104278omplex(aa_fun401690246322682763omplex(aa_fun7501666206714588442omplex(aTP_Lamm_gn(B4),G),R),A3) = groups7754918857620584856omplex(aa_fun9084337531551611917omplex(aa_fun6739043078600750748omplex(aTP_Lamm_gp(A3),G),R),B4) ) ) ) ).

% sum.swap_restrict
tff(fact_6935_sum_Oswap__restrict,axiom,
    ! [A3: set_nat,B4: set_complex,G: fun_na1591161823130732966omplex,R: fun_na1554462459066888140plex_o] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_complex_o(finite3207457112153483333omplex,B4)
       => ( aa_set_nat_complex(groups2073611262835488442omplex(aa_fun4713049822795992587omplex(aa_fun3479084698764006518omplex(aTP_Lamm_gr(B4),G),R)),A3) = groups7754918857620584856omplex(aa_fun1423577888295648489omplex(aa_fun601428455815558228omplex(aTP_Lamm_gt(A3),G),R),B4) ) ) ) ).

% sum.swap_restrict
tff(fact_6936_sum_Oswap__restrict,axiom,
    ! [A3: set_int,B4: set_complex,G: fun_in1307110748760623106omplex,R: fun_in5065666251602127472plex_o] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( aa_set_complex_o(finite3207457112153483333omplex,B4)
       => ( groups3049146728041665814omplex(aa_fun3141780957025892107omplex(aa_fun323909820831547930omplex(aTP_Lamm_gv(B4),G),R),A3) = groups7754918857620584856omplex(aa_fun7450476157521870733omplex(aa_fun6156949752960786844omplex(aTP_Lamm_gx(A3),G),R),B4) ) ) ) ).

% sum.swap_restrict
tff(fact_6937_sum_Oswap__restrict,axiom,
    ! [A3: set_real,B4: set_nat,G: fun_real_fun_nat_nat,R: fun_real_fun_nat_o] :
      ( finite_finite_real(A3)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( groups1935376822645274424al_nat(aa_fun2272639302901571595al_nat(aa_fun7039802393428226646al_nat(aTP_Lamm_gz(B4),G),R),A3) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun7366046364592083119at_nat(aa_fun7718511627822915834at_nat(aTP_Lamm_hb(A3),G),R)),B4) ) ) ) ).

% sum.swap_restrict
tff(fact_6938_sum_Oswap__restrict,axiom,
    ! [A3: set_int,B4: set_nat,G: fun_int_fun_nat_nat,R: fun_int_fun_nat_o] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( groups4541462559716669496nt_nat(aa_fun4650797359919590027nt_nat(aa_fun8303507722728657494nt_nat(aTP_Lamm_hd(B4),G),R),A3) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun3651443648018535471at_nat(aa_fun6116586148035635706at_nat(aTP_Lamm_hf(A3),G),R)),B4) ) ) ) ).

% sum.swap_restrict
tff(fact_6939_sum_Oswap__restrict,axiom,
    ! [A3: set_complex,B4: set_nat,G: fun_co636942507509095880at_nat,R: fun_co5676208753217392844_nat_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A3)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( groups5693394587270226106ex_nat(aa_fun2265263913897893899ex_nat(aa_fun1127489917733235668ex_nat(aTP_Lamm_hh(B4),G),R),A3) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5527826185635525933at_nat(aa_fun4894236392271416310at_nat(aTP_Lamm_hj(A3),G),R)),B4) ) ) ) ).

% sum.swap_restrict
tff(fact_6940_sum_Oswap__restrict,axiom,
    ! [A3: set_real,B4: set_nat,G: fun_re4238438383274954274t_real,R: fun_real_fun_nat_o] :
      ( finite_finite_real(A3)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( groups8097168146408367636l_real(aa_fun6320755270648597735l_real(aa_fun6240130701872165846l_real(aTP_Lamm_hk(B4),G),R),A3) = aa_set_nat_real(groups6591440286371151544t_real(aa_fun6247686994378451851t_real(aa_fun7274880274332377722t_real(aTP_Lamm_hm(A3),G),R)),B4) ) ) ) ).

% sum.swap_restrict
tff(fact_6941_set__bit__integer_Oabs__eq,axiom,
    ! [Xaa: nat,Xb: int] : aa_Cod1783974652533396728nteger(aa_nat6383147911832638789nteger(bit_se2793503036327961859nteger,Xaa),aa_int_Code_integer(code_integer_of_int,Xb)) = aa_int_Code_integer(code_integer_of_int,aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,Xaa),Xb)) ).

% set_bit_integer.abs_eq
tff(fact_6942_mod__sum__eq,axiom,
    ! [F: fun_int_int,Aa: int,A3: set_int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,groups4538972089207619220nt_int(aa_int_fun_int_int(aTP_Lamm_hn(F),Aa),A3)),Aa) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,groups4538972089207619220nt_int(F,A3)),Aa) ).

% mod_sum_eq
tff(fact_6943_mod__sum__eq,axiom,
    ! [F: fun_nat_nat,Aa: 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_ho(F),Aa)),A3)),Aa) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),A3)),Aa) ).

% mod_sum_eq
tff(fact_6944_flip__bit__integer_Oabs__eq,axiom,
    ! [Xaa: nat,Xb: int] : aa_Cod1783974652533396728nteger(aa_nat6383147911832638789nteger(bit_se1345352211410354436nteger,Xaa),aa_int_Code_integer(code_integer_of_int,Xb)) = aa_int_Code_integer(code_integer_of_int,aa_int_int(aa_nat_fun_int_int(bit_se2159334234014336723it_int,Xaa),Xb)) ).

% flip_bit_integer.abs_eq
tff(fact_6945_sum__nonpos,axiom,
    ! [A3: set_real,F: fun_real_real] :
      ( ! [X: real] :
          ( member_real(X,A3)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,X)),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_6946_sum__nonpos,axiom,
    ! [A3: set_int,F: fun_int_real] :
      ( ! [X: int] :
          ( aa_set_int_o(member_int(X),A3)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(F,X)),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_6947_sum__nonpos,axiom,
    ! [A3: set_real,F: fun_real_rat] :
      ( ! [X: real] :
          ( member_real(X,A3)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_real_rat(F,X)),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_6948_sum__nonpos,axiom,
    ! [A3: set_nat,F: fun_nat_rat] :
      ( ! [X: nat] :
          ( aa_set_nat_o(member_nat(X),A3)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X)),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_6949_sum__nonpos,axiom,
    ! [A3: set_int,F: fun_int_rat] :
      ( ! [X: int] :
          ( aa_set_int_o(member_int(X),A3)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,X)),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_6950_sum__nonpos,axiom,
    ! [A3: set_real,F: fun_real_nat] :
      ( ! [X: real] :
          ( member_real(X,A3)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_real_nat(F,X)),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_6951_sum__nonpos,axiom,
    ! [A3: set_int,F: fun_int_nat] :
      ( ! [X: int] :
          ( aa_set_int_o(member_int(X),A3)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,X)),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_6952_sum__nonpos,axiom,
    ! [A3: set_real,F: fun_real_int] :
      ( ! [X: real] :
          ( member_real(X,A3)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(F,X)),zero_zero_int) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups1932886352136224148al_int(F,A3)),zero_zero_int) ) ).

% sum_nonpos
tff(fact_6953_sum__nonpos,axiom,
    ! [A3: set_nat,F: fun_nat_int] :
      ( ! [X: nat] :
          ( aa_set_nat_o(member_nat(X),A3)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,X)),zero_zero_int) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups3539618377306564664at_int(F,A3)),zero_zero_int) ) ).

% sum_nonpos
tff(fact_6954_sum__nonpos,axiom,
    ! [A3: set_int,F: fun_int_int] :
      ( ! [X: int] :
          ( aa_set_int_o(member_int(X),A3)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(F,X)),zero_zero_int) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups4538972089207619220nt_int(F,A3)),zero_zero_int) ) ).

% sum_nonpos
tff(fact_6955_sum__nonneg,axiom,
    ! [A3: set_real,F: fun_real_real] :
      ( ! [X: real] :
          ( member_real(X,A3)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,X)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),groups8097168146408367636l_real(F,A3)) ) ).

% sum_nonneg
tff(fact_6956_sum__nonneg,axiom,
    ! [A3: set_int,F: fun_int_real] :
      ( ! [X: int] :
          ( aa_set_int_o(member_int(X),A3)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(F,X)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),groups8778361861064173332t_real(F,A3)) ) ).

% sum_nonneg
tff(fact_6957_sum__nonneg,axiom,
    ! [A3: set_real,F: fun_real_rat] :
      ( ! [X: real] :
          ( member_real(X,A3)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,X)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),groups1300246762558778688al_rat(F,A3)) ) ).

% sum_nonneg
tff(fact_6958_sum__nonneg,axiom,
    ! [A3: set_nat,F: fun_nat_rat] :
      ( ! [X: nat] :
          ( aa_set_nat_o(member_nat(X),A3)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,X)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),groups2906978787729119204at_rat(F,A3)) ) ).

% sum_nonneg
tff(fact_6959_sum__nonneg,axiom,
    ! [A3: set_int,F: fun_int_rat] :
      ( ! [X: int] :
          ( aa_set_int_o(member_int(X),A3)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,X)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),groups3906332499630173760nt_rat(F,A3)) ) ).

% sum_nonneg
tff(fact_6960_sum__nonneg,axiom,
    ! [A3: set_real,F: fun_real_nat] :
      ( ! [X: real] :
          ( member_real(X,A3)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,X)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),groups1935376822645274424al_nat(F,A3)) ) ).

% sum_nonneg
tff(fact_6961_sum__nonneg,axiom,
    ! [A3: set_int,F: fun_int_nat] :
      ( ! [X: int] :
          ( aa_set_int_o(member_int(X),A3)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F,X)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),groups4541462559716669496nt_nat(F,A3)) ) ).

% sum_nonneg
tff(fact_6962_sum__nonneg,axiom,
    ! [A3: set_real,F: fun_real_int] :
      ( ! [X: real] :
          ( member_real(X,A3)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_real_int(F,X)) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),groups1932886352136224148al_int(F,A3)) ) ).

% sum_nonneg
tff(fact_6963_sum__nonneg,axiom,
    ! [A3: set_nat,F: fun_nat_int] :
      ( ! [X: nat] :
          ( aa_set_nat_o(member_nat(X),A3)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(F,X)) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),groups3539618377306564664at_int(F,A3)) ) ).

% sum_nonneg
tff(fact_6964_sum__nonneg,axiom,
    ! [A3: set_int,F: fun_int_int] :
      ( ! [X: int] :
          ( aa_set_int_o(member_int(X),A3)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(F,X)) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),groups4538972089207619220nt_int(F,A3)) ) ).

% sum_nonneg
tff(fact_6965_sum__mono__inv,axiom,
    ! [F: fun_real_rat,I5: set_real,G: fun_real_rat,I: real] :
      ( ( groups1300246762558778688al_rat(F,I5) = groups1300246762558778688al_rat(G,I5) )
     => ( ! [I2: real] :
            ( member_real(I2,I5)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_real_rat(F,I2)),aa_real_rat(G,I2)) )
       => ( member_real(I,I5)
         => ( finite_finite_real(I5)
           => ( aa_real_rat(F,I) = aa_real_rat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_6966_sum__mono__inv,axiom,
    ! [F: fun_nat_rat,I5: set_nat,G: fun_nat_rat,I: nat] :
      ( ( groups2906978787729119204at_rat(F,I5) = groups2906978787729119204at_rat(G,I5) )
     => ( ! [I2: nat] :
            ( aa_set_nat_o(member_nat(I2),I5)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,I2)),aa_nat_rat(G,I2)) )
       => ( aa_set_nat_o(member_nat(I),I5)
         => ( aa_set_nat_o(finite_finite_nat,I5)
           => ( aa_nat_rat(F,I) = aa_nat_rat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_6967_sum__mono__inv,axiom,
    ! [F: fun_int_rat,I5: set_int,G: fun_int_rat,I: int] :
      ( ( groups3906332499630173760nt_rat(F,I5) = groups3906332499630173760nt_rat(G,I5) )
     => ( ! [I2: int] :
            ( aa_set_int_o(member_int(I2),I5)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,I2)),aa_int_rat(G,I2)) )
       => ( aa_set_int_o(member_int(I),I5)
         => ( aa_set_int_o(finite_finite_int,I5)
           => ( aa_int_rat(F,I) = aa_int_rat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_6968_sum__mono__inv,axiom,
    ! [F: fun_complex_rat,I5: set_complex,G: fun_complex_rat,I: complex] :
      ( ( groups5058264527183730370ex_rat(F,I5) = groups5058264527183730370ex_rat(G,I5) )
     => ( ! [I2: complex] :
            ( aa_set_complex_o(member_complex(I2),I5)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_complex_rat(F,I2)),aa_complex_rat(G,I2)) )
       => ( aa_set_complex_o(member_complex(I),I5)
         => ( aa_set_complex_o(finite3207457112153483333omplex,I5)
           => ( aa_complex_rat(F,I) = aa_complex_rat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_6969_sum__mono__inv,axiom,
    ! [F: fun_real_nat,I5: set_real,G: fun_real_nat,I: real] :
      ( ( groups1935376822645274424al_nat(F,I5) = groups1935376822645274424al_nat(G,I5) )
     => ( ! [I2: real] :
            ( member_real(I2,I5)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_real_nat(F,I2)),aa_real_nat(G,I2)) )
       => ( member_real(I,I5)
         => ( finite_finite_real(I5)
           => ( aa_real_nat(F,I) = aa_real_nat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_6970_sum__mono__inv,axiom,
    ! [F: fun_int_nat,I5: set_int,G: fun_int_nat,I: int] :
      ( ( groups4541462559716669496nt_nat(F,I5) = groups4541462559716669496nt_nat(G,I5) )
     => ( ! [I2: int] :
            ( aa_set_int_o(member_int(I2),I5)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,I2)),aa_int_nat(G,I2)) )
       => ( aa_set_int_o(member_int(I),I5)
         => ( aa_set_int_o(finite_finite_int,I5)
           => ( aa_int_nat(F,I) = aa_int_nat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_6971_sum__mono__inv,axiom,
    ! [F: fun_complex_nat,I5: set_complex,G: fun_complex_nat,I: complex] :
      ( ( groups5693394587270226106ex_nat(F,I5) = groups5693394587270226106ex_nat(G,I5) )
     => ( ! [I2: complex] :
            ( aa_set_complex_o(member_complex(I2),I5)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_complex_nat(F,I2)),aa_complex_nat(G,I2)) )
       => ( aa_set_complex_o(member_complex(I),I5)
         => ( aa_set_complex_o(finite3207457112153483333omplex,I5)
           => ( aa_complex_nat(F,I) = aa_complex_nat(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_6972_sum__mono__inv,axiom,
    ! [F: fun_real_int,I5: set_real,G: fun_real_int,I: real] :
      ( ( groups1932886352136224148al_int(F,I5) = groups1932886352136224148al_int(G,I5) )
     => ( ! [I2: real] :
            ( member_real(I2,I5)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(F,I2)),aa_real_int(G,I2)) )
       => ( member_real(I,I5)
         => ( finite_finite_real(I5)
           => ( aa_real_int(F,I) = aa_real_int(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_6973_sum__mono__inv,axiom,
    ! [F: fun_nat_int,I5: set_nat,G: fun_nat_int,I: nat] :
      ( ( groups3539618377306564664at_int(F,I5) = groups3539618377306564664at_int(G,I5) )
     => ( ! [I2: nat] :
            ( aa_set_nat_o(member_nat(I2),I5)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,I2)),aa_nat_int(G,I2)) )
       => ( aa_set_nat_o(member_nat(I),I5)
         => ( aa_set_nat_o(finite_finite_nat,I5)
           => ( aa_nat_int(F,I) = aa_nat_int(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_6974_sum__mono__inv,axiom,
    ! [F: fun_complex_int,I5: set_complex,G: fun_complex_int,I: complex] :
      ( ( groups5690904116761175830ex_int(F,I5) = groups5690904116761175830ex_int(G,I5) )
     => ( ! [I2: complex] :
            ( aa_set_complex_o(member_complex(I2),I5)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_complex_int(F,I2)),aa_complex_int(G,I2)) )
       => ( aa_set_complex_o(member_complex(I),I5)
         => ( aa_set_complex_o(finite3207457112153483333omplex,I5)
           => ( aa_complex_int(F,I) = aa_complex_int(G,I) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_6975_binomial__eq__0,axiom,
    ! [Na: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),K)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K) = zero_zero_nat ) ) ).

% binomial_eq_0
tff(fact_6976_Suc__times__binomial__eq,axiom,
    ! [Na: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,Na)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,Na)),aa_nat_nat(suc,K))),aa_nat_nat(suc,K)) ).

% Suc_times_binomial_eq
tff(fact_6977_Suc__times__binomial,axiom,
    ! [K: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,Na)),aa_nat_nat(suc,K))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,Na)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K)) ).

% Suc_times_binomial
tff(fact_6978_binomial__symmetric,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K)) ) ) ).

% binomial_symmetric
tff(fact_6979_binomial__le__pow,axiom,
    ! [R2: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,R2),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),R2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Na),R2)) ) ).

% binomial_le_pow
tff(fact_6980_take__bit__tightened__less__eq__int,axiom,
    ! [M: nat,Na: nat,K: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,M),K)),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Na),K)) ) ).

% take_bit_tightened_less_eq_int
tff(fact_6981_take__bit__nonnegative,axiom,
    ! [Na: nat,K: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Na),K)) ).

% take_bit_nonnegative
tff(fact_6982_take__bit__int__less__eq__self__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Na),K)),K)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K) ) ).

% take_bit_int_less_eq_self_iff
tff(fact_6983_signed__take__bit__eq__iff__take__bit__eq,axiom,
    ! [Na: nat,Aa: code_integer,Ba: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(bit_ri6519982836138164636nteger(Na),Aa) = aa_Cod1783974652533396728nteger(bit_ri6519982836138164636nteger(Na),Ba) )
    <=> ( aa_Cod1783974652533396728nteger(aa_nat6383147911832638789nteger(bit_se1745604003318907178nteger,aa_nat_nat(suc,Na)),Aa) = aa_Cod1783974652533396728nteger(aa_nat6383147911832638789nteger(bit_se1745604003318907178nteger,aa_nat_nat(suc,Na)),Ba) ) ) ).

% signed_take_bit_eq_iff_take_bit_eq
tff(fact_6984_signed__take__bit__eq__iff__take__bit__eq,axiom,
    ! [Na: nat,Aa: int,Ba: int] :
      ( ( aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,Na),Aa) = aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,Na),Ba) )
    <=> ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_nat_nat(suc,Na)),Aa) = aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_nat_nat(suc,Na)),Ba) ) ) ).

% signed_take_bit_eq_iff_take_bit_eq
tff(fact_6985_not__take__bit__negative,axiom,
    ! [Na: nat,K: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Na),K)),zero_zero_int) ).

% not_take_bit_negative
tff(fact_6986_take__bit__int__greater__self__iff,axiom,
    ! [K: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Na),K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int) ) ).

% take_bit_int_greater_self_iff
tff(fact_6987_signed__take__bit__take__bit,axiom,
    ! [M: nat,Na: nat,Aa: code_integer] :
      aa_Cod1783974652533396728nteger(bit_ri6519982836138164636nteger(M),aa_Cod1783974652533396728nteger(aa_nat6383147911832638789nteger(bit_se1745604003318907178nteger,Na),Aa)) = aa_Cod1783974652533396728nteger($ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M),aa_nat6383147911832638789nteger(bit_se1745604003318907178nteger,Na),bit_ri6519982836138164636nteger(M)),Aa) ).

% signed_take_bit_take_bit
tff(fact_6988_signed__take__bit__take__bit,axiom,
    ! [M: nat,Na: nat,Aa: int] :
      aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,M),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Na),Aa)) = aa_int_int($ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M),aa_nat_fun_int_int(bit_se2923211474154528505it_int,Na),aa_nat_fun_int_int(bit_ri631733984087533419it_int,M)),Aa) ).

% signed_take_bit_take_bit
tff(fact_6989_take__bit__unset__bit__eq,axiom,
    ! [Na: nat,M: nat,Aa: int] :
      aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Na),aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,M),Aa)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Na),Aa),aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,M),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Na),Aa))) ).

% take_bit_unset_bit_eq
tff(fact_6990_take__bit__unset__bit__eq,axiom,
    ! [Na: nat,M: nat,Aa: code_integer] :
      aa_Cod1783974652533396728nteger(aa_nat6383147911832638789nteger(bit_se1745604003318907178nteger,Na),aa_Cod1783974652533396728nteger(aa_nat6383147911832638789nteger(bit_se8260200283734997820nteger,M),Aa)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M),aa_Cod1783974652533396728nteger(aa_nat6383147911832638789nteger(bit_se1745604003318907178nteger,Na),Aa),aa_Cod1783974652533396728nteger(aa_nat6383147911832638789nteger(bit_se8260200283734997820nteger,M),aa_Cod1783974652533396728nteger(aa_nat6383147911832638789nteger(bit_se1745604003318907178nteger,Na),Aa))) ).

% take_bit_unset_bit_eq
tff(fact_6991_take__bit__unset__bit__eq,axiom,
    ! [Na: nat,M: nat,Aa: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,M),Aa)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),Aa),aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),Aa))) ).

% take_bit_unset_bit_eq
tff(fact_6992_take__bit__set__bit__eq,axiom,
    ! [Na: nat,M: nat,Aa: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,M),Aa)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),Aa),aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),Aa))) ).

% take_bit_set_bit_eq
tff(fact_6993_take__bit__set__bit__eq,axiom,
    ! [Na: nat,M: nat,Aa: code_integer] :
      aa_Cod1783974652533396728nteger(aa_nat6383147911832638789nteger(bit_se1745604003318907178nteger,Na),aa_Cod1783974652533396728nteger(aa_nat6383147911832638789nteger(bit_se2793503036327961859nteger,M),Aa)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M),aa_Cod1783974652533396728nteger(aa_nat6383147911832638789nteger(bit_se1745604003318907178nteger,Na),Aa),aa_Cod1783974652533396728nteger(aa_nat6383147911832638789nteger(bit_se2793503036327961859nteger,M),aa_Cod1783974652533396728nteger(aa_nat6383147911832638789nteger(bit_se1745604003318907178nteger,Na),Aa))) ).

% take_bit_set_bit_eq
tff(fact_6994_take__bit__set__bit__eq,axiom,
    ! [Na: nat,M: nat,Aa: int] :
      aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Na),aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,M),Aa)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Na),Aa),aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,M),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Na),Aa))) ).

% take_bit_set_bit_eq
tff(fact_6995_take__bit__flip__bit__eq,axiom,
    ! [Na: nat,M: nat,Aa: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,M),Aa)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),Aa),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),Aa))) ).

% take_bit_flip_bit_eq
tff(fact_6996_take__bit__flip__bit__eq,axiom,
    ! [Na: nat,M: nat,Aa: code_integer] :
      aa_Cod1783974652533396728nteger(aa_nat6383147911832638789nteger(bit_se1745604003318907178nteger,Na),aa_Cod1783974652533396728nteger(aa_nat6383147911832638789nteger(bit_se1345352211410354436nteger,M),Aa)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M),aa_Cod1783974652533396728nteger(aa_nat6383147911832638789nteger(bit_se1745604003318907178nteger,Na),Aa),aa_Cod1783974652533396728nteger(aa_nat6383147911832638789nteger(bit_se1345352211410354436nteger,M),aa_Cod1783974652533396728nteger(aa_nat6383147911832638789nteger(bit_se1745604003318907178nteger,Na),Aa))) ).

% take_bit_flip_bit_eq
tff(fact_6997_take__bit__flip__bit__eq,axiom,
    ! [Na: nat,M: nat,Aa: int] :
      aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Na),aa_int_int(aa_nat_fun_int_int(bit_se2159334234014336723it_int,M),Aa)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Na),Aa),aa_int_int(aa_nat_fun_int_int(bit_se2159334234014336723it_int,M),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Na),Aa))) ).

% take_bit_flip_bit_eq
tff(fact_6998_pochhammer__pos,axiom,
    ! [Xb: real,Na: 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),Na)) ) ).

% pochhammer_pos
tff(fact_6999_pochhammer__pos,axiom,
    ! [Xb: rat,Na: 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),Na)) ) ).

% pochhammer_pos
tff(fact_7000_pochhammer__pos,axiom,
    ! [Xb: nat,Na: 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),Na)) ) ).

% pochhammer_pos
tff(fact_7001_pochhammer__pos,axiom,
    ! [Xb: int,Na: 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),Na)) ) ).

% pochhammer_pos
tff(fact_7002_pochhammer__neq__0__mono,axiom,
    ! [Aa: real,M: nat,Na: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Aa),M) != zero_zero_real )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
       => ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Aa),Na) != zero_zero_real ) ) ) ).

% pochhammer_neq_0_mono
tff(fact_7003_pochhammer__neq__0__mono,axiom,
    ! [Aa: rat,M: nat,Na: nat] :
      ( ( aa_nat_rat(comm_s4028243227959126397er_rat(Aa),M) != zero_zero_rat )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
       => ( aa_nat_rat(comm_s4028243227959126397er_rat(Aa),Na) != zero_zero_rat ) ) ) ).

% pochhammer_neq_0_mono
tff(fact_7004_pochhammer__eq__0__mono,axiom,
    ! [Aa: real,Na: nat,M: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Aa),Na) = zero_zero_real )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
       => ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Aa),M) = zero_zero_real ) ) ) ).

% pochhammer_eq_0_mono
tff(fact_7005_pochhammer__eq__0__mono,axiom,
    ! [Aa: rat,Na: nat,M: nat] :
      ( ( aa_nat_rat(comm_s4028243227959126397er_rat(Aa),Na) = zero_zero_rat )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
       => ( aa_nat_rat(comm_s4028243227959126397er_rat(Aa),M) = zero_zero_rat ) ) ) ).

% pochhammer_eq_0_mono
tff(fact_7006_plus__integer_Oabs__eq,axiom,
    ! [Xaa: int,Xb: int] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(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_7007_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_7008_one__integer__def,axiom,
    one_one_Code_integer = aa_int_Code_integer(code_integer_of_int,one_one_int) ).

% one_integer_def
tff(fact_7009_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_7010_sum_Ointer__filter,axiom,
    ! [A3: set_real,G: fun_real_real,P: fun_real_o] :
      ( finite_finite_real(A3)
     => ( groups8097168146408367636l_real(G,collect_real(aa_fun5868358341814070780real_o(aTP_Lamm_bl(A3),P))) = groups8097168146408367636l_real(aa_fun4448911596306054750l_real(aTP_Lamm_hp(G),P),A3) ) ) ).

% sum.inter_filter
tff(fact_7011_sum_Ointer__filter,axiom,
    ! [A3: set_int,G: fun_int_real,P: fun_int_o] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( groups8778361861064173332t_real(G,collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_bp(A3),P))) = groups8778361861064173332t_real(aa_fun7506669883404309598t_real(aTP_Lamm_hq(G),P),A3) ) ) ).

% sum.inter_filter
tff(fact_7012_sum_Ointer__filter,axiom,
    ! [A3: set_complex,G: fun_complex_real,P: fun_complex_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A3)
     => ( groups5808333547571424918x_real(G,collect_complex(aa_fun2514317545224092796plex_o(aTP_Lamm_hr(A3),P))) = groups5808333547571424918x_real(aa_fun8261057609414086622x_real(aTP_Lamm_hs(G),P),A3) ) ) ).

% sum.inter_filter
tff(fact_7013_sum_Ointer__filter,axiom,
    ! [A3: set_real,G: fun_real_rat,P: fun_real_o] :
      ( finite_finite_real(A3)
     => ( groups1300246762558778688al_rat(G,collect_real(aa_fun5868358341814070780real_o(aTP_Lamm_bl(A3),P))) = groups1300246762558778688al_rat(aa_fun2004904883527618314al_rat(aTP_Lamm_ht(G),P),A3) ) ) ).

% sum.inter_filter
tff(fact_7014_sum_Ointer__filter,axiom,
    ! [A3: set_nat,G: fun_nat_rat,P: fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( groups2906978787729119204at_rat(G,collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_bo(A3),P))) = groups2906978787729119204at_rat(aa_fun696632391729880074at_rat(aTP_Lamm_hu(G),P),A3) ) ) ).

% sum.inter_filter
tff(fact_7015_sum_Ointer__filter,axiom,
    ! [A3: set_int,G: fun_int_rat,P: fun_int_o] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( groups3906332499630173760nt_rat(G,collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_bp(A3),P))) = groups3906332499630173760nt_rat(aa_fun268837275264525578nt_rat(aTP_Lamm_hv(G),P),A3) ) ) ).

% sum.inter_filter
tff(fact_7016_sum_Ointer__filter,axiom,
    ! [A3: set_complex,G: fun_complex_rat,P: fun_complex_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A3)
     => ( groups5058264527183730370ex_rat(G,collect_complex(aa_fun2514317545224092796plex_o(aTP_Lamm_hr(A3),P))) = groups5058264527183730370ex_rat(aa_fun4010016270681809034ex_rat(aTP_Lamm_hw(G),P),A3) ) ) ).

% sum.inter_filter
tff(fact_7017_sum_Ointer__filter,axiom,
    ! [A3: set_real,G: fun_real_nat,P: fun_real_o] :
      ( finite_finite_real(A3)
     => ( groups1935376822645274424al_nat(G,collect_real(aa_fun5868358341814070780real_o(aTP_Lamm_bl(A3),P))) = groups1935376822645274424al_nat(aa_fun5513133441921015042al_nat(aTP_Lamm_hx(G),P),A3) ) ) ).

% sum.inter_filter
tff(fact_7018_sum_Ointer__filter,axiom,
    ! [A3: set_int,G: fun_int_nat,P: fun_int_o] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( groups4541462559716669496nt_nat(G,collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_bp(A3),P))) = groups4541462559716669496nt_nat(aa_fun3777065833657922306nt_nat(aTP_Lamm_hy(G),P),A3) ) ) ).

% sum.inter_filter
tff(fact_7019_sum_Ointer__filter,axiom,
    ! [A3: set_complex,G: fun_complex_nat,P: fun_complex_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A3)
     => ( groups5693394587270226106ex_nat(G,collect_complex(aa_fun2514317545224092796plex_o(aTP_Lamm_hr(A3),P))) = groups5693394587270226106ex_nat(aa_fun7518244829075205762ex_nat(aTP_Lamm_hz(G),P),A3) ) ) ).

% sum.inter_filter
tff(fact_7020_minus__integer_Oabs__eq,axiom,
    ! [Xaa: int,Xb: int] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,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(minus_minus_int,Xaa),Xb)) ).

% minus_integer.abs_eq
tff(fact_7021_pochhammer__fact,axiom,
    ! [Na: nat] : aa_nat_complex(semiri5044797733671781792omplex,Na) = aa_nat_complex(comm_s2602460028002588243omplex(one_one_complex),Na) ).

% pochhammer_fact
tff(fact_7022_pochhammer__fact,axiom,
    ! [Na: nat] : aa_nat_rat(semiri773545260158071498ct_rat,Na) = aa_nat_rat(comm_s4028243227959126397er_rat(one_one_rat),Na) ).

% pochhammer_fact
tff(fact_7023_pochhammer__fact,axiom,
    ! [Na: nat] : aa_nat_int(semiri1406184849735516958ct_int,Na) = aa_nat_int(comm_s4660882817536571857er_int(one_one_int),Na) ).

% pochhammer_fact
tff(fact_7024_pochhammer__fact,axiom,
    ! [Na: nat] : aa_nat_nat(semiri1408675320244567234ct_nat,Na) = aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,one_one_nat),Na) ).

% pochhammer_fact
tff(fact_7025_pochhammer__fact,axiom,
    ! [Na: nat] : aa_nat_real(semiri2265585572941072030t_real,Na) = aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,one_one_real),Na) ).

% pochhammer_fact
tff(fact_7026_sum__le__included,axiom,
    ! [S3: set_int,Ta: set_int,G: fun_int_real,I: fun_int_int,F: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,S3)
     => ( aa_set_int_o(finite_finite_int,Ta)
       => ( ! [X: int] :
              ( aa_set_int_o(member_int(X),Ta)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(G,X)) )
         => ( ! [X: int] :
                ( aa_set_int_o(member_int(X),S3)
               => ? [Xa: int] :
                    ( aa_set_int_o(member_int(Xa),Ta)
                    & ( aa_int_int(I,Xa) = X )
                    & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(F,X)),aa_int_real(G,Xa)) ) )
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups8778361861064173332t_real(F,S3)),groups8778361861064173332t_real(G,Ta)) ) ) ) ) ).

% sum_le_included
tff(fact_7027_sum__le__included,axiom,
    ! [S3: set_int,Ta: set_complex,G: fun_complex_real,I: fun_complex_int,F: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,S3)
     => ( aa_set_complex_o(finite3207457112153483333omplex,Ta)
       => ( ! [X: complex] :
              ( aa_set_complex_o(member_complex(X),Ta)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(G,X)) )
         => ( ! [X: int] :
                ( aa_set_int_o(member_int(X),S3)
               => ? [Xa: complex] :
                    ( aa_set_complex_o(member_complex(Xa),Ta)
                    & ( aa_complex_int(I,Xa) = X )
                    & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(F,X)),aa_complex_real(G,Xa)) ) )
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups8778361861064173332t_real(F,S3)),groups5808333547571424918x_real(G,Ta)) ) ) ) ) ).

% sum_le_included
tff(fact_7028_sum__le__included,axiom,
    ! [S3: set_complex,Ta: set_int,G: fun_int_real,I: fun_int_complex,F: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S3)
     => ( aa_set_int_o(finite_finite_int,Ta)
       => ( ! [X: int] :
              ( aa_set_int_o(member_int(X),Ta)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(G,X)) )
         => ( ! [X: complex] :
                ( aa_set_complex_o(member_complex(X),S3)
               => ? [Xa: int] :
                    ( aa_set_int_o(member_int(Xa),Ta)
                    & ( aa_int_complex(I,Xa) = X )
                    & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(F,X)),aa_int_real(G,Xa)) ) )
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups5808333547571424918x_real(F,S3)),groups8778361861064173332t_real(G,Ta)) ) ) ) ) ).

% sum_le_included
tff(fact_7029_sum__le__included,axiom,
    ! [S3: set_complex,Ta: set_complex,G: fun_complex_real,I: fun_complex_complex,F: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S3)
     => ( aa_set_complex_o(finite3207457112153483333omplex,Ta)
       => ( ! [X: complex] :
              ( aa_set_complex_o(member_complex(X),Ta)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(G,X)) )
         => ( ! [X: complex] :
                ( aa_set_complex_o(member_complex(X),S3)
               => ? [Xa: complex] :
                    ( aa_set_complex_o(member_complex(Xa),Ta)
                    & ( aa_complex_complex(I,Xa) = X )
                    & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(F,X)),aa_complex_real(G,Xa)) ) )
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups5808333547571424918x_real(F,S3)),groups5808333547571424918x_real(G,Ta)) ) ) ) ) ).

% sum_le_included
tff(fact_7030_sum__le__included,axiom,
    ! [S3: set_nat,Ta: set_nat,G: fun_nat_rat,I: fun_nat_nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,S3)
     => ( aa_set_nat_o(finite_finite_nat,Ta)
       => ( ! [X: nat] :
              ( aa_set_nat_o(member_nat(X),Ta)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(G,X)) )
         => ( ! [X: nat] :
                ( aa_set_nat_o(member_nat(X),S3)
               => ? [Xa: nat] :
                    ( aa_set_nat_o(member_nat(Xa),Ta)
                    & ( aa_nat_nat(I,Xa) = X )
                    & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X)),aa_nat_rat(G,Xa)) ) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups2906978787729119204at_rat(F,S3)),groups2906978787729119204at_rat(G,Ta)) ) ) ) ) ).

% sum_le_included
tff(fact_7031_sum__le__included,axiom,
    ! [S3: set_nat,Ta: set_int,G: fun_int_rat,I: fun_int_nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,S3)
     => ( aa_set_int_o(finite_finite_int,Ta)
       => ( ! [X: int] :
              ( aa_set_int_o(member_int(X),Ta)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(G,X)) )
         => ( ! [X: nat] :
                ( aa_set_nat_o(member_nat(X),S3)
               => ? [Xa: int] :
                    ( aa_set_int_o(member_int(Xa),Ta)
                    & ( aa_int_nat(I,Xa) = X )
                    & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X)),aa_int_rat(G,Xa)) ) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups2906978787729119204at_rat(F,S3)),groups3906332499630173760nt_rat(G,Ta)) ) ) ) ) ).

% sum_le_included
tff(fact_7032_sum__le__included,axiom,
    ! [S3: set_nat,Ta: set_complex,G: fun_complex_rat,I: fun_complex_nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,S3)
     => ( aa_set_complex_o(finite3207457112153483333omplex,Ta)
       => ( ! [X: complex] :
              ( aa_set_complex_o(member_complex(X),Ta)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(G,X)) )
         => ( ! [X: nat] :
                ( aa_set_nat_o(member_nat(X),S3)
               => ? [Xa: complex] :
                    ( aa_set_complex_o(member_complex(Xa),Ta)
                    & ( aa_complex_nat(I,Xa) = X )
                    & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X)),aa_complex_rat(G,Xa)) ) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups2906978787729119204at_rat(F,S3)),groups5058264527183730370ex_rat(G,Ta)) ) ) ) ) ).

% sum_le_included
tff(fact_7033_sum__le__included,axiom,
    ! [S3: set_int,Ta: set_nat,G: fun_nat_rat,I: fun_nat_int,F: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,S3)
     => ( aa_set_nat_o(finite_finite_nat,Ta)
       => ( ! [X: nat] :
              ( aa_set_nat_o(member_nat(X),Ta)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(G,X)) )
         => ( ! [X: int] :
                ( aa_set_int_o(member_int(X),S3)
               => ? [Xa: nat] :
                    ( aa_set_nat_o(member_nat(Xa),Ta)
                    & ( aa_nat_int(I,Xa) = X )
                    & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,X)),aa_nat_rat(G,Xa)) ) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups3906332499630173760nt_rat(F,S3)),groups2906978787729119204at_rat(G,Ta)) ) ) ) ) ).

% sum_le_included
tff(fact_7034_sum__le__included,axiom,
    ! [S3: set_int,Ta: set_int,G: fun_int_rat,I: fun_int_int,F: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,S3)
     => ( aa_set_int_o(finite_finite_int,Ta)
       => ( ! [X: int] :
              ( aa_set_int_o(member_int(X),Ta)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(G,X)) )
         => ( ! [X: int] :
                ( aa_set_int_o(member_int(X),S3)
               => ? [Xa: int] :
                    ( aa_set_int_o(member_int(Xa),Ta)
                    & ( aa_int_int(I,Xa) = X )
                    & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,X)),aa_int_rat(G,Xa)) ) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups3906332499630173760nt_rat(F,S3)),groups3906332499630173760nt_rat(G,Ta)) ) ) ) ) ).

% sum_le_included
tff(fact_7035_sum__le__included,axiom,
    ! [S3: set_int,Ta: set_complex,G: fun_complex_rat,I: fun_complex_int,F: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,S3)
     => ( aa_set_complex_o(finite3207457112153483333omplex,Ta)
       => ( ! [X: complex] :
              ( aa_set_complex_o(member_complex(X),Ta)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(G,X)) )
         => ( ! [X: int] :
                ( aa_set_int_o(member_int(X),S3)
               => ? [Xa: complex] :
                    ( aa_set_complex_o(member_complex(Xa),Ta)
                    & ( aa_complex_int(I,Xa) = X )
                    & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,X)),aa_complex_rat(G,Xa)) ) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups3906332499630173760nt_rat(F,S3)),groups5058264527183730370ex_rat(G,Ta)) ) ) ) ) ).

% sum_le_included
tff(fact_7036_sum__nonneg__eq__0__iff,axiom,
    ! [A3: set_real,F: fun_real_real] :
      ( finite_finite_real(A3)
     => ( ! [X: real] :
            ( member_real(X,A3)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,X)) )
       => ( ( groups8097168146408367636l_real(F,A3) = zero_zero_real )
        <=> ! [X4: real] :
              ( member_real(X4,A3)
             => ( aa_real_real(F,X4) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_7037_sum__nonneg__eq__0__iff,axiom,
    ! [A3: set_int,F: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( ! [X: int] :
            ( aa_set_int_o(member_int(X),A3)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(F,X)) )
       => ( ( groups8778361861064173332t_real(F,A3) = zero_zero_real )
        <=> ! [X4: int] :
              ( aa_set_int_o(member_int(X4),A3)
             => ( aa_int_real(F,X4) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_7038_sum__nonneg__eq__0__iff,axiom,
    ! [A3: set_complex,F: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A3)
     => ( ! [X: complex] :
            ( aa_set_complex_o(member_complex(X),A3)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(F,X)) )
       => ( ( groups5808333547571424918x_real(F,A3) = zero_zero_real )
        <=> ! [X4: complex] :
              ( aa_set_complex_o(member_complex(X4),A3)
             => ( aa_complex_real(F,X4) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_7039_sum__nonneg__eq__0__iff,axiom,
    ! [A3: set_real,F: fun_real_rat] :
      ( finite_finite_real(A3)
     => ( ! [X: real] :
            ( member_real(X,A3)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,X)) )
       => ( ( groups1300246762558778688al_rat(F,A3) = zero_zero_rat )
        <=> ! [X4: real] :
              ( member_real(X4,A3)
             => ( aa_real_rat(F,X4) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_7040_sum__nonneg__eq__0__iff,axiom,
    ! [A3: set_nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( ! [X: nat] :
            ( aa_set_nat_o(member_nat(X),A3)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,X)) )
       => ( ( groups2906978787729119204at_rat(F,A3) = zero_zero_rat )
        <=> ! [X4: nat] :
              ( aa_set_nat_o(member_nat(X4),A3)
             => ( aa_nat_rat(F,X4) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_7041_sum__nonneg__eq__0__iff,axiom,
    ! [A3: set_int,F: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( ! [X: int] :
            ( aa_set_int_o(member_int(X),A3)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,X)) )
       => ( ( groups3906332499630173760nt_rat(F,A3) = zero_zero_rat )
        <=> ! [X4: int] :
              ( aa_set_int_o(member_int(X4),A3)
             => ( aa_int_rat(F,X4) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_7042_sum__nonneg__eq__0__iff,axiom,
    ! [A3: set_complex,F: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A3)
     => ( ! [X: complex] :
            ( aa_set_complex_o(member_complex(X),A3)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,X)) )
       => ( ( groups5058264527183730370ex_rat(F,A3) = zero_zero_rat )
        <=> ! [X4: complex] :
              ( aa_set_complex_o(member_complex(X4),A3)
             => ( aa_complex_rat(F,X4) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_7043_sum__nonneg__eq__0__iff,axiom,
    ! [A3: set_real,F: fun_real_nat] :
      ( finite_finite_real(A3)
     => ( ! [X: real] :
            ( member_real(X,A3)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,X)) )
       => ( ( groups1935376822645274424al_nat(F,A3) = zero_zero_nat )
        <=> ! [X4: real] :
              ( member_real(X4,A3)
             => ( aa_real_nat(F,X4) = zero_zero_nat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_7044_sum__nonneg__eq__0__iff,axiom,
    ! [A3: set_int,F: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( ! [X: int] :
            ( aa_set_int_o(member_int(X),A3)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F,X)) )
       => ( ( groups4541462559716669496nt_nat(F,A3) = zero_zero_nat )
        <=> ! [X4: int] :
              ( aa_set_int_o(member_int(X4),A3)
             => ( aa_int_nat(F,X4) = zero_zero_nat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_7045_sum__nonneg__eq__0__iff,axiom,
    ! [A3: set_complex,F: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A3)
     => ( ! [X: complex] :
            ( aa_set_complex_o(member_complex(X),A3)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_complex_nat(F,X)) )
       => ( ( groups5693394587270226106ex_nat(F,A3) = zero_zero_nat )
        <=> ! [X4: complex] :
              ( aa_set_complex_o(member_complex(X4),A3)
             => ( aa_complex_nat(F,X4) = zero_zero_nat ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_7046_sum__strict__mono__ex1,axiom,
    ! [A3: set_int,F: fun_int_real,G: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( ! [X: int] :
            ( aa_set_int_o(member_int(X),A3)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(F,X)),aa_int_real(G,X)) )
       => ( ? [X3: int] :
              ( aa_set_int_o(member_int(X3),A3)
              & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(F,X3)),aa_int_real(G,X3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,groups8778361861064173332t_real(F,A3)),groups8778361861064173332t_real(G,A3)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_7047_sum__strict__mono__ex1,axiom,
    ! [A3: set_complex,F: fun_complex_real,G: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A3)
     => ( ! [X: complex] :
            ( aa_set_complex_o(member_complex(X),A3)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(F,X)),aa_complex_real(G,X)) )
       => ( ? [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),A3)
              & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(F,X3)),aa_complex_real(G,X3)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,groups5808333547571424918x_real(F,A3)),groups5808333547571424918x_real(G,A3)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_7048_sum__strict__mono__ex1,axiom,
    ! [A3: set_nat,F: fun_nat_rat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( ! [X: nat] :
            ( aa_set_nat_o(member_nat(X),A3)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X)),aa_nat_rat(G,X)) )
       => ( ? [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),A3)
              & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,X3)),aa_nat_rat(G,X3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups2906978787729119204at_rat(F,A3)),groups2906978787729119204at_rat(G,A3)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_7049_sum__strict__mono__ex1,axiom,
    ! [A3: set_int,F: fun_int_rat,G: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( ! [X: int] :
            ( aa_set_int_o(member_int(X),A3)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,X)),aa_int_rat(G,X)) )
       => ( ? [X3: int] :
              ( aa_set_int_o(member_int(X3),A3)
              & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(F,X3)),aa_int_rat(G,X3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups3906332499630173760nt_rat(F,A3)),groups3906332499630173760nt_rat(G,A3)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_7050_sum__strict__mono__ex1,axiom,
    ! [A3: set_complex,F: fun_complex_rat,G: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A3)
     => ( ! [X: complex] :
            ( aa_set_complex_o(member_complex(X),A3)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_complex_rat(F,X)),aa_complex_rat(G,X)) )
       => ( ? [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),A3)
              & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_complex_rat(F,X3)),aa_complex_rat(G,X3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups5058264527183730370ex_rat(F,A3)),groups5058264527183730370ex_rat(G,A3)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_7051_sum__strict__mono__ex1,axiom,
    ! [A3: set_int,F: fun_int_nat,G: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( ! [X: int] :
            ( aa_set_int_o(member_int(X),A3)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,X)),aa_int_nat(G,X)) )
       => ( ? [X3: int] :
              ( aa_set_int_o(member_int(X3),A3)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(F,X3)),aa_int_nat(G,X3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups4541462559716669496nt_nat(F,A3)),groups4541462559716669496nt_nat(G,A3)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_7052_sum__strict__mono__ex1,axiom,
    ! [A3: set_complex,F: fun_complex_nat,G: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A3)
     => ( ! [X: complex] :
            ( aa_set_complex_o(member_complex(X),A3)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_complex_nat(F,X)),aa_complex_nat(G,X)) )
       => ( ? [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),A3)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_complex_nat(F,X3)),aa_complex_nat(G,X3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups5693394587270226106ex_nat(F,A3)),groups5693394587270226106ex_nat(G,A3)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_7053_sum__strict__mono__ex1,axiom,
    ! [A3: set_nat,F: fun_nat_int,G: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( ! [X: nat] :
            ( aa_set_nat_o(member_nat(X),A3)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,X)),aa_nat_int(G,X)) )
       => ( ? [X3: nat] :
              ( aa_set_nat_o(member_nat(X3),A3)
              & aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,X3)),aa_nat_int(G,X3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,groups3539618377306564664at_int(F,A3)),groups3539618377306564664at_int(G,A3)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_7054_sum__strict__mono__ex1,axiom,
    ! [A3: set_complex,F: fun_complex_int,G: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A3)
     => ( ! [X: complex] :
            ( aa_set_complex_o(member_complex(X),A3)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_complex_int(F,X)),aa_complex_int(G,X)) )
       => ( ? [X3: complex] :
              ( aa_set_complex_o(member_complex(X3),A3)
              & aa_int_o(aa_int_fun_int_o(ord_less_int,aa_complex_int(F,X3)),aa_complex_int(G,X3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,groups5690904116761175830ex_int(F,A3)),groups5690904116761175830ex_int(G,A3)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_7055_sum__strict__mono__ex1,axiom,
    ! [A3: set_int,F: fun_int_int,G: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( ! [X: int] :
            ( aa_set_int_o(member_int(X),A3)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(F,X)),aa_int_int(G,X)) )
       => ( ? [X3: int] :
              ( aa_set_int_o(member_int(X3),A3)
              & aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(F,X3)),aa_int_int(G,X3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,groups4538972089207619220nt_int(F,A3)),groups4538972089207619220nt_int(G,A3)) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_7056_sum_Orelated,axiom,
    ! [R: fun_real_fun_real_o,S2: set_int,H: fun_int_real,G: fun_int_real] :
      ( aa_real_o(aa_real_fun_real_o(R,zero_zero_real),zero_zero_real)
     => ( ! [X1: real,Y1: real,X23: real,Y23: real] :
            ( ( aa_real_o(aa_real_fun_real_o(R,X1),X23)
              & aa_real_o(aa_real_fun_real_o(R,Y1),Y23) )
           => aa_real_o(aa_real_fun_real_o(R,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X1),Y1)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X23),Y23)) )
       => ( aa_set_int_o(finite_finite_int,S2)
         => ( ! [X: int] :
                ( aa_set_int_o(member_int(X),S2)
               => aa_real_o(aa_real_fun_real_o(R,aa_int_real(H,X)),aa_int_real(G,X)) )
           => aa_real_o(aa_real_fun_real_o(R,groups8778361861064173332t_real(H,S2)),groups8778361861064173332t_real(G,S2)) ) ) ) ) ).

% sum.related
tff(fact_7057_sum_Orelated,axiom,
    ! [R: fun_real_fun_real_o,S2: set_complex,H: fun_complex_real,G: fun_complex_real] :
      ( aa_real_o(aa_real_fun_real_o(R,zero_zero_real),zero_zero_real)
     => ( ! [X1: real,Y1: real,X23: real,Y23: real] :
            ( ( aa_real_o(aa_real_fun_real_o(R,X1),X23)
              & aa_real_o(aa_real_fun_real_o(R,Y1),Y23) )
           => aa_real_o(aa_real_fun_real_o(R,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X1),Y1)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X23),Y23)) )
       => ( aa_set_complex_o(finite3207457112153483333omplex,S2)
         => ( ! [X: complex] :
                ( aa_set_complex_o(member_complex(X),S2)
               => aa_real_o(aa_real_fun_real_o(R,aa_complex_real(H,X)),aa_complex_real(G,X)) )
           => aa_real_o(aa_real_fun_real_o(R,groups5808333547571424918x_real(H,S2)),groups5808333547571424918x_real(G,S2)) ) ) ) ) ).

% sum.related
tff(fact_7058_sum_Orelated,axiom,
    ! [R: fun_rat_fun_rat_o,S2: set_nat,H: fun_nat_rat,G: fun_nat_rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(R,zero_zero_rat),zero_zero_rat)
     => ( ! [X1: rat,Y1: rat,X23: rat,Y23: rat] :
            ( ( aa_rat_o(aa_rat_fun_rat_o(R,X1),X23)
              & aa_rat_o(aa_rat_fun_rat_o(R,Y1),Y23) )
           => aa_rat_o(aa_rat_fun_rat_o(R,aa_rat_rat(plus_plus_rat(X1),Y1)),aa_rat_rat(plus_plus_rat(X23),Y23)) )
       => ( aa_set_nat_o(finite_finite_nat,S2)
         => ( ! [X: nat] :
                ( aa_set_nat_o(member_nat(X),S2)
               => aa_rat_o(aa_rat_fun_rat_o(R,aa_nat_rat(H,X)),aa_nat_rat(G,X)) )
           => aa_rat_o(aa_rat_fun_rat_o(R,groups2906978787729119204at_rat(H,S2)),groups2906978787729119204at_rat(G,S2)) ) ) ) ) ).

% sum.related
tff(fact_7059_sum_Orelated,axiom,
    ! [R: fun_rat_fun_rat_o,S2: set_int,H: fun_int_rat,G: fun_int_rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(R,zero_zero_rat),zero_zero_rat)
     => ( ! [X1: rat,Y1: rat,X23: rat,Y23: rat] :
            ( ( aa_rat_o(aa_rat_fun_rat_o(R,X1),X23)
              & aa_rat_o(aa_rat_fun_rat_o(R,Y1),Y23) )
           => aa_rat_o(aa_rat_fun_rat_o(R,aa_rat_rat(plus_plus_rat(X1),Y1)),aa_rat_rat(plus_plus_rat(X23),Y23)) )
       => ( aa_set_int_o(finite_finite_int,S2)
         => ( ! [X: int] :
                ( aa_set_int_o(member_int(X),S2)
               => aa_rat_o(aa_rat_fun_rat_o(R,aa_int_rat(H,X)),aa_int_rat(G,X)) )
           => aa_rat_o(aa_rat_fun_rat_o(R,groups3906332499630173760nt_rat(H,S2)),groups3906332499630173760nt_rat(G,S2)) ) ) ) ) ).

% sum.related
tff(fact_7060_sum_Orelated,axiom,
    ! [R: fun_rat_fun_rat_o,S2: set_complex,H: fun_complex_rat,G: fun_complex_rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(R,zero_zero_rat),zero_zero_rat)
     => ( ! [X1: rat,Y1: rat,X23: rat,Y23: rat] :
            ( ( aa_rat_o(aa_rat_fun_rat_o(R,X1),X23)
              & aa_rat_o(aa_rat_fun_rat_o(R,Y1),Y23) )
           => aa_rat_o(aa_rat_fun_rat_o(R,aa_rat_rat(plus_plus_rat(X1),Y1)),aa_rat_rat(plus_plus_rat(X23),Y23)) )
       => ( aa_set_complex_o(finite3207457112153483333omplex,S2)
         => ( ! [X: complex] :
                ( aa_set_complex_o(member_complex(X),S2)
               => aa_rat_o(aa_rat_fun_rat_o(R,aa_complex_rat(H,X)),aa_complex_rat(G,X)) )
           => aa_rat_o(aa_rat_fun_rat_o(R,groups5058264527183730370ex_rat(H,S2)),groups5058264527183730370ex_rat(G,S2)) ) ) ) ) ).

% sum.related
tff(fact_7061_sum_Orelated,axiom,
    ! [R: fun_nat_fun_nat_o,S2: set_int,H: fun_int_nat,G: fun_int_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(R,zero_zero_nat),zero_zero_nat)
     => ( ! [X1: nat,Y1: nat,X23: nat,Y23: nat] :
            ( ( aa_nat_o(aa_nat_fun_nat_o(R,X1),X23)
              & aa_nat_o(aa_nat_fun_nat_o(R,Y1),Y23) )
           => aa_nat_o(aa_nat_fun_nat_o(R,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X1),Y1)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X23),Y23)) )
       => ( aa_set_int_o(finite_finite_int,S2)
         => ( ! [X: int] :
                ( aa_set_int_o(member_int(X),S2)
               => aa_nat_o(aa_nat_fun_nat_o(R,aa_int_nat(H,X)),aa_int_nat(G,X)) )
           => aa_nat_o(aa_nat_fun_nat_o(R,groups4541462559716669496nt_nat(H,S2)),groups4541462559716669496nt_nat(G,S2)) ) ) ) ) ).

% sum.related
tff(fact_7062_sum_Orelated,axiom,
    ! [R: fun_nat_fun_nat_o,S2: set_complex,H: fun_complex_nat,G: fun_complex_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(R,zero_zero_nat),zero_zero_nat)
     => ( ! [X1: nat,Y1: nat,X23: nat,Y23: nat] :
            ( ( aa_nat_o(aa_nat_fun_nat_o(R,X1),X23)
              & aa_nat_o(aa_nat_fun_nat_o(R,Y1),Y23) )
           => aa_nat_o(aa_nat_fun_nat_o(R,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X1),Y1)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X23),Y23)) )
       => ( aa_set_complex_o(finite3207457112153483333omplex,S2)
         => ( ! [X: complex] :
                ( aa_set_complex_o(member_complex(X),S2)
               => aa_nat_o(aa_nat_fun_nat_o(R,aa_complex_nat(H,X)),aa_complex_nat(G,X)) )
           => aa_nat_o(aa_nat_fun_nat_o(R,groups5693394587270226106ex_nat(H,S2)),groups5693394587270226106ex_nat(G,S2)) ) ) ) ) ).

% sum.related
tff(fact_7063_sum_Orelated,axiom,
    ! [R: fun_int_fun_int_o,S2: set_nat,H: fun_nat_int,G: fun_nat_int] :
      ( aa_int_o(aa_int_fun_int_o(R,zero_zero_int),zero_zero_int)
     => ( ! [X1: int,Y1: int,X23: int,Y23: int] :
            ( ( aa_int_o(aa_int_fun_int_o(R,X1),X23)
              & aa_int_o(aa_int_fun_int_o(R,Y1),Y23) )
           => aa_int_o(aa_int_fun_int_o(R,aa_int_int(aa_int_fun_int_int(plus_plus_int,X1),Y1)),aa_int_int(aa_int_fun_int_int(plus_plus_int,X23),Y23)) )
       => ( aa_set_nat_o(finite_finite_nat,S2)
         => ( ! [X: nat] :
                ( aa_set_nat_o(member_nat(X),S2)
               => aa_int_o(aa_int_fun_int_o(R,aa_nat_int(H,X)),aa_nat_int(G,X)) )
           => aa_int_o(aa_int_fun_int_o(R,groups3539618377306564664at_int(H,S2)),groups3539618377306564664at_int(G,S2)) ) ) ) ) ).

% sum.related
tff(fact_7064_sum_Orelated,axiom,
    ! [R: fun_int_fun_int_o,S2: set_complex,H: fun_complex_int,G: fun_complex_int] :
      ( aa_int_o(aa_int_fun_int_o(R,zero_zero_int),zero_zero_int)
     => ( ! [X1: int,Y1: int,X23: int,Y23: int] :
            ( ( aa_int_o(aa_int_fun_int_o(R,X1),X23)
              & aa_int_o(aa_int_fun_int_o(R,Y1),Y23) )
           => aa_int_o(aa_int_fun_int_o(R,aa_int_int(aa_int_fun_int_int(plus_plus_int,X1),Y1)),aa_int_int(aa_int_fun_int_int(plus_plus_int,X23),Y23)) )
       => ( aa_set_complex_o(finite3207457112153483333omplex,S2)
         => ( ! [X: complex] :
                ( aa_set_complex_o(member_complex(X),S2)
               => aa_int_o(aa_int_fun_int_o(R,aa_complex_int(H,X)),aa_complex_int(G,X)) )
           => aa_int_o(aa_int_fun_int_o(R,groups5690904116761175830ex_int(H,S2)),groups5690904116761175830ex_int(G,S2)) ) ) ) ) ).

% sum.related
tff(fact_7065_sum_Orelated,axiom,
    ! [R: fun_int_fun_int_o,S2: set_int,H: fun_int_int,G: fun_int_int] :
      ( aa_int_o(aa_int_fun_int_o(R,zero_zero_int),zero_zero_int)
     => ( ! [X1: int,Y1: int,X23: int,Y23: int] :
            ( ( aa_int_o(aa_int_fun_int_o(R,X1),X23)
              & aa_int_o(aa_int_fun_int_o(R,Y1),Y23) )
           => aa_int_o(aa_int_fun_int_o(R,aa_int_int(aa_int_fun_int_int(plus_plus_int,X1),Y1)),aa_int_int(aa_int_fun_int_int(plus_plus_int,X23),Y23)) )
       => ( aa_set_int_o(finite_finite_int,S2)
         => ( ! [X: int] :
                ( aa_set_int_o(member_int(X),S2)
               => aa_int_o(aa_int_fun_int_o(R,aa_int_int(H,X)),aa_int_int(G,X)) )
           => aa_int_o(aa_int_fun_int_o(R,groups4538972089207619220nt_int(H,S2)),groups4538972089207619220nt_int(G,S2)) ) ) ) ) ).

% sum.related
tff(fact_7066_sum__strict__mono,axiom,
    ! [A3: set_complex,F: fun_complex_real,G: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A3)
     => ( ( A3 != bot_bot_set_complex )
       => ( ! [X: complex] :
              ( aa_set_complex_o(member_complex(X),A3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(F,X)),aa_complex_real(G,X)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,groups5808333547571424918x_real(F,A3)),groups5808333547571424918x_real(G,A3)) ) ) ) ).

% sum_strict_mono
tff(fact_7067_sum__strict__mono,axiom,
    ! [A3: set_real,F: fun_real_real,G: fun_real_real] :
      ( finite_finite_real(A3)
     => ( ( A3 != bot_bot_set_real )
       => ( ! [X: real] :
              ( member_real(X,A3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,X)),aa_real_real(G,X)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,groups8097168146408367636l_real(F,A3)),groups8097168146408367636l_real(G,A3)) ) ) ) ).

% sum_strict_mono
tff(fact_7068_sum__strict__mono,axiom,
    ! [A3: set_int,F: fun_int_real,G: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( ( A3 != bot_bot_set_int )
       => ( ! [X: int] :
              ( aa_set_int_o(member_int(X),A3)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(F,X)),aa_int_real(G,X)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,groups8778361861064173332t_real(F,A3)),groups8778361861064173332t_real(G,A3)) ) ) ) ).

% sum_strict_mono
tff(fact_7069_sum__strict__mono,axiom,
    ! [A3: set_complex,F: fun_complex_rat,G: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A3)
     => ( ( A3 != bot_bot_set_complex )
       => ( ! [X: complex] :
              ( aa_set_complex_o(member_complex(X),A3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_complex_rat(F,X)),aa_complex_rat(G,X)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups5058264527183730370ex_rat(F,A3)),groups5058264527183730370ex_rat(G,A3)) ) ) ) ).

% sum_strict_mono
tff(fact_7070_sum__strict__mono,axiom,
    ! [A3: set_real,F: fun_real_rat,G: fun_real_rat] :
      ( finite_finite_real(A3)
     => ( ( A3 != bot_bot_set_real )
       => ( ! [X: real] :
              ( member_real(X,A3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,X)),aa_real_rat(G,X)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups1300246762558778688al_rat(F,A3)),groups1300246762558778688al_rat(G,A3)) ) ) ) ).

% sum_strict_mono
tff(fact_7071_sum__strict__mono,axiom,
    ! [A3: set_nat,F: fun_nat_rat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( ( A3 != bot_bot_set_nat )
       => ( ! [X: nat] :
              ( aa_set_nat_o(member_nat(X),A3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,X)),aa_nat_rat(G,X)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups2906978787729119204at_rat(F,A3)),groups2906978787729119204at_rat(G,A3)) ) ) ) ).

% sum_strict_mono
tff(fact_7072_sum__strict__mono,axiom,
    ! [A3: set_int,F: fun_int_rat,G: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( ( A3 != bot_bot_set_int )
       => ( ! [X: int] :
              ( aa_set_int_o(member_int(X),A3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(F,X)),aa_int_rat(G,X)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups3906332499630173760nt_rat(F,A3)),groups3906332499630173760nt_rat(G,A3)) ) ) ) ).

% sum_strict_mono
tff(fact_7073_sum__strict__mono,axiom,
    ! [A3: set_complex,F: fun_complex_nat,G: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A3)
     => ( ( A3 != bot_bot_set_complex )
       => ( ! [X: complex] :
              ( aa_set_complex_o(member_complex(X),A3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_complex_nat(F,X)),aa_complex_nat(G,X)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups5693394587270226106ex_nat(F,A3)),groups5693394587270226106ex_nat(G,A3)) ) ) ) ).

% sum_strict_mono
tff(fact_7074_sum__strict__mono,axiom,
    ! [A3: set_real,F: fun_real_nat,G: fun_real_nat] :
      ( finite_finite_real(A3)
     => ( ( A3 != bot_bot_set_real )
       => ( ! [X: real] :
              ( member_real(X,A3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_real_nat(F,X)),aa_real_nat(G,X)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups1935376822645274424al_nat(F,A3)),groups1935376822645274424al_nat(G,A3)) ) ) ) ).

% sum_strict_mono
tff(fact_7075_sum__strict__mono,axiom,
    ! [A3: set_int,F: fun_int_nat,G: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( ( A3 != bot_bot_set_int )
       => ( ! [X: int] :
              ( aa_set_int_o(member_int(X),A3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(F,X)),aa_int_nat(G,X)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups4541462559716669496nt_nat(F,A3)),groups4541462559716669496nt_nat(G,A3)) ) ) ) ).

% sum_strict_mono
tff(fact_7076_zero__less__binomial,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K)) ) ).

% zero_less_binomial
tff(fact_7077_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_real,T3: set_real,S2: set_real,I: fun_real_real,J: fun_real_real,T: set_real,G: fun_real_real,H: fun_real_real] :
      ( finite_finite_real(S4)
     => ( finite_finite_real(T3)
       => ( ! [A2: real] :
              ( member_real(A2,minus_minus_set_real(S2,S4))
             => ( aa_real_real(I,aa_real_real(J,A2)) = A2 ) )
         => ( ! [A2: real] :
                ( member_real(A2,minus_minus_set_real(S2,S4))
               => member_real(aa_real_real(J,A2),minus_minus_set_real(T,T3)) )
           => ( ! [B2: real] :
                  ( member_real(B2,minus_minus_set_real(T,T3))
                 => ( aa_real_real(J,aa_real_real(I,B2)) = B2 ) )
             => ( ! [B2: real] :
                    ( member_real(B2,minus_minus_set_real(T,T3))
                   => member_real(aa_real_real(I,B2),minus_minus_set_real(S2,S4)) )
               => ( ! [A2: real] :
                      ( member_real(A2,S4)
                     => ( aa_real_real(G,A2) = zero_zero_real ) )
                 => ( ! [B2: real] :
                        ( member_real(B2,T3)
                       => ( aa_real_real(H,B2) = zero_zero_real ) )
                   => ( ! [A2: real] :
                          ( member_real(A2,S2)
                         => ( aa_real_real(H,aa_real_real(J,A2)) = aa_real_real(G,A2) ) )
                     => ( groups8097168146408367636l_real(G,S2) = groups8097168146408367636l_real(H,T) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_7078_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_real,T3: set_int,S2: set_real,I: fun_int_real,J: fun_real_int,T: set_int,G: fun_real_real,H: fun_int_real] :
      ( finite_finite_real(S4)
     => ( aa_set_int_o(finite_finite_int,T3)
       => ( ! [A2: real] :
              ( member_real(A2,minus_minus_set_real(S2,S4))
             => ( aa_int_real(I,aa_real_int(J,A2)) = A2 ) )
         => ( ! [A2: real] :
                ( member_real(A2,minus_minus_set_real(S2,S4))
               => aa_set_int_o(member_int(aa_real_int(J,A2)),minus_minus_set_int(T,T3)) )
           => ( ! [B2: int] :
                  ( aa_set_int_o(member_int(B2),minus_minus_set_int(T,T3))
                 => ( aa_real_int(J,aa_int_real(I,B2)) = B2 ) )
             => ( ! [B2: int] :
                    ( aa_set_int_o(member_int(B2),minus_minus_set_int(T,T3))
                   => member_real(aa_int_real(I,B2),minus_minus_set_real(S2,S4)) )
               => ( ! [A2: real] :
                      ( member_real(A2,S4)
                     => ( aa_real_real(G,A2) = zero_zero_real ) )
                 => ( ! [B2: int] :
                        ( aa_set_int_o(member_int(B2),T3)
                       => ( aa_int_real(H,B2) = zero_zero_real ) )
                   => ( ! [A2: real] :
                          ( member_real(A2,S2)
                         => ( aa_int_real(H,aa_real_int(J,A2)) = aa_real_real(G,A2) ) )
                     => ( groups8097168146408367636l_real(G,S2) = groups8778361861064173332t_real(H,T) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_7079_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_real,T3: set_complex,S2: set_real,I: fun_complex_real,J: fun_real_complex,T: set_complex,G: fun_real_real,H: fun_complex_real] :
      ( finite_finite_real(S4)
     => ( aa_set_complex_o(finite3207457112153483333omplex,T3)
       => ( ! [A2: real] :
              ( member_real(A2,minus_minus_set_real(S2,S4))
             => ( aa_complex_real(I,aa_real_complex(J,A2)) = A2 ) )
         => ( ! [A2: real] :
                ( member_real(A2,minus_minus_set_real(S2,S4))
               => aa_set_complex_o(member_complex(aa_real_complex(J,A2)),minus_811609699411566653omplex(T,T3)) )
           => ( ! [B2: complex] :
                  ( aa_set_complex_o(member_complex(B2),minus_811609699411566653omplex(T,T3))
                 => ( aa_real_complex(J,aa_complex_real(I,B2)) = B2 ) )
             => ( ! [B2: complex] :
                    ( aa_set_complex_o(member_complex(B2),minus_811609699411566653omplex(T,T3))
                   => member_real(aa_complex_real(I,B2),minus_minus_set_real(S2,S4)) )
               => ( ! [A2: real] :
                      ( member_real(A2,S4)
                     => ( aa_real_real(G,A2) = zero_zero_real ) )
                 => ( ! [B2: complex] :
                        ( aa_set_complex_o(member_complex(B2),T3)
                       => ( aa_complex_real(H,B2) = zero_zero_real ) )
                   => ( ! [A2: real] :
                          ( member_real(A2,S2)
                         => ( aa_complex_real(H,aa_real_complex(J,A2)) = aa_real_real(G,A2) ) )
                     => ( groups8097168146408367636l_real(G,S2) = groups5808333547571424918x_real(H,T) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_7080_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_int,T3: set_real,S2: set_int,I: fun_real_int,J: fun_int_real,T: set_real,G: fun_int_real,H: fun_real_real] :
      ( aa_set_int_o(finite_finite_int,S4)
     => ( finite_finite_real(T3)
       => ( ! [A2: int] :
              ( aa_set_int_o(member_int(A2),minus_minus_set_int(S2,S4))
             => ( aa_real_int(I,aa_int_real(J,A2)) = A2 ) )
         => ( ! [A2: int] :
                ( aa_set_int_o(member_int(A2),minus_minus_set_int(S2,S4))
               => member_real(aa_int_real(J,A2),minus_minus_set_real(T,T3)) )
           => ( ! [B2: real] :
                  ( member_real(B2,minus_minus_set_real(T,T3))
                 => ( aa_int_real(J,aa_real_int(I,B2)) = B2 ) )
             => ( ! [B2: real] :
                    ( member_real(B2,minus_minus_set_real(T,T3))
                   => aa_set_int_o(member_int(aa_real_int(I,B2)),minus_minus_set_int(S2,S4)) )
               => ( ! [A2: int] :
                      ( aa_set_int_o(member_int(A2),S4)
                     => ( aa_int_real(G,A2) = zero_zero_real ) )
                 => ( ! [B2: real] :
                        ( member_real(B2,T3)
                       => ( aa_real_real(H,B2) = zero_zero_real ) )
                   => ( ! [A2: int] :
                          ( aa_set_int_o(member_int(A2),S2)
                         => ( aa_real_real(H,aa_int_real(J,A2)) = aa_int_real(G,A2) ) )
                     => ( groups8778361861064173332t_real(G,S2) = groups8097168146408367636l_real(H,T) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_7081_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_int,T3: set_int,S2: set_int,I: fun_int_int,J: fun_int_int,T: set_int,G: fun_int_real,H: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,S4)
     => ( aa_set_int_o(finite_finite_int,T3)
       => ( ! [A2: int] :
              ( aa_set_int_o(member_int(A2),minus_minus_set_int(S2,S4))
             => ( aa_int_int(I,aa_int_int(J,A2)) = A2 ) )
         => ( ! [A2: int] :
                ( aa_set_int_o(member_int(A2),minus_minus_set_int(S2,S4))
               => aa_set_int_o(member_int(aa_int_int(J,A2)),minus_minus_set_int(T,T3)) )
           => ( ! [B2: int] :
                  ( aa_set_int_o(member_int(B2),minus_minus_set_int(T,T3))
                 => ( aa_int_int(J,aa_int_int(I,B2)) = B2 ) )
             => ( ! [B2: int] :
                    ( aa_set_int_o(member_int(B2),minus_minus_set_int(T,T3))
                   => aa_set_int_o(member_int(aa_int_int(I,B2)),minus_minus_set_int(S2,S4)) )
               => ( ! [A2: int] :
                      ( aa_set_int_o(member_int(A2),S4)
                     => ( aa_int_real(G,A2) = zero_zero_real ) )
                 => ( ! [B2: int] :
                        ( aa_set_int_o(member_int(B2),T3)
                       => ( aa_int_real(H,B2) = zero_zero_real ) )
                   => ( ! [A2: int] :
                          ( aa_set_int_o(member_int(A2),S2)
                         => ( aa_int_real(H,aa_int_int(J,A2)) = aa_int_real(G,A2) ) )
                     => ( groups8778361861064173332t_real(G,S2) = groups8778361861064173332t_real(H,T) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_7082_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_int,T3: set_complex,S2: set_int,I: fun_complex_int,J: fun_int_complex,T: set_complex,G: fun_int_real,H: fun_complex_real] :
      ( aa_set_int_o(finite_finite_int,S4)
     => ( aa_set_complex_o(finite3207457112153483333omplex,T3)
       => ( ! [A2: int] :
              ( aa_set_int_o(member_int(A2),minus_minus_set_int(S2,S4))
             => ( aa_complex_int(I,aa_int_complex(J,A2)) = A2 ) )
         => ( ! [A2: int] :
                ( aa_set_int_o(member_int(A2),minus_minus_set_int(S2,S4))
               => aa_set_complex_o(member_complex(aa_int_complex(J,A2)),minus_811609699411566653omplex(T,T3)) )
           => ( ! [B2: complex] :
                  ( aa_set_complex_o(member_complex(B2),minus_811609699411566653omplex(T,T3))
                 => ( aa_int_complex(J,aa_complex_int(I,B2)) = B2 ) )
             => ( ! [B2: complex] :
                    ( aa_set_complex_o(member_complex(B2),minus_811609699411566653omplex(T,T3))
                   => aa_set_int_o(member_int(aa_complex_int(I,B2)),minus_minus_set_int(S2,S4)) )
               => ( ! [A2: int] :
                      ( aa_set_int_o(member_int(A2),S4)
                     => ( aa_int_real(G,A2) = zero_zero_real ) )
                 => ( ! [B2: complex] :
                        ( aa_set_complex_o(member_complex(B2),T3)
                       => ( aa_complex_real(H,B2) = zero_zero_real ) )
                   => ( ! [A2: int] :
                          ( aa_set_int_o(member_int(A2),S2)
                         => ( aa_complex_real(H,aa_int_complex(J,A2)) = aa_int_real(G,A2) ) )
                     => ( groups8778361861064173332t_real(G,S2) = groups5808333547571424918x_real(H,T) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_7083_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_complex,T3: set_real,S2: set_complex,I: fun_real_complex,J: fun_complex_real,T: set_real,G: fun_complex_real,H: fun_real_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S4)
     => ( finite_finite_real(T3)
       => ( ! [A2: complex] :
              ( aa_set_complex_o(member_complex(A2),minus_811609699411566653omplex(S2,S4))
             => ( aa_real_complex(I,aa_complex_real(J,A2)) = A2 ) )
         => ( ! [A2: complex] :
                ( aa_set_complex_o(member_complex(A2),minus_811609699411566653omplex(S2,S4))
               => member_real(aa_complex_real(J,A2),minus_minus_set_real(T,T3)) )
           => ( ! [B2: real] :
                  ( member_real(B2,minus_minus_set_real(T,T3))
                 => ( aa_complex_real(J,aa_real_complex(I,B2)) = B2 ) )
             => ( ! [B2: real] :
                    ( member_real(B2,minus_minus_set_real(T,T3))
                   => aa_set_complex_o(member_complex(aa_real_complex(I,B2)),minus_811609699411566653omplex(S2,S4)) )
               => ( ! [A2: complex] :
                      ( aa_set_complex_o(member_complex(A2),S4)
                     => ( aa_complex_real(G,A2) = zero_zero_real ) )
                 => ( ! [B2: real] :
                        ( member_real(B2,T3)
                       => ( aa_real_real(H,B2) = zero_zero_real ) )
                   => ( ! [A2: complex] :
                          ( aa_set_complex_o(member_complex(A2),S2)
                         => ( aa_real_real(H,aa_complex_real(J,A2)) = aa_complex_real(G,A2) ) )
                     => ( groups5808333547571424918x_real(G,S2) = groups8097168146408367636l_real(H,T) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_7084_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_complex,T3: set_int,S2: set_complex,I: fun_int_complex,J: fun_complex_int,T: set_int,G: fun_complex_real,H: fun_int_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S4)
     => ( aa_set_int_o(finite_finite_int,T3)
       => ( ! [A2: complex] :
              ( aa_set_complex_o(member_complex(A2),minus_811609699411566653omplex(S2,S4))
             => ( aa_int_complex(I,aa_complex_int(J,A2)) = A2 ) )
         => ( ! [A2: complex] :
                ( aa_set_complex_o(member_complex(A2),minus_811609699411566653omplex(S2,S4))
               => aa_set_int_o(member_int(aa_complex_int(J,A2)),minus_minus_set_int(T,T3)) )
           => ( ! [B2: int] :
                  ( aa_set_int_o(member_int(B2),minus_minus_set_int(T,T3))
                 => ( aa_complex_int(J,aa_int_complex(I,B2)) = B2 ) )
             => ( ! [B2: int] :
                    ( aa_set_int_o(member_int(B2),minus_minus_set_int(T,T3))
                   => aa_set_complex_o(member_complex(aa_int_complex(I,B2)),minus_811609699411566653omplex(S2,S4)) )
               => ( ! [A2: complex] :
                      ( aa_set_complex_o(member_complex(A2),S4)
                     => ( aa_complex_real(G,A2) = zero_zero_real ) )
                 => ( ! [B2: int] :
                        ( aa_set_int_o(member_int(B2),T3)
                       => ( aa_int_real(H,B2) = zero_zero_real ) )
                   => ( ! [A2: complex] :
                          ( aa_set_complex_o(member_complex(A2),S2)
                         => ( aa_int_real(H,aa_complex_int(J,A2)) = aa_complex_real(G,A2) ) )
                     => ( groups5808333547571424918x_real(G,S2) = groups8778361861064173332t_real(H,T) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_7085_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_complex,T3: set_complex,S2: set_complex,I: fun_complex_complex,J: fun_complex_complex,T: set_complex,G: fun_complex_real,H: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S4)
     => ( aa_set_complex_o(finite3207457112153483333omplex,T3)
       => ( ! [A2: complex] :
              ( aa_set_complex_o(member_complex(A2),minus_811609699411566653omplex(S2,S4))
             => ( aa_complex_complex(I,aa_complex_complex(J,A2)) = A2 ) )
         => ( ! [A2: complex] :
                ( aa_set_complex_o(member_complex(A2),minus_811609699411566653omplex(S2,S4))
               => aa_set_complex_o(member_complex(aa_complex_complex(J,A2)),minus_811609699411566653omplex(T,T3)) )
           => ( ! [B2: complex] :
                  ( aa_set_complex_o(member_complex(B2),minus_811609699411566653omplex(T,T3))
                 => ( aa_complex_complex(J,aa_complex_complex(I,B2)) = B2 ) )
             => ( ! [B2: complex] :
                    ( aa_set_complex_o(member_complex(B2),minus_811609699411566653omplex(T,T3))
                   => aa_set_complex_o(member_complex(aa_complex_complex(I,B2)),minus_811609699411566653omplex(S2,S4)) )
               => ( ! [A2: complex] :
                      ( aa_set_complex_o(member_complex(A2),S4)
                     => ( aa_complex_real(G,A2) = zero_zero_real ) )
                 => ( ! [B2: complex] :
                        ( aa_set_complex_o(member_complex(B2),T3)
                       => ( aa_complex_real(H,B2) = zero_zero_real ) )
                   => ( ! [A2: complex] :
                          ( aa_set_complex_o(member_complex(A2),S2)
                         => ( aa_complex_real(H,aa_complex_complex(J,A2)) = aa_complex_real(G,A2) ) )
                     => ( groups5808333547571424918x_real(G,S2) = groups5808333547571424918x_real(H,T) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_7086_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_real,T3: set_real,S2: set_real,I: fun_real_real,J: fun_real_real,T: set_real,G: fun_real_rat,H: fun_real_rat] :
      ( finite_finite_real(S4)
     => ( finite_finite_real(T3)
       => ( ! [A2: real] :
              ( member_real(A2,minus_minus_set_real(S2,S4))
             => ( aa_real_real(I,aa_real_real(J,A2)) = A2 ) )
         => ( ! [A2: real] :
                ( member_real(A2,minus_minus_set_real(S2,S4))
               => member_real(aa_real_real(J,A2),minus_minus_set_real(T,T3)) )
           => ( ! [B2: real] :
                  ( member_real(B2,minus_minus_set_real(T,T3))
                 => ( aa_real_real(J,aa_real_real(I,B2)) = B2 ) )
             => ( ! [B2: real] :
                    ( member_real(B2,minus_minus_set_real(T,T3))
                   => member_real(aa_real_real(I,B2),minus_minus_set_real(S2,S4)) )
               => ( ! [A2: real] :
                      ( member_real(A2,S4)
                     => ( aa_real_rat(G,A2) = zero_zero_rat ) )
                 => ( ! [B2: real] :
                        ( member_real(B2,T3)
                       => ( aa_real_rat(H,B2) = zero_zero_rat ) )
                   => ( ! [A2: real] :
                          ( member_real(A2,S2)
                         => ( aa_real_rat(H,aa_real_real(J,A2)) = aa_real_rat(G,A2) ) )
                     => ( groups1300246762558778688al_rat(G,S2) = groups1300246762558778688al_rat(H,T) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_7087_Suc__times__binomial__add,axiom,
    ! [Aa: nat,Ba: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,Aa)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),Ba))),aa_nat_nat(suc,Aa))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,Ba)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),Ba))),Aa)) ).

% Suc_times_binomial_add
tff(fact_7088_binomial__Suc__Suc__eq__times,axiom,
    ! [Na: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,Na)),aa_nat_nat(suc,K)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,Na)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K))),aa_nat_nat(suc,K)) ).

% binomial_Suc_Suc_eq_times
tff(fact_7089_choose__mult,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),M)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,M),K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K))) ) ) ) ).

% choose_mult
tff(fact_7090_binomial__absorb__comp,axiom,
    ! [Na: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat)),K)) ).

% binomial_absorb_comp
tff(fact_7091_take__bit__signed__take__bit,axiom,
    ! [M: nat,Na: nat,Aa: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,Na))
     => ( aa_Cod1783974652533396728nteger(aa_nat6383147911832638789nteger(bit_se1745604003318907178nteger,M),aa_Cod1783974652533396728nteger(bit_ri6519982836138164636nteger(Na),Aa)) = aa_Cod1783974652533396728nteger(aa_nat6383147911832638789nteger(bit_se1745604003318907178nteger,M),Aa) ) ) ).

% take_bit_signed_take_bit
tff(fact_7092_take__bit__signed__take__bit,axiom,
    ! [M: nat,Na: nat,Aa: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,Na))
     => ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,M),aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,Na),Aa)) = aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,M),Aa) ) ) ).

% take_bit_signed_take_bit
tff(fact_7093_pochhammer__nonneg,axiom,
    ! [Xb: real,Na: 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),Na)) ) ).

% pochhammer_nonneg
tff(fact_7094_pochhammer__nonneg,axiom,
    ! [Xb: rat,Na: 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),Na)) ) ).

% pochhammer_nonneg
tff(fact_7095_pochhammer__nonneg,axiom,
    ! [Xb: nat,Na: 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),Na)) ) ).

% pochhammer_nonneg
tff(fact_7096_pochhammer__nonneg,axiom,
    ! [Xb: int,Na: 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),Na)) ) ).

% pochhammer_nonneg
tff(fact_7097_sum__nonneg__leq__bound,axiom,
    ! [S3: set_real,F: fun_real_real,B4: real,I: real] :
      ( finite_finite_real(S3)
     => ( ! [I2: real] :
            ( member_real(I2,S3)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,I2)) )
       => ( ( groups8097168146408367636l_real(F,S3) = B4 )
         => ( member_real(I,S3)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,I)),B4) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_7098_sum__nonneg__leq__bound,axiom,
    ! [S3: set_int,F: fun_int_real,B4: real,I: int] :
      ( aa_set_int_o(finite_finite_int,S3)
     => ( ! [I2: int] :
            ( aa_set_int_o(member_int(I2),S3)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(F,I2)) )
       => ( ( groups8778361861064173332t_real(F,S3) = B4 )
         => ( aa_set_int_o(member_int(I),S3)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(F,I)),B4) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_7099_sum__nonneg__leq__bound,axiom,
    ! [S3: set_complex,F: fun_complex_real,B4: real,I: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S3)
     => ( ! [I2: complex] :
            ( aa_set_complex_o(member_complex(I2),S3)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(F,I2)) )
       => ( ( groups5808333547571424918x_real(F,S3) = B4 )
         => ( aa_set_complex_o(member_complex(I),S3)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(F,I)),B4) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_7100_sum__nonneg__leq__bound,axiom,
    ! [S3: set_real,F: fun_real_rat,B4: rat,I: real] :
      ( finite_finite_real(S3)
     => ( ! [I2: real] :
            ( member_real(I2,S3)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,I2)) )
       => ( ( groups1300246762558778688al_rat(F,S3) = B4 )
         => ( member_real(I,S3)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_real_rat(F,I)),B4) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_7101_sum__nonneg__leq__bound,axiom,
    ! [S3: set_nat,F: fun_nat_rat,B4: rat,I: nat] :
      ( aa_set_nat_o(finite_finite_nat,S3)
     => ( ! [I2: nat] :
            ( aa_set_nat_o(member_nat(I2),S3)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,I2)) )
       => ( ( groups2906978787729119204at_rat(F,S3) = B4 )
         => ( aa_set_nat_o(member_nat(I),S3)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,I)),B4) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_7102_sum__nonneg__leq__bound,axiom,
    ! [S3: set_int,F: fun_int_rat,B4: rat,I: int] :
      ( aa_set_int_o(finite_finite_int,S3)
     => ( ! [I2: int] :
            ( aa_set_int_o(member_int(I2),S3)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,I2)) )
       => ( ( groups3906332499630173760nt_rat(F,S3) = B4 )
         => ( aa_set_int_o(member_int(I),S3)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,I)),B4) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_7103_sum__nonneg__leq__bound,axiom,
    ! [S3: set_complex,F: fun_complex_rat,B4: rat,I: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S3)
     => ( ! [I2: complex] :
            ( aa_set_complex_o(member_complex(I2),S3)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,I2)) )
       => ( ( groups5058264527183730370ex_rat(F,S3) = B4 )
         => ( aa_set_complex_o(member_complex(I),S3)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_complex_rat(F,I)),B4) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_7104_sum__nonneg__leq__bound,axiom,
    ! [S3: set_real,F: fun_real_nat,B4: nat,I: real] :
      ( finite_finite_real(S3)
     => ( ! [I2: real] :
            ( member_real(I2,S3)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,I2)) )
       => ( ( groups1935376822645274424al_nat(F,S3) = B4 )
         => ( member_real(I,S3)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_real_nat(F,I)),B4) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_7105_sum__nonneg__leq__bound,axiom,
    ! [S3: set_int,F: fun_int_nat,B4: nat,I: int] :
      ( aa_set_int_o(finite_finite_int,S3)
     => ( ! [I2: int] :
            ( aa_set_int_o(member_int(I2),S3)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F,I2)) )
       => ( ( groups4541462559716669496nt_nat(F,S3) = B4 )
         => ( aa_set_int_o(member_int(I),S3)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,I)),B4) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_7106_sum__nonneg__leq__bound,axiom,
    ! [S3: set_complex,F: fun_complex_nat,B4: nat,I: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S3)
     => ( ! [I2: complex] :
            ( aa_set_complex_o(member_complex(I2),S3)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_complex_nat(F,I2)) )
       => ( ( groups5693394587270226106ex_nat(F,S3) = B4 )
         => ( aa_set_complex_o(member_complex(I),S3)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_complex_nat(F,I)),B4) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_7107_sum__nonneg__0,axiom,
    ! [S3: set_real,F: fun_real_real,I: real] :
      ( finite_finite_real(S3)
     => ( ! [I2: real] :
            ( member_real(I2,S3)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,I2)) )
       => ( ( groups8097168146408367636l_real(F,S3) = zero_zero_real )
         => ( member_real(I,S3)
           => ( aa_real_real(F,I) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_7108_sum__nonneg__0,axiom,
    ! [S3: set_int,F: fun_int_real,I: int] :
      ( aa_set_int_o(finite_finite_int,S3)
     => ( ! [I2: int] :
            ( aa_set_int_o(member_int(I2),S3)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(F,I2)) )
       => ( ( groups8778361861064173332t_real(F,S3) = zero_zero_real )
         => ( aa_set_int_o(member_int(I),S3)
           => ( aa_int_real(F,I) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_7109_sum__nonneg__0,axiom,
    ! [S3: set_complex,F: fun_complex_real,I: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S3)
     => ( ! [I2: complex] :
            ( aa_set_complex_o(member_complex(I2),S3)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(F,I2)) )
       => ( ( groups5808333547571424918x_real(F,S3) = zero_zero_real )
         => ( aa_set_complex_o(member_complex(I),S3)
           => ( aa_complex_real(F,I) = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_7110_sum__nonneg__0,axiom,
    ! [S3: set_real,F: fun_real_rat,I: real] :
      ( finite_finite_real(S3)
     => ( ! [I2: real] :
            ( member_real(I2,S3)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,I2)) )
       => ( ( groups1300246762558778688al_rat(F,S3) = zero_zero_rat )
         => ( member_real(I,S3)
           => ( aa_real_rat(F,I) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_7111_sum__nonneg__0,axiom,
    ! [S3: set_nat,F: fun_nat_rat,I: nat] :
      ( aa_set_nat_o(finite_finite_nat,S3)
     => ( ! [I2: nat] :
            ( aa_set_nat_o(member_nat(I2),S3)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,I2)) )
       => ( ( groups2906978787729119204at_rat(F,S3) = zero_zero_rat )
         => ( aa_set_nat_o(member_nat(I),S3)
           => ( aa_nat_rat(F,I) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_7112_sum__nonneg__0,axiom,
    ! [S3: set_int,F: fun_int_rat,I: int] :
      ( aa_set_int_o(finite_finite_int,S3)
     => ( ! [I2: int] :
            ( aa_set_int_o(member_int(I2),S3)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,I2)) )
       => ( ( groups3906332499630173760nt_rat(F,S3) = zero_zero_rat )
         => ( aa_set_int_o(member_int(I),S3)
           => ( aa_int_rat(F,I) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_7113_sum__nonneg__0,axiom,
    ! [S3: set_complex,F: fun_complex_rat,I: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S3)
     => ( ! [I2: complex] :
            ( aa_set_complex_o(member_complex(I2),S3)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,I2)) )
       => ( ( groups5058264527183730370ex_rat(F,S3) = zero_zero_rat )
         => ( aa_set_complex_o(member_complex(I),S3)
           => ( aa_complex_rat(F,I) = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_7114_sum__nonneg__0,axiom,
    ! [S3: set_real,F: fun_real_nat,I: real] :
      ( finite_finite_real(S3)
     => ( ! [I2: real] :
            ( member_real(I2,S3)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,I2)) )
       => ( ( groups1935376822645274424al_nat(F,S3) = zero_zero_nat )
         => ( member_real(I,S3)
           => ( aa_real_nat(F,I) = zero_zero_nat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_7115_sum__nonneg__0,axiom,
    ! [S3: set_int,F: fun_int_nat,I: int] :
      ( aa_set_int_o(finite_finite_int,S3)
     => ( ! [I2: int] :
            ( aa_set_int_o(member_int(I2),S3)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F,I2)) )
       => ( ( groups4541462559716669496nt_nat(F,S3) = zero_zero_nat )
         => ( aa_set_int_o(member_int(I),S3)
           => ( aa_int_nat(F,I) = zero_zero_nat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_7116_sum__nonneg__0,axiom,
    ! [S3: set_complex,F: fun_complex_nat,I: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S3)
     => ( ! [I2: complex] :
            ( aa_set_complex_o(member_complex(I2),S3)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_complex_nat(F,I2)) )
       => ( ( groups5693394587270226106ex_nat(F,S3) = zero_zero_nat )
         => ( aa_set_complex_o(member_complex(I),S3)
           => ( aa_complex_nat(F,I) = zero_zero_nat ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_7117_pochhammer__0__left,axiom,
    ! [Na: nat] :
      aa_nat_complex(comm_s2602460028002588243omplex(zero_zero_complex),Na) = $ite(Na = zero_zero_nat,one_one_complex,zero_zero_complex) ).

% pochhammer_0_left
tff(fact_7118_pochhammer__0__left,axiom,
    ! [Na: nat] :
      aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,zero_zero_real),Na) = $ite(Na = zero_zero_nat,one_one_real,zero_zero_real) ).

% pochhammer_0_left
tff(fact_7119_pochhammer__0__left,axiom,
    ! [Na: nat] :
      aa_nat_rat(comm_s4028243227959126397er_rat(zero_zero_rat),Na) = $ite(Na = zero_zero_nat,one_one_rat,zero_zero_rat) ).

% pochhammer_0_left
tff(fact_7120_pochhammer__0__left,axiom,
    ! [Na: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,zero_zero_nat),Na) = $ite(Na = zero_zero_nat,one_one_nat,zero_zero_nat) ).

% pochhammer_0_left
tff(fact_7121_pochhammer__0__left,axiom,
    ! [Na: nat] :
      aa_nat_int(comm_s4660882817536571857er_int(zero_zero_int),Na) = $ite(Na = zero_zero_nat,one_one_int,zero_zero_int) ).

% pochhammer_0_left
tff(fact_7122_sum_Osetdiff__irrelevant,axiom,
    ! [A3: set_real,G: fun_real_real] :
      ( finite_finite_real(A3)
     => ( groups8097168146408367636l_real(G,minus_minus_set_real(A3,collect_real(aTP_Lamm_ia(G)))) = groups8097168146408367636l_real(G,A3) ) ) ).

% sum.setdiff_irrelevant
tff(fact_7123_sum_Osetdiff__irrelevant,axiom,
    ! [A3: set_int,G: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( groups8778361861064173332t_real(G,minus_minus_set_int(A3,collect_int(aTP_Lamm_ib(G)))) = groups8778361861064173332t_real(G,A3) ) ) ).

% sum.setdiff_irrelevant
tff(fact_7124_sum_Osetdiff__irrelevant,axiom,
    ! [A3: set_complex,G: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A3)
     => ( groups5808333547571424918x_real(G,minus_811609699411566653omplex(A3,collect_complex(aTP_Lamm_ic(G)))) = groups5808333547571424918x_real(G,A3) ) ) ).

% sum.setdiff_irrelevant
tff(fact_7125_sum_Osetdiff__irrelevant,axiom,
    ! [A3: set_real,G: fun_real_rat] :
      ( finite_finite_real(A3)
     => ( groups1300246762558778688al_rat(G,minus_minus_set_real(A3,collect_real(aTP_Lamm_id(G)))) = groups1300246762558778688al_rat(G,A3) ) ) ).

% sum.setdiff_irrelevant
tff(fact_7126_sum_Osetdiff__irrelevant,axiom,
    ! [A3: set_int,G: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( groups3906332499630173760nt_rat(G,minus_minus_set_int(A3,collect_int(aTP_Lamm_ie(G)))) = groups3906332499630173760nt_rat(G,A3) ) ) ).

% sum.setdiff_irrelevant
tff(fact_7127_sum_Osetdiff__irrelevant,axiom,
    ! [A3: set_complex,G: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A3)
     => ( groups5058264527183730370ex_rat(G,minus_811609699411566653omplex(A3,collect_complex(aTP_Lamm_if(G)))) = groups5058264527183730370ex_rat(G,A3) ) ) ).

% sum.setdiff_irrelevant
tff(fact_7128_sum_Osetdiff__irrelevant,axiom,
    ! [A3: set_real,G: fun_real_nat] :
      ( finite_finite_real(A3)
     => ( groups1935376822645274424al_nat(G,minus_minus_set_real(A3,collect_real(aTP_Lamm_ig(G)))) = groups1935376822645274424al_nat(G,A3) ) ) ).

% sum.setdiff_irrelevant
tff(fact_7129_sum_Osetdiff__irrelevant,axiom,
    ! [A3: set_int,G: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( groups4541462559716669496nt_nat(G,minus_minus_set_int(A3,collect_int(aTP_Lamm_ih(G)))) = groups4541462559716669496nt_nat(G,A3) ) ) ).

% sum.setdiff_irrelevant
tff(fact_7130_sum_Osetdiff__irrelevant,axiom,
    ! [A3: set_complex,G: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A3)
     => ( groups5693394587270226106ex_nat(G,minus_811609699411566653omplex(A3,collect_complex(aTP_Lamm_ii(G)))) = groups5693394587270226106ex_nat(G,A3) ) ) ).

% sum.setdiff_irrelevant
tff(fact_7131_sum_Osetdiff__irrelevant,axiom,
    ! [A3: set_real,G: fun_real_int] :
      ( finite_finite_real(A3)
     => ( groups1932886352136224148al_int(G,minus_minus_set_real(A3,collect_real(aTP_Lamm_ij(G)))) = groups1932886352136224148al_int(G,A3) ) ) ).

% sum.setdiff_irrelevant
tff(fact_7132_sum__pos2,axiom,
    ! [I5: set_real,I: real,F: fun_real_real] :
      ( finite_finite_real(I5)
     => ( member_real(I,I5)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(F,I))
         => ( ! [I2: real] :
                ( member_real(I2,I5)
               => 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_real,zero_zero_real),groups8097168146408367636l_real(F,I5)) ) ) ) ) ).

% sum_pos2
tff(fact_7133_sum__pos2,axiom,
    ! [I5: set_int,I: int,F: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,I5)
     => ( aa_set_int_o(member_int(I),I5)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_int_real(F,I))
         => ( ! [I2: int] :
                ( aa_set_int_o(member_int(I2),I5)
               => 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_real,zero_zero_real),groups8778361861064173332t_real(F,I5)) ) ) ) ) ).

% sum_pos2
tff(fact_7134_sum__pos2,axiom,
    ! [I5: set_complex,I: complex,F: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,I5)
     => ( aa_set_complex_o(member_complex(I),I5)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(F,I))
         => ( ! [I2: complex] :
                ( aa_set_complex_o(member_complex(I2),I5)
               => 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_real,zero_zero_real),groups5808333547571424918x_real(F,I5)) ) ) ) ) ).

% sum_pos2
tff(fact_7135_sum__pos2,axiom,
    ! [I5: set_real,I: real,F: fun_real_rat] :
      ( finite_finite_real(I5)
     => ( member_real(I,I5)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_real_rat(F,I))
         => ( ! [I2: real] :
                ( member_real(I2,I5)
               => 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_rat,zero_zero_rat),groups1300246762558778688al_rat(F,I5)) ) ) ) ) ).

% sum_pos2
tff(fact_7136_sum__pos2,axiom,
    ! [I5: set_nat,I: nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,I5)
     => ( aa_set_nat_o(member_nat(I),I5)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(F,I))
         => ( ! [I2: nat] :
                ( aa_set_nat_o(member_nat(I2),I5)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,I2)) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),groups2906978787729119204at_rat(F,I5)) ) ) ) ) ).

% sum_pos2
tff(fact_7137_sum__pos2,axiom,
    ! [I5: set_int,I: int,F: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,I5)
     => ( aa_set_int_o(member_int(I),I5)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_int_rat(F,I))
         => ( ! [I2: int] :
                ( aa_set_int_o(member_int(I2),I5)
               => 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_rat,zero_zero_rat),groups3906332499630173760nt_rat(F,I5)) ) ) ) ) ).

% sum_pos2
tff(fact_7138_sum__pos2,axiom,
    ! [I5: set_complex,I: complex,F: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,I5)
     => ( aa_set_complex_o(member_complex(I),I5)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_complex_rat(F,I))
         => ( ! [I2: complex] :
                ( aa_set_complex_o(member_complex(I2),I5)
               => 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_rat,zero_zero_rat),groups5058264527183730370ex_rat(F,I5)) ) ) ) ) ).

% sum_pos2
tff(fact_7139_sum__pos2,axiom,
    ! [I5: set_real,I: real,F: fun_real_nat] :
      ( finite_finite_real(I5)
     => ( member_real(I,I5)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_real_nat(F,I))
         => ( ! [I2: real] :
                ( member_real(I2,I5)
               => 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_nat,zero_zero_nat),groups1935376822645274424al_nat(F,I5)) ) ) ) ) ).

% sum_pos2
tff(fact_7140_sum__pos2,axiom,
    ! [I5: set_int,I: int,F: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,I5)
     => ( aa_set_int_o(member_int(I),I5)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_int_nat(F,I))
         => ( ! [I2: int] :
                ( aa_set_int_o(member_int(I2),I5)
               => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F,I2)) )
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),groups4541462559716669496nt_nat(F,I5)) ) ) ) ) ).

% sum_pos2
tff(fact_7141_sum__pos2,axiom,
    ! [I5: set_complex,I: complex,F: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,I5)
     => ( aa_set_complex_o(member_complex(I),I5)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_complex_nat(F,I))
         => ( ! [I2: complex] :
                ( aa_set_complex_o(member_complex(I2),I5)
               => 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_nat,zero_zero_nat),groups5693394587270226106ex_nat(F,I5)) ) ) ) ) ).

% sum_pos2
tff(fact_7142_sum__pos,axiom,
    ! [I5: set_complex,F: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,I5)
     => ( ( I5 != bot_bot_set_complex )
       => ( ! [I2: complex] :
              ( aa_set_complex_o(member_complex(I2),I5)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(F,I2)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),groups5808333547571424918x_real(F,I5)) ) ) ) ).

% sum_pos
tff(fact_7143_sum__pos,axiom,
    ! [I5: set_real,F: fun_real_real] :
      ( finite_finite_real(I5)
     => ( ( I5 != bot_bot_set_real )
       => ( ! [I2: real] :
              ( member_real(I2,I5)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(F,I2)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),groups8097168146408367636l_real(F,I5)) ) ) ) ).

% sum_pos
tff(fact_7144_sum__pos,axiom,
    ! [I5: set_int,F: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,I5)
     => ( ( I5 != bot_bot_set_int )
       => ( ! [I2: int] :
              ( aa_set_int_o(member_int(I2),I5)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_int_real(F,I2)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),groups8778361861064173332t_real(F,I5)) ) ) ) ).

% sum_pos
tff(fact_7145_sum__pos,axiom,
    ! [I5: set_complex,F: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,I5)
     => ( ( I5 != bot_bot_set_complex )
       => ( ! [I2: complex] :
              ( aa_set_complex_o(member_complex(I2),I5)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_complex_rat(F,I2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),groups5058264527183730370ex_rat(F,I5)) ) ) ) ).

% sum_pos
tff(fact_7146_sum__pos,axiom,
    ! [I5: set_real,F: fun_real_rat] :
      ( finite_finite_real(I5)
     => ( ( I5 != bot_bot_set_real )
       => ( ! [I2: real] :
              ( member_real(I2,I5)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_real_rat(F,I2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),groups1300246762558778688al_rat(F,I5)) ) ) ) ).

% sum_pos
tff(fact_7147_sum__pos,axiom,
    ! [I5: set_nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,I5)
     => ( ( I5 != bot_bot_set_nat )
       => ( ! [I2: nat] :
              ( aa_set_nat_o(member_nat(I2),I5)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(F,I2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),groups2906978787729119204at_rat(F,I5)) ) ) ) ).

% sum_pos
tff(fact_7148_sum__pos,axiom,
    ! [I5: set_int,F: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,I5)
     => ( ( I5 != bot_bot_set_int )
       => ( ! [I2: int] :
              ( aa_set_int_o(member_int(I2),I5)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_int_rat(F,I2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),groups3906332499630173760nt_rat(F,I5)) ) ) ) ).

% sum_pos
tff(fact_7149_sum__pos,axiom,
    ! [I5: set_complex,F: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,I5)
     => ( ( I5 != bot_bot_set_complex )
       => ( ! [I2: complex] :
              ( aa_set_complex_o(member_complex(I2),I5)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_complex_nat(F,I2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),groups5693394587270226106ex_nat(F,I5)) ) ) ) ).

% sum_pos
tff(fact_7150_sum__pos,axiom,
    ! [I5: set_real,F: fun_real_nat] :
      ( finite_finite_real(I5)
     => ( ( I5 != bot_bot_set_real )
       => ( ! [I2: real] :
              ( member_real(I2,I5)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_real_nat(F,I2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),groups1935376822645274424al_nat(F,I5)) ) ) ) ).

% sum_pos
tff(fact_7151_sum__pos,axiom,
    ! [I5: set_int,F: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,I5)
     => ( ( I5 != bot_bot_set_int )
       => ( ! [I2: int] :
              ( aa_set_int_o(member_int(I2),I5)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_int_nat(F,I2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),groups4541462559716669496nt_nat(F,I5)) ) ) ) ).

% sum_pos
tff(fact_7152_sum_Osame__carrier,axiom,
    ! [C2: set_real,A3: set_real,B4: set_real,G: fun_real_real,H: fun_real_real] :
      ( finite_finite_real(C2)
     => ( ord_less_eq_set_real(A3,C2)
       => ( ord_less_eq_set_real(B4,C2)
         => ( ! [A2: real] :
                ( member_real(A2,minus_minus_set_real(C2,A3))
               => ( aa_real_real(G,A2) = zero_zero_real ) )
           => ( ! [B2: real] :
                  ( member_real(B2,minus_minus_set_real(C2,B4))
                 => ( aa_real_real(H,B2) = zero_zero_real ) )
             => ( ( groups8097168146408367636l_real(G,A3) = groups8097168146408367636l_real(H,B4) )
              <=> ( groups8097168146408367636l_real(G,C2) = groups8097168146408367636l_real(H,C2) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_7153_sum_Osame__carrier,axiom,
    ! [C2: set_complex,A3: set_complex,B4: set_complex,G: fun_complex_real,H: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,C2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A3),C2)
       => ( aa_set_complex_o(ord_le211207098394363844omplex(B4),C2)
         => ( ! [A2: complex] :
                ( aa_set_complex_o(member_complex(A2),minus_811609699411566653omplex(C2,A3))
               => ( aa_complex_real(G,A2) = zero_zero_real ) )
           => ( ! [B2: complex] :
                  ( aa_set_complex_o(member_complex(B2),minus_811609699411566653omplex(C2,B4))
                 => ( aa_complex_real(H,B2) = zero_zero_real ) )
             => ( ( groups5808333547571424918x_real(G,A3) = groups5808333547571424918x_real(H,B4) )
              <=> ( groups5808333547571424918x_real(G,C2) = groups5808333547571424918x_real(H,C2) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_7154_sum_Osame__carrier,axiom,
    ! [C2: set_real,A3: set_real,B4: set_real,G: fun_real_rat,H: fun_real_rat] :
      ( finite_finite_real(C2)
     => ( ord_less_eq_set_real(A3,C2)
       => ( ord_less_eq_set_real(B4,C2)
         => ( ! [A2: real] :
                ( member_real(A2,minus_minus_set_real(C2,A3))
               => ( aa_real_rat(G,A2) = zero_zero_rat ) )
           => ( ! [B2: real] :
                  ( member_real(B2,minus_minus_set_real(C2,B4))
                 => ( aa_real_rat(H,B2) = zero_zero_rat ) )
             => ( ( groups1300246762558778688al_rat(G,A3) = groups1300246762558778688al_rat(H,B4) )
              <=> ( groups1300246762558778688al_rat(G,C2) = groups1300246762558778688al_rat(H,C2) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_7155_sum_Osame__carrier,axiom,
    ! [C2: set_complex,A3: set_complex,B4: set_complex,G: fun_complex_rat,H: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,C2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A3),C2)
       => ( aa_set_complex_o(ord_le211207098394363844omplex(B4),C2)
         => ( ! [A2: complex] :
                ( aa_set_complex_o(member_complex(A2),minus_811609699411566653omplex(C2,A3))
               => ( aa_complex_rat(G,A2) = zero_zero_rat ) )
           => ( ! [B2: complex] :
                  ( aa_set_complex_o(member_complex(B2),minus_811609699411566653omplex(C2,B4))
                 => ( aa_complex_rat(H,B2) = zero_zero_rat ) )
             => ( ( groups5058264527183730370ex_rat(G,A3) = groups5058264527183730370ex_rat(H,B4) )
              <=> ( groups5058264527183730370ex_rat(G,C2) = groups5058264527183730370ex_rat(H,C2) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_7156_sum_Osame__carrier,axiom,
    ! [C2: set_real,A3: set_real,B4: set_real,G: fun_real_nat,H: fun_real_nat] :
      ( finite_finite_real(C2)
     => ( ord_less_eq_set_real(A3,C2)
       => ( ord_less_eq_set_real(B4,C2)
         => ( ! [A2: real] :
                ( member_real(A2,minus_minus_set_real(C2,A3))
               => ( aa_real_nat(G,A2) = zero_zero_nat ) )
           => ( ! [B2: real] :
                  ( member_real(B2,minus_minus_set_real(C2,B4))
                 => ( aa_real_nat(H,B2) = zero_zero_nat ) )
             => ( ( groups1935376822645274424al_nat(G,A3) = groups1935376822645274424al_nat(H,B4) )
              <=> ( groups1935376822645274424al_nat(G,C2) = groups1935376822645274424al_nat(H,C2) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_7157_sum_Osame__carrier,axiom,
    ! [C2: set_complex,A3: set_complex,B4: set_complex,G: fun_complex_nat,H: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,C2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A3),C2)
       => ( aa_set_complex_o(ord_le211207098394363844omplex(B4),C2)
         => ( ! [A2: complex] :
                ( aa_set_complex_o(member_complex(A2),minus_811609699411566653omplex(C2,A3))
               => ( aa_complex_nat(G,A2) = zero_zero_nat ) )
           => ( ! [B2: complex] :
                  ( aa_set_complex_o(member_complex(B2),minus_811609699411566653omplex(C2,B4))
                 => ( aa_complex_nat(H,B2) = zero_zero_nat ) )
             => ( ( groups5693394587270226106ex_nat(G,A3) = groups5693394587270226106ex_nat(H,B4) )
              <=> ( groups5693394587270226106ex_nat(G,C2) = groups5693394587270226106ex_nat(H,C2) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_7158_sum_Osame__carrier,axiom,
    ! [C2: set_real,A3: set_real,B4: set_real,G: fun_real_int,H: fun_real_int] :
      ( finite_finite_real(C2)
     => ( ord_less_eq_set_real(A3,C2)
       => ( ord_less_eq_set_real(B4,C2)
         => ( ! [A2: real] :
                ( member_real(A2,minus_minus_set_real(C2,A3))
               => ( aa_real_int(G,A2) = zero_zero_int ) )
           => ( ! [B2: real] :
                  ( member_real(B2,minus_minus_set_real(C2,B4))
                 => ( aa_real_int(H,B2) = zero_zero_int ) )
             => ( ( groups1932886352136224148al_int(G,A3) = groups1932886352136224148al_int(H,B4) )
              <=> ( groups1932886352136224148al_int(G,C2) = groups1932886352136224148al_int(H,C2) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_7159_sum_Osame__carrier,axiom,
    ! [C2: set_complex,A3: set_complex,B4: set_complex,G: fun_complex_int,H: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,C2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A3),C2)
       => ( aa_set_complex_o(ord_le211207098394363844omplex(B4),C2)
         => ( ! [A2: complex] :
                ( aa_set_complex_o(member_complex(A2),minus_811609699411566653omplex(C2,A3))
               => ( aa_complex_int(G,A2) = zero_zero_int ) )
           => ( ! [B2: complex] :
                  ( aa_set_complex_o(member_complex(B2),minus_811609699411566653omplex(C2,B4))
                 => ( aa_complex_int(H,B2) = zero_zero_int ) )
             => ( ( groups5690904116761175830ex_int(G,A3) = groups5690904116761175830ex_int(H,B4) )
              <=> ( groups5690904116761175830ex_int(G,C2) = groups5690904116761175830ex_int(H,C2) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_7160_sum_Osame__carrier,axiom,
    ! [C2: set_nat,A3: set_nat,B4: set_nat,G: fun_nat_rat,H: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,C2)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A3),C2)
       => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),C2)
         => ( ! [A2: nat] :
                ( aa_set_nat_o(member_nat(A2),minus_minus_set_nat(C2,A3))
               => ( aa_nat_rat(G,A2) = zero_zero_rat ) )
           => ( ! [B2: nat] :
                  ( aa_set_nat_o(member_nat(B2),minus_minus_set_nat(C2,B4))
                 => ( aa_nat_rat(H,B2) = zero_zero_rat ) )
             => ( ( groups2906978787729119204at_rat(G,A3) = groups2906978787729119204at_rat(H,B4) )
              <=> ( groups2906978787729119204at_rat(G,C2) = groups2906978787729119204at_rat(H,C2) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_7161_sum_Osame__carrier,axiom,
    ! [C2: set_nat,A3: set_nat,B4: set_nat,G: fun_nat_int,H: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,C2)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A3),C2)
       => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),C2)
         => ( ! [A2: nat] :
                ( aa_set_nat_o(member_nat(A2),minus_minus_set_nat(C2,A3))
               => ( aa_nat_int(G,A2) = zero_zero_int ) )
           => ( ! [B2: nat] :
                  ( aa_set_nat_o(member_nat(B2),minus_minus_set_nat(C2,B4))
                 => ( aa_nat_int(H,B2) = zero_zero_int ) )
             => ( ( groups3539618377306564664at_int(G,A3) = groups3539618377306564664at_int(H,B4) )
              <=> ( groups3539618377306564664at_int(G,C2) = groups3539618377306564664at_int(H,C2) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_7162_sum_Osame__carrierI,axiom,
    ! [C2: set_real,A3: set_real,B4: set_real,G: fun_real_real,H: fun_real_real] :
      ( finite_finite_real(C2)
     => ( ord_less_eq_set_real(A3,C2)
       => ( ord_less_eq_set_real(B4,C2)
         => ( ! [A2: real] :
                ( member_real(A2,minus_minus_set_real(C2,A3))
               => ( aa_real_real(G,A2) = zero_zero_real ) )
           => ( ! [B2: real] :
                  ( member_real(B2,minus_minus_set_real(C2,B4))
                 => ( aa_real_real(H,B2) = zero_zero_real ) )
             => ( ( groups8097168146408367636l_real(G,C2) = groups8097168146408367636l_real(H,C2) )
               => ( groups8097168146408367636l_real(G,A3) = groups8097168146408367636l_real(H,B4) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_7163_sum_Osame__carrierI,axiom,
    ! [C2: set_complex,A3: set_complex,B4: set_complex,G: fun_complex_real,H: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,C2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A3),C2)
       => ( aa_set_complex_o(ord_le211207098394363844omplex(B4),C2)
         => ( ! [A2: complex] :
                ( aa_set_complex_o(member_complex(A2),minus_811609699411566653omplex(C2,A3))
               => ( aa_complex_real(G,A2) = zero_zero_real ) )
           => ( ! [B2: complex] :
                  ( aa_set_complex_o(member_complex(B2),minus_811609699411566653omplex(C2,B4))
                 => ( aa_complex_real(H,B2) = zero_zero_real ) )
             => ( ( groups5808333547571424918x_real(G,C2) = groups5808333547571424918x_real(H,C2) )
               => ( groups5808333547571424918x_real(G,A3) = groups5808333547571424918x_real(H,B4) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_7164_sum_Osame__carrierI,axiom,
    ! [C2: set_real,A3: set_real,B4: set_real,G: fun_real_rat,H: fun_real_rat] :
      ( finite_finite_real(C2)
     => ( ord_less_eq_set_real(A3,C2)
       => ( ord_less_eq_set_real(B4,C2)
         => ( ! [A2: real] :
                ( member_real(A2,minus_minus_set_real(C2,A3))
               => ( aa_real_rat(G,A2) = zero_zero_rat ) )
           => ( ! [B2: real] :
                  ( member_real(B2,minus_minus_set_real(C2,B4))
                 => ( aa_real_rat(H,B2) = zero_zero_rat ) )
             => ( ( groups1300246762558778688al_rat(G,C2) = groups1300246762558778688al_rat(H,C2) )
               => ( groups1300246762558778688al_rat(G,A3) = groups1300246762558778688al_rat(H,B4) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_7165_sum_Osame__carrierI,axiom,
    ! [C2: set_complex,A3: set_complex,B4: set_complex,G: fun_complex_rat,H: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,C2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A3),C2)
       => ( aa_set_complex_o(ord_le211207098394363844omplex(B4),C2)
         => ( ! [A2: complex] :
                ( aa_set_complex_o(member_complex(A2),minus_811609699411566653omplex(C2,A3))
               => ( aa_complex_rat(G,A2) = zero_zero_rat ) )
           => ( ! [B2: complex] :
                  ( aa_set_complex_o(member_complex(B2),minus_811609699411566653omplex(C2,B4))
                 => ( aa_complex_rat(H,B2) = zero_zero_rat ) )
             => ( ( groups5058264527183730370ex_rat(G,C2) = groups5058264527183730370ex_rat(H,C2) )
               => ( groups5058264527183730370ex_rat(G,A3) = groups5058264527183730370ex_rat(H,B4) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_7166_sum_Osame__carrierI,axiom,
    ! [C2: set_real,A3: set_real,B4: set_real,G: fun_real_nat,H: fun_real_nat] :
      ( finite_finite_real(C2)
     => ( ord_less_eq_set_real(A3,C2)
       => ( ord_less_eq_set_real(B4,C2)
         => ( ! [A2: real] :
                ( member_real(A2,minus_minus_set_real(C2,A3))
               => ( aa_real_nat(G,A2) = zero_zero_nat ) )
           => ( ! [B2: real] :
                  ( member_real(B2,minus_minus_set_real(C2,B4))
                 => ( aa_real_nat(H,B2) = zero_zero_nat ) )
             => ( ( groups1935376822645274424al_nat(G,C2) = groups1935376822645274424al_nat(H,C2) )
               => ( groups1935376822645274424al_nat(G,A3) = groups1935376822645274424al_nat(H,B4) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_7167_sum_Osame__carrierI,axiom,
    ! [C2: set_complex,A3: set_complex,B4: set_complex,G: fun_complex_nat,H: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,C2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A3),C2)
       => ( aa_set_complex_o(ord_le211207098394363844omplex(B4),C2)
         => ( ! [A2: complex] :
                ( aa_set_complex_o(member_complex(A2),minus_811609699411566653omplex(C2,A3))
               => ( aa_complex_nat(G,A2) = zero_zero_nat ) )
           => ( ! [B2: complex] :
                  ( aa_set_complex_o(member_complex(B2),minus_811609699411566653omplex(C2,B4))
                 => ( aa_complex_nat(H,B2) = zero_zero_nat ) )
             => ( ( groups5693394587270226106ex_nat(G,C2) = groups5693394587270226106ex_nat(H,C2) )
               => ( groups5693394587270226106ex_nat(G,A3) = groups5693394587270226106ex_nat(H,B4) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_7168_sum_Osame__carrierI,axiom,
    ! [C2: set_real,A3: set_real,B4: set_real,G: fun_real_int,H: fun_real_int] :
      ( finite_finite_real(C2)
     => ( ord_less_eq_set_real(A3,C2)
       => ( ord_less_eq_set_real(B4,C2)
         => ( ! [A2: real] :
                ( member_real(A2,minus_minus_set_real(C2,A3))
               => ( aa_real_int(G,A2) = zero_zero_int ) )
           => ( ! [B2: real] :
                  ( member_real(B2,minus_minus_set_real(C2,B4))
                 => ( aa_real_int(H,B2) = zero_zero_int ) )
             => ( ( groups1932886352136224148al_int(G,C2) = groups1932886352136224148al_int(H,C2) )
               => ( groups1932886352136224148al_int(G,A3) = groups1932886352136224148al_int(H,B4) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_7169_sum_Osame__carrierI,axiom,
    ! [C2: set_complex,A3: set_complex,B4: set_complex,G: fun_complex_int,H: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,C2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A3),C2)
       => ( aa_set_complex_o(ord_le211207098394363844omplex(B4),C2)
         => ( ! [A2: complex] :
                ( aa_set_complex_o(member_complex(A2),minus_811609699411566653omplex(C2,A3))
               => ( aa_complex_int(G,A2) = zero_zero_int ) )
           => ( ! [B2: complex] :
                  ( aa_set_complex_o(member_complex(B2),minus_811609699411566653omplex(C2,B4))
                 => ( aa_complex_int(H,B2) = zero_zero_int ) )
             => ( ( groups5690904116761175830ex_int(G,C2) = groups5690904116761175830ex_int(H,C2) )
               => ( groups5690904116761175830ex_int(G,A3) = groups5690904116761175830ex_int(H,B4) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_7170_sum_Osame__carrierI,axiom,
    ! [C2: set_nat,A3: set_nat,B4: set_nat,G: fun_nat_rat,H: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,C2)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A3),C2)
       => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),C2)
         => ( ! [A2: nat] :
                ( aa_set_nat_o(member_nat(A2),minus_minus_set_nat(C2,A3))
               => ( aa_nat_rat(G,A2) = zero_zero_rat ) )
           => ( ! [B2: nat] :
                  ( aa_set_nat_o(member_nat(B2),minus_minus_set_nat(C2,B4))
                 => ( aa_nat_rat(H,B2) = zero_zero_rat ) )
             => ( ( groups2906978787729119204at_rat(G,C2) = groups2906978787729119204at_rat(H,C2) )
               => ( groups2906978787729119204at_rat(G,A3) = groups2906978787729119204at_rat(H,B4) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_7171_sum_Osame__carrierI,axiom,
    ! [C2: set_nat,A3: set_nat,B4: set_nat,G: fun_nat_int,H: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,C2)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A3),C2)
       => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),C2)
         => ( ! [A2: nat] :
                ( aa_set_nat_o(member_nat(A2),minus_minus_set_nat(C2,A3))
               => ( aa_nat_int(G,A2) = zero_zero_int ) )
           => ( ! [B2: nat] :
                  ( aa_set_nat_o(member_nat(B2),minus_minus_set_nat(C2,B4))
                 => ( aa_nat_int(H,B2) = zero_zero_int ) )
             => ( ( groups3539618377306564664at_int(G,C2) = groups3539618377306564664at_int(H,C2) )
               => ( groups3539618377306564664at_int(G,A3) = groups3539618377306564664at_int(H,B4) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_7172_sum_Omono__neutral__left,axiom,
    ! [T: set_complex,S2: set_complex,G: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T)
       => ( ! [X: complex] :
              ( aa_set_complex_o(member_complex(X),minus_811609699411566653omplex(T,S2))
             => ( aa_complex_real(G,X) = zero_zero_real ) )
         => ( groups5808333547571424918x_real(G,S2) = groups5808333547571424918x_real(G,T) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_7173_sum_Omono__neutral__left,axiom,
    ! [T: set_complex,S2: set_complex,G: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T)
       => ( ! [X: complex] :
              ( aa_set_complex_o(member_complex(X),minus_811609699411566653omplex(T,S2))
             => ( aa_complex_rat(G,X) = zero_zero_rat ) )
         => ( groups5058264527183730370ex_rat(G,S2) = groups5058264527183730370ex_rat(G,T) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_7174_sum_Omono__neutral__left,axiom,
    ! [T: set_complex,S2: set_complex,G: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T)
       => ( ! [X: complex] :
              ( aa_set_complex_o(member_complex(X),minus_811609699411566653omplex(T,S2))
             => ( aa_complex_nat(G,X) = zero_zero_nat ) )
         => ( groups5693394587270226106ex_nat(G,S2) = groups5693394587270226106ex_nat(G,T) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_7175_sum_Omono__neutral__left,axiom,
    ! [T: set_complex,S2: set_complex,G: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T)
       => ( ! [X: complex] :
              ( aa_set_complex_o(member_complex(X),minus_811609699411566653omplex(T,S2))
             => ( aa_complex_int(G,X) = zero_zero_int ) )
         => ( groups5690904116761175830ex_int(G,S2) = groups5690904116761175830ex_int(G,T) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_7176_sum_Omono__neutral__left,axiom,
    ! [T: set_nat,S2: set_nat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,T)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S2),T)
       => ( ! [X: nat] :
              ( aa_set_nat_o(member_nat(X),minus_minus_set_nat(T,S2))
             => ( aa_nat_rat(G,X) = zero_zero_rat ) )
         => ( groups2906978787729119204at_rat(G,S2) = groups2906978787729119204at_rat(G,T) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_7177_sum_Omono__neutral__left,axiom,
    ! [T: set_nat,S2: set_nat,G: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,T)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S2),T)
       => ( ! [X: nat] :
              ( aa_set_nat_o(member_nat(X),minus_minus_set_nat(T,S2))
             => ( aa_nat_int(G,X) = zero_zero_int ) )
         => ( groups3539618377306564664at_int(G,S2) = groups3539618377306564664at_int(G,T) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_7178_sum_Omono__neutral__left,axiom,
    ! [T: set_int,S2: set_int,G: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,T)
     => ( aa_set_int_o(ord_less_eq_set_int(S2),T)
       => ( ! [X: int] :
              ( aa_set_int_o(member_int(X),minus_minus_set_int(T,S2))
             => ( aa_int_real(G,X) = zero_zero_real ) )
         => ( groups8778361861064173332t_real(G,S2) = groups8778361861064173332t_real(G,T) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_7179_sum_Omono__neutral__left,axiom,
    ! [T: set_int,S2: set_int,G: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,T)
     => ( aa_set_int_o(ord_less_eq_set_int(S2),T)
       => ( ! [X: int] :
              ( aa_set_int_o(member_int(X),minus_minus_set_int(T,S2))
             => ( aa_int_rat(G,X) = zero_zero_rat ) )
         => ( groups3906332499630173760nt_rat(G,S2) = groups3906332499630173760nt_rat(G,T) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_7180_sum_Omono__neutral__left,axiom,
    ! [T: set_int,S2: set_int,G: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,T)
     => ( aa_set_int_o(ord_less_eq_set_int(S2),T)
       => ( ! [X: int] :
              ( aa_set_int_o(member_int(X),minus_minus_set_int(T,S2))
             => ( aa_int_nat(G,X) = zero_zero_nat ) )
         => ( groups4541462559716669496nt_nat(G,S2) = groups4541462559716669496nt_nat(G,T) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_7181_sum_Omono__neutral__left,axiom,
    ! [T: set_int,S2: set_int,G: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,T)
     => ( aa_set_int_o(ord_less_eq_set_int(S2),T)
       => ( ! [X: int] :
              ( aa_set_int_o(member_int(X),minus_minus_set_int(T,S2))
             => ( aa_int_int(G,X) = zero_zero_int ) )
         => ( groups4538972089207619220nt_int(G,S2) = groups4538972089207619220nt_int(G,T) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_7182_sum_Omono__neutral__right,axiom,
    ! [T: set_complex,S2: set_complex,G: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T)
       => ( ! [X: complex] :
              ( aa_set_complex_o(member_complex(X),minus_811609699411566653omplex(T,S2))
             => ( aa_complex_real(G,X) = zero_zero_real ) )
         => ( groups5808333547571424918x_real(G,T) = groups5808333547571424918x_real(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_7183_sum_Omono__neutral__right,axiom,
    ! [T: set_complex,S2: set_complex,G: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T)
       => ( ! [X: complex] :
              ( aa_set_complex_o(member_complex(X),minus_811609699411566653omplex(T,S2))
             => ( aa_complex_rat(G,X) = zero_zero_rat ) )
         => ( groups5058264527183730370ex_rat(G,T) = groups5058264527183730370ex_rat(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_7184_sum_Omono__neutral__right,axiom,
    ! [T: set_complex,S2: set_complex,G: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T)
       => ( ! [X: complex] :
              ( aa_set_complex_o(member_complex(X),minus_811609699411566653omplex(T,S2))
             => ( aa_complex_nat(G,X) = zero_zero_nat ) )
         => ( groups5693394587270226106ex_nat(G,T) = groups5693394587270226106ex_nat(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_7185_sum_Omono__neutral__right,axiom,
    ! [T: set_complex,S2: set_complex,G: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T)
       => ( ! [X: complex] :
              ( aa_set_complex_o(member_complex(X),minus_811609699411566653omplex(T,S2))
             => ( aa_complex_int(G,X) = zero_zero_int ) )
         => ( groups5690904116761175830ex_int(G,T) = groups5690904116761175830ex_int(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_7186_sum_Omono__neutral__right,axiom,
    ! [T: set_nat,S2: set_nat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,T)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S2),T)
       => ( ! [X: nat] :
              ( aa_set_nat_o(member_nat(X),minus_minus_set_nat(T,S2))
             => ( aa_nat_rat(G,X) = zero_zero_rat ) )
         => ( groups2906978787729119204at_rat(G,T) = groups2906978787729119204at_rat(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_7187_sum_Omono__neutral__right,axiom,
    ! [T: set_nat,S2: set_nat,G: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,T)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S2),T)
       => ( ! [X: nat] :
              ( aa_set_nat_o(member_nat(X),minus_minus_set_nat(T,S2))
             => ( aa_nat_int(G,X) = zero_zero_int ) )
         => ( groups3539618377306564664at_int(G,T) = groups3539618377306564664at_int(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_7188_sum_Omono__neutral__right,axiom,
    ! [T: set_int,S2: set_int,G: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,T)
     => ( aa_set_int_o(ord_less_eq_set_int(S2),T)
       => ( ! [X: int] :
              ( aa_set_int_o(member_int(X),minus_minus_set_int(T,S2))
             => ( aa_int_real(G,X) = zero_zero_real ) )
         => ( groups8778361861064173332t_real(G,T) = groups8778361861064173332t_real(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_7189_sum_Omono__neutral__right,axiom,
    ! [T: set_int,S2: set_int,G: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,T)
     => ( aa_set_int_o(ord_less_eq_set_int(S2),T)
       => ( ! [X: int] :
              ( aa_set_int_o(member_int(X),minus_minus_set_int(T,S2))
             => ( aa_int_rat(G,X) = zero_zero_rat ) )
         => ( groups3906332499630173760nt_rat(G,T) = groups3906332499630173760nt_rat(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_7190_sum_Omono__neutral__right,axiom,
    ! [T: set_int,S2: set_int,G: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,T)
     => ( aa_set_int_o(ord_less_eq_set_int(S2),T)
       => ( ! [X: int] :
              ( aa_set_int_o(member_int(X),minus_minus_set_int(T,S2))
             => ( aa_int_nat(G,X) = zero_zero_nat ) )
         => ( groups4541462559716669496nt_nat(G,T) = groups4541462559716669496nt_nat(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_7191_sum_Omono__neutral__right,axiom,
    ! [T: set_int,S2: set_int,G: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,T)
     => ( aa_set_int_o(ord_less_eq_set_int(S2),T)
       => ( ! [X: int] :
              ( aa_set_int_o(member_int(X),minus_minus_set_int(T,S2))
             => ( aa_int_int(G,X) = zero_zero_int ) )
         => ( groups4538972089207619220nt_int(G,T) = groups4538972089207619220nt_int(G,S2) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_7192_sum_Omono__neutral__cong__left,axiom,
    ! [T: set_real,S2: set_real,H: fun_real_real,G: fun_real_real] :
      ( finite_finite_real(T)
     => ( ord_less_eq_set_real(S2,T)
       => ( ! [X: real] :
              ( member_real(X,minus_minus_set_real(T,S2))
             => ( aa_real_real(H,X) = zero_zero_real ) )
         => ( ! [X: real] :
                ( member_real(X,S2)
               => ( aa_real_real(G,X) = aa_real_real(H,X) ) )
           => ( groups8097168146408367636l_real(G,S2) = groups8097168146408367636l_real(H,T) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_7193_sum_Omono__neutral__cong__left,axiom,
    ! [T: set_complex,S2: set_complex,H: fun_complex_real,G: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T)
       => ( ! [X: complex] :
              ( aa_set_complex_o(member_complex(X),minus_811609699411566653omplex(T,S2))
             => ( aa_complex_real(H,X) = zero_zero_real ) )
         => ( ! [X: complex] :
                ( aa_set_complex_o(member_complex(X),S2)
               => ( aa_complex_real(G,X) = aa_complex_real(H,X) ) )
           => ( groups5808333547571424918x_real(G,S2) = groups5808333547571424918x_real(H,T) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_7194_sum_Omono__neutral__cong__left,axiom,
    ! [T: set_real,S2: set_real,H: fun_real_rat,G: fun_real_rat] :
      ( finite_finite_real(T)
     => ( ord_less_eq_set_real(S2,T)
       => ( ! [X: real] :
              ( member_real(X,minus_minus_set_real(T,S2))
             => ( aa_real_rat(H,X) = zero_zero_rat ) )
         => ( ! [X: real] :
                ( member_real(X,S2)
               => ( aa_real_rat(G,X) = aa_real_rat(H,X) ) )
           => ( groups1300246762558778688al_rat(G,S2) = groups1300246762558778688al_rat(H,T) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_7195_sum_Omono__neutral__cong__left,axiom,
    ! [T: set_complex,S2: set_complex,H: fun_complex_rat,G: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T)
       => ( ! [X: complex] :
              ( aa_set_complex_o(member_complex(X),minus_811609699411566653omplex(T,S2))
             => ( aa_complex_rat(H,X) = zero_zero_rat ) )
         => ( ! [X: complex] :
                ( aa_set_complex_o(member_complex(X),S2)
               => ( aa_complex_rat(G,X) = aa_complex_rat(H,X) ) )
           => ( groups5058264527183730370ex_rat(G,S2) = groups5058264527183730370ex_rat(H,T) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_7196_sum_Omono__neutral__cong__left,axiom,
    ! [T: set_real,S2: set_real,H: fun_real_nat,G: fun_real_nat] :
      ( finite_finite_real(T)
     => ( ord_less_eq_set_real(S2,T)
       => ( ! [X: real] :
              ( member_real(X,minus_minus_set_real(T,S2))
             => ( aa_real_nat(H,X) = zero_zero_nat ) )
         => ( ! [X: real] :
                ( member_real(X,S2)
               => ( aa_real_nat(G,X) = aa_real_nat(H,X) ) )
           => ( groups1935376822645274424al_nat(G,S2) = groups1935376822645274424al_nat(H,T) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_7197_sum_Omono__neutral__cong__left,axiom,
    ! [T: set_complex,S2: set_complex,H: fun_complex_nat,G: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T)
       => ( ! [X: complex] :
              ( aa_set_complex_o(member_complex(X),minus_811609699411566653omplex(T,S2))
             => ( aa_complex_nat(H,X) = zero_zero_nat ) )
         => ( ! [X: complex] :
                ( aa_set_complex_o(member_complex(X),S2)
               => ( aa_complex_nat(G,X) = aa_complex_nat(H,X) ) )
           => ( groups5693394587270226106ex_nat(G,S2) = groups5693394587270226106ex_nat(H,T) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_7198_sum_Omono__neutral__cong__left,axiom,
    ! [T: set_real,S2: set_real,H: fun_real_int,G: fun_real_int] :
      ( finite_finite_real(T)
     => ( ord_less_eq_set_real(S2,T)
       => ( ! [X: real] :
              ( member_real(X,minus_minus_set_real(T,S2))
             => ( aa_real_int(H,X) = zero_zero_int ) )
         => ( ! [X: real] :
                ( member_real(X,S2)
               => ( aa_real_int(G,X) = aa_real_int(H,X) ) )
           => ( groups1932886352136224148al_int(G,S2) = groups1932886352136224148al_int(H,T) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_7199_sum_Omono__neutral__cong__left,axiom,
    ! [T: set_complex,S2: set_complex,H: fun_complex_int,G: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T)
       => ( ! [X: complex] :
              ( aa_set_complex_o(member_complex(X),minus_811609699411566653omplex(T,S2))
             => ( aa_complex_int(H,X) = zero_zero_int ) )
         => ( ! [X: complex] :
                ( aa_set_complex_o(member_complex(X),S2)
               => ( aa_complex_int(G,X) = aa_complex_int(H,X) ) )
           => ( groups5690904116761175830ex_int(G,S2) = groups5690904116761175830ex_int(H,T) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_7200_sum_Omono__neutral__cong__left,axiom,
    ! [T: set_nat,S2: set_nat,H: fun_nat_rat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,T)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S2),T)
       => ( ! [X: nat] :
              ( aa_set_nat_o(member_nat(X),minus_minus_set_nat(T,S2))
             => ( aa_nat_rat(H,X) = zero_zero_rat ) )
         => ( ! [X: nat] :
                ( aa_set_nat_o(member_nat(X),S2)
               => ( aa_nat_rat(G,X) = aa_nat_rat(H,X) ) )
           => ( groups2906978787729119204at_rat(G,S2) = groups2906978787729119204at_rat(H,T) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_7201_sum_Omono__neutral__cong__left,axiom,
    ! [T: set_nat,S2: set_nat,H: fun_nat_int,G: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,T)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S2),T)
       => ( ! [X: nat] :
              ( aa_set_nat_o(member_nat(X),minus_minus_set_nat(T,S2))
             => ( aa_nat_int(H,X) = zero_zero_int ) )
         => ( ! [X: nat] :
                ( aa_set_nat_o(member_nat(X),S2)
               => ( aa_nat_int(G,X) = aa_nat_int(H,X) ) )
           => ( groups3539618377306564664at_int(G,S2) = groups3539618377306564664at_int(H,T) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_7202_sum_Omono__neutral__cong__right,axiom,
    ! [T: set_real,S2: set_real,G: fun_real_real,H: fun_real_real] :
      ( finite_finite_real(T)
     => ( ord_less_eq_set_real(S2,T)
       => ( ! [X: real] :
              ( member_real(X,minus_minus_set_real(T,S2))
             => ( aa_real_real(G,X) = zero_zero_real ) )
         => ( ! [X: real] :
                ( member_real(X,S2)
               => ( aa_real_real(G,X) = aa_real_real(H,X) ) )
           => ( groups8097168146408367636l_real(G,T) = groups8097168146408367636l_real(H,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_7203_sum_Omono__neutral__cong__right,axiom,
    ! [T: set_complex,S2: set_complex,G: fun_complex_real,H: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T)
       => ( ! [X: complex] :
              ( aa_set_complex_o(member_complex(X),minus_811609699411566653omplex(T,S2))
             => ( aa_complex_real(G,X) = zero_zero_real ) )
         => ( ! [X: complex] :
                ( aa_set_complex_o(member_complex(X),S2)
               => ( aa_complex_real(G,X) = aa_complex_real(H,X) ) )
           => ( groups5808333547571424918x_real(G,T) = groups5808333547571424918x_real(H,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_7204_sum_Omono__neutral__cong__right,axiom,
    ! [T: set_real,S2: set_real,G: fun_real_rat,H: fun_real_rat] :
      ( finite_finite_real(T)
     => ( ord_less_eq_set_real(S2,T)
       => ( ! [X: real] :
              ( member_real(X,minus_minus_set_real(T,S2))
             => ( aa_real_rat(G,X) = zero_zero_rat ) )
         => ( ! [X: real] :
                ( member_real(X,S2)
               => ( aa_real_rat(G,X) = aa_real_rat(H,X) ) )
           => ( groups1300246762558778688al_rat(G,T) = groups1300246762558778688al_rat(H,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_7205_sum_Omono__neutral__cong__right,axiom,
    ! [T: set_complex,S2: set_complex,G: fun_complex_rat,H: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T)
       => ( ! [X: complex] :
              ( aa_set_complex_o(member_complex(X),minus_811609699411566653omplex(T,S2))
             => ( aa_complex_rat(G,X) = zero_zero_rat ) )
         => ( ! [X: complex] :
                ( aa_set_complex_o(member_complex(X),S2)
               => ( aa_complex_rat(G,X) = aa_complex_rat(H,X) ) )
           => ( groups5058264527183730370ex_rat(G,T) = groups5058264527183730370ex_rat(H,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_7206_sum_Omono__neutral__cong__right,axiom,
    ! [T: set_real,S2: set_real,G: fun_real_nat,H: fun_real_nat] :
      ( finite_finite_real(T)
     => ( ord_less_eq_set_real(S2,T)
       => ( ! [X: real] :
              ( member_real(X,minus_minus_set_real(T,S2))
             => ( aa_real_nat(G,X) = zero_zero_nat ) )
         => ( ! [X: real] :
                ( member_real(X,S2)
               => ( aa_real_nat(G,X) = aa_real_nat(H,X) ) )
           => ( groups1935376822645274424al_nat(G,T) = groups1935376822645274424al_nat(H,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_7207_sum_Omono__neutral__cong__right,axiom,
    ! [T: set_complex,S2: set_complex,G: fun_complex_nat,H: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T)
       => ( ! [X: complex] :
              ( aa_set_complex_o(member_complex(X),minus_811609699411566653omplex(T,S2))
             => ( aa_complex_nat(G,X) = zero_zero_nat ) )
         => ( ! [X: complex] :
                ( aa_set_complex_o(member_complex(X),S2)
               => ( aa_complex_nat(G,X) = aa_complex_nat(H,X) ) )
           => ( groups5693394587270226106ex_nat(G,T) = groups5693394587270226106ex_nat(H,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_7208_sum_Omono__neutral__cong__right,axiom,
    ! [T: set_real,S2: set_real,G: fun_real_int,H: fun_real_int] :
      ( finite_finite_real(T)
     => ( ord_less_eq_set_real(S2,T)
       => ( ! [X: real] :
              ( member_real(X,minus_minus_set_real(T,S2))
             => ( aa_real_int(G,X) = zero_zero_int ) )
         => ( ! [X: real] :
                ( member_real(X,S2)
               => ( aa_real_int(G,X) = aa_real_int(H,X) ) )
           => ( groups1932886352136224148al_int(G,T) = groups1932886352136224148al_int(H,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_7209_sum_Omono__neutral__cong__right,axiom,
    ! [T: set_complex,S2: set_complex,G: fun_complex_int,H: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T)
       => ( ! [X: complex] :
              ( aa_set_complex_o(member_complex(X),minus_811609699411566653omplex(T,S2))
             => ( aa_complex_int(G,X) = zero_zero_int ) )
         => ( ! [X: complex] :
                ( aa_set_complex_o(member_complex(X),S2)
               => ( aa_complex_int(G,X) = aa_complex_int(H,X) ) )
           => ( groups5690904116761175830ex_int(G,T) = groups5690904116761175830ex_int(H,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_7210_sum_Omono__neutral__cong__right,axiom,
    ! [T: set_nat,S2: set_nat,G: fun_nat_rat,H: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,T)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S2),T)
       => ( ! [X: nat] :
              ( aa_set_nat_o(member_nat(X),minus_minus_set_nat(T,S2))
             => ( aa_nat_rat(G,X) = zero_zero_rat ) )
         => ( ! [X: nat] :
                ( aa_set_nat_o(member_nat(X),S2)
               => ( aa_nat_rat(G,X) = aa_nat_rat(H,X) ) )
           => ( groups2906978787729119204at_rat(G,T) = groups2906978787729119204at_rat(H,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_7211_sum_Omono__neutral__cong__right,axiom,
    ! [T: set_nat,S2: set_nat,G: fun_nat_int,H: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,T)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S2),T)
       => ( ! [X: nat] :
              ( aa_set_nat_o(member_nat(X),minus_minus_set_nat(T,S2))
             => ( aa_nat_int(G,X) = zero_zero_int ) )
         => ( ! [X: nat] :
                ( aa_set_nat_o(member_nat(X),S2)
               => ( aa_nat_int(G,X) = aa_nat_int(H,X) ) )
           => ( groups3539618377306564664at_int(G,T) = groups3539618377306564664at_int(H,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_7212_sum_Osubset__diff,axiom,
    ! [B4: set_complex,A3: set_complex,G: fun_complex_real] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(B4),A3)
     => ( aa_set_complex_o(finite3207457112153483333omplex,A3)
       => ( groups5808333547571424918x_real(G,A3) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups5808333547571424918x_real(G,minus_811609699411566653omplex(A3,B4))),groups5808333547571424918x_real(G,B4)) ) ) ) ).

% sum.subset_diff
tff(fact_7213_sum_Osubset__diff,axiom,
    ! [B4: set_complex,A3: set_complex,G: fun_complex_rat] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(B4),A3)
     => ( aa_set_complex_o(finite3207457112153483333omplex,A3)
       => ( groups5058264527183730370ex_rat(G,A3) = aa_rat_rat(plus_plus_rat(groups5058264527183730370ex_rat(G,minus_811609699411566653omplex(A3,B4))),groups5058264527183730370ex_rat(G,B4)) ) ) ) ).

% sum.subset_diff
tff(fact_7214_sum_Osubset__diff,axiom,
    ! [B4: set_complex,A3: set_complex,G: fun_complex_nat] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(B4),A3)
     => ( aa_set_complex_o(finite3207457112153483333omplex,A3)
       => ( groups5693394587270226106ex_nat(G,A3) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups5693394587270226106ex_nat(G,minus_811609699411566653omplex(A3,B4))),groups5693394587270226106ex_nat(G,B4)) ) ) ) ).

% sum.subset_diff
tff(fact_7215_sum_Osubset__diff,axiom,
    ! [B4: set_complex,A3: set_complex,G: fun_complex_int] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(B4),A3)
     => ( aa_set_complex_o(finite3207457112153483333omplex,A3)
       => ( groups5690904116761175830ex_int(G,A3) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups5690904116761175830ex_int(G,minus_811609699411566653omplex(A3,B4))),groups5690904116761175830ex_int(G,B4)) ) ) ) ).

% sum.subset_diff
tff(fact_7216_sum_Osubset__diff,axiom,
    ! [B4: set_nat,A3: set_nat,G: fun_nat_rat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),A3)
     => ( aa_set_nat_o(finite_finite_nat,A3)
       => ( groups2906978787729119204at_rat(G,A3) = aa_rat_rat(plus_plus_rat(groups2906978787729119204at_rat(G,minus_minus_set_nat(A3,B4))),groups2906978787729119204at_rat(G,B4)) ) ) ) ).

% sum.subset_diff
tff(fact_7217_sum_Osubset__diff,axiom,
    ! [B4: set_nat,A3: set_nat,G: fun_nat_int] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),A3)
     => ( aa_set_nat_o(finite_finite_nat,A3)
       => ( groups3539618377306564664at_int(G,A3) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,minus_minus_set_nat(A3,B4))),groups3539618377306564664at_int(G,B4)) ) ) ) ).

% sum.subset_diff
tff(fact_7218_sum_Osubset__diff,axiom,
    ! [B4: set_int,A3: set_int,G: fun_int_real] :
      ( aa_set_int_o(ord_less_eq_set_int(B4),A3)
     => ( aa_set_int_o(finite_finite_int,A3)
       => ( groups8778361861064173332t_real(G,A3) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups8778361861064173332t_real(G,minus_minus_set_int(A3,B4))),groups8778361861064173332t_real(G,B4)) ) ) ) ).

% sum.subset_diff
tff(fact_7219_sum_Osubset__diff,axiom,
    ! [B4: set_int,A3: set_int,G: fun_int_rat] :
      ( aa_set_int_o(ord_less_eq_set_int(B4),A3)
     => ( aa_set_int_o(finite_finite_int,A3)
       => ( groups3906332499630173760nt_rat(G,A3) = aa_rat_rat(plus_plus_rat(groups3906332499630173760nt_rat(G,minus_minus_set_int(A3,B4))),groups3906332499630173760nt_rat(G,B4)) ) ) ) ).

% sum.subset_diff
tff(fact_7220_sum_Osubset__diff,axiom,
    ! [B4: set_int,A3: set_int,G: fun_int_nat] :
      ( aa_set_int_o(ord_less_eq_set_int(B4),A3)
     => ( aa_set_int_o(finite_finite_int,A3)
       => ( groups4541462559716669496nt_nat(G,A3) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups4541462559716669496nt_nat(G,minus_minus_set_int(A3,B4))),groups4541462559716669496nt_nat(G,B4)) ) ) ) ).

% sum.subset_diff
tff(fact_7221_sum_Osubset__diff,axiom,
    ! [B4: set_int,A3: set_int,G: fun_int_int] :
      ( aa_set_int_o(ord_less_eq_set_int(B4),A3)
     => ( aa_set_int_o(finite_finite_int,A3)
       => ( groups4538972089207619220nt_int(G,A3) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups4538972089207619220nt_int(G,minus_minus_set_int(A3,B4))),groups4538972089207619220nt_int(G,B4)) ) ) ) ).

% sum.subset_diff
tff(fact_7222_sum__diff,axiom,
    ! [A3: set_complex,B4: set_complex,F: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A3)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(B4),A3)
       => ( groups5808333547571424918x_real(F,minus_811609699411566653omplex(A3,B4)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups5808333547571424918x_real(F,A3)),groups5808333547571424918x_real(F,B4)) ) ) ) ).

% sum_diff
tff(fact_7223_sum__diff,axiom,
    ! [A3: set_complex,B4: set_complex,F: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A3)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(B4),A3)
       => ( groups5058264527183730370ex_rat(F,minus_811609699411566653omplex(A3,B4)) = aa_rat_rat(minus_minus_rat(groups5058264527183730370ex_rat(F,A3)),groups5058264527183730370ex_rat(F,B4)) ) ) ) ).

% sum_diff
tff(fact_7224_sum__diff,axiom,
    ! [A3: set_complex,B4: set_complex,F: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A3)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(B4),A3)
       => ( groups5690904116761175830ex_int(F,minus_811609699411566653omplex(A3,B4)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,groups5690904116761175830ex_int(F,A3)),groups5690904116761175830ex_int(F,B4)) ) ) ) ).

% sum_diff
tff(fact_7225_sum__diff,axiom,
    ! [A3: set_nat,B4: set_nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),A3)
       => ( groups2906978787729119204at_rat(F,minus_minus_set_nat(A3,B4)) = aa_rat_rat(minus_minus_rat(groups2906978787729119204at_rat(F,A3)),groups2906978787729119204at_rat(F,B4)) ) ) ) ).

% sum_diff
tff(fact_7226_sum__diff,axiom,
    ! [A3: set_nat,B4: set_nat,F: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),A3)
       => ( groups3539618377306564664at_int(F,minus_minus_set_nat(A3,B4)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,groups3539618377306564664at_int(F,A3)),groups3539618377306564664at_int(F,B4)) ) ) ) ).

% sum_diff
tff(fact_7227_sum__diff,axiom,
    ! [A3: set_int,B4: set_int,F: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( aa_set_int_o(ord_less_eq_set_int(B4),A3)
       => ( groups8778361861064173332t_real(F,minus_minus_set_int(A3,B4)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups8778361861064173332t_real(F,A3)),groups8778361861064173332t_real(F,B4)) ) ) ) ).

% sum_diff
tff(fact_7228_sum__diff,axiom,
    ! [A3: set_int,B4: set_int,F: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( aa_set_int_o(ord_less_eq_set_int(B4),A3)
       => ( groups3906332499630173760nt_rat(F,minus_minus_set_int(A3,B4)) = aa_rat_rat(minus_minus_rat(groups3906332499630173760nt_rat(F,A3)),groups3906332499630173760nt_rat(F,B4)) ) ) ) ).

% sum_diff
tff(fact_7229_sum__diff,axiom,
    ! [A3: set_int,B4: set_int,F: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( aa_set_int_o(ord_less_eq_set_int(B4),A3)
       => ( groups4538972089207619220nt_int(F,minus_minus_set_int(A3,B4)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,groups4538972089207619220nt_int(F,A3)),groups4538972089207619220nt_int(F,B4)) ) ) ) ).

% sum_diff
tff(fact_7230_sum__diff,axiom,
    ! [A3: set_complex,B4: set_complex,F: fun_complex_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A3)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(B4),A3)
       => ( groups7754918857620584856omplex(F,minus_811609699411566653omplex(A3,B4)) = aa_complex_complex(minus_minus_complex(groups7754918857620584856omplex(F,A3)),groups7754918857620584856omplex(F,B4)) ) ) ) ).

% sum_diff
tff(fact_7231_sum__diff,axiom,
    ! [A3: set_nat,B4: set_nat,F: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),A3)
       => ( aa_set_nat_real(groups6591440286371151544t_real(F),minus_minus_set_nat(A3,B4)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_set_nat_real(groups6591440286371151544t_real(F),A3)),aa_set_nat_real(groups6591440286371151544t_real(F),B4)) ) ) ) ).

% sum_diff
tff(fact_7232_binomial__absorption,axiom,
    ! [K: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),aa_nat_nat(suc,K))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat)),K)) ).

% binomial_absorption
tff(fact_7233_binomial__fact__lemma,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,K)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K)))),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K)) = aa_nat_nat(semiri1408675320244567234ct_nat,Na) ) ) ).

% binomial_fact_lemma
tff(fact_7234_pochhammer__rec,axiom,
    ! [Aa: complex,Na: nat] : aa_nat_complex(comm_s2602460028002588243omplex(Aa),aa_nat_nat(suc,Na)) = aa_complex_complex(times_times_complex(Aa),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(Aa),one_one_complex)),Na)) ).

% pochhammer_rec
tff(fact_7235_pochhammer__rec,axiom,
    ! [Aa: real,Na: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Aa),aa_nat_nat(suc,Na)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),one_one_real)),Na)) ).

% pochhammer_rec
tff(fact_7236_pochhammer__rec,axiom,
    ! [Aa: rat,Na: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(Aa),aa_nat_nat(suc,Na)) = aa_rat_rat(times_times_rat(Aa),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(plus_plus_rat(Aa),one_one_rat)),Na)) ).

% pochhammer_rec
tff(fact_7237_pochhammer__rec,axiom,
    ! [Aa: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Aa),aa_nat_nat(suc,Na)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),one_one_nat)),Na)) ).

% pochhammer_rec
tff(fact_7238_pochhammer__rec,axiom,
    ! [Aa: int,Na: nat] : aa_nat_int(comm_s4660882817536571857er_int(Aa),aa_nat_nat(suc,Na)) = aa_int_int(aa_int_fun_int_int(times_times_int,Aa),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),one_one_int)),Na)) ).

% pochhammer_rec
tff(fact_7239_take__bit__Suc__bit0,axiom,
    ! [Na: nat,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_nat_nat(suc,Na)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),aa_num_nat(numeral_numeral_nat,K))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% take_bit_Suc_bit0
tff(fact_7240_take__bit__Suc__bit0,axiom,
    ! [Na: nat,K: num] : aa_Cod1783974652533396728nteger(aa_nat6383147911832638789nteger(bit_se1745604003318907178nteger,aa_nat_nat(suc,Na)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,K))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_nat6383147911832638789nteger(bit_se1745604003318907178nteger,Na),aa_num_Code_integer(numera6620942414471956472nteger,K))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) ).

% take_bit_Suc_bit0
tff(fact_7241_take__bit__Suc__bit0,axiom,
    ! [Na: nat,K: num] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_nat_nat(suc,Na)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,K))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Na),aa_num_int(numeral_numeral_int,K))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% take_bit_Suc_bit0
tff(fact_7242_binomial__maximum_H,axiom,
    ! [Na: nat,K: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)),Na)) ).

% binomial_maximum'
tff(fact_7243_binomial__mono,axiom,
    ! [K: nat,K5: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),K5)
     => ( 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))),K5)),Na)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K5)) ) ) ).

% binomial_mono
tff(fact_7244_binomial__antimono,axiom,
    ! [K: nat,K5: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),K5)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),K)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K5),Na)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K5)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K)) ) ) ) ).

% binomial_antimono
tff(fact_7245_binomial__maximum,axiom,
    ! [Na: nat,K: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% binomial_maximum
tff(fact_7246_sum__mono2,axiom,
    ! [B4: set_real,A3: set_real,F: fun_real_real] :
      ( finite_finite_real(B4)
     => ( ord_less_eq_set_real(A3,B4)
       => ( ! [B2: real] :
              ( member_real(B2,minus_minus_set_real(B4,A3))
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,B2)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups8097168146408367636l_real(F,A3)),groups8097168146408367636l_real(F,B4)) ) ) ) ).

% sum_mono2
tff(fact_7247_sum__mono2,axiom,
    ! [B4: set_complex,A3: set_complex,F: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A3),B4)
       => ( ! [B2: complex] :
              ( aa_set_complex_o(member_complex(B2),minus_811609699411566653omplex(B4,A3))
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(F,B2)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups5808333547571424918x_real(F,A3)),groups5808333547571424918x_real(F,B4)) ) ) ) ).

% sum_mono2
tff(fact_7248_sum__mono2,axiom,
    ! [B4: set_real,A3: set_real,F: fun_real_rat] :
      ( finite_finite_real(B4)
     => ( ord_less_eq_set_real(A3,B4)
       => ( ! [B2: real] :
              ( member_real(B2,minus_minus_set_real(B4,A3))
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,B2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups1300246762558778688al_rat(F,A3)),groups1300246762558778688al_rat(F,B4)) ) ) ) ).

% sum_mono2
tff(fact_7249_sum__mono2,axiom,
    ! [B4: set_complex,A3: set_complex,F: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A3),B4)
       => ( ! [B2: complex] :
              ( aa_set_complex_o(member_complex(B2),minus_811609699411566653omplex(B4,A3))
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,B2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups5058264527183730370ex_rat(F,A3)),groups5058264527183730370ex_rat(F,B4)) ) ) ) ).

% sum_mono2
tff(fact_7250_sum__mono2,axiom,
    ! [B4: set_nat,A3: set_nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,B4)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A3),B4)
       => ( ! [B2: nat] :
              ( aa_set_nat_o(member_nat(B2),minus_minus_set_nat(B4,A3))
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,B2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups2906978787729119204at_rat(F,A3)),groups2906978787729119204at_rat(F,B4)) ) ) ) ).

% sum_mono2
tff(fact_7251_sum__mono2,axiom,
    ! [B4: set_real,A3: set_real,F: fun_real_nat] :
      ( finite_finite_real(B4)
     => ( ord_less_eq_set_real(A3,B4)
       => ( ! [B2: real] :
              ( member_real(B2,minus_minus_set_real(B4,A3))
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,B2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups1935376822645274424al_nat(F,A3)),groups1935376822645274424al_nat(F,B4)) ) ) ) ).

% sum_mono2
tff(fact_7252_sum__mono2,axiom,
    ! [B4: set_complex,A3: set_complex,F: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A3),B4)
       => ( ! [B2: complex] :
              ( aa_set_complex_o(member_complex(B2),minus_811609699411566653omplex(B4,A3))
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_complex_nat(F,B2)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups5693394587270226106ex_nat(F,A3)),groups5693394587270226106ex_nat(F,B4)) ) ) ) ).

% sum_mono2
tff(fact_7253_sum__mono2,axiom,
    ! [B4: set_real,A3: set_real,F: fun_real_int] :
      ( finite_finite_real(B4)
     => ( ord_less_eq_set_real(A3,B4)
       => ( ! [B2: real] :
              ( member_real(B2,minus_minus_set_real(B4,A3))
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_real_int(F,B2)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups1932886352136224148al_int(F,A3)),groups1932886352136224148al_int(F,B4)) ) ) ) ).

% sum_mono2
tff(fact_7254_sum__mono2,axiom,
    ! [B4: set_complex,A3: set_complex,F: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A3),B4)
       => ( ! [B2: complex] :
              ( aa_set_complex_o(member_complex(B2),minus_811609699411566653omplex(B4,A3))
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_complex_int(F,B2)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups5690904116761175830ex_int(F,A3)),groups5690904116761175830ex_int(F,B4)) ) ) ) ).

% sum_mono2
tff(fact_7255_sum__mono2,axiom,
    ! [B4: set_nat,A3: set_nat,F: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,B4)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A3),B4)
       => ( ! [B2: nat] :
              ( aa_set_nat_o(member_nat(B2),minus_minus_set_nat(B4,A3))
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(F,B2)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups3539618377306564664at_int(F,A3)),groups3539618377306564664at_int(F,B4)) ) ) ) ).

% sum_mono2
tff(fact_7256_binomial__le__pow2,axiom,
    ! [Na: nat,K: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) ).

% binomial_le_pow2
tff(fact_7257_take__bit__nat__eq__self__iff,axiom,
    ! [Na: nat,M: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),M) = M )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) ) ).

% take_bit_nat_eq_self_iff
tff(fact_7258_take__bit__nat__less__exp,axiom,
    ! [Na: nat,M: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) ).

% take_bit_nat_less_exp
tff(fact_7259_take__bit__nat__eq__self,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),M) = M ) ) ).

% take_bit_nat_eq_self
tff(fact_7260_choose__reduce__nat,axiom,
    ! [Na: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat)),K)) ) ) ) ).

% choose_reduce_nat
tff(fact_7261_times__binomial__minus1__eq,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))) ) ) ).

% times_binomial_minus1_eq
tff(fact_7262_take__bit__int__less__exp,axiom,
    ! [Na: nat,K: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Na),K)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)) ).

% take_bit_int_less_exp
tff(fact_7263_binomial__altdef__nat,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Na)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,K)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K)))) ) ) ).

% binomial_altdef_nat
tff(fact_7264_take__bit__eq__0__iff,axiom,
    ! [Na: nat,Aa: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),Aa) = zero_zero_nat )
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)),Aa) ) ).

% take_bit_eq_0_iff
tff(fact_7265_take__bit__eq__0__iff,axiom,
    ! [Na: nat,Aa: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_nat6383147911832638789nteger(bit_se1745604003318907178nteger,Na),Aa) = zero_z3403309356797280102nteger )
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Na)),Aa) ) ).

% take_bit_eq_0_iff
tff(fact_7266_take__bit__eq__0__iff,axiom,
    ! [Na: nat,Aa: int] :
      ( ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Na),Aa) = zero_zero_int )
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)),Aa) ) ).

% take_bit_eq_0_iff
tff(fact_7267_sum__strict__mono2,axiom,
    ! [B4: set_real,A3: set_real,Ba: real,F: fun_real_real] :
      ( finite_finite_real(B4)
     => ( ord_less_eq_set_real(A3,B4)
       => ( member_real(Ba,minus_minus_set_real(B4,A3))
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(F,Ba))
           => ( ! [X: real] :
                  ( member_real(X,B4)
                 => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,X)) )
             => aa_real_o(aa_real_fun_real_o(ord_less_real,groups8097168146408367636l_real(F,A3)),groups8097168146408367636l_real(F,B4)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_7268_sum__strict__mono2,axiom,
    ! [B4: set_complex,A3: set_complex,Ba: complex,F: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A3),B4)
       => ( aa_set_complex_o(member_complex(Ba),minus_811609699411566653omplex(B4,A3))
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(F,Ba))
           => ( ! [X: complex] :
                  ( aa_set_complex_o(member_complex(X),B4)
                 => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(F,X)) )
             => aa_real_o(aa_real_fun_real_o(ord_less_real,groups5808333547571424918x_real(F,A3)),groups5808333547571424918x_real(F,B4)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_7269_sum__strict__mono2,axiom,
    ! [B4: set_real,A3: set_real,Ba: real,F: fun_real_rat] :
      ( finite_finite_real(B4)
     => ( ord_less_eq_set_real(A3,B4)
       => ( member_real(Ba,minus_minus_set_real(B4,A3))
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_real_rat(F,Ba))
           => ( ! [X: real] :
                  ( member_real(X,B4)
                 => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,X)) )
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups1300246762558778688al_rat(F,A3)),groups1300246762558778688al_rat(F,B4)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_7270_sum__strict__mono2,axiom,
    ! [B4: set_complex,A3: set_complex,Ba: complex,F: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A3),B4)
       => ( aa_set_complex_o(member_complex(Ba),minus_811609699411566653omplex(B4,A3))
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_complex_rat(F,Ba))
           => ( ! [X: complex] :
                  ( aa_set_complex_o(member_complex(X),B4)
                 => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,X)) )
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups5058264527183730370ex_rat(F,A3)),groups5058264527183730370ex_rat(F,B4)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_7271_sum__strict__mono2,axiom,
    ! [B4: set_nat,A3: set_nat,Ba: nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,B4)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A3),B4)
       => ( aa_set_nat_o(member_nat(Ba),minus_minus_set_nat(B4,A3))
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(F,Ba))
           => ( ! [X: nat] :
                  ( aa_set_nat_o(member_nat(X),B4)
                 => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,X)) )
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups2906978787729119204at_rat(F,A3)),groups2906978787729119204at_rat(F,B4)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_7272_sum__strict__mono2,axiom,
    ! [B4: set_real,A3: set_real,Ba: real,F: fun_real_nat] :
      ( finite_finite_real(B4)
     => ( ord_less_eq_set_real(A3,B4)
       => ( member_real(Ba,minus_minus_set_real(B4,A3))
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_real_nat(F,Ba))
           => ( ! [X: real] :
                  ( member_real(X,B4)
                 => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,X)) )
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups1935376822645274424al_nat(F,A3)),groups1935376822645274424al_nat(F,B4)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_7273_sum__strict__mono2,axiom,
    ! [B4: set_complex,A3: set_complex,Ba: complex,F: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A3),B4)
       => ( aa_set_complex_o(member_complex(Ba),minus_811609699411566653omplex(B4,A3))
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_complex_nat(F,Ba))
           => ( ! [X: complex] :
                  ( aa_set_complex_o(member_complex(X),B4)
                 => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_complex_nat(F,X)) )
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups5693394587270226106ex_nat(F,A3)),groups5693394587270226106ex_nat(F,B4)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_7274_sum__strict__mono2,axiom,
    ! [B4: set_real,A3: set_real,Ba: real,F: fun_real_int] :
      ( finite_finite_real(B4)
     => ( ord_less_eq_set_real(A3,B4)
       => ( member_real(Ba,minus_minus_set_real(B4,A3))
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_real_int(F,Ba))
           => ( ! [X: real] :
                  ( member_real(X,B4)
                 => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_real_int(F,X)) )
             => aa_int_o(aa_int_fun_int_o(ord_less_int,groups1932886352136224148al_int(F,A3)),groups1932886352136224148al_int(F,B4)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_7275_sum__strict__mono2,axiom,
    ! [B4: set_complex,A3: set_complex,Ba: complex,F: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A3),B4)
       => ( aa_set_complex_o(member_complex(Ba),minus_811609699411566653omplex(B4,A3))
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_complex_int(F,Ba))
           => ( ! [X: complex] :
                  ( aa_set_complex_o(member_complex(X),B4)
                 => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_complex_int(F,X)) )
             => aa_int_o(aa_int_fun_int_o(ord_less_int,groups5690904116761175830ex_int(F,A3)),groups5690904116761175830ex_int(F,B4)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_7276_sum__strict__mono2,axiom,
    ! [B4: set_nat,A3: set_nat,Ba: nat,F: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,B4)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A3),B4)
       => ( aa_set_nat_o(member_nat(Ba),minus_minus_set_nat(B4,A3))
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(F,Ba))
           => ( ! [X: nat] :
                  ( aa_set_nat_o(member_nat(X),B4)
                 => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(F,X)) )
             => aa_int_o(aa_int_fun_int_o(ord_less_int,groups3539618377306564664at_int(F,A3)),groups3539618377306564664at_int(F,B4)) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_7277_binomial__less__binomial__Suc,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),aa_nat_nat(suc,K))) ) ).

% binomial_less_binomial_Suc
tff(fact_7278_binomial__strict__mono,axiom,
    ! [K: nat,K5: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),K5)
     => ( 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))),K5)),Na)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K5)) ) ) ).

% binomial_strict_mono
tff(fact_7279_binomial__strict__antimono,axiom,
    ! [K: nat,K5: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),K5)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K5),Na)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K5)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K)) ) ) ) ).

% binomial_strict_antimono
tff(fact_7280_central__binomial__odd,axiom,
    ! [Na: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ).

% central_binomial_odd
tff(fact_7281_binomial__addition__formula,axiom,
    ! [Na: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),aa_nat_nat(suc,K)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat)),aa_nat_nat(suc,K))),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat)),K)) ) ) ).

% binomial_addition_formula
tff(fact_7282_take__bit__nat__less__self__iff,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),M)),M)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)),M) ) ).

% take_bit_nat_less_self_iff
tff(fact_7283_take__bit__Suc__minus__bit0,axiom,
    ! [Na: nat,K: num] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_nat_nat(suc,Na)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,K)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Na),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% take_bit_Suc_minus_bit0
tff(fact_7284_convex__sum__bound__le,axiom,
    ! [I5: set_real,Xb: fun_re6978948838062957830nteger,Aa: fun_re6978948838062957830nteger,Ba: code_integer,Delta: code_integer] :
      ( ! [I2: real] :
          ( member_real(I2,I5)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_real_Code_integer(Xb,I2)) )
     => ( ( groups7713935264441627589nteger(Xb,I5) = one_one_Code_integer )
       => ( ! [I2: real] :
              ( member_real(I2,I5)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_real_Code_integer(Aa,I2)),Ba))),Delta) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,groups7713935264441627589nteger(aa_fun7923231501384339402nteger(aTP_Lamm_ik(Xb),Aa),I5)),Ba))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_7285_convex__sum__bound__le,axiom,
    ! [I5: set_nat,Xb: fun_nat_Code_integer,Aa: fun_nat_Code_integer,Ba: code_integer,Delta: code_integer] :
      ( ! [I2: nat] :
          ( aa_set_nat_o(member_nat(I2),I5)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(Xb,I2)) )
     => ( ( groups7501900531339628137nteger(Xb,I5) = one_one_Code_integer )
       => ( ! [I2: nat] :
              ( aa_set_nat_o(member_nat(I2),I5)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_nat_Code_integer(Aa,I2)),Ba))),Delta) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,groups7501900531339628137nteger(aa_fun589076786913273746nteger(aTP_Lamm_il(Xb),Aa),I5)),Ba))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_7286_convex__sum__bound__le,axiom,
    ! [I5: set_int,Xb: fun_int_Code_integer,Aa: fun_int_Code_integer,Ba: code_integer,Delta: code_integer] :
      ( ! [I2: int] :
          ( aa_set_int_o(member_int(I2),I5)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_int_Code_integer(Xb,I2)) )
     => ( ( groups7873554091576472773nteger(Xb,I5) = one_one_Code_integer )
       => ( ! [I2: int] :
              ( aa_set_int_o(member_int(I2),I5)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_int_Code_integer(Aa,I2)),Ba))),Delta) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,groups7873554091576472773nteger(aa_fun5304498221708291146nteger(aTP_Lamm_im(Xb),Aa),I5)),Ba))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_7287_convex__sum__bound__le,axiom,
    ! [I5: set_real,Xb: fun_real_real,Aa: fun_real_real,Ba: real,Delta: real] :
      ( ! [I2: real] :
          ( member_real(I2,I5)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(Xb,I2)) )
     => ( ( groups8097168146408367636l_real(Xb,I5) = one_one_real )
       => ( ! [I2: real] :
              ( member_real(I2,I5)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(Aa,I2)),Ba))),Delta) )
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups8097168146408367636l_real(aa_fun6946558556880649704l_real(aTP_Lamm_in(Xb),Aa),I5)),Ba))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_7288_convex__sum__bound__le,axiom,
    ! [I5: set_int,Xb: fun_int_real,Aa: fun_int_real,Ba: real,Delta: real] :
      ( ! [I2: int] :
          ( aa_set_int_o(member_int(I2),I5)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(Xb,I2)) )
     => ( ( groups8778361861064173332t_real(Xb,I5) = one_one_real )
       => ( ! [I2: int] :
              ( aa_set_int_o(member_int(I2),I5)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(Aa,I2)),Ba))),Delta) )
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups8778361861064173332t_real(aa_fun3252798400008884712t_real(aTP_Lamm_io(Xb),Aa),I5)),Ba))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_7289_convex__sum__bound__le,axiom,
    ! [I5: set_real,Xb: fun_real_rat,Aa: fun_real_rat,Ba: rat,Delta: rat] :
      ( ! [I2: real] :
          ( member_real(I2,I5)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(Xb,I2)) )
     => ( ( groups1300246762558778688al_rat(Xb,I5) = one_one_rat )
       => ( ! [I2: real] :
              ( member_real(I2,I5)
             => 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(Aa,I2)),Ba))),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_ip(Xb),Aa),I5)),Ba))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_7290_convex__sum__bound__le,axiom,
    ! [I5: set_nat,Xb: fun_nat_rat,Aa: fun_nat_rat,Ba: rat,Delta: rat] :
      ( ! [I2: nat] :
          ( aa_set_nat_o(member_nat(I2),I5)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(Xb,I2)) )
     => ( ( groups2906978787729119204at_rat(Xb,I5) = one_one_rat )
       => ( ! [I2: nat] :
              ( aa_set_nat_o(member_nat(I2),I5)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(aa_nat_rat(Aa,I2)),Ba))),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_iq(Xb),Aa),I5)),Ba))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_7291_convex__sum__bound__le,axiom,
    ! [I5: set_int,Xb: fun_int_rat,Aa: fun_int_rat,Ba: rat,Delta: rat] :
      ( ! [I2: int] :
          ( aa_set_int_o(member_int(I2),I5)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(Xb,I2)) )
     => ( ( groups3906332499630173760nt_rat(Xb,I5) = one_one_rat )
       => ( ! [I2: int] :
              ( aa_set_int_o(member_int(I2),I5)
             => 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(Aa,I2)),Ba))),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(groups3906332499630173760nt_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_ir(Xb),Aa),I5)),Ba))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_7292_convex__sum__bound__le,axiom,
    ! [I5: set_real,Xb: fun_real_int,Aa: fun_real_int,Ba: int,Delta: int] :
      ( ! [I2: real] :
          ( member_real(I2,I5)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_real_int(Xb,I2)) )
     => ( ( groups1932886352136224148al_int(Xb,I5) = one_one_int )
       => ( ! [I2: real] :
              ( member_real(I2,I5)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_real_int(Aa,I2)),Ba))),Delta) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,groups1932886352136224148al_int(aa_fun8270430325945277160al_int(aTP_Lamm_is(Xb),Aa),I5)),Ba))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_7293_convex__sum__bound__le,axiom,
    ! [I5: set_nat,Xb: fun_nat_int,Aa: fun_nat_int,Ba: int,Delta: int] :
      ( ! [I2: nat] :
          ( aa_set_nat_o(member_nat(I2),I5)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(Xb,I2)) )
     => ( ( groups3539618377306564664at_int(Xb,I5) = one_one_int )
       => ( ! [I2: nat] :
              ( aa_set_nat_o(member_nat(I2),I5)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(Aa,I2)),Ba))),Delta) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,groups3539618377306564664at_int(aa_fun5565043122120104496at_int(aTP_Lamm_it(Xb),Aa),I5)),Ba))),Delta) ) ) ) ).

% convex_sum_bound_le
tff(fact_7294_take__bit__int__less__self__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Na),K)),K)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)),K) ) ).

% take_bit_int_less_self_iff
tff(fact_7295_take__bit__int__greater__eq__self__iff,axiom,
    ! [K: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Na),K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)) ) ).

% take_bit_int_greater_eq_self_iff
tff(fact_7296_choose__two,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% choose_two
tff(fact_7297_take__bit__int__eq__self__iff,axiom,
    ! [Na: nat,K: int] :
      ( ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Na),K) = K )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)) ) ) ).

% take_bit_int_eq_self_iff
tff(fact_7298_take__bit__int__eq__self,axiom,
    ! [K: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na))
       => ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Na),K) = K ) ) ) ).

% take_bit_int_eq_self
tff(fact_7299_take__bit__Suc__minus__1__eq,axiom,
    ! [Na: nat] : aa_Cod1783974652533396728nteger(aa_nat6383147911832638789nteger(bit_se1745604003318907178nteger,aa_nat_nat(suc,Na)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_nat_nat(suc,Na))),one_one_Code_integer) ).

% take_bit_Suc_minus_1_eq
tff(fact_7300_take__bit__Suc__minus__1__eq,axiom,
    ! [Na: nat] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_nat_nat(suc,Na)),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,Na))),one_one_int) ).

% take_bit_Suc_minus_1_eq
tff(fact_7301_take__bit__Suc__bit1,axiom,
    ! [Na: nat,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_nat_nat(suc,Na)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),aa_num_nat(numeral_numeral_nat,K))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat) ).

% take_bit_Suc_bit1
tff(fact_7302_take__bit__Suc__bit1,axiom,
    ! [Na: nat,K: num] : aa_Cod1783974652533396728nteger(aa_nat6383147911832638789nteger(bit_se1745604003318907178nteger,aa_nat_nat(suc,Na)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,K))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_nat6383147911832638789nteger(bit_se1745604003318907178nteger,Na),aa_num_Code_integer(numera6620942414471956472nteger,K))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))),one_one_Code_integer) ).

% take_bit_Suc_bit1
tff(fact_7303_take__bit__Suc__bit1,axiom,
    ! [Na: nat,K: num] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_nat_nat(suc,Na)),aa_num_int(numeral_numeral_int,aa_num_num(bit1,K))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Na),aa_num_int(numeral_numeral_int,K))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).

% take_bit_Suc_bit1
tff(fact_7304_take__bit__numeral__minus__1__eq,axiom,
    ! [K: num] : aa_Cod1783974652533396728nteger(aa_nat6383147911832638789nteger(bit_se1745604003318907178nteger,aa_num_nat(numeral_numeral_nat,K)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(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_7305_take__bit__numeral__minus__1__eq,axiom,
    ! [K: num] : aa_int_int(aa_nat_fun_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_7306_take__bit__Suc,axiom,
    ! [Na: nat,Aa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_nat_nat(suc,Na)),Aa) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Aa),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,Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% take_bit_Suc
tff(fact_7307_take__bit__Suc,axiom,
    ! [Na: nat,Aa: code_integer] : aa_Cod1783974652533396728nteger(aa_nat6383147911832638789nteger(bit_se1745604003318907178nteger,aa_nat_nat(suc,Na)),Aa) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_nat6383147911832638789nteger(bit_se1745604003318907178nteger,Na),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Aa),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Aa),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))) ).

% take_bit_Suc
tff(fact_7308_take__bit__Suc,axiom,
    ! [Na: nat,Aa: int] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_nat_nat(suc,Na)),Aa) = 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_nat_fun_int_int(bit_se2923211474154528505it_int,Na),aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),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,Aa),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ).

% take_bit_Suc
tff(fact_7309_take__bit__int__less__eq,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Na),K)),aa_int_int(aa_int_fun_int_int(minus_minus_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na))) ) ) ).

% take_bit_int_less_eq
tff(fact_7310_take__bit__int__greater__eq,axiom,
    ! [K: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na))),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Na),K)) ) ).

% take_bit_int_greater_eq
tff(fact_7311_signed__take__bit__eq__take__bit__shift,axiom,
    ! [Na: nat,K: int] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,Na),K) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_nat_nat(suc,Na)),aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)) ).

% signed_take_bit_eq_take_bit_shift
tff(fact_7312_stable__imp__take__bit__eq,axiom,
    ! [Aa: nat,Na: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = Aa )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),Aa) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Aa),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)),one_one_nat)) ) ) ).

% stable_imp_take_bit_eq
tff(fact_7313_stable__imp__take__bit__eq,axiom,
    ! [Aa: code_integer,Na: nat] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Aa),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) = Aa )
     => ( aa_Cod1783974652533396728nteger(aa_nat6383147911832638789nteger(bit_se1745604003318907178nteger,Na),Aa) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Aa),zero_z3403309356797280102nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Na)),one_one_Code_integer)) ) ) ).

% stable_imp_take_bit_eq
tff(fact_7314_stable__imp__take__bit__eq,axiom,
    ! [Aa: int,Na: nat] :
      ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = Aa )
     => ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Na),Aa) = $ite(aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Aa),zero_zero_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)),one_one_int)) ) ) ).

% stable_imp_take_bit_eq
tff(fact_7315_take__bit__minus__small__eq,axiom,
    ! [K: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na))
       => ( aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Na),aa_int_int(uminus_uminus_int,K)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)),K) ) ) ) ).

% take_bit_minus_small_eq
tff(fact_7316_take__bit__Suc__minus__bit1,axiom,
    ! [Na: nat,K: num] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_nat_nat(suc,Na)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Na),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_7317_central__binomial__lower__bound,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))),Na)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Na)))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)),Na))) ) ).

% central_binomial_lower_bound
tff(fact_7318_pochhammer__double,axiom,
    ! [Z: complex,Na: nat] : aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),Z)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)))),aa_nat_complex(comm_s2602460028002588243omplex(Z),Na))),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(Z),aa_complex_complex(divide1717551699836669952omplex(one_one_complex),numera6690914467698888265omplex(aa_num_num(bit0,one))))),Na)) ).

% pochhammer_double
tff(fact_7319_pochhammer__double,axiom,
    ! [Z: rat,Na: 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))),Na)) = aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)))),aa_nat_rat(comm_s4028243227959126397er_rat(Z),Na))),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(plus_plus_rat(Z),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))),Na)) ).

% pochhammer_double
tff(fact_7320_pochhammer__double,axiom,
    ! [Z: real,Na: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Z)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)))),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z),Na))),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Na)) ).

% pochhammer_double
tff(fact_7321_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_7322_take__bit__numeral__bit1,axiom,
    ! [L: num,K: num] : aa_Cod1783974652533396728nteger(aa_nat6383147911832638789nteger(bit_se1745604003318907178nteger,aa_num_nat(numeral_numeral_nat,L)),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit1,K))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_nat6383147911832638789nteger(bit_se1745604003318907178nteger,aa_num_nat(pred_numeral,L)),aa_num_Code_integer(numera6620942414471956472nteger,K))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)))),one_one_Code_integer) ).

% take_bit_numeral_bit1
tff(fact_7323_take__bit__numeral__bit1,axiom,
    ! [L: num,K: num] : aa_int_int(aa_nat_fun_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(aa_nat_fun_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_7324_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(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xb)),aa_real_real(aa_rea7340605660300375871l_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_7325_sqrt__sum__squares__half__less,axiom,
    ! [Xb: real,U: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_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,Y),aa_real_real(aa_rea7340605660300375871l_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),Y)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(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,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),U) ) ) ) ) ).

% sqrt_sum_squares_half_less
tff(fact_7326_Code__Numeral_Opositive__def,axiom,
    code_positive = numera6620942414471956472nteger ).

% Code_Numeral.positive_def
tff(fact_7327_of__nat__eq__iff,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,M) = aa_nat_real(semiri5074537144036343181t_real,Na) )
    <=> ( M = Na ) ) ).

% of_nat_eq_iff
tff(fact_7328_of__nat__eq__iff,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,M) = aa_nat_int(semiri1314217659103216013at_int,Na) )
    <=> ( M = Na ) ) ).

% of_nat_eq_iff
tff(fact_7329_of__nat__eq__iff,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa_nat_Code_integer(semiri4939895301339042750nteger,M) = aa_nat_Code_integer(semiri4939895301339042750nteger,Na) )
    <=> ( M = Na ) ) ).

% of_nat_eq_iff
tff(fact_7330_abs__of__nat,axiom,
    ! [Na: nat] : aa_rat_rat(abs_abs_rat,aa_nat_rat(semiri681578069525770553at_rat,Na)) = aa_nat_rat(semiri681578069525770553at_rat,Na) ).

% abs_of_nat
tff(fact_7331_abs__of__nat,axiom,
    ! [Na: nat] : aa_real_real(abs_abs_real,aa_nat_real(semiri5074537144036343181t_real,Na)) = aa_nat_real(semiri5074537144036343181t_real,Na) ).

% abs_of_nat
tff(fact_7332_abs__of__nat,axiom,
    ! [Na: nat] : aa_int_int(abs_abs_int,aa_nat_int(semiri1314217659103216013at_int,Na)) = aa_nat_int(semiri1314217659103216013at_int,Na) ).

% abs_of_nat
tff(fact_7333_abs__of__nat,axiom,
    ! [Na: nat] : aa_Cod1783974652533396728nteger(abs_abs_Code_integer,aa_nat_Code_integer(semiri4939895301339042750nteger,Na)) = aa_nat_Code_integer(semiri4939895301339042750nteger,Na) ).

% abs_of_nat
tff(fact_7334_negative__eq__positive,axiom,
    ! [Na: nat,M: nat] :
      ( ( aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,Na)) = aa_nat_int(semiri1314217659103216013at_int,M) )
    <=> ( ( Na = zero_zero_nat )
        & ( M = zero_zero_nat ) ) ) ).

% negative_eq_positive
tff(fact_7335_negative__zle,axiom,
    ! [Na: nat,M: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,Na))),aa_nat_int(semiri1314217659103216013at_int,M)) ).

% negative_zle
tff(fact_7336_real__sqrt__less__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,Xb)),aa_real_real(sqrt,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y) ) ).

% real_sqrt_less_iff
tff(fact_7337_real__sqrt__le__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,Xb)),aa_real_real(sqrt,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y) ) ).

% real_sqrt_le_iff
tff(fact_7338_exp__less__mono,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(exp_real,Xb)),aa_real_real(exp_real,Y)) ) ).

% exp_less_mono
tff(fact_7339_exp__less__cancel__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(exp_real,Xb)),aa_real_real(exp_real,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y) ) ).

% exp_less_cancel_iff
tff(fact_7340_exp__le__cancel__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,Xb)),aa_real_real(exp_real,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y) ) ).

% exp_le_cancel_iff
tff(fact_7341_euclidean__size__of__nat,axiom,
    ! [Na: nat] : aa_Code_integer_nat(euclid6377331345833325938nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Na)) = Na ).

% euclidean_size_of_nat
tff(fact_7342_euclidean__size__of__nat,axiom,
    ! [Na: nat] : aa_int_nat(euclid4774559944035922753ze_int,aa_nat_int(semiri1314217659103216013at_int,Na)) = Na ).

% euclidean_size_of_nat
tff(fact_7343_euclidean__size__of__nat,axiom,
    ! [Na: nat] : aa_nat_nat(euclid4777050414544973029ze_nat,aa_nat_nat(semiri1316708129612266289at_nat,Na)) = Na ).

% euclidean_size_of_nat
tff(fact_7344_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( aa_nat_rat(semiri681578069525770553at_rat,M) = zero_zero_rat )
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_7345_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,M) = zero_zero_nat )
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_7346_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,M) = zero_zero_real )
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_7347_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,M) = zero_zero_int )
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_7348_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( aa_nat_Code_integer(semiri4939895301339042750nteger,M) = zero_z3403309356797280102nteger )
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_7349_of__nat__0__eq__iff,axiom,
    ! [Na: nat] :
      ( ( zero_zero_rat = aa_nat_rat(semiri681578069525770553at_rat,Na) )
    <=> ( zero_zero_nat = Na ) ) ).

% of_nat_0_eq_iff
tff(fact_7350_of__nat__0__eq__iff,axiom,
    ! [Na: nat] :
      ( ( zero_zero_nat = aa_nat_nat(semiri1316708129612266289at_nat,Na) )
    <=> ( zero_zero_nat = Na ) ) ).

% of_nat_0_eq_iff
tff(fact_7351_of__nat__0__eq__iff,axiom,
    ! [Na: nat] :
      ( ( zero_zero_real = aa_nat_real(semiri5074537144036343181t_real,Na) )
    <=> ( zero_zero_nat = Na ) ) ).

% of_nat_0_eq_iff
tff(fact_7352_of__nat__0__eq__iff,axiom,
    ! [Na: nat] :
      ( ( zero_zero_int = aa_nat_int(semiri1314217659103216013at_int,Na) )
    <=> ( zero_zero_nat = Na ) ) ).

% of_nat_0_eq_iff
tff(fact_7353_of__nat__0__eq__iff,axiom,
    ! [Na: nat] :
      ( ( zero_z3403309356797280102nteger = aa_nat_Code_integer(semiri4939895301339042750nteger,Na) )
    <=> ( zero_zero_nat = Na ) ) ).

% of_nat_0_eq_iff
tff(fact_7354_of__nat__0,axiom,
    aa_nat_rat(semiri681578069525770553at_rat,zero_zero_nat) = zero_zero_rat ).

% of_nat_0
tff(fact_7355_of__nat__0,axiom,
    aa_nat_nat(semiri1316708129612266289at_nat,zero_zero_nat) = zero_zero_nat ).

% of_nat_0
tff(fact_7356_of__nat__0,axiom,
    aa_nat_real(semiri5074537144036343181t_real,zero_zero_nat) = zero_zero_real ).

% of_nat_0
tff(fact_7357_of__nat__0,axiom,
    aa_nat_int(semiri1314217659103216013at_int,zero_zero_nat) = zero_zero_int ).

% of_nat_0
tff(fact_7358_of__nat__0,axiom,
    aa_nat_Code_integer(semiri4939895301339042750nteger,zero_zero_nat) = zero_z3403309356797280102nteger ).

% of_nat_0
tff(fact_7359_of__nat__less__iff,axiom,
    ! [M: nat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ).

% of_nat_less_iff
tff(fact_7360_of__nat__less__iff,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ).

% of_nat_less_iff
tff(fact_7361_of__nat__less__iff,axiom,
    ! [M: nat,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ).

% of_nat_less_iff
tff(fact_7362_of__nat__less__iff,axiom,
    ! [M: nat,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ).

% of_nat_less_iff
tff(fact_7363_of__nat__less__iff,axiom,
    ! [M: nat,Na: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ).

% of_nat_less_iff
tff(fact_7364_of__nat__le__iff,axiom,
    ! [M: nat,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ).

% of_nat_le_iff
tff(fact_7365_of__nat__le__iff,axiom,
    ! [M: nat,Na: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ).

% of_nat_le_iff
tff(fact_7366_of__nat__le__iff,axiom,
    ! [M: nat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ).

% of_nat_le_iff
tff(fact_7367_of__nat__le__iff,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ).

% of_nat_le_iff
tff(fact_7368_of__nat__le__iff,axiom,
    ! [M: nat,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ).

% of_nat_le_iff
tff(fact_7369_of__nat__numeral,axiom,
    ! [Na: num] : aa_nat_nat(semiri1316708129612266289at_nat,aa_num_nat(numeral_numeral_nat,Na)) = aa_num_nat(numeral_numeral_nat,Na) ).

% of_nat_numeral
tff(fact_7370_of__nat__numeral,axiom,
    ! [Na: num] : aa_nat_Extended_enat(semiri4216267220026989637d_enat,aa_num_nat(numeral_numeral_nat,Na)) = numera1916890842035813515d_enat(Na) ).

% of_nat_numeral
tff(fact_7371_of__nat__numeral,axiom,
    ! [Na: num] : aa_nat_real(semiri5074537144036343181t_real,aa_num_nat(numeral_numeral_nat,Na)) = aa_num_real(numeral_numeral_real,Na) ).

% of_nat_numeral
tff(fact_7372_of__nat__numeral,axiom,
    ! [Na: num] : aa_nat_int(semiri1314217659103216013at_int,aa_num_nat(numeral_numeral_nat,Na)) = aa_num_int(numeral_numeral_int,Na) ).

% of_nat_numeral
tff(fact_7373_of__nat__numeral,axiom,
    ! [Na: num] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_num_nat(numeral_numeral_nat,Na)) = aa_num_Code_integer(numera6620942414471956472nteger,Na) ).

% of_nat_numeral
tff(fact_7374_of__nat__add,axiom,
    ! [M: nat,Na: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na)) = aa_rat_rat(plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,Na)) ).

% of_nat_add
tff(fact_7375_of__nat__add,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,Na)) ).

% of_nat_add
tff(fact_7376_of__nat__add,axiom,
    ! [M: nat,Na: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,Na)) ).

% of_nat_add
tff(fact_7377_of__nat__add,axiom,
    ! [M: nat,Na: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,Na)) ).

% of_nat_add
tff(fact_7378_of__nat__add,axiom,
    ! [M: nat,Na: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,Na)) ).

% of_nat_add
tff(fact_7379_of__nat__mult,axiom,
    ! [M: nat,Na: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na)) = aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,Na)) ).

% of_nat_mult
tff(fact_7380_of__nat__mult,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,Na)) ).

% of_nat_mult
tff(fact_7381_of__nat__mult,axiom,
    ! [M: nat,Na: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,Na)) ).

% of_nat_mult
tff(fact_7382_of__nat__mult,axiom,
    ! [M: nat,Na: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,Na)) ).

% of_nat_mult
tff(fact_7383_of__nat__mult,axiom,
    ! [M: nat,Na: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,Na)) ).

% of_nat_mult
tff(fact_7384_exp__zero,axiom,
    aa_complex_complex(exp_complex,zero_zero_complex) = one_one_complex ).

% exp_zero
tff(fact_7385_exp__zero,axiom,
    aa_real_real(exp_real,zero_zero_real) = one_one_real ).

% exp_zero
tff(fact_7386_of__nat__1,axiom,
    aa_nat_complex(semiri8010041392384452111omplex,one_one_nat) = one_one_complex ).

% of_nat_1
tff(fact_7387_of__nat__1,axiom,
    aa_nat_rat(semiri681578069525770553at_rat,one_one_nat) = one_one_rat ).

% of_nat_1
tff(fact_7388_of__nat__1,axiom,
    aa_nat_nat(semiri1316708129612266289at_nat,one_one_nat) = one_one_nat ).

% of_nat_1
tff(fact_7389_of__nat__1,axiom,
    aa_nat_real(semiri5074537144036343181t_real,one_one_nat) = one_one_real ).

% of_nat_1
tff(fact_7390_of__nat__1,axiom,
    aa_nat_int(semiri1314217659103216013at_int,one_one_nat) = one_one_int ).

% of_nat_1
tff(fact_7391_of__nat__1,axiom,
    aa_nat_Code_integer(semiri4939895301339042750nteger,one_one_nat) = one_one_Code_integer ).

% of_nat_1
tff(fact_7392_of__nat__1__eq__iff,axiom,
    ! [Na: nat] :
      ( ( one_one_complex = aa_nat_complex(semiri8010041392384452111omplex,Na) )
    <=> ( Na = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_7393_of__nat__1__eq__iff,axiom,
    ! [Na: nat] :
      ( ( one_one_rat = aa_nat_rat(semiri681578069525770553at_rat,Na) )
    <=> ( Na = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_7394_of__nat__1__eq__iff,axiom,
    ! [Na: nat] :
      ( ( one_one_nat = aa_nat_nat(semiri1316708129612266289at_nat,Na) )
    <=> ( Na = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_7395_of__nat__1__eq__iff,axiom,
    ! [Na: nat] :
      ( ( one_one_real = aa_nat_real(semiri5074537144036343181t_real,Na) )
    <=> ( Na = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_7396_of__nat__1__eq__iff,axiom,
    ! [Na: nat] :
      ( ( one_one_int = aa_nat_int(semiri1314217659103216013at_int,Na) )
    <=> ( Na = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_7397_of__nat__1__eq__iff,axiom,
    ! [Na: nat] :
      ( ( one_one_Code_integer = aa_nat_Code_integer(semiri4939895301339042750nteger,Na) )
    <=> ( Na = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_7398_of__nat__eq__1__iff,axiom,
    ! [Na: nat] :
      ( ( aa_nat_complex(semiri8010041392384452111omplex,Na) = one_one_complex )
    <=> ( Na = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_7399_of__nat__eq__1__iff,axiom,
    ! [Na: nat] :
      ( ( aa_nat_rat(semiri681578069525770553at_rat,Na) = one_one_rat )
    <=> ( Na = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_7400_of__nat__eq__1__iff,axiom,
    ! [Na: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,Na) = one_one_nat )
    <=> ( Na = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_7401_of__nat__eq__1__iff,axiom,
    ! [Na: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,Na) = one_one_real )
    <=> ( Na = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_7402_of__nat__eq__1__iff,axiom,
    ! [Na: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,Na) = one_one_int )
    <=> ( Na = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_7403_of__nat__eq__1__iff,axiom,
    ! [Na: nat] :
      ( ( aa_nat_Code_integer(semiri4939895301339042750nteger,Na) = one_one_Code_integer )
    <=> ( Na = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_7404_negative__zless,axiom,
    ! [Na: nat,M: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,Na)))),aa_nat_int(semiri1314217659103216013at_int,M)) ).

% negative_zless
tff(fact_7405_real__sqrt__lt__0__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,Xb)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real) ) ).

% real_sqrt_lt_0_iff
tff(fact_7406_real__sqrt__gt__0__iff,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(sqrt,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y) ) ).

% real_sqrt_gt_0_iff
tff(fact_7407_real__sqrt__ge__0__iff,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(sqrt,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y) ) ).

% real_sqrt_ge_0_iff
tff(fact_7408_real__sqrt__le__0__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,Xb)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real) ) ).

% real_sqrt_le_0_iff
tff(fact_7409_real__sqrt__gt__1__iff,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(sqrt,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Y) ) ).

% real_sqrt_gt_1_iff
tff(fact_7410_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_7411_real__sqrt__ge__1__iff,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(sqrt,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Y) ) ).

% real_sqrt_ge_1_iff
tff(fact_7412_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_7413_pred__numeral__simps_I1_J,axiom,
    aa_num_nat(pred_numeral,one) = zero_zero_nat ).

% pred_numeral_simps(1)
tff(fact_7414_Suc__eq__numeral,axiom,
    ! [Na: nat,K: num] :
      ( ( aa_nat_nat(suc,Na) = aa_num_nat(numeral_numeral_nat,K) )
    <=> ( Na = aa_num_nat(pred_numeral,K) ) ) ).

% Suc_eq_numeral
tff(fact_7415_eq__numeral__Suc,axiom,
    ! [K: num,Na: nat] :
      ( ( aa_num_nat(numeral_numeral_nat,K) = aa_nat_nat(suc,Na) )
    <=> ( aa_num_nat(pred_numeral,K) = Na ) ) ).

% eq_numeral_Suc
tff(fact_7416_of__nat__fact,axiom,
    ! [Na: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(semiri1408675320244567234ct_nat,Na)) = aa_nat_int(semiri1406184849735516958ct_int,Na) ).

% of_nat_fact
tff(fact_7417_of__nat__fact,axiom,
    ! [Na: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(semiri1408675320244567234ct_nat,Na)) = aa_nat_Code_integer(semiri3624122377584611663nteger,Na) ).

% of_nat_fact
tff(fact_7418_of__nat__fact,axiom,
    ! [Na: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Na)) = aa_nat_nat(semiri1408675320244567234ct_nat,Na) ).

% of_nat_fact
tff(fact_7419_of__nat__fact,axiom,
    ! [Na: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(semiri1408675320244567234ct_nat,Na)) = aa_nat_real(semiri2265585572941072030t_real,Na) ).

% of_nat_fact
tff(fact_7420_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_7421_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_7422_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_7423_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_7424_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_7425_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,M)),zero_zero_real)
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_7426_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),zero_z3403309356797280102nteger)
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_7427_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),zero_zero_rat)
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_7428_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),zero_zero_nat)
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_7429_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,M)),zero_zero_int)
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_7430_of__nat__Suc,axiom,
    ! [M: nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,M)) = aa_complex_complex(plus_plus_complex(one_one_complex),aa_nat_complex(semiri8010041392384452111omplex,M)) ).

% of_nat_Suc
tff(fact_7431_of__nat__Suc,axiom,
    ! [M: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,M)) = aa_rat_rat(plus_plus_rat(one_one_rat),aa_nat_rat(semiri681578069525770553at_rat,M)) ).

% of_nat_Suc
tff(fact_7432_of__nat__Suc,axiom,
    ! [M: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(suc,M)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(semiri1316708129612266289at_nat,M)) ).

% of_nat_Suc
tff(fact_7433_of__nat__Suc,axiom,
    ! [M: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,M)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,M)) ).

% of_nat_Suc
tff(fact_7434_of__nat__Suc,axiom,
    ! [M: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,M)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_nat_int(semiri1314217659103216013at_int,M)) ).

% of_nat_Suc
tff(fact_7435_of__nat__Suc,axiom,
    ! [M: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(suc,M)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,one_one_Code_integer),aa_nat_Code_integer(semiri4939895301339042750nteger,M)) ).

% of_nat_Suc
tff(fact_7436_fact__Suc,axiom,
    ! [Na: nat] : aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(suc,Na)) = aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,Na))),aa_nat_rat(semiri773545260158071498ct_rat,Na)) ).

% fact_Suc
tff(fact_7437_fact__Suc,axiom,
    ! [Na: nat] : aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(suc,Na)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,Na))),aa_nat_int(semiri1406184849735516958ct_int,Na)) ).

% fact_Suc
tff(fact_7438_fact__Suc,axiom,
    ! [Na: nat] : aa_nat_Code_integer(semiri3624122377584611663nteger,aa_nat_nat(suc,Na)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(suc,Na))),aa_nat_Code_integer(semiri3624122377584611663nteger,Na)) ).

% fact_Suc
tff(fact_7439_fact__Suc,axiom,
    ! [Na: nat] : aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(suc,Na)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(suc,Na))),aa_nat_nat(semiri1408675320244567234ct_nat,Na)) ).

% fact_Suc
tff(fact_7440_fact__Suc,axiom,
    ! [Na: nat] : aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(suc,Na)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Na))),aa_nat_real(semiri2265585572941072030t_real,Na)) ).

% fact_Suc
tff(fact_7441_less__numeral__Suc,axiom,
    ! [K: num,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(suc,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(pred_numeral,K)),Na) ) ).

% less_numeral_Suc
tff(fact_7442_less__Suc__numeral,axiom,
    ! [Na: nat,K: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Na)),aa_num_nat(numeral_numeral_nat,K))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_num_nat(pred_numeral,K)) ) ).

% less_Suc_numeral
tff(fact_7443_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_7444_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_7445_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_7446_le__numeral__Suc,axiom,
    ! [K: num,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(suc,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(pred_numeral,K)),Na) ) ).

% le_numeral_Suc
tff(fact_7447_le__Suc__numeral,axiom,
    ! [Na: nat,K: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Na)),aa_num_nat(numeral_numeral_nat,K))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),aa_num_nat(pred_numeral,K)) ) ).

% le_Suc_numeral
tff(fact_7448_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_7449_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_7450_diff__numeral__Suc,axiom,
    ! [K: num,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(suc,Na)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_num_nat(pred_numeral,K)),Na) ).

% diff_numeral_Suc
tff(fact_7451_diff__Suc__numeral,axiom,
    ! [Na: nat,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,Na)),aa_num_nat(numeral_numeral_nat,K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_num_nat(pred_numeral,K)) ).

% diff_Suc_numeral
tff(fact_7452_exp__ln,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_real(exp_real,aa_real_real(ln_ln_real,Xb)) = Xb ) ) ).

% exp_ln
tff(fact_7453_exp__ln__iff,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(exp_real,aa_real_real(ln_ln_real,Xb)) = Xb )
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb) ) ).

% exp_ln_iff
tff(fact_7454_max__numeral__Suc,axiom,
    ! [K: num,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(suc,Na)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(pred_numeral,K)),Na)) ).

% max_numeral_Suc
tff(fact_7455_max__Suc__numeral,axiom,
    ! [Na: nat,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(suc,Na)),aa_num_nat(numeral_numeral_nat,K)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Na),aa_num_nat(pred_numeral,K))) ).

% max_Suc_numeral
tff(fact_7456_of__nat__0__less__iff,axiom,
    ! [Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(semiri681578069525770553at_rat,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ).

% of_nat_0_less_iff
tff(fact_7457_of__nat__0__less__iff,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(semiri1316708129612266289at_nat,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ).

% of_nat_0_less_iff
tff(fact_7458_of__nat__0__less__iff,axiom,
    ! [Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(semiri5074537144036343181t_real,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ).

% of_nat_0_less_iff
tff(fact_7459_of__nat__0__less__iff,axiom,
    ! [Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(semiri1314217659103216013at_int,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ).

% of_nat_0_less_iff
tff(fact_7460_of__nat__0__less__iff,axiom,
    ! [Na: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(semiri4939895301339042750nteger,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ).

% of_nat_0_less_iff
tff(fact_7461_sum_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_rat,M: nat,Na: nat] :
      groups2906978787729119204at_rat(G,set_or1269000886237332187st_nat(M,aa_nat_nat(suc,Na))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Na)),M),zero_zero_rat,aa_rat_rat(plus_plus_rat(groups2906978787729119204at_rat(G,set_or1269000886237332187st_nat(M,Na))),aa_nat_rat(G,aa_nat_nat(suc,Na)))) ).

% sum.cl_ivl_Suc
tff(fact_7462_sum_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_int,M: nat,Na: nat] :
      groups3539618377306564664at_int(G,set_or1269000886237332187st_nat(M,aa_nat_nat(suc,Na))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Na)),M),zero_zero_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,set_or1269000886237332187st_nat(M,Na))),aa_nat_int(G,aa_nat_nat(suc,Na)))) ).

% sum.cl_ivl_Suc
tff(fact_7463_sum_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_nat,M: nat,Na: nat] :
      aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(M,aa_nat_nat(suc,Na))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Na)),M),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(M,Na))),aa_nat_nat(G,aa_nat_nat(suc,Na)))) ).

% sum.cl_ivl_Suc
tff(fact_7464_sum_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_real,M: nat,Na: nat] :
      aa_set_nat_real(groups6591440286371151544t_real(G),set_or1269000886237332187st_nat(M,aa_nat_nat(suc,Na))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Na)),M),zero_zero_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(G),set_or1269000886237332187st_nat(M,Na))),aa_nat_real(G,aa_nat_nat(suc,Na)))) ).

% sum.cl_ivl_Suc
tff(fact_7465_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [Ba: nat,W2: 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,Ba)),W2)),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,Ba),W2)),Xb) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_7466_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [Ba: nat,W2: 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,Ba)),W2)),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,Ba),W2)),Xb) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_7467_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [Ba: nat,W2: 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,Ba)),W2)),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,Ba),W2)),Xb) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_7468_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [Ba: nat,W2: 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,Ba)),W2)),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,Ba),W2)),Xb) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_7469_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [Ba: nat,W2: nat,Xb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,Ba)),W2)),aa_nat_Code_integer(semiri4939895301339042750nteger,Xb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Ba),W2)),Xb) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_7470_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [Xb: nat,Ba: nat,W2: 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,Ba)),W2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Ba),W2)) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_7471_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [Xb: nat,Ba: nat,W2: 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,Ba)),W2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Ba),W2)) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_7472_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [Xb: nat,Ba: nat,W2: 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,Ba)),W2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Ba),W2)) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_7473_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [Xb: nat,Ba: nat,W2: 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,Ba)),W2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Ba),W2)) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_7474_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [Xb: nat,Ba: nat,W2: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Xb)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,Ba)),W2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Ba),W2)) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_7475_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [Xb: nat,Ba: nat,W2: 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,Ba)),W2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Ba),W2)) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_7476_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [Xb: nat,Ba: nat,W2: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Xb)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,Ba)),W2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Ba),W2)) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_7477_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [Xb: nat,Ba: nat,W2: 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,Ba)),W2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Ba),W2)) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_7478_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [Xb: nat,Ba: nat,W2: 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,Ba)),W2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Ba),W2)) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_7479_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [Xb: nat,Ba: nat,W2: 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,Ba)),W2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Ba),W2)) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_7480_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [Ba: nat,W2: 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,Ba)),W2)),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,Ba),W2)),Xb) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_7481_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [Ba: nat,W2: nat,Xb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,Ba)),W2)),aa_nat_Code_integer(semiri4939895301339042750nteger,Xb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Ba),W2)),Xb) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_7482_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [Ba: nat,W2: 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,Ba)),W2)),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,Ba),W2)),Xb) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_7483_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [Ba: nat,W2: 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,Ba)),W2)),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,Ba),W2)),Xb) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_7484_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [Ba: nat,W2: 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,Ba)),W2)),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,Ba),W2)),Xb) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_7485_numeral__less__real__of__nat__iff,axiom,
    ! [W2: num,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,W2)),aa_nat_real(semiri5074537144036343181t_real,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,W2)),Na) ) ).

% numeral_less_real_of_nat_iff
tff(fact_7486_real__of__nat__less__numeral__iff,axiom,
    ! [Na: nat,W2: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_num_real(numeral_numeral_real,W2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_num_nat(numeral_numeral_nat,W2)) ) ).

% real_of_nat_less_numeral_iff
tff(fact_7487_numeral__le__real__of__nat__iff,axiom,
    ! [Na: num,M: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,Na)),aa_nat_real(semiri5074537144036343181t_real,M))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,Na)),M) ) ).

% numeral_le_real_of_nat_iff
tff(fact_7488_sum__zero__power,axiom,
    ! [C: fun_nat_complex,A3: set_nat] :
      aa_set_nat_complex(groups2073611262835488442omplex(aTP_Lamm_iu(C)),A3) = $ite(
        ( aa_set_nat_o(finite_finite_nat,A3)
        & aa_set_nat_o(member_nat(zero_zero_nat),A3) ),
        aa_nat_complex(C,zero_zero_nat),
        zero_zero_complex ) ).

% sum_zero_power
tff(fact_7489_sum__zero__power,axiom,
    ! [C: fun_nat_rat,A3: set_nat] :
      groups2906978787729119204at_rat(aTP_Lamm_iv(C),A3) = $ite(
        ( aa_set_nat_o(finite_finite_nat,A3)
        & aa_set_nat_o(member_nat(zero_zero_nat),A3) ),
        aa_nat_rat(C,zero_zero_nat),
        zero_zero_rat ) ).

% sum_zero_power
tff(fact_7490_sum__zero__power,axiom,
    ! [C: fun_nat_real,A3: set_nat] :
      aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_iw(C)),A3) = $ite(
        ( aa_set_nat_o(finite_finite_nat,A3)
        & aa_set_nat_o(member_nat(zero_zero_nat),A3) ),
        aa_nat_real(C,zero_zero_nat),
        zero_zero_real ) ).

% sum_zero_power
tff(fact_7491_of__nat__zero__less__power__iff,axiom,
    ! [Xb: nat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,Xb)),Na))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Xb)
        | ( Na = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_7492_of__nat__zero__less__power__iff,axiom,
    ! [Xb: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,Xb)),Na))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Xb)
        | ( Na = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_7493_of__nat__zero__less__power__iff,axiom,
    ! [Xb: nat,Na: 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)),Na))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Xb)
        | ( Na = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_7494_of__nat__zero__less__power__iff,axiom,
    ! [Xb: nat,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,Xb)),Na))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Xb)
        | ( Na = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_7495_of__nat__zero__less__power__iff,axiom,
    ! [Xb: nat,Na: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(power_8256067586552552935nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,Xb)),Na))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Xb)
        | ( Na = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_7496_sum__zero__power_H,axiom,
    ! [C: fun_nat_complex,D: fun_nat_complex,A3: set_nat] :
      aa_set_nat_complex(groups2073611262835488442omplex(aa_fun2955221450537884980omplex(aTP_Lamm_ix(C),D)),A3) = $ite(
        ( aa_set_nat_o(finite_finite_nat,A3)
        & aa_set_nat_o(member_nat(zero_zero_nat),A3) ),
        aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(C,zero_zero_nat)),aa_nat_complex(D,zero_zero_nat)),
        zero_zero_complex ) ).

% sum_zero_power'
tff(fact_7497_sum__zero__power_H,axiom,
    ! [C: fun_nat_rat,D: fun_nat_rat,A3: set_nat] :
      groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_iy(C),D),A3) = $ite(
        ( aa_set_nat_o(finite_finite_nat,A3)
        & aa_set_nat_o(member_nat(zero_zero_nat),A3) ),
        aa_rat_rat(divide_divide_rat(aa_nat_rat(C,zero_zero_nat)),aa_nat_rat(D,zero_zero_nat)),
        zero_zero_rat ) ).

% sum_zero_power'
tff(fact_7498_sum__zero__power_H,axiom,
    ! [C: fun_nat_real,D: fun_nat_real,A3: set_nat] :
      aa_set_nat_real(groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_iz(C),D)),A3) = $ite(
        ( aa_set_nat_o(finite_finite_nat,A3)
        & aa_set_nat_o(member_nat(zero_zero_nat),A3) ),
        aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(C,zero_zero_nat)),aa_nat_real(D,zero_zero_nat)),
        zero_zero_real ) ).

% sum_zero_power'
tff(fact_7499_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [Xb: nat,I: num,Na: 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,I)),Na))
    <=> 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,I)),Na)) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_7500_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [Xb: nat,I: num,Na: 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,I)),Na))
    <=> 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,I)),Na)) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_7501_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [Xb: nat,I: num,Na: 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,I)),Na))
    <=> 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,I)),Na)) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_7502_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [Xb: nat,I: num,Na: 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,I)),Na))
    <=> 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,I)),Na)) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_7503_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [Xb: nat,I: num,Na: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Xb)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,I)),Na))
    <=> 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,I)),Na)) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_7504_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,Na: 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,I)),Na)),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,I)),Na)),Xb) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_7505_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,Na: 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,I)),Na)),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,I)),Na)),Xb) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_7506_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,Na: 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,I)),Na)),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,I)),Na)),Xb) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_7507_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,Na: 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,I)),Na)),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,I)),Na)),Xb) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_7508_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,Na: nat,Xb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,I)),Na)),aa_nat_Code_integer(semiri4939895301339042750nteger,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,I)),Na)),Xb) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_7509_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [Xb: nat,I: num,Na: 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,I)),Na))
    <=> 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,I)),Na)) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_7510_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [Xb: nat,I: num,Na: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Xb)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,I)),Na))
    <=> 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,I)),Na)) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_7511_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [Xb: nat,I: num,Na: 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,I)),Na))
    <=> 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,I)),Na)) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_7512_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [Xb: nat,I: num,Na: 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,I)),Na))
    <=> 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,I)),Na)) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_7513_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [Xb: nat,I: num,Na: 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,I)),Na))
    <=> 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,I)),Na)) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_7514_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,Na: 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,I)),Na)),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,I)),Na)),Xb) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_7515_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,Na: nat,Xb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,I)),Na)),aa_nat_Code_integer(semiri4939895301339042750nteger,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,I)),Na)),Xb) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_7516_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,Na: 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,I)),Na)),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,I)),Na)),Xb) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_7517_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,Na: 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,I)),Na)),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,I)),Na)),Xb) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_7518_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,Na: 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,I)),Na)),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,I)),Na)),Xb) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_7519_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_7520_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_7521_real__sqrt__less__mono,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,Xb)),aa_real_real(sqrt,Y)) ) ).

% real_sqrt_less_mono
tff(fact_7522_exp__less__cancel,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(exp_real,Xb)),aa_real_real(exp_real,Y))
     => aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y) ) ).

% exp_less_cancel
tff(fact_7523_real__sqrt__le__mono,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,Xb)),aa_real_real(sqrt,Y)) ) ).

% real_sqrt_le_mono
tff(fact_7524_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_7525_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_7526_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_7527_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_7528_mult__of__nat__commute,axiom,
    ! [Xb: nat,Y: rat] : aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,Xb)),Y) = aa_rat_rat(times_times_rat(Y),aa_nat_rat(semiri681578069525770553at_rat,Xb)) ).

% mult_of_nat_commute
tff(fact_7529_mult__of__nat__commute,axiom,
    ! [Xb: nat,Y: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Xb)),Y) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Y),aa_nat_nat(semiri1316708129612266289at_nat,Xb)) ).

% mult_of_nat_commute
tff(fact_7530_mult__of__nat__commute,axiom,
    ! [Xb: nat,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Xb)),Y) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),aa_nat_real(semiri5074537144036343181t_real,Xb)) ).

% mult_of_nat_commute
tff(fact_7531_mult__of__nat__commute,axiom,
    ! [Xb: nat,Y: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Xb)),Y) = aa_int_int(aa_int_fun_int_int(times_times_int,Y),aa_nat_int(semiri1314217659103216013at_int,Xb)) ).

% mult_of_nat_commute
tff(fact_7532_mult__of__nat__commute,axiom,
    ! [Xb: nat,Y: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Xb)),Y) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Y),aa_nat_Code_integer(semiri4939895301339042750nteger,Xb)) ).

% mult_of_nat_commute
tff(fact_7533_exp__not__eq__zero,axiom,
    ! [Xb: real] : aa_real_real(exp_real,Xb) != zero_zero_real ).

% exp_not_eq_zero
tff(fact_7534_pochhammer__of__nat,axiom,
    ! [Xb: nat,Na: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_nat_real(semiri5074537144036343181t_real,Xb)),Na) = aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Xb),Na)) ).

% pochhammer_of_nat
tff(fact_7535_pochhammer__of__nat,axiom,
    ! [Xb: nat,Na: nat] : aa_nat_int(comm_s4660882817536571857er_int(aa_nat_int(semiri1314217659103216013at_int,Xb)),Na) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Xb),Na)) ).

% pochhammer_of_nat
tff(fact_7536_pochhammer__of__nat,axiom,
    ! [Xb: nat,Na: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,Xb)),Na) = aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Xb),Na)) ).

% pochhammer_of_nat
tff(fact_7537_of__nat__less__of__int__iff,axiom,
    ! [Na: nat,Xb: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,Na)),aa_int_rat(ring_1_of_int_rat,Xb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Na)),Xb) ) ).

% of_nat_less_of_int_iff
tff(fact_7538_of__nat__less__of__int__iff,axiom,
    ! [Na: nat,Xb: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_int_real(ring_1_of_int_real,Xb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Na)),Xb) ) ).

% of_nat_less_of_int_iff
tff(fact_7539_of__nat__less__of__int__iff,axiom,
    ! [Na: nat,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Na)),aa_int_int(ring_1_of_int_int,Xb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Na)),Xb) ) ).

% of_nat_less_of_int_iff
tff(fact_7540_of__nat__less__of__int__iff,axiom,
    ! [Na: nat,Xb: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Na)),aa_int_Code_integer(ring_18347121197199848620nteger,Xb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Na)),Xb) ) ).

% of_nat_less_of_int_iff
tff(fact_7541_exp__divide__power__eq,axiom,
    ! [Na: nat,Xb: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_complex(power_power_complex(aa_complex_complex(exp_complex,aa_complex_complex(divide1717551699836669952omplex(Xb),aa_nat_complex(semiri8010041392384452111omplex,Na)))),Na) = aa_complex_complex(exp_complex,Xb) ) ) ).

% exp_divide_power_eq
tff(fact_7542_exp__divide__power__eq,axiom,
    ! [Na: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),aa_nat_real(semiri5074537144036343181t_real,Na)))),Na) = aa_real_real(exp_real,Xb) ) ) ).

% exp_divide_power_eq
tff(fact_7543_real__sqrt__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,zero_zero_real),aa_real_real(sqrt,Xb)) ) ).

% real_sqrt_gt_zero
tff(fact_7544_not__exp__less__zero,axiom,
    ! [Xb: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(exp_real,Xb)),zero_zero_real) ).

% not_exp_less_zero
tff(fact_7545_exp__gt__zero,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(exp_real,Xb)) ).

% exp_gt_zero
tff(fact_7546_exp__total,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
     => ? [X: real] : aa_real_real(exp_real,X) = Y ) ).

% exp_total
tff(fact_7547_real__sqrt__ge__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_eq_real,zero_zero_real),aa_real_real(sqrt,Xb)) ) ).

% real_sqrt_ge_zero
tff(fact_7548_real__sqrt__eq__zero__cancel,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( ( aa_real_real(sqrt,Xb) = zero_zero_real )
       => ( Xb = zero_zero_real ) ) ) ).

% real_sqrt_eq_zero_cancel
tff(fact_7549_not__exp__le__zero,axiom,
    ! [Xb: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,Xb)),zero_zero_real) ).

% not_exp_le_zero
tff(fact_7550_exp__ge__zero,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(exp_real,Xb)) ).

% exp_ge_zero
tff(fact_7551_of__nat__0__le__iff,axiom,
    ! [Na: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(semiri5074537144036343181t_real,Na)) ).

% of_nat_0_le_iff
tff(fact_7552_of__nat__0__le__iff,axiom,
    ! [Na: nat] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(semiri4939895301339042750nteger,Na)) ).

% of_nat_0_le_iff
tff(fact_7553_of__nat__0__le__iff,axiom,
    ! [Na: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(semiri681578069525770553at_rat,Na)) ).

% of_nat_0_le_iff
tff(fact_7554_of__nat__0__le__iff,axiom,
    ! [Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(semiri1316708129612266289at_nat,Na)) ).

% of_nat_0_le_iff
tff(fact_7555_of__nat__0__le__iff,axiom,
    ! [Na: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(semiri1314217659103216013at_int,Na)) ).

% of_nat_0_le_iff
tff(fact_7556_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)
     => ( ! [X: nat] :
            ( aa_set_nat_o(member_nat(aa_nat_nat(suc,X)),A3)
           => ( aa_nat_nat(F,aa_nat_nat(suc,X)) = aa_nat_nat(G,aa_nat_nat(suc,X)) ) )
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(F),A3) = aa_set_nat_nat(groups3542108847815614940at_nat(G),A3) ) ) ) ).

% sum_cong_Suc
tff(fact_7557_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)
     => ( ! [X: nat] :
            ( aa_set_nat_o(member_nat(aa_nat_nat(suc,X)),A3)
           => ( aa_nat_real(F,aa_nat_nat(suc,X)) = aa_nat_real(G,aa_nat_nat(suc,X)) ) )
       => ( aa_set_nat_real(groups6591440286371151544t_real(F),A3) = aa_set_nat_real(groups6591440286371151544t_real(G),A3) ) ) ) ).

% sum_cong_Suc
tff(fact_7558_of__nat__less__0__iff,axiom,
    ! [M: nat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),zero_zero_rat) ).

% of_nat_less_0_iff
tff(fact_7559_of__nat__less__0__iff,axiom,
    ! [M: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),zero_zero_nat) ).

% of_nat_less_0_iff
tff(fact_7560_of__nat__less__0__iff,axiom,
    ! [M: nat] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,M)),zero_zero_real) ).

% of_nat_less_0_iff
tff(fact_7561_of__nat__less__0__iff,axiom,
    ! [M: nat] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M)),zero_zero_int) ).

% of_nat_less_0_iff
tff(fact_7562_of__nat__less__0__iff,axiom,
    ! [M: nat] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),zero_z3403309356797280102nteger) ).

% of_nat_less_0_iff
tff(fact_7563_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_7564_of__nat__neq__0,axiom,
    ! [Na: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,Na)) != zero_zero_rat ).

% of_nat_neq_0
tff(fact_7565_of__nat__neq__0,axiom,
    ! [Na: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(suc,Na)) != zero_zero_nat ).

% of_nat_neq_0
tff(fact_7566_of__nat__neq__0,axiom,
    ! [Na: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Na)) != zero_zero_real ).

% of_nat_neq_0
tff(fact_7567_of__nat__neq__0,axiom,
    ! [Na: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,Na)) != zero_zero_int ).

% of_nat_neq_0
tff(fact_7568_of__nat__neq__0,axiom,
    ! [Na: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(suc,Na)) != zero_z3403309356797280102nteger ).

% of_nat_neq_0
tff(fact_7569_div__mult2__eq_H,axiom,
    ! [Aa: nat,M: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,Na))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Aa),aa_nat_nat(semiri1316708129612266289at_nat,M))),aa_nat_nat(semiri1316708129612266289at_nat,Na)) ).

% div_mult2_eq'
tff(fact_7570_div__mult2__eq_H,axiom,
    ! [Aa: int,M: nat,Na: nat] : aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,Na))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),aa_nat_int(semiri1314217659103216013at_int,M))),aa_nat_int(semiri1314217659103216013at_int,Na)) ).

% div_mult2_eq'
tff(fact_7571_div__mult2__eq_H,axiom,
    ! [Aa: code_integer,M: nat,Na: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Aa),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,Na))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Aa),aa_nat_Code_integer(semiri4939895301339042750nteger,M))),aa_nat_Code_integer(semiri4939895301339042750nteger,Na)) ).

% div_mult2_eq'
tff(fact_7572_of__nat__less__imp__less,axiom,
    ! [M: nat,Na: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,Na))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ).

% of_nat_less_imp_less
tff(fact_7573_of__nat__less__imp__less,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,Na))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ).

% of_nat_less_imp_less
tff(fact_7574_of__nat__less__imp__less,axiom,
    ! [M: nat,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,Na))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ).

% of_nat_less_imp_less
tff(fact_7575_of__nat__less__imp__less,axiom,
    ! [M: nat,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,Na))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ).

% of_nat_less_imp_less
tff(fact_7576_of__nat__less__imp__less,axiom,
    ! [M: nat,Na: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,Na))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ).

% of_nat_less_imp_less
tff(fact_7577_less__imp__of__nat__less,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,Na)) ) ).

% less_imp_of_nat_less
tff(fact_7578_less__imp__of__nat__less,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,Na)) ) ).

% less_imp_of_nat_less
tff(fact_7579_less__imp__of__nat__less,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,Na)) ) ).

% less_imp_of_nat_less
tff(fact_7580_less__imp__of__nat__less,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,Na)) ) ).

% less_imp_of_nat_less
tff(fact_7581_less__imp__of__nat__less,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,Na)) ) ).

% less_imp_of_nat_less
tff(fact_7582_of__nat__mono,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,I)),aa_nat_real(semiri5074537144036343181t_real,J)) ) ).

% of_nat_mono
tff(fact_7583_of__nat__mono,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,I)),aa_nat_Code_integer(semiri4939895301339042750nteger,J)) ) ).

% of_nat_mono
tff(fact_7584_of__nat__mono,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,I)),aa_nat_rat(semiri681578069525770553at_rat,J)) ) ).

% of_nat_mono
tff(fact_7585_of__nat__mono,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,I)),aa_nat_nat(semiri1316708129612266289at_nat,J)) ) ).

% of_nat_mono
tff(fact_7586_of__nat__mono,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,I)),aa_nat_int(semiri1314217659103216013at_int,J)) ) ).

% of_nat_mono
tff(fact_7587_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,Na)) ).

% unique_euclidean_semiring_with_nat_class.of_nat_div
tff(fact_7588_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
    ! [M: nat,Na: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,Na)) ).

% unique_euclidean_semiring_with_nat_class.of_nat_div
tff(fact_7589_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
    ! [M: nat,Na: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,Na)) ).

% unique_euclidean_semiring_with_nat_class.of_nat_div
tff(fact_7590_of__nat__dvd__iff,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),Na) ) ).

% of_nat_dvd_iff
tff(fact_7591_of__nat__dvd__iff,axiom,
    ! [M: nat,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),Na) ) ).

% of_nat_dvd_iff
tff(fact_7592_of__nat__dvd__iff,axiom,
    ! [M: nat,Na: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),Na) ) ).

% of_nat_dvd_iff
tff(fact_7593_int__ops_I1_J,axiom,
    aa_nat_int(semiri1314217659103216013at_int,zero_zero_nat) = zero_zero_int ).

% int_ops(1)
tff(fact_7594_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_7595_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_7596_nat__int__comparison_I2_J,axiom,
    ! [Aa: nat,Ba: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),Ba)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Aa)),aa_nat_int(semiri1314217659103216013at_int,Ba)) ) ).

% nat_int_comparison(2)
tff(fact_7597_zle__int,axiom,
    ! [M: nat,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ).

% zle_int
tff(fact_7598_nat__int__comparison_I3_J,axiom,
    ! [Aa: nat,Ba: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),Ba)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,Aa)),aa_nat_int(semiri1314217659103216013at_int,Ba)) ) ).

% nat_int_comparison(3)
tff(fact_7599_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_7600_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_7601_of__nat__mod,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,Na)) ).

% of_nat_mod
tff(fact_7602_of__nat__mod,axiom,
    ! [M: nat,Na: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Na)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,Na)) ).

% of_nat_mod
tff(fact_7603_of__nat__mod,axiom,
    ! [M: nat,Na: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Na)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,Na)) ).

% of_nat_mod
tff(fact_7604_int__ops_I2_J,axiom,
    aa_nat_int(semiri1314217659103216013at_int,one_one_nat) = one_one_int ).

% int_ops(2)
tff(fact_7605_zle__iff__zadd,axiom,
    ! [W2: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W2),Z)
    <=> ? [N4: nat] : Z = aa_int_int(aa_int_fun_int_int(plus_plus_int,W2),aa_nat_int(semiri1314217659103216013at_int,N4)) ) ).

% zle_iff_zadd
tff(fact_7606_not__int__zless__negative,axiom,
    ! [Na: nat,M: nat] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Na)),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,M))) ).

% not_int_zless_negative
tff(fact_7607_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_7608_of__nat__max,axiom,
    ! [Xb: nat,Y: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xb),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(semiri1316708129612266289at_nat,Xb)),aa_nat_nat(semiri1316708129612266289at_nat,Y)) ).

% of_nat_max
tff(fact_7609_of__nat__max,axiom,
    ! [Xb: nat,Y: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_nat_real(semiri5074537144036343181t_real,Xb)),aa_nat_real(semiri5074537144036343181t_real,Y)) ).

% of_nat_max
tff(fact_7610_of__nat__max,axiom,
    ! [Xb: nat,Y: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xb),Y)) = aa_int_int(ord_max_int(aa_nat_int(semiri1314217659103216013at_int,Xb)),aa_nat_int(semiri1314217659103216013at_int,Y)) ).

% of_nat_max
tff(fact_7611_of__nat__max,axiom,
    ! [Xb: nat,Y: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xb),Y)) = ord_max_Code_integer(aa_nat_Code_integer(semiri4939895301339042750nteger,Xb),aa_nat_Code_integer(semiri4939895301339042750nteger,Y)) ).

% of_nat_max
tff(fact_7612_sum__subtractf__nat,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,G: fun_Pr2602117396471556196at_nat,F: fun_Pr2602117396471556196at_nat] :
      ( ! [X: product_prod_nat_nat] :
          ( member8440522571783428010at_nat(X,A3)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_Pro4243173770044397973at_nat(G,X)),aa_Pro4243173770044397973at_nat(F,X)) )
     => ( groups977919841031483927at_nat(aa_fun2351638177499875132at_nat(aTP_Lamm_ja(G),F),A3) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups977919841031483927at_nat(F,A3)),groups977919841031483927at_nat(G,A3)) ) ) ).

% sum_subtractf_nat
tff(fact_7613_sum__subtractf__nat,axiom,
    ! [A3: set_real,G: fun_real_nat,F: fun_real_nat] :
      ( ! [X: real] :
          ( member_real(X,A3)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_real_nat(G,X)),aa_real_nat(F,X)) )
     => ( groups1935376822645274424al_nat(aa_fun211602392931233840al_nat(aTP_Lamm_jb(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_7614_sum__subtractf__nat,axiom,
    ! [A3: set_set_nat,G: fun_set_nat_nat,F: fun_set_nat_nat] :
      ( ! [X: set_nat] :
          ( member_set_nat(X,A3)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(G,X)),aa_set_nat_nat(F,X)) )
     => ( groups8294997508430121362at_nat(aa_fun3386010016858254564at_nat(aTP_Lamm_jc(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_7615_sum__subtractf__nat,axiom,
    ! [A3: set_int,G: fun_int_nat,F: fun_int_nat] :
      ( ! [X: int] :
          ( aa_set_int_o(member_int(X),A3)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(G,X)),aa_int_nat(F,X)) )
     => ( groups4541462559716669496nt_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_jd(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_7616_sum__subtractf__nat,axiom,
    ! [A3: set_nat,G: fun_nat_nat,F: fun_nat_nat] :
      ( ! [X: nat] :
          ( aa_set_nat_o(member_nat(X),A3)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(G,X)),aa_nat_nat(F,X)) )
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_je(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_7617_sum_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: fun_nat_nat,M: nat,Na: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(aa_nat_nat(suc,M),aa_nat_nat(suc,Na))) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_jf(G)),set_or1269000886237332187st_nat(M,Na)) ).

% sum.shift_bounds_cl_Suc_ivl
tff(fact_7618_sum_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: fun_nat_real,M: nat,Na: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),set_or1269000886237332187st_nat(aa_nat_nat(suc,M),aa_nat_nat(suc,Na))) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_jg(G)),set_or1269000886237332187st_nat(M,Na)) ).

% sum.shift_bounds_cl_Suc_ivl
tff(fact_7619_sum_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: fun_nat_nat,M: nat,K: nat,Na: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),K))) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_jh(G),K)),set_or1269000886237332187st_nat(M,Na)) ).

% sum.shift_bounds_cl_nat_ivl
tff(fact_7620_sum_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: fun_nat_real,M: nat,K: nat,Na: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),K))) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_ji(G),K)),set_or1269000886237332187st_nat(M,Na)) ).

% sum.shift_bounds_cl_nat_ivl
tff(fact_7621_nat__less__as__int,axiom,
    ! [X3: nat,Xa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Xa)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,X3)),aa_nat_int(semiri1314217659103216013at_int,Xa)) ) ).

% nat_less_as_int
tff(fact_7622_nat__leq__as__int,axiom,
    ! [X3: nat,Xa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Xa)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,X3)),aa_nat_int(semiri1314217659103216013at_int,Xa)) ) ).

% nat_leq_as_int
tff(fact_7623_sum__eq__Suc0__iff,axiom,
    ! [A3: set_int,F: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( ( groups4541462559716669496nt_nat(F,A3) = aa_nat_nat(suc,zero_zero_nat) )
      <=> ? [X4: int] :
            ( aa_set_int_o(member_int(X4),A3)
            & ( aa_int_nat(F,X4) = aa_nat_nat(suc,zero_zero_nat) )
            & ! [Xa3: int] :
                ( aa_set_int_o(member_int(Xa3),A3)
               => ( ( X4 != Xa3 )
                 => ( aa_int_nat(F,Xa3) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
tff(fact_7624_sum__eq__Suc0__iff,axiom,
    ! [A3: set_complex,F: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A3)
     => ( ( groups5693394587270226106ex_nat(F,A3) = aa_nat_nat(suc,zero_zero_nat) )
      <=> ? [X4: complex] :
            ( aa_set_complex_o(member_complex(X4),A3)
            & ( aa_complex_nat(F,X4) = aa_nat_nat(suc,zero_zero_nat) )
            & ! [Xa3: complex] :
                ( aa_set_complex_o(member_complex(Xa3),A3)
               => ( ( X4 != Xa3 )
                 => ( aa_complex_nat(F,Xa3) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
tff(fact_7625_sum__eq__Suc0__iff,axiom,
    ! [A3: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( ( aa_set_nat_nat(groups3542108847815614940at_nat(F),A3) = aa_nat_nat(suc,zero_zero_nat) )
      <=> ? [X4: nat] :
            ( aa_set_nat_o(member_nat(X4),A3)
            & ( aa_nat_nat(F,X4) = aa_nat_nat(suc,zero_zero_nat) )
            & ! [Xa3: nat] :
                ( aa_set_nat_o(member_nat(Xa3),A3)
               => ( ( X4 != Xa3 )
                 => ( aa_nat_nat(F,Xa3) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
tff(fact_7626_sum__SucD,axiom,
    ! [F: fun_nat_nat,A3: set_nat,Na: nat] :
      ( ( aa_set_nat_nat(groups3542108847815614940at_nat(F),A3) = aa_nat_nat(suc,Na) )
     => ? [X: nat] :
          ( aa_set_nat_o(member_nat(X),A3)
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(F,X)) ) ) ).

% sum_SucD
tff(fact_7627_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_7628_sum__eq__1__iff,axiom,
    ! [A3: set_int,F: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( ( groups4541462559716669496nt_nat(F,A3) = one_one_nat )
      <=> ? [X4: int] :
            ( aa_set_int_o(member_int(X4),A3)
            & ( aa_int_nat(F,X4) = one_one_nat )
            & ! [Xa3: int] :
                ( aa_set_int_o(member_int(Xa3),A3)
               => ( ( X4 != Xa3 )
                 => ( aa_int_nat(F,Xa3) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_1_iff
tff(fact_7629_sum__eq__1__iff,axiom,
    ! [A3: set_complex,F: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A3)
     => ( ( groups5693394587270226106ex_nat(F,A3) = one_one_nat )
      <=> ? [X4: complex] :
            ( aa_set_complex_o(member_complex(X4),A3)
            & ( aa_complex_nat(F,X4) = one_one_nat )
            & ! [Xa3: complex] :
                ( aa_set_complex_o(member_complex(Xa3),A3)
               => ( ( X4 != Xa3 )
                 => ( aa_complex_nat(F,Xa3) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_1_iff
tff(fact_7630_sum__eq__1__iff,axiom,
    ! [A3: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( ( aa_set_nat_nat(groups3542108847815614940at_nat(F),A3) = one_one_nat )
      <=> ? [X4: nat] :
            ( aa_set_nat_o(member_nat(X4),A3)
            & ( aa_nat_nat(F,X4) = one_one_nat )
            & ! [Xa3: nat] :
                ( aa_set_nat_o(member_nat(Xa3),A3)
               => ( ( X4 != Xa3 )
                 => ( aa_nat_nat(F,Xa3) = zero_zero_nat ) ) ) ) ) ) ).

% sum_eq_1_iff
tff(fact_7631_real__div__sqrt,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),aa_real_real(sqrt,Xb)) = aa_real_real(sqrt,Xb) ) ) ).

% real_div_sqrt
tff(fact_7632_ex__less__of__nat__mult,axiom,
    ! [Xb: rat,Y: 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,Y),aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,N)),Xb)) ) ).

% ex_less_of_nat_mult
tff(fact_7633_ex__less__of__nat__mult,axiom,
    ! [Xb: real,Y: 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,Y),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),Xb)) ) ).

% ex_less_of_nat_mult
tff(fact_7634_sqrt__add__le__add__sqrt,axiom,
    ! [Xb: real,Y: 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),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(sqrt,Xb)),aa_real_real(sqrt,Y))) ) ) ).

% sqrt_add_le_add_sqrt
tff(fact_7635_exp__ge__add__one__self,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xb)),aa_real_real(exp_real,Xb)) ).

% exp_ge_add_one_self
tff(fact_7636_le__real__sqrt__sumsq,axiom,
    ! [Xb: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Xb)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Y)))) ).

% le_real_sqrt_sumsq
tff(fact_7637_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_7638_exp__minus__inverse,axiom,
    ! [Xb: real] : aa_real_real(aa_rea7340605660300375871l_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_7639_of__nat__diff,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
     => ( aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na)) = aa_rat_rat(minus_minus_rat(aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,Na)) ) ) ).

% of_nat_diff
tff(fact_7640_of__nat__diff,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
     => ( aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,Na)) ) ) ).

% of_nat_diff
tff(fact_7641_of__nat__diff,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
     => ( aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,Na)) ) ) ).

% of_nat_diff
tff(fact_7642_of__nat__diff,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
     => ( aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,Na)) ) ) ).

% of_nat_diff
tff(fact_7643_of__nat__diff,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
     => ( aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,Na)) ) ) ).

% of_nat_diff
tff(fact_7644_reals__Archimedean3,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ! [Y4: real] :
        ? [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,Y4),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),Xb)) ) ).

% reals_Archimedean3
tff(fact_7645_fact__le__power,axiom,
    ! [Na: nat] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(semiri3624122377584611663nteger,Na)),aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Na),Na))) ).

% fact_le_power
tff(fact_7646_fact__le__power,axiom,
    ! [Na: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri773545260158071498ct_rat,Na)),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Na),Na))) ).

% fact_le_power
tff(fact_7647_fact__le__power,axiom,
    ! [Na: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1406184849735516958ct_int,Na)),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Na),Na))) ).

% fact_le_power
tff(fact_7648_fact__le__power,axiom,
    ! [Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Na)),aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Na),Na))) ).

% fact_le_power
tff(fact_7649_fact__le__power,axiom,
    ! [Na: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri2265585572941072030t_real,Na)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Na),Na))) ).

% fact_le_power
tff(fact_7650_int__cases4,axiom,
    ! [M: int] :
      ( ! [N: nat] : M != aa_nat_int(semiri1314217659103216013at_int,N)
     => ~ ! [N: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
           => ( M != aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N)) ) ) ) ).

% int_cases4
tff(fact_7651_real__of__nat__div4,axiom,
    ! [Na: 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,Na),Xb))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_nat_real(semiri5074537144036343181t_real,Xb))) ).

% real_of_nat_div4
tff(fact_7652_int__zle__neg,axiom,
    ! [Na: nat,M: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,Na)),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,M)))
    <=> ( ( Na = zero_zero_nat )
        & ( M = zero_zero_nat ) ) ) ).

% int_zle_neg
tff(fact_7653_int__Suc,axiom,
    ! [Na: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,Na)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Na)),one_one_int) ).

% int_Suc
tff(fact_7654_int__ops_I4_J,axiom,
    ! [Aa: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,Aa)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Aa)),one_one_int) ).

% int_ops(4)
tff(fact_7655_zless__iff__Suc__zadd,axiom,
    ! [W2: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,W2),Z)
    <=> ? [N4: nat] : Z = aa_int_int(aa_int_fun_int_int(plus_plus_int,W2),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N4))) ) ).

% zless_iff_Suc_zadd
tff(fact_7656_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_7657_negative__zle__0,axiom,
    ! [Na: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,Na))),zero_zero_int) ).

% negative_zle_0
tff(fact_7658_sum__power__add,axiom,
    ! [Xb: complex,M: nat,I5: set_nat] : aa_set_nat_complex(groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aTP_Lamm_jj(Xb),M)),I5) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(Xb),M)),aa_set_nat_complex(groups2073611262835488442omplex(power_power_complex(Xb)),I5)) ).

% sum_power_add
tff(fact_7659_sum__power__add,axiom,
    ! [Xb: rat,M: nat,I5: set_nat] : groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aTP_Lamm_jk(Xb),M),I5) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(Xb),M)),groups2906978787729119204at_rat(power_power_rat(Xb),I5)) ).

% sum_power_add
tff(fact_7660_sum__power__add,axiom,
    ! [Xb: int,M: nat,I5: set_nat] : groups3539618377306564664at_int(aa_nat_fun_nat_int(aTP_Lamm_jl(Xb),M),I5) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(Xb),M)),groups3539618377306564664at_int(power_power_int(Xb),I5)) ).

% sum_power_add
tff(fact_7661_sum__power__add,axiom,
    ! [Xb: real,M: nat,I5: set_nat] : aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_jm(Xb),M)),I5) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),M)),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,Xb)),I5)) ).

% sum_power_add
tff(fact_7662_sum_OatLeastAtMost__rev,axiom,
    ! [G: fun_nat_nat,Na: nat,M: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(Na,M)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_jn(G),Na),M)),set_or1269000886237332187st_nat(Na,M)) ).

% sum.atLeastAtMost_rev
tff(fact_7663_sum_OatLeastAtMost__rev,axiom,
    ! [G: fun_nat_real,Na: nat,M: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),set_or1269000886237332187st_nat(Na,M)) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_nat4085805313368488148t_real(aTP_Lamm_jo(G),Na),M)),set_or1269000886237332187st_nat(Na,M)) ).

% sum.atLeastAtMost_rev
tff(fact_7664_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_7665_sum__nth__roots,axiom,
    ! [Na: nat,C: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),Na)
     => ( groups7754918857620584856omplex(aTP_Lamm_bz,collect_complex(aa_com5808801942499186519plex_o(aTP_Lamm_dd(Na),C))) = zero_zero_complex ) ) ).

% sum_nth_roots
tff(fact_7666_sum__roots__unity,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),Na)
     => ( groups7754918857620584856omplex(aTP_Lamm_bz,collect_complex(aa_nat_fun_complex_o(aTP_Lamm_cs,Na))) = zero_zero_complex ) ) ).

% sum_roots_unity
tff(fact_7667_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_7668_exp__ge__one__plus__x__over__n__power__n,axiom,
    ! [Na: 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,Na))),Xb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),aa_nat_real(semiri5074537144036343181t_real,Na)))),Na)),aa_real_real(exp_real,Xb)) ) ) ).

% exp_ge_one_plus_x_over_n_power_n
tff(fact_7669_sum__diff__nat,axiom,
    ! [B4: set_complex,A3: set_complex,F: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(B4),A3)
       => ( groups5693394587270226106ex_nat(F,minus_811609699411566653omplex(A3,B4)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups5693394587270226106ex_nat(F,A3)),groups5693394587270226106ex_nat(F,B4)) ) ) ) ).

% sum_diff_nat
tff(fact_7670_sum__diff__nat,axiom,
    ! [B4: set_int,A3: set_int,F: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,B4)
     => ( aa_set_int_o(ord_less_eq_set_int(B4),A3)
       => ( groups4541462559716669496nt_nat(F,minus_minus_set_int(A3,B4)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups4541462559716669496nt_nat(F,A3)),groups4541462559716669496nt_nat(F,B4)) ) ) ) ).

% sum_diff_nat
tff(fact_7671_sum__diff__nat,axiom,
    ! [B4: set_nat,A3: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,B4)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),A3)
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(F),minus_minus_set_nat(A3,B4)) = 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(F),B4)) ) ) ) ).

% sum_diff_nat
tff(fact_7672_exp__ge__one__minus__x__over__n__power__n,axiom,
    ! [Xb: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_nat_real(semiri5074537144036343181t_real,Na))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),aa_nat_real(semiri5074537144036343181t_real,Na)))),Na)),aa_real_real(exp_real,aa_real_real(uminus_uminus_real,Xb))) ) ) ).

% exp_ge_one_minus_x_over_n_power_n
tff(fact_7673_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(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xb)),aa_real_real(exp_real,Xb)) ) ).

% exp_ge_add_one_self_aux
tff(fact_7674_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,set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),K)) = groups2906978787729119204at_rat(F,set_or1269000886237332187st_nat(zero_zero_nat,K)) ) ) ).

% sum_shift_lb_Suc0_0
tff(fact_7675_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,set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),K)) = groups3539618377306564664at_int(F,set_or1269000886237332187st_nat(zero_zero_nat,K)) ) ) ).

% sum_shift_lb_Suc0_0
tff(fact_7676_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),set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),K)) = aa_set_nat_nat(groups3542108847815614940at_nat(F),set_or1269000886237332187st_nat(zero_zero_nat,K)) ) ) ).

% sum_shift_lb_Suc0_0
tff(fact_7677_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),set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),K)) = aa_set_nat_real(groups6591440286371151544t_real(F),set_or1269000886237332187st_nat(zero_zero_nat,K)) ) ) ).

% sum_shift_lb_Suc0_0
tff(fact_7678_double__gauss__sum,axiom,
    ! [Na: nat] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_set_nat_complex(groups2073611262835488442omplex(semiri8010041392384452111omplex),set_or1269000886237332187st_nat(zero_zero_nat,Na))) = aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,Na)),aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,Na)),one_one_complex)) ).

% double_gauss_sum
tff(fact_7679_double__gauss__sum,axiom,
    ! [Na: nat] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),groups2906978787729119204at_rat(semiri681578069525770553at_rat,set_or1269000886237332187st_nat(zero_zero_nat,Na))) = aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,Na)),aa_rat_rat(plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,Na)),one_one_rat)) ).

% double_gauss_sum
tff(fact_7680_double__gauss__sum,axiom,
    ! [Na: nat] : times_7803423173614009249d_enat(numera1916890842035813515d_enat(aa_num_num(bit0,one)),groups7108830773950497114d_enat(semiri4216267220026989637d_enat,set_or1269000886237332187st_nat(zero_zero_nat,Na))) = times_7803423173614009249d_enat(aa_nat_Extended_enat(semiri4216267220026989637d_enat,Na),plus_p3455044024723400733d_enat(aa_nat_Extended_enat(semiri4216267220026989637d_enat,Na),one_on7984719198319812577d_enat)) ).

% double_gauss_sum
tff(fact_7681_double__gauss__sum,axiom,
    ! [Na: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),groups3539618377306564664at_int(semiri1314217659103216013at_int,set_or1269000886237332187st_nat(zero_zero_nat,Na))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Na)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Na)),one_one_int)) ).

% double_gauss_sum
tff(fact_7682_double__gauss__sum,axiom,
    ! [Na: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),groups7501900531339628137nteger(semiri4939895301339042750nteger,set_or1269000886237332187st_nat(zero_zero_nat,Na))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Na)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Na)),one_one_Code_integer)) ).

% double_gauss_sum
tff(fact_7683_double__gauss__sum,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_set_nat_nat(groups3542108847815614940at_nat(semiri1316708129612266289at_nat),set_or1269000886237332187st_nat(zero_zero_nat,Na))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Na)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,Na)),one_one_nat)) ).

% double_gauss_sum
tff(fact_7684_double__gauss__sum,axiom,
    ! [Na: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_set_nat_real(groups6591440286371151544t_real(semiri5074537144036343181t_real),set_or1269000886237332187st_nat(zero_zero_nat,Na))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,Na)),one_one_real)) ).

% double_gauss_sum
tff(fact_7685_double__arith__series,axiom,
    ! [Aa: complex,D: complex,Na: nat] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_jp(Aa),D)),set_or1269000886237332187st_nat(zero_zero_nat,Na))) = aa_complex_complex(times_times_complex(aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,Na)),one_one_complex)),aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),Aa)),aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,Na)),D))) ).

% double_arith_series
tff(fact_7686_double__arith__series,axiom,
    ! [Aa: rat,D: rat,Na: 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_jq(Aa),D),set_or1269000886237332187st_nat(zero_zero_nat,Na))) = aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,Na)),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))),Aa)),aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,Na)),D))) ).

% double_arith_series
tff(fact_7687_double__arith__series,axiom,
    ! [Aa: extended_enat,D: extended_enat,Na: nat] : times_7803423173614009249d_enat(numera1916890842035813515d_enat(aa_num_num(bit0,one)),groups7108830773950497114d_enat(aa_Ext1337733060012393797d_enat(aTP_Lamm_jr(Aa),D),set_or1269000886237332187st_nat(zero_zero_nat,Na))) = times_7803423173614009249d_enat(plus_p3455044024723400733d_enat(aa_nat_Extended_enat(semiri4216267220026989637d_enat,Na),one_on7984719198319812577d_enat),plus_p3455044024723400733d_enat(times_7803423173614009249d_enat(numera1916890842035813515d_enat(aa_num_num(bit0,one)),Aa),times_7803423173614009249d_enat(aa_nat_Extended_enat(semiri4216267220026989637d_enat,Na),D))) ).

% double_arith_series
tff(fact_7688_double__arith__series,axiom,
    ! [Aa: int,D: int,Na: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_js(Aa),D),set_or1269000886237332187st_nat(zero_zero_nat,Na))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Na)),one_one_int)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Aa)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Na)),D))) ).

% double_arith_series
tff(fact_7689_double__arith__series,axiom,
    ! [Aa: code_integer,D: code_integer,Na: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),groups7501900531339628137nteger(aa_Cod1578151519121547077nteger(aTP_Lamm_jt(Aa),D),set_or1269000886237332187st_nat(zero_zero_nat,Na))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Na)),one_one_Code_integer)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Aa)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Na)),D))) ).

% double_arith_series
tff(fact_7690_double__arith__series,axiom,
    ! [Aa: nat,D: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ju(Aa),D)),set_or1269000886237332187st_nat(zero_zero_nat,Na))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,Na)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Aa)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Na)),D))) ).

% double_arith_series
tff(fact_7691_double__arith__series,axiom,
    ! [Aa: real,D: real,Na: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_jv(Aa),D)),set_or1269000886237332187st_nat(zero_zero_nat,Na))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,Na)),one_one_real)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Aa)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Na)),D))) ).

% double_arith_series
tff(fact_7692_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_rat,Na: nat] : groups2906978787729119204at_rat(G,set_or1269000886237332187st_nat(zero_zero_nat,aa_nat_nat(suc,Na))) = aa_rat_rat(plus_plus_rat(groups2906978787729119204at_rat(G,set_or1269000886237332187st_nat(zero_zero_nat,Na))),aa_nat_rat(G,aa_nat_nat(suc,Na))) ).

% sum.atLeast0_atMost_Suc
tff(fact_7693_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_int,Na: nat] : groups3539618377306564664at_int(G,set_or1269000886237332187st_nat(zero_zero_nat,aa_nat_nat(suc,Na))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,set_or1269000886237332187st_nat(zero_zero_nat,Na))),aa_nat_int(G,aa_nat_nat(suc,Na))) ).

% sum.atLeast0_atMost_Suc
tff(fact_7694_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_nat,Na: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(zero_zero_nat,aa_nat_nat(suc,Na))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(zero_zero_nat,Na))),aa_nat_nat(G,aa_nat_nat(suc,Na))) ).

% sum.atLeast0_atMost_Suc
tff(fact_7695_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_real,Na: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),set_or1269000886237332187st_nat(zero_zero_nat,aa_nat_nat(suc,Na))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(G),set_or1269000886237332187st_nat(zero_zero_nat,Na))),aa_nat_real(G,aa_nat_nat(suc,Na))) ).

% sum.atLeast0_atMost_Suc
tff(fact_7696_lemma__exp__total,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Y)
     => ? [X: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y),one_one_real))
          & ( aa_real_real(exp_real,X) = Y ) ) ) ).

% lemma_exp_total
tff(fact_7697_ln__ge__iff,axiom,
    ! [Xb: real,Y: 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,Y),aa_real_real(ln_ln_real,Xb))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,Y)),Xb) ) ) ).

% ln_ge_iff
tff(fact_7698_sum_Onat__ivl__Suc_H,axiom,
    ! [M: nat,Na: nat,G: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,Na))
     => ( groups2906978787729119204at_rat(G,set_or1269000886237332187st_nat(M,aa_nat_nat(suc,Na))) = aa_rat_rat(plus_plus_rat(aa_nat_rat(G,aa_nat_nat(suc,Na))),groups2906978787729119204at_rat(G,set_or1269000886237332187st_nat(M,Na))) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_7699_sum_Onat__ivl__Suc_H,axiom,
    ! [M: nat,Na: nat,G: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,Na))
     => ( groups3539618377306564664at_int(G,set_or1269000886237332187st_nat(M,aa_nat_nat(suc,Na))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,aa_nat_nat(suc,Na))),groups3539618377306564664at_int(G,set_or1269000886237332187st_nat(M,Na))) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_7700_sum_Onat__ivl__Suc_H,axiom,
    ! [M: nat,Na: nat,G: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,Na))
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(M,aa_nat_nat(suc,Na))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G,aa_nat_nat(suc,Na))),aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(M,Na))) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_7701_sum_Onat__ivl__Suc_H,axiom,
    ! [M: nat,Na: nat,G: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,Na))
     => ( aa_set_nat_real(groups6591440286371151544t_real(G),set_or1269000886237332187st_nat(M,aa_nat_nat(suc,Na))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(G,aa_nat_nat(suc,Na))),aa_set_nat_real(groups6591440286371151544t_real(G),set_or1269000886237332187st_nat(M,Na))) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_7702_sum_OatLeast__Suc__atMost,axiom,
    ! [M: nat,Na: nat,G: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( groups2906978787729119204at_rat(G,set_or1269000886237332187st_nat(M,Na)) = aa_rat_rat(plus_plus_rat(aa_nat_rat(G,M)),groups2906978787729119204at_rat(G,set_or1269000886237332187st_nat(aa_nat_nat(suc,M),Na))) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_7703_sum_OatLeast__Suc__atMost,axiom,
    ! [M: nat,Na: nat,G: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( groups3539618377306564664at_int(G,set_or1269000886237332187st_nat(M,Na)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,M)),groups3539618377306564664at_int(G,set_or1269000886237332187st_nat(aa_nat_nat(suc,M),Na))) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_7704_sum_OatLeast__Suc__atMost,axiom,
    ! [M: nat,Na: nat,G: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(M,Na)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G,M)),aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(aa_nat_nat(suc,M),Na))) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_7705_sum_OatLeast__Suc__atMost,axiom,
    ! [M: nat,Na: nat,G: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( aa_set_nat_real(groups6591440286371151544t_real(G),set_or1269000886237332187st_nat(M,Na)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(G,M)),aa_set_nat_real(groups6591440286371151544t_real(G),set_or1269000886237332187st_nat(aa_nat_nat(suc,M),Na))) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_7706_mod__mult2__eq_H,axiom,
    ! [Aa: nat,M: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Aa),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,Na))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Aa),aa_nat_nat(semiri1316708129612266289at_nat,M))),aa_nat_nat(semiri1316708129612266289at_nat,Na)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Aa),aa_nat_nat(semiri1316708129612266289at_nat,M))) ).

% mod_mult2_eq'
tff(fact_7707_mod__mult2__eq_H,axiom,
    ! [Aa: int,M: nat,Na: nat] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,Na))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,Aa),aa_nat_int(semiri1314217659103216013at_int,M))),aa_nat_int(semiri1314217659103216013at_int,Na)))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Aa),aa_nat_int(semiri1314217659103216013at_int,M))) ).

% mod_mult2_eq'
tff(fact_7708_mod__mult2__eq_H,axiom,
    ! [Aa: code_integer,M: nat,Na: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Aa),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,Na))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Aa),aa_nat_Code_integer(semiri4939895301339042750nteger,M))),aa_nat_Code_integer(semiri4939895301339042750nteger,Na)))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Aa),aa_nat_Code_integer(semiri4939895301339042750nteger,M))) ).

% mod_mult2_eq'
tff(fact_7709_ln__x__over__x__mono,axiom,
    ! [Xb: real,Y: 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),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,Y)),Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,Xb)),Xb)) ) ) ).

% ln_x_over_x_mono
tff(fact_7710_pochhammer__rec_H,axiom,
    ! [Z: rat,Na: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(Z),aa_nat_nat(suc,Na)) = aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(Z),aa_nat_rat(semiri681578069525770553at_rat,Na))),aa_nat_rat(comm_s4028243227959126397er_rat(Z),Na)) ).

% pochhammer_rec'
tff(fact_7711_pochhammer__rec_H,axiom,
    ! [Z: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),aa_nat_nat(suc,Na)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Z),aa_nat_nat(semiri1316708129612266289at_nat,Na))),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),Na)) ).

% pochhammer_rec'
tff(fact_7712_pochhammer__rec_H,axiom,
    ! [Z: real,Na: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z),aa_nat_nat(suc,Na)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),aa_nat_real(semiri5074537144036343181t_real,Na))),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z),Na)) ).

% pochhammer_rec'
tff(fact_7713_pochhammer__rec_H,axiom,
    ! [Z: int,Na: nat] : aa_nat_int(comm_s4660882817536571857er_int(Z),aa_nat_nat(suc,Na)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),aa_nat_int(semiri1314217659103216013at_int,Na))),aa_nat_int(comm_s4660882817536571857er_int(Z),Na)) ).

% pochhammer_rec'
tff(fact_7714_pochhammer__rec_H,axiom,
    ! [Z: code_integer,Na: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(Z),aa_nat_nat(suc,Na)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Z),aa_nat_Code_integer(semiri4939895301339042750nteger,Na))),aa_nat_Code_integer(comm_s8582702949713902594nteger(Z),Na)) ).

% pochhammer_rec'
tff(fact_7715_pochhammer__Suc,axiom,
    ! [Aa: rat,Na: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(Aa),aa_nat_nat(suc,Na)) = aa_rat_rat(times_times_rat(aa_nat_rat(comm_s4028243227959126397er_rat(Aa),Na)),aa_rat_rat(plus_plus_rat(Aa),aa_nat_rat(semiri681578069525770553at_rat,Na))) ).

% pochhammer_Suc
tff(fact_7716_pochhammer__Suc,axiom,
    ! [Aa: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Aa),aa_nat_nat(suc,Na)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Aa),Na)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Aa),aa_nat_nat(semiri1316708129612266289at_nat,Na))) ).

% pochhammer_Suc
tff(fact_7717_pochhammer__Suc,axiom,
    ! [Aa: real,Na: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Aa),aa_nat_nat(suc,Na)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Aa),Na)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),aa_nat_real(semiri5074537144036343181t_real,Na))) ).

% pochhammer_Suc
tff(fact_7718_pochhammer__Suc,axiom,
    ! [Aa: int,Na: nat] : aa_nat_int(comm_s4660882817536571857er_int(Aa),aa_nat_nat(suc,Na)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(comm_s4660882817536571857er_int(Aa),Na)),aa_int_int(aa_int_fun_int_int(plus_plus_int,Aa),aa_nat_int(semiri1314217659103216013at_int,Na))) ).

% pochhammer_Suc
tff(fact_7719_pochhammer__Suc,axiom,
    ! [Aa: code_integer,Na: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(Aa),aa_nat_nat(suc,Na)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(comm_s8582702949713902594nteger(Aa),Na)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Aa),aa_nat_Code_integer(semiri4939895301339042750nteger,Na))) ).

% pochhammer_Suc
tff(fact_7720_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [Na: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),K)
     => ( aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,Na))),K) = zero_zero_rat ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_7721_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [Na: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),K)
     => ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,Na))),K) = zero_zero_real ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_7722_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [Na: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),K)
     => ( aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,Na))),K) = zero_zero_int ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_7723_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [Na: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),K)
     => ( aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Na))),K) = zero_z3403309356797280102nteger ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_7724_pochhammer__of__nat__eq__0__iff,axiom,
    ! [Na: nat,K: nat] :
      ( ( aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,Na))),K) = zero_zero_rat )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),K) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_7725_pochhammer__of__nat__eq__0__iff,axiom,
    ! [Na: 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,Na))),K) = zero_zero_real )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),K) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_7726_pochhammer__of__nat__eq__0__iff,axiom,
    ! [Na: nat,K: nat] :
      ( ( aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,Na))),K) = zero_zero_int )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),K) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_7727_pochhammer__of__nat__eq__0__iff,axiom,
    ! [Na: nat,K: nat] :
      ( ( aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Na))),K) = zero_z3403309356797280102nteger )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),K) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_7728_pochhammer__eq__0__iff,axiom,
    ! [Aa: rat,Na: nat] :
      ( ( aa_nat_rat(comm_s4028243227959126397er_rat(Aa),Na) = zero_zero_rat )
    <=> ? [K3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K3),Na)
          & ( Aa = aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,K3)) ) ) ) ).

% pochhammer_eq_0_iff
tff(fact_7729_pochhammer__eq__0__iff,axiom,
    ! [Aa: real,Na: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Aa),Na) = zero_zero_real )
    <=> ? [K3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K3),Na)
          & ( Aa = aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,K3)) ) ) ) ).

% pochhammer_eq_0_iff
tff(fact_7730_field__char__0__class_Oof__nat__div,axiom,
    ! [M: nat,Na: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Na)))),aa_nat_rat(semiri681578069525770553at_rat,Na)) ).

% field_char_0_class.of_nat_div
tff(fact_7731_field__char__0__class_Oof__nat__div,axiom,
    ! [M: nat,Na: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Na)))),aa_nat_real(semiri5074537144036343181t_real,Na)) ).

% field_char_0_class.of_nat_div
tff(fact_7732_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => ( aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,Na))),K) != zero_zero_rat ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_7733_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,Na))),K) != zero_zero_real ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_7734_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => ( aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,Na))),K) != zero_zero_int ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_7735_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => ( aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Na))),K) != zero_z3403309356797280102nteger ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_7736_pochhammer__product_H,axiom,
    ! [Z: rat,Na: nat,M: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(Z),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),M)) = aa_rat_rat(times_times_rat(aa_nat_rat(comm_s4028243227959126397er_rat(Z),Na)),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(plus_plus_rat(Z),aa_nat_rat(semiri681578069525770553at_rat,Na))),M)) ).

% pochhammer_product'
tff(fact_7737_pochhammer__product_H,axiom,
    ! [Z: nat,Na: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),M)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),Na)),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Z),aa_nat_nat(semiri1316708129612266289at_nat,Na))),M)) ).

% pochhammer_product'
tff(fact_7738_pochhammer__product_H,axiom,
    ! [Z: real,Na: nat,M: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),M)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z),Na)),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),aa_nat_real(semiri5074537144036343181t_real,Na))),M)) ).

% pochhammer_product'
tff(fact_7739_pochhammer__product_H,axiom,
    ! [Z: int,Na: nat,M: nat] : aa_nat_int(comm_s4660882817536571857er_int(Z),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),M)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(comm_s4660882817536571857er_int(Z),Na)),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),aa_nat_int(semiri1314217659103216013at_int,Na))),M)) ).

% pochhammer_product'
tff(fact_7740_pochhammer__product_H,axiom,
    ! [Z: code_integer,Na: nat,M: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(Z),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),M)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(comm_s8582702949713902594nteger(Z),Na)),aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Z),aa_nat_Code_integer(semiri4939895301339042750nteger,Na))),M)) ).

% pochhammer_product'
tff(fact_7741_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_7742_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_7743_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_7744_nat__less__real__le,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,Na)),one_one_real)),aa_nat_real(semiri5074537144036343181t_real,M)) ) ).

% nat_less_real_le
tff(fact_7745_nat__le__real__less,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,M)),one_one_real)) ) ).

% nat_le_real_less
tff(fact_7746_zmult__zless__mono2__lemma,axiom,
    ! [I: int,J: int,K: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,I),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,K)),I)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,K)),J)) ) ) ).

% zmult_zless_mono2_lemma
tff(fact_7747_not__zle__0__negative,axiom,
    ! [Na: nat] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,Na)))) ).

% not_zle_0_negative
tff(fact_7748_negative__zless__0,axiom,
    ! [Na: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,Na)))),zero_zero_int) ).

% negative_zless_0
tff(fact_7749_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_7750_int__ops_I6_J,axiom,
    ! [Aa: nat,Ba: nat] :
      aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Aa),Ba)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Aa)),aa_nat_int(semiri1314217659103216013at_int,Ba)),zero_zero_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,Aa)),aa_nat_int(semiri1314217659103216013at_int,Ba))) ).

% int_ops(6)
tff(fact_7751_sum_OSuc__reindex__ivl,axiom,
    ! [M: nat,Na: nat,G: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( aa_rat_rat(plus_plus_rat(groups2906978787729119204at_rat(G,set_or1269000886237332187st_nat(M,Na))),aa_nat_rat(G,aa_nat_nat(suc,Na))) = aa_rat_rat(plus_plus_rat(aa_nat_rat(G,M)),groups2906978787729119204at_rat(aTP_Lamm_jw(G),set_or1269000886237332187st_nat(M,Na))) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_7752_sum_OSuc__reindex__ivl,axiom,
    ! [M: nat,Na: nat,G: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,set_or1269000886237332187st_nat(M,Na))),aa_nat_int(G,aa_nat_nat(suc,Na))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,M)),groups3539618377306564664at_int(aTP_Lamm_jx(G),set_or1269000886237332187st_nat(M,Na))) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_7753_sum_OSuc__reindex__ivl,axiom,
    ! [M: nat,Na: nat,G: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(M,Na))),aa_nat_nat(G,aa_nat_nat(suc,Na))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G,M)),aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_jf(G)),set_or1269000886237332187st_nat(M,Na))) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_7754_sum_OSuc__reindex__ivl,axiom,
    ! [M: nat,Na: nat,G: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(G),set_or1269000886237332187st_nat(M,Na))),aa_nat_real(G,aa_nat_nat(suc,Na))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(G,M)),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_jg(G)),set_or1269000886237332187st_nat(M,Na))) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_7755_sum__Suc__diff,axiom,
    ! [M: nat,Na: nat,F: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,Na))
     => ( groups2906978787729119204at_rat(aTP_Lamm_jy(F),set_or1269000886237332187st_nat(M,Na)) = aa_rat_rat(minus_minus_rat(aa_nat_rat(F,aa_nat_nat(suc,Na))),aa_nat_rat(F,M)) ) ) ).

% sum_Suc_diff
tff(fact_7756_sum__Suc__diff,axiom,
    ! [M: nat,Na: nat,F: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,Na))
     => ( groups3539618377306564664at_int(aTP_Lamm_jz(F),set_or1269000886237332187st_nat(M,Na)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,aa_nat_nat(suc,Na))),aa_nat_int(F,M)) ) ) ).

% sum_Suc_diff
tff(fact_7757_sum__Suc__diff,axiom,
    ! [M: nat,Na: nat,F: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,Na))
     => ( aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_ka(F)),set_or1269000886237332187st_nat(M,Na)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(F,aa_nat_nat(suc,Na))),aa_nat_real(F,M)) ) ) ).

% sum_Suc_diff
tff(fact_7758_gauss__sum,axiom,
    ! [Na: nat] : groups3539618377306564664at_int(semiri1314217659103216013at_int,set_or1269000886237332187st_nat(zero_zero_nat,Na)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Na)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Na)),one_one_int))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% gauss_sum
tff(fact_7759_gauss__sum,axiom,
    ! [Na: nat] : groups7501900531339628137nteger(semiri4939895301339042750nteger,set_or1269000886237332187st_nat(zero_zero_nat,Na)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Na)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Na)),one_one_Code_integer))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) ).

% gauss_sum
tff(fact_7760_gauss__sum,axiom,
    ! [Na: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(semiri1316708129612266289at_nat),set_or1269000886237332187st_nat(zero_zero_nat,Na)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Na)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,Na)),one_one_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% gauss_sum
tff(fact_7761_arith__series,axiom,
    ! [Aa: int,D: int,Na: nat] : groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_js(Aa),D),set_or1269000886237332187st_nat(zero_zero_nat,Na)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Na)),one_one_int)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Aa)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Na)),D)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% arith_series
tff(fact_7762_arith__series,axiom,
    ! [Aa: code_integer,D: code_integer,Na: nat] : groups7501900531339628137nteger(aa_Cod1578151519121547077nteger(aTP_Lamm_jt(Aa),D),set_or1269000886237332187st_nat(zero_zero_nat,Na)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Na)),one_one_Code_integer)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Aa)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Na)),D)))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) ).

% arith_series
tff(fact_7763_arith__series,axiom,
    ! [Aa: nat,D: nat,Na: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ju(Aa),D)),set_or1269000886237332187st_nat(zero_zero_nat,Na)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,Na)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Aa)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Na)),D)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% arith_series
tff(fact_7764_double__gauss__sum__from__Suc__0,axiom,
    ! [Na: nat] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_set_nat_complex(groups2073611262835488442omplex(semiri8010041392384452111omplex),set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),Na))) = aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,Na)),aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,Na)),one_one_complex)) ).

% double_gauss_sum_from_Suc_0
tff(fact_7765_double__gauss__sum__from__Suc__0,axiom,
    ! [Na: nat] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),groups2906978787729119204at_rat(semiri681578069525770553at_rat,set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),Na))) = aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,Na)),aa_rat_rat(plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,Na)),one_one_rat)) ).

% double_gauss_sum_from_Suc_0
tff(fact_7766_double__gauss__sum__from__Suc__0,axiom,
    ! [Na: nat] : times_7803423173614009249d_enat(numera1916890842035813515d_enat(aa_num_num(bit0,one)),groups7108830773950497114d_enat(semiri4216267220026989637d_enat,set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),Na))) = times_7803423173614009249d_enat(aa_nat_Extended_enat(semiri4216267220026989637d_enat,Na),plus_p3455044024723400733d_enat(aa_nat_Extended_enat(semiri4216267220026989637d_enat,Na),one_on7984719198319812577d_enat)) ).

% double_gauss_sum_from_Suc_0
tff(fact_7767_double__gauss__sum__from__Suc__0,axiom,
    ! [Na: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),groups3539618377306564664at_int(semiri1314217659103216013at_int,set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),Na))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Na)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Na)),one_one_int)) ).

% double_gauss_sum_from_Suc_0
tff(fact_7768_double__gauss__sum__from__Suc__0,axiom,
    ! [Na: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),groups7501900531339628137nteger(semiri4939895301339042750nteger,set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),Na))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Na)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Na)),one_one_Code_integer)) ).

% double_gauss_sum_from_Suc_0
tff(fact_7769_double__gauss__sum__from__Suc__0,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_set_nat_nat(groups3542108847815614940at_nat(semiri1316708129612266289at_nat),set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),Na))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Na)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,Na)),one_one_nat)) ).

% double_gauss_sum_from_Suc_0
tff(fact_7770_double__gauss__sum__from__Suc__0,axiom,
    ! [Na: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_set_nat_real(groups6591440286371151544t_real(semiri5074537144036343181t_real),set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),Na))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,Na)),one_one_real)) ).

% double_gauss_sum_from_Suc_0
tff(fact_7771_sum__gp__offset,axiom,
    ! [Xb: complex,M: nat,Na: nat] :
      aa_set_nat_complex(groups2073611262835488442omplex(power_power_complex(Xb)),set_or1269000886237332187st_nat(M,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na))) = $ite(Xb = one_one_complex,aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,Na)),one_one_complex),aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(Xb),M)),aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(power_power_complex(Xb),aa_nat_nat(suc,Na))))),aa_complex_complex(minus_minus_complex(one_one_complex),Xb))) ).

% sum_gp_offset
tff(fact_7772_sum__gp__offset,axiom,
    ! [Xb: rat,M: nat,Na: nat] :
      groups2906978787729119204at_rat(power_power_rat(Xb),set_or1269000886237332187st_nat(M,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na))) = $ite(Xb = one_one_rat,aa_rat_rat(plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,Na)),one_one_rat),aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(Xb),M)),aa_rat_rat(minus_minus_rat(one_one_rat),aa_nat_rat(power_power_rat(Xb),aa_nat_nat(suc,Na))))),aa_rat_rat(minus_minus_rat(one_one_rat),Xb))) ).

% sum_gp_offset
tff(fact_7773_sum__gp__offset,axiom,
    ! [Xb: real,M: nat,Na: nat] :
      aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,Xb)),set_or1269000886237332187st_nat(M,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na))) = $ite(Xb = one_one_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,Na)),one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),M)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_nat_nat(suc,Na))))),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),Xb))) ).

% sum_gp_offset
tff(fact_7774_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_7775_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_7776_fact__numeral,axiom,
    ! [K: num] : semiri4449623510593786356d_enat(aa_num_nat(numeral_numeral_nat,K)) = times_7803423173614009249d_enat(numera1916890842035813515d_enat(K),semiri4449623510593786356d_enat(aa_num_nat(pred_numeral,K))) ).

% fact_numeral
tff(fact_7777_fact__numeral,axiom,
    ! [K: num] : aa_nat_Code_integer(semiri3624122377584611663nteger,aa_num_nat(numeral_numeral_nat,K)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,K)),aa_nat_Code_integer(semiri3624122377584611663nteger,aa_num_nat(pred_numeral,K))) ).

% fact_numeral
tff(fact_7778_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_7779_fact__numeral,axiom,
    ! [K: num] : aa_nat_real(semiri2265585572941072030t_real,aa_num_nat(numeral_numeral_nat,K)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,K)),aa_nat_real(semiri2265585572941072030t_real,aa_num_nat(pred_numeral,K))) ).

% fact_numeral
tff(fact_7780_sum__atLeastAtMost__code,axiom,
    ! [F: fun_nat_rat,Aa: nat,Ba: nat] : groups2906978787729119204at_rat(F,set_or1269000886237332187st_nat(Aa,Ba)) = aa_rat_rat(set_fo1949268297981939178at_rat(aTP_Lamm_kb(F),Aa,Ba),zero_zero_rat) ).

% sum_atLeastAtMost_code
tff(fact_7781_sum__atLeastAtMost__code,axiom,
    ! [F: fun_nat_int,Aa: nat,Ba: nat] : groups3539618377306564664at_int(F,set_or1269000886237332187st_nat(Aa,Ba)) = aa_int_int(aa_nat_fun_int_int(set_fo2581907887559384638at_int(aTP_Lamm_kc(F),Aa),Ba),zero_zero_int) ).

% sum_atLeastAtMost_code
tff(fact_7782_sum__atLeastAtMost__code,axiom,
    ! [F: fun_nat_nat,Aa: nat,Ba: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(F),set_or1269000886237332187st_nat(Aa,Ba)) = aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(aTP_Lamm_kd(F),Aa),Ba),zero_zero_nat) ).

% sum_atLeastAtMost_code
tff(fact_7783_sum__atLeastAtMost__code,axiom,
    ! [F: fun_nat_real,Aa: nat,Ba: nat] : aa_set_nat_real(groups6591440286371151544t_real(F),set_or1269000886237332187st_nat(Aa,Ba)) = aa_real_real(aa_nat_fun_real_real(set_fo3111899725591712190t_real(aTP_Lamm_ke(F),Aa),Ba),zero_zero_real) ).

% sum_atLeastAtMost_code
tff(fact_7784_real__less__rsqrt,axiom,
    ! [Xb: real,Y: 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)))),Y)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(sqrt,Y)) ) ).

% real_less_rsqrt
tff(fact_7785_sqrt__le__D,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,Xb)),Y)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% sqrt_le_D
tff(fact_7786_real__le__rsqrt,axiom,
    ! [Xb: real,Y: 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)))),Y)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(sqrt,Y)) ) ).

% real_le_rsqrt
tff(fact_7787_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_7788_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_7789_nat__approx__posE,axiom,
    ! [E2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E2)
     => ~ ! [N: nat] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N)))),E2) ) ).

% nat_approx_posE
tff(fact_7790_gauss__sum__from__Suc__0,axiom,
    ! [Na: nat] : groups3539618377306564664at_int(semiri1314217659103216013at_int,set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),Na)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Na)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Na)),one_one_int))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% gauss_sum_from_Suc_0
tff(fact_7791_gauss__sum__from__Suc__0,axiom,
    ! [Na: nat] : groups7501900531339628137nteger(semiri4939895301339042750nteger,set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),Na)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Na)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Na)),one_one_Code_integer))),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))) ).

% gauss_sum_from_Suc_0
tff(fact_7792_gauss__sum__from__Suc__0,axiom,
    ! [Na: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(semiri1316708129612266289at_nat),set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Na)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,Na)),one_one_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% gauss_sum_from_Suc_0
tff(fact_7793_of__nat__less__two__power,axiom,
    ! [Na: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,Na)),aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),Na)) ).

% of_nat_less_two_power
tff(fact_7794_of__nat__less__two__power,axiom,
    ! [Na: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Na)) ).

% of_nat_less_two_power
tff(fact_7795_of__nat__less__two__power,axiom,
    ! [Na: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Na)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)) ).

% of_nat_less_two_power
tff(fact_7796_of__nat__less__two__power,axiom,
    ! [Na: nat] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Na)),aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Na)) ).

% of_nat_less_two_power
tff(fact_7797_inverse__of__nat__le,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
     => ( ( Na != zero_zero_nat )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,M))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,Na))) ) ) ).

% inverse_of_nat_le
tff(fact_7798_inverse__of__nat__le,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
     => ( ( Na != zero_zero_nat )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(one_one_rat),aa_nat_rat(semiri681578069525770553at_rat,M))),aa_rat_rat(divide_divide_rat(one_one_rat),aa_nat_rat(semiri681578069525770553at_rat,Na))) ) ) ).

% inverse_of_nat_le
tff(fact_7799_sum_Oub__add__nat,axiom,
    ! [M: nat,Na: nat,G: fun_nat_rat,P3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),one_one_nat))
     => ( groups2906978787729119204at_rat(G,set_or1269000886237332187st_nat(M,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),P3))) = aa_rat_rat(plus_plus_rat(groups2906978787729119204at_rat(G,set_or1269000886237332187st_nat(M,Na))),groups2906978787729119204at_rat(G,set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),P3)))) ) ) ).

% sum.ub_add_nat
tff(fact_7800_sum_Oub__add__nat,axiom,
    ! [M: nat,Na: nat,G: fun_nat_int,P3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),one_one_nat))
     => ( groups3539618377306564664at_int(G,set_or1269000886237332187st_nat(M,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),P3))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,set_or1269000886237332187st_nat(M,Na))),groups3539618377306564664at_int(G,set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),P3)))) ) ) ).

% sum.ub_add_nat
tff(fact_7801_sum_Oub__add__nat,axiom,
    ! [M: nat,Na: nat,G: fun_nat_nat,P3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),one_one_nat))
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(M,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),P3))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(M,Na))),aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),P3)))) ) ) ).

% sum.ub_add_nat
tff(fact_7802_sum_Oub__add__nat,axiom,
    ! [M: nat,Na: nat,G: fun_nat_real,P3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),one_one_nat))
     => ( aa_set_nat_real(groups6591440286371151544t_real(G),set_or1269000886237332187st_nat(M,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),P3))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(G),set_or1269000886237332187st_nat(M,Na))),aa_set_nat_real(groups6591440286371151544t_real(G),set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),P3)))) ) ) ).

% sum.ub_add_nat
tff(fact_7803_binomial__ge__n__over__k__pow__k,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_nat_real(semiri5074537144036343181t_real,K))),K)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K))) ) ).

% binomial_ge_n_over_k_pow_k
tff(fact_7804_binomial__ge__n__over__k__pow__k,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_rat_rat(divide_divide_rat(aa_nat_rat(semiri681578069525770553at_rat,Na)),aa_nat_rat(semiri681578069525770553at_rat,K))),K)),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K))) ) ).

% binomial_ge_n_over_k_pow_k
tff(fact_7805_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)
       => ( ! [M3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M3)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,M3)),Xb)),C) )
         => ( Xb = zero_zero_real ) ) ) ) ).

% real_archimedian_rdiv_eq_0
tff(fact_7806_pochhammer__product,axiom,
    ! [M: nat,Na: nat,Z: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( aa_nat_rat(comm_s4028243227959126397er_rat(Z),Na) = aa_rat_rat(times_times_rat(aa_nat_rat(comm_s4028243227959126397er_rat(Z),M)),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(plus_plus_rat(Z),aa_nat_rat(semiri681578069525770553at_rat,M))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),M))) ) ) ).

% pochhammer_product
tff(fact_7807_pochhammer__product,axiom,
    ! [M: nat,Na: nat,Z: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),Na) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Z),M)),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Z),aa_nat_nat(semiri1316708129612266289at_nat,M))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),M))) ) ) ).

% pochhammer_product
tff(fact_7808_pochhammer__product,axiom,
    ! [M: nat,Na: nat,Z: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z),Na) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z),M)),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),aa_nat_real(semiri5074537144036343181t_real,M))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),M))) ) ) ).

% pochhammer_product
tff(fact_7809_pochhammer__product,axiom,
    ! [M: nat,Na: nat,Z: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( aa_nat_int(comm_s4660882817536571857er_int(Z),Na) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(comm_s4660882817536571857er_int(Z),M)),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),aa_nat_int(semiri1314217659103216013at_int,M))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),M))) ) ) ).

% pochhammer_product
tff(fact_7810_pochhammer__product,axiom,
    ! [M: nat,Na: nat,Z: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( aa_nat_Code_integer(comm_s8582702949713902594nteger(Z),Na) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(comm_s8582702949713902594nteger(Z),M)),aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Z),aa_nat_Code_integer(semiri4939895301339042750nteger,M))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),M))) ) ) ).

% pochhammer_product
tff(fact_7811_sum__count__set,axiom,
    ! [Xs: list_complex,X5: set_complex] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(set_complex2(Xs)),X5)
     => ( aa_set_complex_o(finite3207457112153483333omplex,X5)
       => ( groups5693394587270226106ex_nat(count_list_complex(Xs),X5) = size_s3451745648224563538omplex(Xs) ) ) ) ).

% sum_count_set
tff(fact_7812_sum__count__set,axiom,
    ! [Xs: list_VEBT_VEBT,X5: set_VEBT_VEBT] :
      ( ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(Xs),X5)
     => ( finite5795047828879050333T_VEBT(X5)
       => ( groups771621172384141258BT_nat(count_list_VEBT_VEBT(Xs),X5) = size_s6755466524823107622T_VEBT(Xs) ) ) ) ).

% sum_count_set
tff(fact_7813_sum__count__set,axiom,
    ! [Xs: list_o,X5: set_o] :
      ( ord_less_eq_set_o(set_o2(Xs),X5)
     => ( finite_finite_o(X5)
       => ( groups8507830703676809646_o_nat(count_list_o(Xs),X5) = size_size_list_o(Xs) ) ) ) ).

% sum_count_set
tff(fact_7814_sum__count__set,axiom,
    ! [Xs: list_int,X5: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(set_int2(Xs)),X5)
     => ( aa_set_int_o(finite_finite_int,X5)
       => ( groups4541462559716669496nt_nat(count_list_int(Xs),X5) = size_size_list_int(Xs) ) ) ) ).

% sum_count_set
tff(fact_7815_sum__count__set,axiom,
    ! [Xs: list_nat,X5: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,set_nat2(Xs)),X5)
     => ( aa_set_nat_o(finite_finite_nat,X5)
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(count_list_nat(Xs)),X5) = size_size_list_nat(Xs) ) ) ) ).

% sum_count_set
tff(fact_7816_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_7817_zdiff__int__split,axiom,
    ! [P: fun_int_o,Xb: nat,Y: nat] :
      ( aa_int_o(P,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xb),Y)))
    <=> ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),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,Y))) )
        & ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
         => aa_int_o(P,zero_zero_int) ) ) ) ).

% zdiff_int_split
tff(fact_7818_real__of__nat__div2,axiom,
    ! [Na: nat,Xb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_nat_real(semiri5074537144036343181t_real,Xb))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),Xb)))) ).

% real_of_nat_div2
tff(fact_7819_real__of__nat__div3,axiom,
    ! [Na: nat,Xb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_nat_real(semiri5074537144036343181t_real,Xb))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),Xb)))),one_one_real) ).

% real_of_nat_div3
tff(fact_7820_ln__realpow,axiom,
    ! [Xb: real,Na: 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),Na)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_real_real(ln_ln_real,Xb)) ) ) ).

% ln_realpow
tff(fact_7821_sum__gp,axiom,
    ! [Xb: complex,M: nat,Na: nat] :
      aa_set_nat_complex(groups2073611262835488442omplex(power_power_complex(Xb)),set_or1269000886237332187st_nat(M,Na)) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M),
        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,Na),one_one_nat)),M)),aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(Xb),M)),aa_nat_complex(power_power_complex(Xb),aa_nat_nat(suc,Na)))),aa_complex_complex(minus_minus_complex(one_one_complex),Xb))) ) ).

% sum_gp
tff(fact_7822_sum__gp,axiom,
    ! [Xb: rat,M: nat,Na: nat] :
      groups2906978787729119204at_rat(power_power_rat(Xb),set_or1269000886237332187st_nat(M,Na)) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M),
        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,Na),one_one_nat)),M)),aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_nat_rat(power_power_rat(Xb),M)),aa_nat_rat(power_power_rat(Xb),aa_nat_nat(suc,Na)))),aa_rat_rat(minus_minus_rat(one_one_rat),Xb))) ) ).

% sum_gp
tff(fact_7823_sum__gp,axiom,
    ! [Xb: real,M: nat,Na: nat] :
      aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,Xb)),set_or1269000886237332187st_nat(M,Na)) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M),
        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,Na),one_one_nat)),M)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_nat_nat(suc,Na)))),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),Xb))) ) ).

% sum_gp
tff(fact_7824_real__sqrt__unique,axiom,
    ! [Y: real,Xb: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),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),Y)
       => ( aa_real_real(sqrt,Xb) = Y ) ) ) ).

% real_sqrt_unique
tff(fact_7825_real__le__lsqrt,axiom,
    ! [Xb: real,Y: 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),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,Xb)),Y) ) ) ) ).

% real_le_lsqrt
tff(fact_7826_lemma__real__divide__sqrt__less,axiom,
    ! [U: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),U)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,U),aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),U) ) ).

% lemma_real_divide_sqrt_less
tff(fact_7827_real__sqrt__sum__squares__triangle__ineq,axiom,
    ! [Aa: real,C: real,Ba: real,D: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),C)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Ba),D)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ba),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,C),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,D),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ).

% real_sqrt_sum_squares_triangle_ineq
tff(fact_7828_real__sqrt__sum__squares__ge2,axiom,
    ! [Y: real,Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(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,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% real_sqrt_sum_squares_ge2
tff(fact_7829_real__sqrt__sum__squares__ge1,axiom,
    ! [Xb: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_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,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% real_sqrt_sum_squares_ge1
tff(fact_7830_exp__half__le2,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),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_7831_sqrt__ge__absD,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),aa_real_real(sqrt,Y))
     => 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)))),Y) ) ).

% sqrt_ge_absD
tff(fact_7832_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_7833_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_7834_pochhammer__absorb__comp,axiom,
    ! [R2: real,K: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,R2),aa_nat_real(semiri5074537144036343181t_real,K))),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(uminus_uminus_real,R2)),K)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,R2)),one_one_real)),K)) ).

% pochhammer_absorb_comp
tff(fact_7835_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_7836_pochhammer__absorb__comp,axiom,
    ! [R2: code_integer,K: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(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(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,R2)),one_one_Code_integer)),K)) ).

% pochhammer_absorb_comp
tff(fact_7837_fact__num__eq__if,axiom,
    ! [M: nat] :
      aa_nat_complex(semiri5044797733671781792omplex,M) = $ite(M = zero_zero_nat,one_one_complex,aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,M)),aa_nat_complex(semiri5044797733671781792omplex,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))) ).

% fact_num_eq_if
tff(fact_7838_fact__num__eq__if,axiom,
    ! [M: nat] :
      aa_nat_rat(semiri773545260158071498ct_rat,M) = $ite(M = zero_zero_nat,one_one_rat,aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))) ).

% fact_num_eq_if
tff(fact_7839_fact__num__eq__if,axiom,
    ! [M: nat] :
      aa_nat_int(semiri1406184849735516958ct_int,M) = $ite(M = zero_zero_nat,one_one_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))) ).

% fact_num_eq_if
tff(fact_7840_fact__num__eq__if,axiom,
    ! [M: nat] :
      aa_nat_Code_integer(semiri3624122377584611663nteger,M) = $ite(M = zero_zero_nat,one_one_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri3624122377584611663nteger,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))) ).

% fact_num_eq_if
tff(fact_7841_fact__num__eq__if,axiom,
    ! [M: nat] :
      aa_nat_nat(semiri1408675320244567234ct_nat,M) = $ite(M = zero_zero_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))) ).

% fact_num_eq_if
tff(fact_7842_fact__num__eq__if,axiom,
    ! [M: nat] :
      aa_nat_real(semiri2265585572941072030t_real,M) = $ite(M = zero_zero_nat,one_one_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))) ).

% fact_num_eq_if
tff(fact_7843_fact__reduce,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_rat(semiri773545260158071498ct_rat,Na) = aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,Na)),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat))) ) ) ).

% fact_reduce
tff(fact_7844_fact__reduce,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_int(semiri1406184849735516958ct_int,Na) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Na)),aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat))) ) ) ).

% fact_reduce
tff(fact_7845_fact__reduce,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_Code_integer(semiri3624122377584611663nteger,Na) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Na)),aa_nat_Code_integer(semiri3624122377584611663nteger,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat))) ) ) ).

% fact_reduce
tff(fact_7846_fact__reduce,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_nat(semiri1408675320244567234ct_nat,Na) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Na)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat))) ) ) ).

% fact_reduce
tff(fact_7847_fact__reduce,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_real(semiri2265585572941072030t_real,Na) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat))) ) ) ).

% fact_reduce
tff(fact_7848_pochhammer__same,axiom,
    ! [Na: nat] : aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,Na))),Na) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Na)),aa_nat_complex(semiri5044797733671781792omplex,Na)) ).

% pochhammer_same
tff(fact_7849_pochhammer__same,axiom,
    ! [Na: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,Na))),Na) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Na)),aa_nat_rat(semiri773545260158071498ct_rat,Na)) ).

% pochhammer_same
tff(fact_7850_pochhammer__same,axiom,
    ! [Na: nat] : aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,Na))),Na) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),Na)),aa_nat_int(semiri1406184849735516958ct_int,Na)) ).

% pochhammer_same
tff(fact_7851_pochhammer__same,axiom,
    ! [Na: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Na))),Na) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Na)),aa_nat_Code_integer(semiri3624122377584611663nteger,Na)) ).

% pochhammer_same
tff(fact_7852_pochhammer__same,axiom,
    ! [Na: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,Na))),Na) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Na)),aa_nat_real(semiri2265585572941072030t_real,Na)) ).

% pochhammer_same
tff(fact_7853_fact__binomial,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => ( aa_rat_rat(times_times_rat(aa_nat_rat(semiri773545260158071498ct_rat,K)),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K))) = aa_rat_rat(divide_divide_rat(aa_nat_rat(semiri773545260158071498ct_rat,Na)),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K))) ) ) ).

% fact_binomial
tff(fact_7854_fact__binomial,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri2265585572941072030t_real,K)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri2265585572941072030t_real,Na)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K))) ) ) ).

% fact_binomial
tff(fact_7855_binomial__fact,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => ( aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K)) = aa_rat_rat(divide_divide_rat(aa_nat_rat(semiri773545260158071498ct_rat,Na)),aa_rat_rat(times_times_rat(aa_nat_rat(semiri773545260158071498ct_rat,K)),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K)))) ) ) ).

% binomial_fact
tff(fact_7856_binomial__fact,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => ( aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri2265585572941072030t_real,Na)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri2265585572941072030t_real,K)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K)))) ) ) ).

% binomial_fact
tff(fact_7857_linear__plus__1__le__power,axiom,
    ! [Xb: real,Na: 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(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Na)),Xb)),one_one_real)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),one_one_real)),Na)) ) ).

% linear_plus_1_le_power
tff(fact_7858_sum__natinterval__diff,axiom,
    ! [F: fun_nat_rat,M: nat,Na: nat] :
      groups2906978787729119204at_rat(aTP_Lamm_kf(F),set_or1269000886237332187st_nat(M,Na)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na),aa_rat_rat(minus_minus_rat(aa_nat_rat(F,M)),aa_nat_rat(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),one_one_nat))),zero_zero_rat) ).

% sum_natinterval_diff
tff(fact_7859_sum__natinterval__diff,axiom,
    ! [F: fun_nat_int,M: nat,Na: nat] :
      groups3539618377306564664at_int(aTP_Lamm_kg(F),set_or1269000886237332187st_nat(M,Na)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,M)),aa_nat_int(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),one_one_nat))),zero_zero_int) ).

% sum_natinterval_diff
tff(fact_7860_sum__natinterval__diff,axiom,
    ! [F: fun_nat_real,M: nat,Na: nat] :
      aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_kh(F)),set_or1269000886237332187st_nat(M,Na)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(F,M)),aa_nat_real(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),one_one_nat))),zero_zero_real) ).

% sum_natinterval_diff
tff(fact_7861_sum__telescope_H_H,axiom,
    ! [M: nat,Na: nat,F: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( groups2906978787729119204at_rat(aTP_Lamm_ki(F),set_or1269000886237332187st_nat(aa_nat_nat(suc,M),Na)) = aa_rat_rat(minus_minus_rat(aa_nat_rat(F,Na)),aa_nat_rat(F,M)) ) ) ).

% sum_telescope''
tff(fact_7862_sum__telescope_H_H,axiom,
    ! [M: nat,Na: nat,F: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( groups3539618377306564664at_int(aTP_Lamm_kj(F),set_or1269000886237332187st_nat(aa_nat_nat(suc,M),Na)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,Na)),aa_nat_int(F,M)) ) ) ).

% sum_telescope''
tff(fact_7863_sum__telescope_H_H,axiom,
    ! [M: nat,Na: nat,F: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_kk(F)),set_or1269000886237332187st_nat(aa_nat_nat(suc,M),Na)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(F,Na)),aa_nat_real(F,M)) ) ) ).

% sum_telescope''
tff(fact_7864_Bernoulli__inequality,axiom,
    ! [Xb: real,Na: 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(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Na)),Xb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xb)),Na)) ) ).

% Bernoulli_inequality
tff(fact_7865_real__less__lsqrt,axiom,
    ! [Xb: real,Y: 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),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,Xb)),Y) ) ) ) ).

% real_less_lsqrt
tff(fact_7866_sqrt__sum__squares__le__sum,axiom,
    ! [Xb: real,Y: 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),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(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,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y)) ) ) ).

% sqrt_sum_squares_le_sum
tff(fact_7867_sqrt__sum__squares__le__sum__abs,axiom,
    ! [Xb: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(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,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,Xb)),aa_real_real(abs_abs_real,Y))) ).

% sqrt_sum_squares_le_sum_abs
tff(fact_7868_real__sqrt__ge__abs2,axiom,
    ! [Y: real,Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Y)),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(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,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% real_sqrt_ge_abs2
tff(fact_7869_real__sqrt__ge__abs1,axiom,
    ! [Xb: real,Y: 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(aa_rea7340605660300375871l_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,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% real_sqrt_ge_abs1
tff(fact_7870_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(aa_rea7340605660300375871l_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_7871_pochhammer__minus_H,axiom,
    ! [Ba: complex,K: nat] : aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(minus_minus_complex(Ba),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,Ba)),K)) ).

% pochhammer_minus'
tff(fact_7872_pochhammer__minus_H,axiom,
    ! [Ba: rat,K: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(minus_minus_rat(Ba),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,Ba)),K)) ).

% pochhammer_minus'
tff(fact_7873_pochhammer__minus_H,axiom,
    ! [Ba: real,K: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Ba),aa_nat_real(semiri5074537144036343181t_real,K))),one_one_real)),K) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),K)),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(uminus_uminus_real,Ba)),K)) ).

% pochhammer_minus'
tff(fact_7874_pochhammer__minus_H,axiom,
    ! [Ba: 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,Ba),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,Ba)),K)) ).

% pochhammer_minus'
tff(fact_7875_pochhammer__minus_H,axiom,
    ! [Ba: code_integer,K: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,Ba),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,Ba)),K)) ).

% pochhammer_minus'
tff(fact_7876_pochhammer__minus,axiom,
    ! [Ba: complex,K: nat] : aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,Ba)),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(Ba),aa_nat_complex(semiri8010041392384452111omplex,K))),one_one_complex)),K)) ).

% pochhammer_minus
tff(fact_7877_pochhammer__minus,axiom,
    ! [Ba: rat,K: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,Ba)),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(Ba),aa_nat_rat(semiri681578069525770553at_rat,K))),one_one_rat)),K)) ).

% pochhammer_minus
tff(fact_7878_pochhammer__minus,axiom,
    ! [Ba: real,K: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(uminus_uminus_real,Ba)),K) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),K)),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Ba),aa_nat_real(semiri5074537144036343181t_real,K))),one_one_real)),K)) ).

% pochhammer_minus
tff(fact_7879_pochhammer__minus,axiom,
    ! [Ba: int,K: nat] : aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,Ba)),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,Ba),aa_nat_int(semiri1314217659103216013at_int,K))),one_one_int)),K)) ).

% pochhammer_minus
tff(fact_7880_pochhammer__minus,axiom,
    ! [Ba: code_integer,K: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Ba)),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(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,Ba),aa_nat_Code_integer(semiri4939895301339042750nteger,K))),one_one_Code_integer)),K)) ).

% pochhammer_minus
tff(fact_7881_mask__eq__sum__exp,axiom,
    ! [Na: nat] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)),one_one_int) = groups3539618377306564664at_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),collect_nat(aa_nat_fun_nat_o(aTP_Lamm_au,Na))) ).

% mask_eq_sum_exp
tff(fact_7882_mask__eq__sum__exp,axiom,
    ! [Na: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Na)),one_one_Code_integer) = groups7501900531339628137nteger(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),collect_nat(aa_nat_fun_nat_o(aTP_Lamm_au,Na))) ).

% mask_eq_sum_exp
tff(fact_7883_mask__eq__sum__exp,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)),one_one_nat) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),collect_nat(aa_nat_fun_nat_o(aTP_Lamm_au,Na))) ).

% mask_eq_sum_exp
tff(fact_7884_sum__gp__multiplied,axiom,
    ! [M: nat,Na: nat,Xb: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(one_one_complex),Xb)),aa_set_nat_complex(groups2073611262835488442omplex(power_power_complex(Xb)),set_or1269000886237332187st_nat(M,Na))) = aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(Xb),M)),aa_nat_complex(power_power_complex(Xb),aa_nat_nat(suc,Na))) ) ) ).

% sum_gp_multiplied
tff(fact_7885_sum__gp__multiplied,axiom,
    ! [M: nat,Na: nat,Xb: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(one_one_rat),Xb)),groups2906978787729119204at_rat(power_power_rat(Xb),set_or1269000886237332187st_nat(M,Na))) = aa_rat_rat(minus_minus_rat(aa_nat_rat(power_power_rat(Xb),M)),aa_nat_rat(power_power_rat(Xb),aa_nat_nat(suc,Na))) ) ) ).

% sum_gp_multiplied
tff(fact_7886_sum__gp__multiplied,axiom,
    ! [M: nat,Na: nat,Xb: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),Xb)),groups3539618377306564664at_int(power_power_int(Xb),set_or1269000886237332187st_nat(M,Na))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(Xb),M)),aa_nat_int(power_power_int(Xb),aa_nat_nat(suc,Na))) ) ) ).

% sum_gp_multiplied
tff(fact_7887_sum__gp__multiplied,axiom,
    ! [M: nat,Na: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),Xb)),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,Xb)),set_or1269000886237332187st_nat(M,Na))) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_nat_nat(suc,Na))) ) ) ).

% sum_gp_multiplied
tff(fact_7888_sum_Oin__pairs,axiom,
    ! [G: fun_nat_rat,M: nat,Na: nat] : groups2906978787729119204at_rat(G,set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)))) = groups2906978787729119204at_rat(aTP_Lamm_kl(G),set_or1269000886237332187st_nat(M,Na)) ).

% sum.in_pairs
tff(fact_7889_sum_Oin__pairs,axiom,
    ! [G: fun_nat_int,M: nat,Na: nat] : groups3539618377306564664at_int(G,set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)))) = groups3539618377306564664at_int(aTP_Lamm_km(G),set_or1269000886237332187st_nat(M,Na)) ).

% sum.in_pairs
tff(fact_7890_sum_Oin__pairs,axiom,
    ! [G: fun_nat_nat,M: nat,Na: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)))) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_kn(G)),set_or1269000886237332187st_nat(M,Na)) ).

% sum.in_pairs
tff(fact_7891_sum_Oin__pairs,axiom,
    ! [G: fun_nat_real,M: nat,Na: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)))) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_ko(G)),set_or1269000886237332187st_nat(M,Na)) ).

% sum.in_pairs
tff(fact_7892_arsinh__real__aux,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_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_7893_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(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_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_7894_real__sqrt__sum__squares__mult__ge__zero,axiom,
    ! [Xb: real,Y: real,Xaa: real,Ya: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(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,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xaa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ).

% real_sqrt_sum_squares_mult_ge_zero
tff(fact_7895_real__sqrt__power__even,axiom,
    ! [Na: 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))),Na)
     => ( 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)),Na) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ) ).

% real_sqrt_power_even
tff(fact_7896_arith__geo__mean__sqrt,axiom,
    ! [Xb: real,Y: 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),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Y))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ).

% arith_geo_mean_sqrt
tff(fact_7897_mask__eq__sum__exp__nat,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)),aa_nat_nat(suc,zero_zero_nat)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),collect_nat(aa_nat_fun_nat_o(aTP_Lamm_au,Na))) ).

% mask_eq_sum_exp_nat
tff(fact_7898_gauss__sum__nat,axiom,
    ! [Na: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_cc),set_or1269000886237332187st_nat(zero_zero_nat,Na)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),aa_nat_nat(suc,Na))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% gauss_sum_nat
tff(fact_7899_fact__code,axiom,
    ! [Na: nat] : aa_nat_int(semiri1406184849735516958ct_int,Na) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na),one_one_nat)) ).

% fact_code
tff(fact_7900_fact__code,axiom,
    ! [Na: nat] : aa_nat_Code_integer(semiri3624122377584611663nteger,Na) = aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na),one_one_nat)) ).

% fact_code
tff(fact_7901_fact__code,axiom,
    ! [Na: nat] : aa_nat_nat(semiri1408675320244567234ct_nat,Na) = aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na),one_one_nat)) ).

% fact_code
tff(fact_7902_fact__code,axiom,
    ! [Na: nat] : aa_nat_real(semiri2265585572941072030t_real,Na) = aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na),one_one_nat)) ).

% fact_code
tff(fact_7903_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(aa_rea7340605660300375871l_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(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Xb))) ) ) ).

% real_exp_bound_lemma
tff(fact_7904_cos__x__y__le__one,axiom,
    ! [Xb: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_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,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),one_one_real) ).

% cos_x_y_le_one
tff(fact_7905_real__sqrt__sum__squares__less,axiom,
    ! [Xb: real,U: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xb)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,U),aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,U),aa_real_real(sqrt,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(aa_rea7340605660300375871l_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,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),U) ) ) ).

% real_sqrt_sum_squares_less
tff(fact_7906_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(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_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_7907_pochhammer__code,axiom,
    ! [Aa: complex,Na: nat] :
      aa_nat_complex(comm_s2602460028002588243omplex(Aa),Na) = $ite(Na = zero_zero_nat,one_one_complex,aa_complex_complex(aa_nat3829984526574172007omplex(set_fo1517530859248394432omplex(aTP_Lamm_kp(Aa),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat)),one_one_complex)) ).

% pochhammer_code
tff(fact_7908_pochhammer__code,axiom,
    ! [Aa: rat,Na: nat] :
      aa_nat_rat(comm_s4028243227959126397er_rat(Aa),Na) = $ite(Na = zero_zero_nat,one_one_rat,aa_rat_rat(set_fo1949268297981939178at_rat(aTP_Lamm_kq(Aa),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat)),one_one_rat)) ).

% pochhammer_code
tff(fact_7909_pochhammer__code,axiom,
    ! [Aa: real,Na: nat] :
      aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Aa),Na) = $ite(Na = zero_zero_nat,one_one_real,aa_real_real(aa_nat_fun_real_real(set_fo3111899725591712190t_real(aTP_Lamm_kr(Aa),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat)),one_one_real)) ).

% pochhammer_code
tff(fact_7910_pochhammer__code,axiom,
    ! [Aa: int,Na: nat] :
      aa_nat_int(comm_s4660882817536571857er_int(Aa),Na) = $ite(Na = zero_zero_nat,one_one_int,aa_int_int(aa_nat_fun_int_int(set_fo2581907887559384638at_int(aTP_Lamm_ks(Aa),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat)),one_one_int)) ).

% pochhammer_code
tff(fact_7911_pochhammer__code,axiom,
    ! [Aa: code_integer,Na: nat] :
      aa_nat_Code_integer(comm_s8582702949713902594nteger(Aa),Na) = $ite(Na = zero_zero_nat,one_one_Code_integer,set_fo1084959871951514735nteger(aTP_Lamm_kt(Aa),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat),one_one_Code_integer)) ).

% pochhammer_code
tff(fact_7912_pochhammer__code,axiom,
    ! [Aa: nat,Na: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Aa),Na) = $ite(Na = zero_zero_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(aTP_Lamm_ku(Aa),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat)),one_one_nat)) ).

% pochhammer_code
tff(fact_7913_arith__series__nat,axiom,
    ! [Aa: nat,D: nat,Na: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_kv(Aa),D)),set_or1269000886237332187st_nat(zero_zero_nat,Na)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,Na)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Aa)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),D)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% arith_series_nat
tff(fact_7914_Bernoulli__inequality__even,axiom,
    ! [Na: 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))),Na)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Na)),Xb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xb)),Na)) ) ).

% Bernoulli_inequality_even
tff(fact_7915_Sum__Icc__nat,axiom,
    ! [M: nat,Na: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_cc),set_or1269000886237332187st_nat(M,Na)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),one_one_nat))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% Sum_Icc_nat
tff(fact_7916_of__nat__code__if,axiom,
    ! [Na: nat] :
      aa_nat_complex(semiri8010041392384452111omplex,Na) = $ite(Na = zero_zero_nat,zero_zero_complex,aa_Pro82899191565270131omplex(produc1917071388513777916omplex(aTP_Lamm_kw),divmod_nat(Na,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% of_nat_code_if
tff(fact_7917_of__nat__code__if,axiom,
    ! [Na: nat] :
      aa_nat_rat(semiri681578069525770553at_rat,Na) = $ite(Na = zero_zero_nat,zero_zero_rat,produc6207742614233964070at_rat(aTP_Lamm_kx,divmod_nat(Na,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% of_nat_code_if
tff(fact_7918_of__nat__code__if,axiom,
    ! [Na: nat] :
      aa_nat_nat(semiri1316708129612266289at_nat,Na) = $ite(Na = zero_zero_nat,zero_zero_nat,aa_Pro4243173770044397973at_nat(produc6842872674320459806at_nat(aTP_Lamm_ky),divmod_nat(Na,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% of_nat_code_if
tff(fact_7919_of__nat__code__if,axiom,
    ! [Na: nat] :
      aa_nat_Extended_enat(semiri4216267220026989637d_enat,Na) = $ite(Na = zero_zero_nat,zero_z5237406670263579293d_enat,produc2676513652042109336d_enat(aTP_Lamm_kz,divmod_nat(Na,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% of_nat_code_if
tff(fact_7920_of__nat__code__if,axiom,
    ! [Na: nat] :
      aa_nat_real(semiri5074537144036343181t_real,Na) = $ite(Na = zero_zero_nat,zero_zero_real,aa_Pro6168199777849092593t_real(produc1703576794950452218t_real(aTP_Lamm_la),divmod_nat(Na,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% of_nat_code_if
tff(fact_7921_of__nat__code__if,axiom,
    ! [Na: nat] :
      aa_nat_int(semiri1314217659103216013at_int,Na) = $ite(Na = zero_zero_nat,zero_zero_int,aa_Pro4240683299535347697at_int(produc6840382203811409530at_int(aTP_Lamm_lb),divmod_nat(Na,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% of_nat_code_if
tff(fact_7922_of__nat__code__if,axiom,
    ! [Na: nat] :
      aa_nat_Code_integer(semiri4939895301339042750nteger,Na) = $ite(Na = zero_zero_nat,zero_z3403309356797280102nteger,produc1830744345554046123nteger(aTP_Lamm_lc,divmod_nat(Na,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% of_nat_code_if
tff(fact_7923_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_ld,Xb)) ) ).

% monoseq_arctan_series
tff(fact_7924_lemma__termdiff3,axiom,
    ! [H: real,Z: real,K4: real,Na: nat] :
      ( ( H != zero_zero_real )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,Z)),K4)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),H))),K4)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),H)),Na)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),Na))),H)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(suc,zero_zero_nat))))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(suc,zero_zero_nat))))),aa_nat_real(aa_real_fun_nat_real(power_power_real,K4),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(real_V7735802525324610683m_real,H))) ) ) ) ).

% lemma_termdiff3
tff(fact_7925_lemma__termdiff3,axiom,
    ! [H: complex,Z: complex,K4: real,Na: nat] :
      ( ( H != zero_zero_complex )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Z)),K4)
       => ( 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))),K4)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(aa_complex_complex(plus_plus_complex(Z),H)),Na)),aa_nat_complex(power_power_complex(Z),Na))),H)),aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,Na)),aa_nat_complex(power_power_complex(Z),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(suc,zero_zero_nat))))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(suc,zero_zero_nat))))),aa_nat_real(aa_real_fun_nat_real(power_power_real,K4),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_complex_real(real_V1022390504157884413omplex,H))) ) ) ) ).

% lemma_termdiff3
tff(fact_7926_pochhammer__times__pochhammer__half,axiom,
    ! [Z: complex,Na: nat] : aa_complex_complex(times_times_complex(aa_nat_complex(comm_s2602460028002588243omplex(Z),aa_nat_nat(suc,Na))),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(Z),aa_complex_complex(divide1717551699836669952omplex(one_one_complex),numera6690914467698888265omplex(aa_num_num(bit0,one))))),aa_nat_nat(suc,Na))) = aa_set_nat_complex(groups6464643781859351333omplex(aTP_Lamm_le(Z)),set_or1269000886237332187st_nat(zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)),one_one_nat))) ).

% pochhammer_times_pochhammer_half
tff(fact_7927_pochhammer__times__pochhammer__half,axiom,
    ! [Z: rat,Na: nat] : aa_rat_rat(times_times_rat(aa_nat_rat(comm_s4028243227959126397er_rat(Z),aa_nat_nat(suc,Na))),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(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,Na))) = groups73079841787564623at_rat(aTP_Lamm_lf(Z),set_or1269000886237332187st_nat(zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)),one_one_nat))) ).

% pochhammer_times_pochhammer_half
tff(fact_7928_pochhammer__times__pochhammer__half,axiom,
    ! [Z: real,Na: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Z),aa_nat_nat(suc,Na))),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_nat_nat(suc,Na))) = aa_set_nat_real(groups129246275422532515t_real(aa_real_fun_nat_real(aTP_Lamm_lg,Z)),set_or1269000886237332187st_nat(zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)),one_one_nat))) ).

% pochhammer_times_pochhammer_half
tff(fact_7929_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_lh,Xb)) ) ) ) ).

% ln_series
tff(fact_7930_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_li,Xb)) ) ) ).

% arctan_series
tff(fact_7931_prod_Oneutral__const,axiom,
    ! [A3: set_nat] : aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_lj),A3) = one_one_nat ).

% prod.neutral_const
tff(fact_7932_prod_Oneutral__const,axiom,
    ! [A3: set_nat] : groups705719431365010083at_int(aTP_Lamm_lk,A3) = one_one_int ).

% prod.neutral_const
tff(fact_7933_prod_Oneutral__const,axiom,
    ! [A3: set_int] : groups1705073143266064639nt_int(aTP_Lamm_ll,A3) = one_one_int ).

% prod.neutral_const
tff(fact_7934_prod__zero__iff,axiom,
    ! [A3: set_nat,F: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( ( aa_set_nat_real(groups129246275422532515t_real(F),A3) = zero_zero_real )
      <=> ? [X4: nat] :
            ( aa_set_nat_o(member_nat(X4),A3)
            & ( aa_nat_real(F,X4) = zero_zero_real ) ) ) ) ).

% prod_zero_iff
tff(fact_7935_prod__zero__iff,axiom,
    ! [A3: set_int,F: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( ( groups2316167850115554303t_real(F,A3) = zero_zero_real )
      <=> ? [X4: int] :
            ( aa_set_int_o(member_int(X4),A3)
            & ( aa_int_real(F,X4) = zero_zero_real ) ) ) ) ).

% prod_zero_iff
tff(fact_7936_prod__zero__iff,axiom,
    ! [A3: set_complex,F: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A3)
     => ( ( groups766887009212190081x_real(F,A3) = zero_zero_real )
      <=> ? [X4: complex] :
            ( aa_set_complex_o(member_complex(X4),A3)
            & ( aa_complex_real(F,X4) = zero_zero_real ) ) ) ) ).

% prod_zero_iff
tff(fact_7937_prod__zero__iff,axiom,
    ! [A3: set_nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( ( groups73079841787564623at_rat(F,A3) = zero_zero_rat )
      <=> ? [X4: nat] :
            ( aa_set_nat_o(member_nat(X4),A3)
            & ( aa_nat_rat(F,X4) = zero_zero_rat ) ) ) ) ).

% prod_zero_iff
tff(fact_7938_prod__zero__iff,axiom,
    ! [A3: set_int,F: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( ( groups1072433553688619179nt_rat(F,A3) = zero_zero_rat )
      <=> ? [X4: int] :
            ( aa_set_int_o(member_int(X4),A3)
            & ( aa_int_rat(F,X4) = zero_zero_rat ) ) ) ) ).

% prod_zero_iff
tff(fact_7939_prod__zero__iff,axiom,
    ! [A3: set_complex,F: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A3)
     => ( ( groups225925009352817453ex_rat(F,A3) = zero_zero_rat )
      <=> ? [X4: complex] :
            ( aa_set_complex_o(member_complex(X4),A3)
            & ( aa_complex_rat(F,X4) = zero_zero_rat ) ) ) ) ).

% prod_zero_iff
tff(fact_7940_prod__zero__iff,axiom,
    ! [A3: set_int,F: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( ( groups1707563613775114915nt_nat(F,A3) = zero_zero_nat )
      <=> ? [X4: int] :
            ( aa_set_int_o(member_int(X4),A3)
            & ( aa_int_nat(F,X4) = zero_zero_nat ) ) ) ) ).

% prod_zero_iff
tff(fact_7941_prod__zero__iff,axiom,
    ! [A3: set_complex,F: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A3)
     => ( ( groups861055069439313189ex_nat(F,A3) = zero_zero_nat )
      <=> ? [X4: complex] :
            ( aa_set_complex_o(member_complex(X4),A3)
            & ( aa_complex_nat(F,X4) = zero_zero_nat ) ) ) ) ).

% prod_zero_iff
tff(fact_7942_prod__zero__iff,axiom,
    ! [A3: set_complex,F: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A3)
     => ( ( groups858564598930262913ex_int(F,A3) = zero_zero_int )
      <=> ? [X4: complex] :
            ( aa_set_complex_o(member_complex(X4),A3)
            & ( aa_complex_int(F,X4) = zero_zero_int ) ) ) ) ).

% prod_zero_iff
tff(fact_7943_prod__zero__iff,axiom,
    ! [A3: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( ( aa_set_nat_nat(groups708209901874060359at_nat(F),A3) = zero_zero_nat )
      <=> ? [X4: nat] :
            ( aa_set_nat_o(member_nat(X4),A3)
            & ( aa_nat_nat(F,X4) = zero_zero_nat ) ) ) ) ).

% prod_zero_iff
tff(fact_7944_prod_Oempty,axiom,
    ! [G: fun_real_complex] : groups713298508707869441omplex(G,bot_bot_set_real) = one_one_complex ).

% prod.empty
tff(fact_7945_prod_Oempty,axiom,
    ! [G: fun_real_real] : groups1681761925125756287l_real(G,bot_bot_set_real) = one_one_real ).

% prod.empty
tff(fact_7946_prod_Oempty,axiom,
    ! [G: fun_real_rat] : groups4061424788464935467al_rat(G,bot_bot_set_real) = one_one_rat ).

% prod.empty
tff(fact_7947_prod_Oempty,axiom,
    ! [G: fun_real_nat] : groups4696554848551431203al_nat(G,bot_bot_set_real) = one_one_nat ).

% prod.empty
tff(fact_7948_prod_Oempty,axiom,
    ! [G: fun_real_int] : groups4694064378042380927al_int(G,bot_bot_set_real) = one_one_int ).

% prod.empty
tff(fact_7949_prod_Oempty,axiom,
    ! [G: fun_nat_complex] : aa_set_nat_complex(groups6464643781859351333omplex(G),bot_bot_set_nat) = one_one_complex ).

% prod.empty
tff(fact_7950_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_7951_prod_Oempty,axiom,
    ! [G: fun_nat_rat] : groups73079841787564623at_rat(G,bot_bot_set_nat) = one_one_rat ).

% prod.empty
tff(fact_7952_prod_Oempty,axiom,
    ! [G: fun_int_complex] : groups7440179247065528705omplex(G,bot_bot_set_int) = one_one_complex ).

% prod.empty
tff(fact_7953_prod_Oempty,axiom,
    ! [G: fun_int_real] : groups2316167850115554303t_real(G,bot_bot_set_int) = one_one_real ).

% prod.empty
tff(fact_7954_prod_Oinfinite,axiom,
    ! [A3: set_nat,G: fun_nat_complex] :
      ( ~ aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_nat_complex(groups6464643781859351333omplex(G),A3) = one_one_complex ) ) ).

% prod.infinite
tff(fact_7955_prod_Oinfinite,axiom,
    ! [A3: set_int,G: fun_int_complex] :
      ( ~ aa_set_int_o(finite_finite_int,A3)
     => ( groups7440179247065528705omplex(G,A3) = one_one_complex ) ) ).

% prod.infinite
tff(fact_7956_prod_Oinfinite,axiom,
    ! [A3: set_complex,G: fun_complex_complex] :
      ( ~ aa_set_complex_o(finite3207457112153483333omplex,A3)
     => ( groups3708469109370488835omplex(G,A3) = one_one_complex ) ) ).

% prod.infinite
tff(fact_7957_prod_Oinfinite,axiom,
    ! [A3: set_nat,G: fun_nat_real] :
      ( ~ aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_nat_real(groups129246275422532515t_real(G),A3) = one_one_real ) ) ).

% prod.infinite
tff(fact_7958_prod_Oinfinite,axiom,
    ! [A3: set_int,G: fun_int_real] :
      ( ~ aa_set_int_o(finite_finite_int,A3)
     => ( groups2316167850115554303t_real(G,A3) = one_one_real ) ) ).

% prod.infinite
tff(fact_7959_prod_Oinfinite,axiom,
    ! [A3: set_complex,G: fun_complex_real] :
      ( ~ aa_set_complex_o(finite3207457112153483333omplex,A3)
     => ( groups766887009212190081x_real(G,A3) = one_one_real ) ) ).

% prod.infinite
tff(fact_7960_prod_Oinfinite,axiom,
    ! [A3: set_nat,G: fun_nat_rat] :
      ( ~ aa_set_nat_o(finite_finite_nat,A3)
     => ( groups73079841787564623at_rat(G,A3) = one_one_rat ) ) ).

% prod.infinite
tff(fact_7961_prod_Oinfinite,axiom,
    ! [A3: set_int,G: fun_int_rat] :
      ( ~ aa_set_int_o(finite_finite_int,A3)
     => ( groups1072433553688619179nt_rat(G,A3) = one_one_rat ) ) ).

% prod.infinite
tff(fact_7962_prod_Oinfinite,axiom,
    ! [A3: set_complex,G: fun_complex_rat] :
      ( ~ aa_set_complex_o(finite3207457112153483333omplex,A3)
     => ( groups225925009352817453ex_rat(G,A3) = one_one_rat ) ) ).

% prod.infinite
tff(fact_7963_prod_Oinfinite,axiom,
    ! [A3: set_int,G: fun_int_nat] :
      ( ~ aa_set_int_o(finite_finite_int,A3)
     => ( groups1707563613775114915nt_nat(G,A3) = one_one_nat ) ) ).

% prod.infinite
tff(fact_7964_dvd__prod__eqI,axiom,
    ! [A3: set_real,Aa: real,Ba: nat,F: fun_real_nat] :
      ( finite_finite_real(A3)
     => ( member_real(Aa,A3)
       => ( ( Ba = aa_real_nat(F,Aa) )
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Ba),groups4696554848551431203al_nat(F,A3)) ) ) ) ).

% dvd_prod_eqI
tff(fact_7965_dvd__prod__eqI,axiom,
    ! [A3: set_int,Aa: int,Ba: nat,F: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( aa_set_int_o(member_int(Aa),A3)
       => ( ( Ba = aa_int_nat(F,Aa) )
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Ba),groups1707563613775114915nt_nat(F,A3)) ) ) ) ).

% dvd_prod_eqI
tff(fact_7966_dvd__prod__eqI,axiom,
    ! [A3: set_complex,Aa: complex,Ba: nat,F: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A3)
     => ( aa_set_complex_o(member_complex(Aa),A3)
       => ( ( Ba = aa_complex_nat(F,Aa) )
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Ba),groups861055069439313189ex_nat(F,A3)) ) ) ) ).

% dvd_prod_eqI
tff(fact_7967_dvd__prod__eqI,axiom,
    ! [A3: set_real,Aa: real,Ba: int,F: fun_real_int] :
      ( finite_finite_real(A3)
     => ( member_real(Aa,A3)
       => ( ( Ba = aa_real_int(F,Aa) )
         => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Ba),groups4694064378042380927al_int(F,A3)) ) ) ) ).

% dvd_prod_eqI
tff(fact_7968_dvd__prod__eqI,axiom,
    ! [A3: set_complex,Aa: complex,Ba: int,F: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A3)
     => ( aa_set_complex_o(member_complex(Aa),A3)
       => ( ( Ba = aa_complex_int(F,Aa) )
         => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Ba),groups858564598930262913ex_int(F,A3)) ) ) ) ).

% dvd_prod_eqI
tff(fact_7969_dvd__prod__eqI,axiom,
    ! [A3: set_real,Aa: real,Ba: code_integer,F: fun_re6978948838062957830nteger] :
      ( finite_finite_real(A3)
     => ( member_real(Aa,A3)
       => ( ( Ba = aa_real_Code_integer(F,Aa) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Ba),groups6225526099057966256nteger(F,A3)) ) ) ) ).

% dvd_prod_eqI
tff(fact_7970_dvd__prod__eqI,axiom,
    ! [A3: set_nat,Aa: nat,Ba: code_integer,F: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_nat_o(member_nat(Aa),A3)
       => ( ( Ba = aa_nat_Code_integer(F,Aa) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Ba),groups3455450783089532116nteger(F,A3)) ) ) ) ).

% dvd_prod_eqI
tff(fact_7971_dvd__prod__eqI,axiom,
    ! [A3: set_int,Aa: int,Ba: code_integer,F: fun_int_Code_integer] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( aa_set_int_o(member_int(Aa),A3)
       => ( ( Ba = aa_int_Code_integer(F,Aa) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Ba),groups3827104343326376752nteger(F,A3)) ) ) ) ).

% dvd_prod_eqI
tff(fact_7972_dvd__prod__eqI,axiom,
    ! [A3: set_complex,Aa: complex,Ba: code_integer,F: fun_co8153928105989009288nteger] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A3)
     => ( aa_set_complex_o(member_complex(Aa),A3)
       => ( ( Ba = aa_com4217951221296830281nteger(F,Aa) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Ba),groups8682486955453173170nteger(F,A3)) ) ) ) ).

% dvd_prod_eqI
tff(fact_7973_dvd__prod__eqI,axiom,
    ! [A3: set_nat,Aa: nat,Ba: nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_nat_o(member_nat(Aa),A3)
       => ( ( Ba = aa_nat_nat(F,Aa) )
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Ba),aa_set_nat_nat(groups708209901874060359at_nat(F),A3)) ) ) ) ).

% dvd_prod_eqI
tff(fact_7974_dvd__prodI,axiom,
    ! [A3: set_real,Aa: real,F: fun_real_nat] :
      ( finite_finite_real(A3)
     => ( member_real(Aa,A3)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_real_nat(F,Aa)),groups4696554848551431203al_nat(F,A3)) ) ) ).

% dvd_prodI
tff(fact_7975_dvd__prodI,axiom,
    ! [A3: set_int,Aa: int,F: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( aa_set_int_o(member_int(Aa),A3)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_int_nat(F,Aa)),groups1707563613775114915nt_nat(F,A3)) ) ) ).

% dvd_prodI
tff(fact_7976_dvd__prodI,axiom,
    ! [A3: set_complex,Aa: complex,F: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A3)
     => ( aa_set_complex_o(member_complex(Aa),A3)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_complex_nat(F,Aa)),groups861055069439313189ex_nat(F,A3)) ) ) ).

% dvd_prodI
tff(fact_7977_dvd__prodI,axiom,
    ! [A3: set_real,Aa: real,F: fun_real_int] :
      ( finite_finite_real(A3)
     => ( member_real(Aa,A3)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_real_int(F,Aa)),groups4694064378042380927al_int(F,A3)) ) ) ).

% dvd_prodI
tff(fact_7978_dvd__prodI,axiom,
    ! [A3: set_complex,Aa: complex,F: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A3)
     => ( aa_set_complex_o(member_complex(Aa),A3)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_complex_int(F,Aa)),groups858564598930262913ex_int(F,A3)) ) ) ).

% dvd_prodI
tff(fact_7979_dvd__prodI,axiom,
    ! [A3: set_real,Aa: real,F: fun_re6978948838062957830nteger] :
      ( finite_finite_real(A3)
     => ( member_real(Aa,A3)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_real_Code_integer(F,Aa)),groups6225526099057966256nteger(F,A3)) ) ) ).

% dvd_prodI
tff(fact_7980_dvd__prodI,axiom,
    ! [A3: set_nat,Aa: nat,F: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_nat_o(member_nat(Aa),A3)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(F,Aa)),groups3455450783089532116nteger(F,A3)) ) ) ).

% dvd_prodI
tff(fact_7981_dvd__prodI,axiom,
    ! [A3: set_int,Aa: int,F: fun_int_Code_integer] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( aa_set_int_o(member_int(Aa),A3)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_int_Code_integer(F,Aa)),groups3827104343326376752nteger(F,A3)) ) ) ).

% dvd_prodI
tff(fact_7982_dvd__prodI,axiom,
    ! [A3: set_complex,Aa: complex,F: fun_co8153928105989009288nteger] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A3)
     => ( aa_set_complex_o(member_complex(Aa),A3)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_com4217951221296830281nteger(F,Aa)),groups8682486955453173170nteger(F,A3)) ) ) ).

% dvd_prodI
tff(fact_7983_dvd__prodI,axiom,
    ! [A3: set_nat,Aa: nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_nat_o(member_nat(Aa),A3)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(F,Aa)),aa_set_nat_nat(groups708209901874060359at_nat(F),A3)) ) ) ).

% dvd_prodI
tff(fact_7984_norm__fact,axiom,
    ! [Na: nat] : aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(semiri5044797733671781792omplex,Na)) = aa_nat_real(semiri2265585572941072030t_real,Na) ).

% norm_fact
tff(fact_7985_norm__fact,axiom,
    ! [Na: nat] : aa_real_real(real_V7735802525324610683m_real,aa_nat_real(semiri2265585572941072030t_real,Na)) = aa_nat_real(semiri2265585572941072030t_real,Na) ).

% norm_fact
tff(fact_7986_prod_Odelta_H,axiom,
    ! [S2: set_real,Aa: real,Ba: fun_real_complex] :
      ( finite_finite_real(S2)
     => ( groups713298508707869441omplex(aa_fun2233226938935547116omplex(aTP_Lamm_lm(Aa),Ba),S2) = $ite(member_real(Aa,S2),aa_real_complex(Ba,Aa),one_one_complex) ) ) ).

% prod.delta'
tff(fact_7987_prod_Odelta_H,axiom,
    ! [S2: set_nat,Aa: nat,Ba: fun_nat_complex] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( aa_set_nat_complex(groups6464643781859351333omplex(aa_fun2955221450537884980omplex(aTP_Lamm_ln(Aa),Ba)),S2) = $ite(aa_set_nat_o(member_nat(Aa),S2),aa_nat_complex(Ba,Aa),one_one_complex) ) ) ).

% prod.delta'
tff(fact_7988_prod_Odelta_H,axiom,
    ! [S2: set_int,Aa: int,Ba: fun_int_complex] :
      ( aa_set_int_o(finite_finite_int,S2)
     => ( groups7440179247065528705omplex(aa_fun3088484340408535532omplex(aTP_Lamm_lo(Aa),Ba),S2) = $ite(aa_set_int_o(member_int(Aa),S2),aa_int_complex(Ba,Aa),one_one_complex) ) ) ).

% prod.delta'
tff(fact_7989_prod_Odelta_H,axiom,
    ! [S2: set_complex,Aa: complex,Ba: fun_complex_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S2)
     => ( groups3708469109370488835omplex(aa_fun3187328805393345264omplex(aTP_Lamm_lp(Aa),Ba),S2) = $ite(aa_set_complex_o(member_complex(Aa),S2),aa_complex_complex(Ba,Aa),one_one_complex) ) ) ).

% prod.delta'
tff(fact_7990_prod_Odelta_H,axiom,
    ! [S2: set_real,Aa: real,Ba: fun_real_real] :
      ( finite_finite_real(S2)
     => ( groups1681761925125756287l_real(aa_fun6946558556880649704l_real(aTP_Lamm_lq(Aa),Ba),S2) = $ite(member_real(Aa,S2),aa_real_real(Ba,Aa),one_one_real) ) ) ).

% prod.delta'
tff(fact_7991_prod_Odelta_H,axiom,
    ! [S2: set_nat,Aa: nat,Ba: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( aa_set_nat_real(groups129246275422532515t_real(aa_fun1839206346889086768t_real(aTP_Lamm_lr(Aa),Ba)),S2) = $ite(aa_set_nat_o(member_nat(Aa),S2),aa_nat_real(Ba,Aa),one_one_real) ) ) ).

% prod.delta'
tff(fact_7992_prod_Odelta_H,axiom,
    ! [S2: set_int,Aa: int,Ba: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,S2)
     => ( groups2316167850115554303t_real(aa_fun3252798400008884712t_real(aTP_Lamm_ls(Aa),Ba),S2) = $ite(aa_set_int_o(member_int(Aa),S2),aa_int_real(Ba,Aa),one_one_real) ) ) ).

% prod.delta'
tff(fact_7993_prod_Odelta_H,axiom,
    ! [S2: set_complex,Aa: complex,Ba: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S2)
     => ( groups766887009212190081x_real(aa_fun8933909628229591020x_real(aTP_Lamm_lt(Aa),Ba),S2) = $ite(aa_set_complex_o(member_complex(Aa),S2),aa_complex_real(Ba,Aa),one_one_real) ) ) ).

% prod.delta'
tff(fact_7994_prod_Odelta_H,axiom,
    ! [S2: set_real,Aa: real,Ba: fun_real_rat] :
      ( finite_finite_real(S2)
     => ( groups4061424788464935467al_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_lu(Aa),Ba),S2) = $ite(member_real(Aa,S2),aa_real_rat(Ba,Aa),one_one_rat) ) ) ).

% prod.delta'
tff(fact_7995_prod_Odelta_H,axiom,
    ! [S2: set_nat,Aa: nat,Ba: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( groups73079841787564623at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_lv(Aa),Ba),S2) = $ite(aa_set_nat_o(member_nat(Aa),S2),aa_nat_rat(Ba,Aa),one_one_rat) ) ) ).

% prod.delta'
tff(fact_7996_prod_Odelta,axiom,
    ! [S2: set_real,Aa: real,Ba: fun_real_complex] :
      ( finite_finite_real(S2)
     => ( groups713298508707869441omplex(aa_fun2233226938935547116omplex(aTP_Lamm_lw(Aa),Ba),S2) = $ite(member_real(Aa,S2),aa_real_complex(Ba,Aa),one_one_complex) ) ) ).

% prod.delta
tff(fact_7997_prod_Odelta,axiom,
    ! [S2: set_nat,Aa: nat,Ba: fun_nat_complex] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( aa_set_nat_complex(groups6464643781859351333omplex(aa_fun2955221450537884980omplex(aTP_Lamm_lx(Aa),Ba)),S2) = $ite(aa_set_nat_o(member_nat(Aa),S2),aa_nat_complex(Ba,Aa),one_one_complex) ) ) ).

% prod.delta
tff(fact_7998_prod_Odelta,axiom,
    ! [S2: set_int,Aa: int,Ba: fun_int_complex] :
      ( aa_set_int_o(finite_finite_int,S2)
     => ( groups7440179247065528705omplex(aa_fun3088484340408535532omplex(aTP_Lamm_ly(Aa),Ba),S2) = $ite(aa_set_int_o(member_int(Aa),S2),aa_int_complex(Ba,Aa),one_one_complex) ) ) ).

% prod.delta
tff(fact_7999_prod_Odelta,axiom,
    ! [S2: set_complex,Aa: complex,Ba: fun_complex_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S2)
     => ( groups3708469109370488835omplex(aa_fun3187328805393345264omplex(aTP_Lamm_lz(Aa),Ba),S2) = $ite(aa_set_complex_o(member_complex(Aa),S2),aa_complex_complex(Ba,Aa),one_one_complex) ) ) ).

% prod.delta
tff(fact_8000_prod_Odelta,axiom,
    ! [S2: set_real,Aa: real,Ba: fun_real_real] :
      ( finite_finite_real(S2)
     => ( groups1681761925125756287l_real(aa_fun6946558556880649704l_real(aTP_Lamm_ma(Aa),Ba),S2) = $ite(member_real(Aa,S2),aa_real_real(Ba,Aa),one_one_real) ) ) ).

% prod.delta
tff(fact_8001_prod_Odelta,axiom,
    ! [S2: set_nat,Aa: nat,Ba: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( aa_set_nat_real(groups129246275422532515t_real(aa_fun1839206346889086768t_real(aTP_Lamm_mb(Aa),Ba)),S2) = $ite(aa_set_nat_o(member_nat(Aa),S2),aa_nat_real(Ba,Aa),one_one_real) ) ) ).

% prod.delta
tff(fact_8002_prod_Odelta,axiom,
    ! [S2: set_int,Aa: int,Ba: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,S2)
     => ( groups2316167850115554303t_real(aa_fun3252798400008884712t_real(aTP_Lamm_mc(Aa),Ba),S2) = $ite(aa_set_int_o(member_int(Aa),S2),aa_int_real(Ba,Aa),one_one_real) ) ) ).

% prod.delta
tff(fact_8003_prod_Odelta,axiom,
    ! [S2: set_complex,Aa: complex,Ba: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S2)
     => ( groups766887009212190081x_real(aa_fun8933909628229591020x_real(aTP_Lamm_md(Aa),Ba),S2) = $ite(aa_set_complex_o(member_complex(Aa),S2),aa_complex_real(Ba,Aa),one_one_real) ) ) ).

% prod.delta
tff(fact_8004_prod_Odelta,axiom,
    ! [S2: set_real,Aa: real,Ba: fun_real_rat] :
      ( finite_finite_real(S2)
     => ( groups4061424788464935467al_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_me(Aa),Ba),S2) = $ite(member_real(Aa,S2),aa_real_rat(Ba,Aa),one_one_rat) ) ) ).

% prod.delta
tff(fact_8005_prod_Odelta,axiom,
    ! [S2: set_nat,Aa: nat,Ba: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( groups73079841787564623at_rat(aa_fun765758790203053448at_rat(aTP_Lamm_mf(Aa),Ba),S2) = $ite(aa_set_nat_o(member_nat(Aa),S2),aa_nat_rat(Ba,Aa),one_one_rat) ) ) ).

% prod.delta
tff(fact_8006_powser__zero,axiom,
    ! [F: fun_nat_complex] : suminf_complex(aTP_Lamm_iu(F)) = aa_nat_complex(F,zero_zero_nat) ).

% powser_zero
tff(fact_8007_powser__zero,axiom,
    ! [F: fun_nat_real] : suminf_real(aTP_Lamm_iw(F)) = aa_nat_real(F,zero_zero_nat) ).

% powser_zero
tff(fact_8008_prod_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_complex,M: nat,Na: nat] :
      aa_set_nat_complex(groups6464643781859351333omplex(G),set_or1269000886237332187st_nat(M,aa_nat_nat(suc,Na))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Na)),M),one_one_complex,aa_complex_complex(times_times_complex(aa_set_nat_complex(groups6464643781859351333omplex(G),set_or1269000886237332187st_nat(M,Na))),aa_nat_complex(G,aa_nat_nat(suc,Na)))) ).

% prod.cl_ivl_Suc
tff(fact_8009_prod_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_real,M: nat,Na: nat] :
      aa_set_nat_real(groups129246275422532515t_real(G),set_or1269000886237332187st_nat(M,aa_nat_nat(suc,Na))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Na)),M),one_one_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_set_nat_real(groups129246275422532515t_real(G),set_or1269000886237332187st_nat(M,Na))),aa_nat_real(G,aa_nat_nat(suc,Na)))) ).

% prod.cl_ivl_Suc
tff(fact_8010_prod_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_rat,M: nat,Na: nat] :
      groups73079841787564623at_rat(G,set_or1269000886237332187st_nat(M,aa_nat_nat(suc,Na))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Na)),M),one_one_rat,aa_rat_rat(times_times_rat(groups73079841787564623at_rat(G,set_or1269000886237332187st_nat(M,Na))),aa_nat_rat(G,aa_nat_nat(suc,Na)))) ).

% prod.cl_ivl_Suc
tff(fact_8011_prod_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_nat,M: nat,Na: nat] :
      aa_set_nat_nat(groups708209901874060359at_nat(G),set_or1269000886237332187st_nat(M,aa_nat_nat(suc,Na))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Na)),M),one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups708209901874060359at_nat(G),set_or1269000886237332187st_nat(M,Na))),aa_nat_nat(G,aa_nat_nat(suc,Na)))) ).

% prod.cl_ivl_Suc
tff(fact_8012_prod_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_int,M: nat,Na: nat] :
      groups705719431365010083at_int(G,set_or1269000886237332187st_nat(M,aa_nat_nat(suc,Na))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Na)),M),one_one_int,aa_int_int(aa_int_fun_int_int(times_times_int,groups705719431365010083at_int(G,set_or1269000886237332187st_nat(M,Na))),aa_nat_int(G,aa_nat_nat(suc,Na)))) ).

% prod.cl_ivl_Suc
tff(fact_8013_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_real_complex,A3: set_real] :
      ( ( groups713298508707869441omplex(G,A3) != one_one_complex )
     => ~ ! [A2: real] :
            ( member_real(A2,A3)
           => ( aa_real_complex(G,A2) = one_one_complex ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_8014_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_nat_complex,A3: set_nat] :
      ( ( aa_set_nat_complex(groups6464643781859351333omplex(G),A3) != one_one_complex )
     => ~ ! [A2: nat] :
            ( aa_set_nat_o(member_nat(A2),A3)
           => ( aa_nat_complex(G,A2) = one_one_complex ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_8015_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_int_complex,A3: set_int] :
      ( ( groups7440179247065528705omplex(G,A3) != one_one_complex )
     => ~ ! [A2: int] :
            ( aa_set_int_o(member_int(A2),A3)
           => ( aa_int_complex(G,A2) = one_one_complex ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_8016_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_real_real,A3: set_real] :
      ( ( groups1681761925125756287l_real(G,A3) != one_one_real )
     => ~ ! [A2: real] :
            ( member_real(A2,A3)
           => ( aa_real_real(G,A2) = one_one_real ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_8017_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 )
     => ~ ! [A2: nat] :
            ( aa_set_nat_o(member_nat(A2),A3)
           => ( aa_nat_real(G,A2) = one_one_real ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_8018_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_int_real,A3: set_int] :
      ( ( groups2316167850115554303t_real(G,A3) != one_one_real )
     => ~ ! [A2: int] :
            ( aa_set_int_o(member_int(A2),A3)
           => ( aa_int_real(G,A2) = one_one_real ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_8019_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_real_rat,A3: set_real] :
      ( ( groups4061424788464935467al_rat(G,A3) != one_one_rat )
     => ~ ! [A2: real] :
            ( member_real(A2,A3)
           => ( aa_real_rat(G,A2) = one_one_rat ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_8020_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_nat_rat,A3: set_nat] :
      ( ( groups73079841787564623at_rat(G,A3) != one_one_rat )
     => ~ ! [A2: nat] :
            ( aa_set_nat_o(member_nat(A2),A3)
           => ( aa_nat_rat(G,A2) = one_one_rat ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_8021_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_int_rat,A3: set_int] :
      ( ( groups1072433553688619179nt_rat(G,A3) != one_one_rat )
     => ~ ! [A2: int] :
            ( aa_set_int_o(member_int(A2),A3)
           => ( aa_int_rat(G,A2) = one_one_rat ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_8022_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_real_nat,A3: set_real] :
      ( ( groups4696554848551431203al_nat(G,A3) != one_one_nat )
     => ~ ! [A2: real] :
            ( member_real(A2,A3)
           => ( aa_real_nat(G,A2) = one_one_nat ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_8023_prod_Oneutral,axiom,
    ! [A3: set_nat,G: fun_nat_nat] :
      ( ! [X: nat] :
          ( aa_set_nat_o(member_nat(X),A3)
         => ( aa_nat_nat(G,X) = one_one_nat ) )
     => ( aa_set_nat_nat(groups708209901874060359at_nat(G),A3) = one_one_nat ) ) ).

% prod.neutral
tff(fact_8024_prod_Oneutral,axiom,
    ! [A3: set_nat,G: fun_nat_int] :
      ( ! [X: nat] :
          ( aa_set_nat_o(member_nat(X),A3)
         => ( aa_nat_int(G,X) = one_one_int ) )
     => ( groups705719431365010083at_int(G,A3) = one_one_int ) ) ).

% prod.neutral
tff(fact_8025_prod_Oneutral,axiom,
    ! [A3: set_int,G: fun_int_int] :
      ( ! [X: int] :
          ( aa_set_int_o(member_int(X),A3)
         => ( aa_int_int(G,X) = one_one_int ) )
     => ( groups1705073143266064639nt_int(G,A3) = one_one_int ) ) ).

% prod.neutral
tff(fact_8026_prod_Oswap__restrict,axiom,
    ! [A3: set_real,B4: set_nat,G: fun_real_fun_nat_nat,R: fun_real_fun_nat_o] :
      ( finite_finite_real(A3)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( groups4696554848551431203al_nat(aa_fun2272639302901571595al_nat(aa_fun7039802393428226646al_nat(aTP_Lamm_mg(B4),G),R),A3) = aa_set_nat_nat(groups708209901874060359at_nat(aa_fun7366046364592083119at_nat(aa_fun7718511627822915834at_nat(aTP_Lamm_mh(A3),G),R)),B4) ) ) ) ).

% prod.swap_restrict
tff(fact_8027_prod_Oswap__restrict,axiom,
    ! [A3: set_int,B4: set_nat,G: fun_int_fun_nat_nat,R: fun_int_fun_nat_o] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( groups1707563613775114915nt_nat(aa_fun4650797359919590027nt_nat(aa_fun8303507722728657494nt_nat(aTP_Lamm_mi(B4),G),R),A3) = aa_set_nat_nat(groups708209901874060359at_nat(aa_fun3651443648018535471at_nat(aa_fun6116586148035635706at_nat(aTP_Lamm_mj(A3),G),R)),B4) ) ) ) ).

% prod.swap_restrict
tff(fact_8028_prod_Oswap__restrict,axiom,
    ! [A3: set_complex,B4: set_nat,G: fun_co636942507509095880at_nat,R: fun_co5676208753217392844_nat_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A3)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( groups861055069439313189ex_nat(aa_fun2265263913897893899ex_nat(aa_fun1127489917733235668ex_nat(aTP_Lamm_mk(B4),G),R),A3) = aa_set_nat_nat(groups708209901874060359at_nat(aa_fun5527826185635525933at_nat(aa_fun4894236392271416310at_nat(aTP_Lamm_ml(A3),G),R)),B4) ) ) ) ).

% prod.swap_restrict
tff(fact_8029_prod_Oswap__restrict,axiom,
    ! [A3: set_real,B4: set_nat,G: fun_real_fun_nat_int,R: fun_real_fun_nat_o] :
      ( finite_finite_real(A3)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( groups4694064378042380927al_int(aa_fun7318160320247150695al_int(aa_fun4230461673125498710al_int(aTP_Lamm_mm(B4),G),R),A3) = groups705719431365010083at_int(aa_fun3188195345082886411at_int(aa_fun8006801000864960506at_int(aTP_Lamm_mo(A3),G),R),B4) ) ) ) ).

% prod.swap_restrict
tff(fact_8030_prod_Oswap__restrict,axiom,
    ! [A3: set_complex,B4: set_nat,G: fun_co7370231399432144932at_int,R: fun_co5676208753217392844_nat_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A3)
     => ( aa_set_nat_o(finite_finite_nat,B4)
       => ( groups858564598930262913ex_int(aa_fun7310784931243472999ex_int(aa_fun7210183792663747796ex_int(aTP_Lamm_mp(B4),G),R),A3) = groups705719431365010083at_int(aa_fun1349975166126329225at_int(aa_fun7503395680861046006at_int(aTP_Lamm_mr(A3),G),R),B4) ) ) ) ).

% prod.swap_restrict
tff(fact_8031_prod_Oswap__restrict,axiom,
    ! [A3: set_real,B4: set_int,G: fun_real_fun_int_int,R: fun_real_fun_int_o] :
      ( finite_finite_real(A3)
     => ( aa_set_int_o(finite_finite_int,B4)
       => ( groups4694064378042380927al_int(aa_fun7555737461897725707al_int(aa_fun1679075639499870878al_int(aTP_Lamm_ms(B4),G),R),A3) = groups1705073143266064639nt_int(aa_fun3996843141751676939nt_int(aa_fun129049606478405534nt_int(aTP_Lamm_mt(A3),G),R),B4) ) ) ) ).

% prod.swap_restrict
tff(fact_8032_prod_Oswap__restrict,axiom,
    ! [A3: set_complex,B4: set_int,G: fun_co333780937270390912nt_int,R: fun_co3431063785546946928_int_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A3)
     => ( aa_set_int_o(finite_finite_int,B4)
       => ( groups858564598930262913ex_int(aa_fun5055871773053899019ex_int(aa_fun7361901416012365340ex_int(aTP_Lamm_mu(B4),G),R),A3) = groups1705073143266064639nt_int(aa_fun2158622962795119753nt_int(aa_fun607866698265288346nt_int(aTP_Lamm_mv(A3),G),R),B4) ) ) ) ).

% prod.swap_restrict
tff(fact_8033_prod_Oswap__restrict,axiom,
    ! [A3: set_nat,B4: set_real,G: fun_nat_fun_real_nat,R: fun_nat_fun_real_o] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( finite_finite_real(B4)
       => ( aa_set_nat_nat(groups708209901874060359at_nat(aa_fun6477086565944264623at_nat(aa_fun888394026836741882at_nat(aTP_Lamm_mx(B4),G),R)),A3) = groups4696554848551431203al_nat(aa_fun1455215566758156043al_nat(aa_fun1180509458258223190al_nat(aTP_Lamm_na(A3),G),R),B4) ) ) ) ).

% prod.swap_restrict
tff(fact_8034_prod_Oswap__restrict,axiom,
    ! [A3: set_nat,B4: set_int,G: fun_nat_fun_int_nat,R: fun_nat_fun_int_o] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_int_o(finite_finite_int,B4)
       => ( aa_set_nat_nat(groups708209901874060359at_nat(aa_fun8666973529875159599at_nat(aa_fun3399039426249668090at_nat(aTP_Lamm_nb(B4),G),R)),A3) = groups1707563613775114915nt_nat(aa_fun442955204921438347nt_nat(aa_fun5585961000942689878nt_nat(aTP_Lamm_nd(A3),G),R),B4) ) ) ) ).

% prod.swap_restrict
tff(fact_8035_prod_Oswap__restrict,axiom,
    ! [A3: set_nat,B4: set_complex,G: fun_na8457291176974909128ex_nat,R: fun_na1554462459066888140plex_o] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_complex_o(finite3207457112153483333omplex,B4)
       => ( aa_set_nat_nat(groups708209901874060359at_nat(aa_fun1031781741659083821at_nat(aa_fun7518049653624490486at_nat(aTP_Lamm_ne(B4),G),R)),A3) = groups861055069439313189ex_nat(aa_fun1649587401935388427ex_nat(aa_fun3336366540202158548ex_nat(aTP_Lamm_ng(A3),G),R),B4) ) ) ) ).

% prod.swap_restrict
tff(fact_8036_mod__prod__eq,axiom,
    ! [F: fun_nat_nat,Aa: 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_ho(F),Aa)),A3)),Aa) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_set_nat_nat(groups708209901874060359at_nat(F),A3)),Aa) ).

% mod_prod_eq
tff(fact_8037_mod__prod__eq,axiom,
    ! [F: fun_nat_int,Aa: int,A3: set_nat] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,groups705719431365010083at_int(aa_int_fun_nat_int(aTP_Lamm_nh(F),Aa),A3)),Aa) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,groups705719431365010083at_int(F,A3)),Aa) ).

% mod_prod_eq
tff(fact_8038_mod__prod__eq,axiom,
    ! [F: fun_int_int,Aa: int,A3: set_int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,groups1705073143266064639nt_int(aa_int_fun_int_int(aTP_Lamm_hn(F),Aa),A3)),Aa) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,groups1705073143266064639nt_int(F,A3)),Aa) ).

% mod_prod_eq
tff(fact_8039_prod__nonneg,axiom,
    ! [A3: set_nat,F: fun_nat_nat] :
      ( ! [X: nat] :
          ( aa_set_nat_o(member_nat(X),A3)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,X)) )
     => 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_8040_prod__nonneg,axiom,
    ! [A3: set_nat,F: fun_nat_int] :
      ( ! [X: nat] :
          ( aa_set_nat_o(member_nat(X),A3)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(F,X)) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),groups705719431365010083at_int(F,A3)) ) ).

% prod_nonneg
tff(fact_8041_prod__nonneg,axiom,
    ! [A3: set_int,F: fun_int_int] :
      ( ! [X: int] :
          ( aa_set_int_o(member_int(X),A3)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(F,X)) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),groups1705073143266064639nt_int(F,A3)) ) ).

% prod_nonneg
tff(fact_8042_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_8043_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_8044_prod__mono,axiom,
    ! [A3: set_int,F: fun_int_real,G: fun_int_real] :
      ( ! [I2: int] :
          ( aa_set_int_o(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_8045_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_8046_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_8047_prod__mono,axiom,
    ! [A3: set_int,F: fun_int_rat,G: fun_int_rat] :
      ( ! [I2: int] :
          ( aa_set_int_o(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_8048_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_8049_prod__mono,axiom,
    ! [A3: set_int,F: fun_int_nat,G: fun_int_nat] :
      ( ! [I2: int] :
          ( aa_set_int_o(member_int(I2),A3)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F,I2))
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,I2)),aa_int_nat(G,I2)) ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups1707563613775114915nt_nat(F,A3)),groups1707563613775114915nt_nat(G,A3)) ) ).

% prod_mono
tff(fact_8050_prod__mono,axiom,
    ! [A3: set_real,F: fun_real_int,G: fun_real_int] :
      ( ! [I2: real] :
          ( member_real(I2,A3)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_real_int(F,I2))
            & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(F,I2)),aa_real_int(G,I2)) ) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups4694064378042380927al_int(F,A3)),groups4694064378042380927al_int(G,A3)) ) ).

% prod_mono
tff(fact_8051_prod__mono,axiom,
    ! [A3: set_nat,F: fun_nat_nat,G: fun_nat_nat] :
      ( ! [I2: nat] :
          ( aa_set_nat_o(member_nat(I2),A3)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,I2))
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,I2)),aa_nat_nat(G,I2)) ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(groups708209901874060359at_nat(F),A3)),aa_set_nat_nat(groups708209901874060359at_nat(G),A3)) ) ).

% prod_mono
tff(fact_8052_prod__pos,axiom,
    ! [A3: set_nat,F: fun_nat_nat] :
      ( ! [X: nat] :
          ( aa_set_nat_o(member_nat(X),A3)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(F,X)) )
     => 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_8053_prod__pos,axiom,
    ! [A3: set_nat,F: fun_nat_int] :
      ( ! [X: nat] :
          ( aa_set_nat_o(member_nat(X),A3)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(F,X)) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),groups705719431365010083at_int(F,A3)) ) ).

% prod_pos
tff(fact_8054_prod__pos,axiom,
    ! [A3: set_int,F: fun_int_int] :
      ( ! [X: int] :
          ( aa_set_int_o(member_int(X),A3)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(F,X)) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),groups1705073143266064639nt_int(F,A3)) ) ).

% prod_pos
tff(fact_8055_prod__ge__1,axiom,
    ! [A3: set_real,F: fun_real_real] :
      ( ! [X: real] :
          ( member_real(X,A3)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(F,X)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),groups1681761925125756287l_real(F,A3)) ) ).

% prod_ge_1
tff(fact_8056_prod__ge__1,axiom,
    ! [A3: set_nat,F: fun_nat_real] :
      ( ! [X: nat] :
          ( aa_set_nat_o(member_nat(X),A3)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_nat_real(F,X)) )
     => 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_8057_prod__ge__1,axiom,
    ! [A3: set_int,F: fun_int_real] :
      ( ! [X: int] :
          ( aa_set_int_o(member_int(X),A3)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_int_real(F,X)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),groups2316167850115554303t_real(F,A3)) ) ).

% prod_ge_1
tff(fact_8058_prod__ge__1,axiom,
    ! [A3: set_real,F: fun_real_rat] :
      ( ! [X: real] :
          ( member_real(X,A3)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_real_rat(F,X)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),groups4061424788464935467al_rat(F,A3)) ) ).

% prod_ge_1
tff(fact_8059_prod__ge__1,axiom,
    ! [A3: set_nat,F: fun_nat_rat] :
      ( ! [X: nat] :
          ( aa_set_nat_o(member_nat(X),A3)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_nat_rat(F,X)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),groups73079841787564623at_rat(F,A3)) ) ).

% prod_ge_1
tff(fact_8060_prod__ge__1,axiom,
    ! [A3: set_int,F: fun_int_rat] :
      ( ! [X: int] :
          ( aa_set_int_o(member_int(X),A3)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_int_rat(F,X)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),groups1072433553688619179nt_rat(F,A3)) ) ).

% prod_ge_1
tff(fact_8061_prod__ge__1,axiom,
    ! [A3: set_real,F: fun_real_nat] :
      ( ! [X: real] :
          ( member_real(X,A3)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),aa_real_nat(F,X)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),groups4696554848551431203al_nat(F,A3)) ) ).

% prod_ge_1
tff(fact_8062_prod__ge__1,axiom,
    ! [A3: set_int,F: fun_int_nat] :
      ( ! [X: int] :
          ( aa_set_int_o(member_int(X),A3)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),aa_int_nat(F,X)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),groups1707563613775114915nt_nat(F,A3)) ) ).

% prod_ge_1
tff(fact_8063_prod__ge__1,axiom,
    ! [A3: set_real,F: fun_real_int] :
      ( ! [X: real] :
          ( member_real(X,A3)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_real_int(F,X)) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),groups4694064378042380927al_int(F,A3)) ) ).

% prod_ge_1
tff(fact_8064_prod__ge__1,axiom,
    ! [A3: set_nat,F: fun_nat_nat] :
      ( ! [X: nat] :
          ( aa_set_nat_o(member_nat(X),A3)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),aa_nat_nat(F,X)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),aa_set_nat_nat(groups708209901874060359at_nat(F),A3)) ) ).

% prod_ge_1
tff(fact_8065_prod__zero,axiom,
    ! [A3: set_nat,F: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( ? [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),A3)
            & ( aa_nat_real(F,X3) = zero_zero_real ) )
       => ( aa_set_nat_real(groups129246275422532515t_real(F),A3) = zero_zero_real ) ) ) ).

% prod_zero
tff(fact_8066_prod__zero,axiom,
    ! [A3: set_int,F: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( ? [X3: int] :
            ( aa_set_int_o(member_int(X3),A3)
            & ( aa_int_real(F,X3) = zero_zero_real ) )
       => ( groups2316167850115554303t_real(F,A3) = zero_zero_real ) ) ) ).

% prod_zero
tff(fact_8067_prod__zero,axiom,
    ! [A3: set_complex,F: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A3)
     => ( ? [X3: complex] :
            ( aa_set_complex_o(member_complex(X3),A3)
            & ( aa_complex_real(F,X3) = zero_zero_real ) )
       => ( groups766887009212190081x_real(F,A3) = zero_zero_real ) ) ) ).

% prod_zero
tff(fact_8068_prod__zero,axiom,
    ! [A3: set_nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( ? [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),A3)
            & ( aa_nat_rat(F,X3) = zero_zero_rat ) )
       => ( groups73079841787564623at_rat(F,A3) = zero_zero_rat ) ) ) ).

% prod_zero
tff(fact_8069_prod__zero,axiom,
    ! [A3: set_int,F: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( ? [X3: int] :
            ( aa_set_int_o(member_int(X3),A3)
            & ( aa_int_rat(F,X3) = zero_zero_rat ) )
       => ( groups1072433553688619179nt_rat(F,A3) = zero_zero_rat ) ) ) ).

% prod_zero
tff(fact_8070_prod__zero,axiom,
    ! [A3: set_complex,F: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A3)
     => ( ? [X3: complex] :
            ( aa_set_complex_o(member_complex(X3),A3)
            & ( aa_complex_rat(F,X3) = zero_zero_rat ) )
       => ( groups225925009352817453ex_rat(F,A3) = zero_zero_rat ) ) ) ).

% prod_zero
tff(fact_8071_prod__zero,axiom,
    ! [A3: set_int,F: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( ? [X3: int] :
            ( aa_set_int_o(member_int(X3),A3)
            & ( aa_int_nat(F,X3) = zero_zero_nat ) )
       => ( groups1707563613775114915nt_nat(F,A3) = zero_zero_nat ) ) ) ).

% prod_zero
tff(fact_8072_prod__zero,axiom,
    ! [A3: set_complex,F: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A3)
     => ( ? [X3: complex] :
            ( aa_set_complex_o(member_complex(X3),A3)
            & ( aa_complex_nat(F,X3) = zero_zero_nat ) )
       => ( groups861055069439313189ex_nat(F,A3) = zero_zero_nat ) ) ) ).

% prod_zero
tff(fact_8073_prod__zero,axiom,
    ! [A3: set_complex,F: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A3)
     => ( ? [X3: complex] :
            ( aa_set_complex_o(member_complex(X3),A3)
            & ( aa_complex_int(F,X3) = zero_zero_int ) )
       => ( groups858564598930262913ex_int(F,A3) = zero_zero_int ) ) ) ).

% prod_zero
tff(fact_8074_prod__zero,axiom,
    ! [A3: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( ? [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),A3)
            & ( aa_nat_nat(F,X3) = zero_zero_nat ) )
       => ( aa_set_nat_nat(groups708209901874060359at_nat(F),A3) = zero_zero_nat ) ) ) ).

% prod_zero
tff(fact_8075_complex__mod__minus__le__complex__mod,axiom,
    ! [Xb: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_complex_real(real_V1022390504157884413omplex,Xb))),aa_complex_real(real_V1022390504157884413omplex,Xb)) ).

% complex_mod_minus_le_complex_mod
tff(fact_8076_complex__mod__triangle__ineq2,axiom,
    ! [Ba: complex,Aa: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(Ba),Aa))),aa_complex_real(real_V1022390504157884413omplex,Ba))),aa_complex_real(real_V1022390504157884413omplex,Aa)) ).

% complex_mod_triangle_ineq2
tff(fact_8077_prod_Ointer__filter,axiom,
    ! [A3: set_real,G: fun_real_complex,P: fun_real_o] :
      ( finite_finite_real(A3)
     => ( groups713298508707869441omplex(G,collect_real(aa_fun5868358341814070780real_o(aTP_Lamm_bl(A3),P))) = groups713298508707869441omplex(aa_fun1098856492951973088omplex(aTP_Lamm_ni(G),P),A3) ) ) ).

% prod.inter_filter
tff(fact_8078_prod_Ointer__filter,axiom,
    ! [A3: set_nat,G: fun_nat_complex,P: fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_nat_complex(groups6464643781859351333omplex(G),collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_bo(A3),P))) = aa_set_nat_complex(groups6464643781859351333omplex(aa_fun1400689748890728928omplex(aTP_Lamm_nj(G),P)),A3) ) ) ).

% prod.inter_filter
tff(fact_8079_prod_Ointer__filter,axiom,
    ! [A3: set_int,G: fun_int_complex,P: fun_int_o] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( groups7440179247065528705omplex(G,collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_bp(A3),P))) = groups7440179247065528705omplex(aa_fun6439487526662521568omplex(aTP_Lamm_nk(G),P),A3) ) ) ).

% prod.inter_filter
tff(fact_8080_prod_Ointer__filter,axiom,
    ! [A3: set_complex,G: fun_complex_complex,P: fun_complex_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A3)
     => ( groups3708469109370488835omplex(G,collect_complex(aa_fun2514317545224092796plex_o(aTP_Lamm_hr(A3),P))) = groups3708469109370488835omplex(aa_fun3681620399087435360omplex(aTP_Lamm_nl(G),P),A3) ) ) ).

% prod.inter_filter
tff(fact_8081_prod_Ointer__filter,axiom,
    ! [A3: set_real,G: fun_real_real,P: fun_real_o] :
      ( finite_finite_real(A3)
     => ( groups1681761925125756287l_real(G,collect_real(aa_fun5868358341814070780real_o(aTP_Lamm_bl(A3),P))) = groups1681761925125756287l_real(aa_fun4448911596306054750l_real(aTP_Lamm_nm(G),P),A3) ) ) ).

% prod.inter_filter
tff(fact_8082_prod_Ointer__filter,axiom,
    ! [A3: set_nat,G: fun_nat_real,P: fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_set_nat_real(groups129246275422532515t_real(G),collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_bo(A3),P))) = aa_set_nat_real(groups129246275422532515t_real(aa_fun5510454223943551838t_real(aTP_Lamm_nn(G),P)),A3) ) ) ).

% prod.inter_filter
tff(fact_8083_prod_Ointer__filter,axiom,
    ! [A3: set_int,G: fun_int_real,P: fun_int_o] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( groups2316167850115554303t_real(G,collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_bp(A3),P))) = groups2316167850115554303t_real(aa_fun7506669883404309598t_real(aTP_Lamm_no(G),P),A3) ) ) ).

% prod.inter_filter
tff(fact_8084_prod_Ointer__filter,axiom,
    ! [A3: set_complex,G: fun_complex_real,P: fun_complex_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A3)
     => ( groups766887009212190081x_real(G,collect_complex(aa_fun2514317545224092796plex_o(aTP_Lamm_hr(A3),P))) = groups766887009212190081x_real(aa_fun8261057609414086622x_real(aTP_Lamm_np(G),P),A3) ) ) ).

% prod.inter_filter
tff(fact_8085_prod_Ointer__filter,axiom,
    ! [A3: set_real,G: fun_real_rat,P: fun_real_o] :
      ( finite_finite_real(A3)
     => ( groups4061424788464935467al_rat(G,collect_real(aa_fun5868358341814070780real_o(aTP_Lamm_bl(A3),P))) = groups4061424788464935467al_rat(aa_fun2004904883527618314al_rat(aTP_Lamm_nq(G),P),A3) ) ) ).

% prod.inter_filter
tff(fact_8086_prod_Ointer__filter,axiom,
    ! [A3: set_nat,G: fun_nat_rat,P: fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( groups73079841787564623at_rat(G,collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_bo(A3),P))) = groups73079841787564623at_rat(aa_fun696632391729880074at_rat(aTP_Lamm_nr(G),P),A3) ) ) ).

% prod.inter_filter
tff(fact_8087_prod_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: fun_nat_nat,M: nat,Na: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),set_or1269000886237332187st_nat(aa_nat_nat(suc,M),aa_nat_nat(suc,Na))) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_jf(G)),set_or1269000886237332187st_nat(M,Na)) ).

% prod.shift_bounds_cl_Suc_ivl
tff(fact_8088_prod_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: fun_nat_int,M: nat,Na: nat] : groups705719431365010083at_int(G,set_or1269000886237332187st_nat(aa_nat_nat(suc,M),aa_nat_nat(suc,Na))) = groups705719431365010083at_int(aTP_Lamm_jx(G),set_or1269000886237332187st_nat(M,Na)) ).

% prod.shift_bounds_cl_Suc_ivl
tff(fact_8089_prod_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: fun_nat_nat,M: nat,K: nat,Na: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),K))) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_jh(G),K)),set_or1269000886237332187st_nat(M,Na)) ).

% prod.shift_bounds_cl_nat_ivl
tff(fact_8090_prod_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: fun_nat_int,M: nat,K: nat,Na: nat] : groups705719431365010083at_int(G,set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),K))) = groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_ns(G),K),set_or1269000886237332187st_nat(M,Na)) ).

% prod.shift_bounds_cl_nat_ivl
tff(fact_8091_prod__le__1,axiom,
    ! [A3: set_real,F: fun_real_real] :
      ( ! [X: real] :
          ( member_real(X,A3)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,X))
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,X)),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_8092_prod__le__1,axiom,
    ! [A3: set_nat,F: fun_nat_real] :
      ( ! [X: nat] :
          ( aa_set_nat_o(member_nat(X),A3)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,X))
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,X)),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_8093_prod__le__1,axiom,
    ! [A3: set_int,F: fun_int_real] :
      ( ! [X: int] :
          ( aa_set_int_o(member_int(X),A3)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(F,X))
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(F,X)),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_8094_prod__le__1,axiom,
    ! [A3: set_real,F: fun_real_rat] :
      ( ! [X: real] :
          ( member_real(X,A3)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,X))
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_real_rat(F,X)),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_8095_prod__le__1,axiom,
    ! [A3: set_nat,F: fun_nat_rat] :
      ( ! [X: nat] :
          ( aa_set_nat_o(member_nat(X),A3)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,X))
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X)),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_8096_prod__le__1,axiom,
    ! [A3: set_int,F: fun_int_rat] :
      ( ! [X: int] :
          ( aa_set_int_o(member_int(X),A3)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,X))
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,X)),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_8097_prod__le__1,axiom,
    ! [A3: set_real,F: fun_real_nat] :
      ( ! [X: real] :
          ( member_real(X,A3)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_real_nat(F,X))
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_real_nat(F,X)),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_8098_prod__le__1,axiom,
    ! [A3: set_int,F: fun_int_nat] :
      ( ! [X: int] :
          ( aa_set_int_o(member_int(X),A3)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_int_nat(F,X))
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,X)),one_one_nat) ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,groups1707563613775114915nt_nat(F,A3)),one_one_nat) ) ).

% prod_le_1
tff(fact_8099_prod__le__1,axiom,
    ! [A3: set_real,F: fun_real_int] :
      ( ! [X: real] :
          ( member_real(X,A3)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_real_int(F,X))
            & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(F,X)),one_one_int) ) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups4694064378042380927al_int(F,A3)),one_one_int) ) ).

% prod_le_1
tff(fact_8100_prod__le__1,axiom,
    ! [A3: set_nat,F: fun_nat_nat] :
      ( ! [X: nat] :
          ( aa_set_nat_o(member_nat(X),A3)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,X))
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,X)),one_one_nat) ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(groups708209901874060359at_nat(F),A3)),one_one_nat) ) ).

% prod_le_1
tff(fact_8101_prod_Orelated,axiom,
    ! [R: fun_co1148497455491383662plex_o,S2: set_nat,H: fun_nat_complex,G: fun_nat_complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(R,one_one_complex),one_one_complex)
     => ( ! [X1: complex,Y1: complex,X23: complex,Y23: complex] :
            ( ( aa_complex_o(aa_com5808801942499186519plex_o(R,X1),X23)
              & aa_complex_o(aa_com5808801942499186519plex_o(R,Y1),Y23) )
           => aa_complex_o(aa_com5808801942499186519plex_o(R,aa_complex_complex(times_times_complex(X1),Y1)),aa_complex_complex(times_times_complex(X23),Y23)) )
       => ( aa_set_nat_o(finite_finite_nat,S2)
         => ( ! [X: nat] :
                ( aa_set_nat_o(member_nat(X),S2)
               => aa_complex_o(aa_com5808801942499186519plex_o(R,aa_nat_complex(H,X)),aa_nat_complex(G,X)) )
           => aa_complex_o(aa_com5808801942499186519plex_o(R,aa_set_nat_complex(groups6464643781859351333omplex(H),S2)),aa_set_nat_complex(groups6464643781859351333omplex(G),S2)) ) ) ) ) ).

% prod.related
tff(fact_8102_prod_Orelated,axiom,
    ! [R: fun_co1148497455491383662plex_o,S2: set_int,H: fun_int_complex,G: fun_int_complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(R,one_one_complex),one_one_complex)
     => ( ! [X1: complex,Y1: complex,X23: complex,Y23: complex] :
            ( ( aa_complex_o(aa_com5808801942499186519plex_o(R,X1),X23)
              & aa_complex_o(aa_com5808801942499186519plex_o(R,Y1),Y23) )
           => aa_complex_o(aa_com5808801942499186519plex_o(R,aa_complex_complex(times_times_complex(X1),Y1)),aa_complex_complex(times_times_complex(X23),Y23)) )
       => ( aa_set_int_o(finite_finite_int,S2)
         => ( ! [X: int] :
                ( aa_set_int_o(member_int(X),S2)
               => aa_complex_o(aa_com5808801942499186519plex_o(R,aa_int_complex(H,X)),aa_int_complex(G,X)) )
           => aa_complex_o(aa_com5808801942499186519plex_o(R,groups7440179247065528705omplex(H,S2)),groups7440179247065528705omplex(G,S2)) ) ) ) ) ).

% prod.related
tff(fact_8103_prod_Orelated,axiom,
    ! [R: fun_co1148497455491383662plex_o,S2: set_complex,H: fun_complex_complex,G: fun_complex_complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(R,one_one_complex),one_one_complex)
     => ( ! [X1: complex,Y1: complex,X23: complex,Y23: complex] :
            ( ( aa_complex_o(aa_com5808801942499186519plex_o(R,X1),X23)
              & aa_complex_o(aa_com5808801942499186519plex_o(R,Y1),Y23) )
           => aa_complex_o(aa_com5808801942499186519plex_o(R,aa_complex_complex(times_times_complex(X1),Y1)),aa_complex_complex(times_times_complex(X23),Y23)) )
       => ( aa_set_complex_o(finite3207457112153483333omplex,S2)
         => ( ! [X: complex] :
                ( aa_set_complex_o(member_complex(X),S2)
               => aa_complex_o(aa_com5808801942499186519plex_o(R,aa_complex_complex(H,X)),aa_complex_complex(G,X)) )
           => aa_complex_o(aa_com5808801942499186519plex_o(R,groups3708469109370488835omplex(H,S2)),groups3708469109370488835omplex(G,S2)) ) ) ) ) ).

% prod.related
tff(fact_8104_prod_Orelated,axiom,
    ! [R: fun_real_fun_real_o,S2: set_nat,H: fun_nat_real,G: fun_nat_real] :
      ( aa_real_o(aa_real_fun_real_o(R,one_one_real),one_one_real)
     => ( ! [X1: real,Y1: real,X23: real,Y23: real] :
            ( ( aa_real_o(aa_real_fun_real_o(R,X1),X23)
              & aa_real_o(aa_real_fun_real_o(R,Y1),Y23) )
           => aa_real_o(aa_real_fun_real_o(R,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X1),Y1)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X23),Y23)) )
       => ( aa_set_nat_o(finite_finite_nat,S2)
         => ( ! [X: nat] :
                ( aa_set_nat_o(member_nat(X),S2)
               => aa_real_o(aa_real_fun_real_o(R,aa_nat_real(H,X)),aa_nat_real(G,X)) )
           => aa_real_o(aa_real_fun_real_o(R,aa_set_nat_real(groups129246275422532515t_real(H),S2)),aa_set_nat_real(groups129246275422532515t_real(G),S2)) ) ) ) ) ).

% prod.related
tff(fact_8105_prod_Orelated,axiom,
    ! [R: fun_real_fun_real_o,S2: set_int,H: fun_int_real,G: fun_int_real] :
      ( aa_real_o(aa_real_fun_real_o(R,one_one_real),one_one_real)
     => ( ! [X1: real,Y1: real,X23: real,Y23: real] :
            ( ( aa_real_o(aa_real_fun_real_o(R,X1),X23)
              & aa_real_o(aa_real_fun_real_o(R,Y1),Y23) )
           => aa_real_o(aa_real_fun_real_o(R,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X1),Y1)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X23),Y23)) )
       => ( aa_set_int_o(finite_finite_int,S2)
         => ( ! [X: int] :
                ( aa_set_int_o(member_int(X),S2)
               => aa_real_o(aa_real_fun_real_o(R,aa_int_real(H,X)),aa_int_real(G,X)) )
           => aa_real_o(aa_real_fun_real_o(R,groups2316167850115554303t_real(H,S2)),groups2316167850115554303t_real(G,S2)) ) ) ) ) ).

% prod.related
tff(fact_8106_prod_Orelated,axiom,
    ! [R: fun_real_fun_real_o,S2: set_complex,H: fun_complex_real,G: fun_complex_real] :
      ( aa_real_o(aa_real_fun_real_o(R,one_one_real),one_one_real)
     => ( ! [X1: real,Y1: real,X23: real,Y23: real] :
            ( ( aa_real_o(aa_real_fun_real_o(R,X1),X23)
              & aa_real_o(aa_real_fun_real_o(R,Y1),Y23) )
           => aa_real_o(aa_real_fun_real_o(R,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X1),Y1)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X23),Y23)) )
       => ( aa_set_complex_o(finite3207457112153483333omplex,S2)
         => ( ! [X: complex] :
                ( aa_set_complex_o(member_complex(X),S2)
               => aa_real_o(aa_real_fun_real_o(R,aa_complex_real(H,X)),aa_complex_real(G,X)) )
           => aa_real_o(aa_real_fun_real_o(R,groups766887009212190081x_real(H,S2)),groups766887009212190081x_real(G,S2)) ) ) ) ) ).

% prod.related
tff(fact_8107_prod_Orelated,axiom,
    ! [R: fun_rat_fun_rat_o,S2: set_nat,H: fun_nat_rat,G: fun_nat_rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(R,one_one_rat),one_one_rat)
     => ( ! [X1: rat,Y1: rat,X23: rat,Y23: rat] :
            ( ( aa_rat_o(aa_rat_fun_rat_o(R,X1),X23)
              & aa_rat_o(aa_rat_fun_rat_o(R,Y1),Y23) )
           => aa_rat_o(aa_rat_fun_rat_o(R,aa_rat_rat(times_times_rat(X1),Y1)),aa_rat_rat(times_times_rat(X23),Y23)) )
       => ( aa_set_nat_o(finite_finite_nat,S2)
         => ( ! [X: nat] :
                ( aa_set_nat_o(member_nat(X),S2)
               => aa_rat_o(aa_rat_fun_rat_o(R,aa_nat_rat(H,X)),aa_nat_rat(G,X)) )
           => aa_rat_o(aa_rat_fun_rat_o(R,groups73079841787564623at_rat(H,S2)),groups73079841787564623at_rat(G,S2)) ) ) ) ) ).

% prod.related
tff(fact_8108_prod_Orelated,axiom,
    ! [R: fun_rat_fun_rat_o,S2: set_int,H: fun_int_rat,G: fun_int_rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(R,one_one_rat),one_one_rat)
     => ( ! [X1: rat,Y1: rat,X23: rat,Y23: rat] :
            ( ( aa_rat_o(aa_rat_fun_rat_o(R,X1),X23)
              & aa_rat_o(aa_rat_fun_rat_o(R,Y1),Y23) )
           => aa_rat_o(aa_rat_fun_rat_o(R,aa_rat_rat(times_times_rat(X1),Y1)),aa_rat_rat(times_times_rat(X23),Y23)) )
       => ( aa_set_int_o(finite_finite_int,S2)
         => ( ! [X: int] :
                ( aa_set_int_o(member_int(X),S2)
               => aa_rat_o(aa_rat_fun_rat_o(R,aa_int_rat(H,X)),aa_int_rat(G,X)) )
           => aa_rat_o(aa_rat_fun_rat_o(R,groups1072433553688619179nt_rat(H,S2)),groups1072433553688619179nt_rat(G,S2)) ) ) ) ) ).

% prod.related
tff(fact_8109_prod_Orelated,axiom,
    ! [R: fun_rat_fun_rat_o,S2: set_complex,H: fun_complex_rat,G: fun_complex_rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(R,one_one_rat),one_one_rat)
     => ( ! [X1: rat,Y1: rat,X23: rat,Y23: rat] :
            ( ( aa_rat_o(aa_rat_fun_rat_o(R,X1),X23)
              & aa_rat_o(aa_rat_fun_rat_o(R,Y1),Y23) )
           => aa_rat_o(aa_rat_fun_rat_o(R,aa_rat_rat(times_times_rat(X1),Y1)),aa_rat_rat(times_times_rat(X23),Y23)) )
       => ( aa_set_complex_o(finite3207457112153483333omplex,S2)
         => ( ! [X: complex] :
                ( aa_set_complex_o(member_complex(X),S2)
               => aa_rat_o(aa_rat_fun_rat_o(R,aa_complex_rat(H,X)),aa_complex_rat(G,X)) )
           => aa_rat_o(aa_rat_fun_rat_o(R,groups225925009352817453ex_rat(H,S2)),groups225925009352817453ex_rat(G,S2)) ) ) ) ) ).

% prod.related
tff(fact_8110_prod_Orelated,axiom,
    ! [R: fun_nat_fun_nat_o,S2: set_int,H: fun_int_nat,G: fun_int_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(R,one_one_nat),one_one_nat)
     => ( ! [X1: nat,Y1: nat,X23: nat,Y23: nat] :
            ( ( aa_nat_o(aa_nat_fun_nat_o(R,X1),X23)
              & aa_nat_o(aa_nat_fun_nat_o(R,Y1),Y23) )
           => aa_nat_o(aa_nat_fun_nat_o(R,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X1),Y1)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X23),Y23)) )
       => ( aa_set_int_o(finite_finite_int,S2)
         => ( ! [X: int] :
                ( aa_set_int_o(member_int(X),S2)
               => aa_nat_o(aa_nat_fun_nat_o(R,aa_int_nat(H,X)),aa_int_nat(G,X)) )
           => aa_nat_o(aa_nat_fun_nat_o(R,groups1707563613775114915nt_nat(H,S2)),groups1707563613775114915nt_nat(G,S2)) ) ) ) ) ).

% prod.related
tff(fact_8111_prod__dvd__prod__subset,axiom,
    ! [B4: set_complex,A3: set_complex,F: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A3),B4)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,groups861055069439313189ex_nat(F,A3)),groups861055069439313189ex_nat(F,B4)) ) ) ).

% prod_dvd_prod_subset
tff(fact_8112_prod__dvd__prod__subset,axiom,
    ! [B4: set_complex,A3: set_complex,F: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A3),B4)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,groups858564598930262913ex_int(F,A3)),groups858564598930262913ex_int(F,B4)) ) ) ).

% prod_dvd_prod_subset
tff(fact_8113_prod__dvd__prod__subset,axiom,
    ! [B4: set_nat,A3: set_nat,F: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,B4)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A3),B4)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,groups3455450783089532116nteger(F,A3)),groups3455450783089532116nteger(F,B4)) ) ) ).

% prod_dvd_prod_subset
tff(fact_8114_prod__dvd__prod__subset,axiom,
    ! [B4: set_complex,A3: set_complex,F: fun_co8153928105989009288nteger] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A3),B4)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,groups8682486955453173170nteger(F,A3)),groups8682486955453173170nteger(F,B4)) ) ) ).

% prod_dvd_prod_subset
tff(fact_8115_prod__dvd__prod__subset,axiom,
    ! [B4: set_int,A3: set_int,F: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,B4)
     => ( aa_set_int_o(ord_less_eq_set_int(A3),B4)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,groups1707563613775114915nt_nat(F,A3)),groups1707563613775114915nt_nat(F,B4)) ) ) ).

% prod_dvd_prod_subset
tff(fact_8116_prod__dvd__prod__subset,axiom,
    ! [B4: set_int,A3: set_int,F: fun_int_Code_integer] :
      ( aa_set_int_o(finite_finite_int,B4)
     => ( aa_set_int_o(ord_less_eq_set_int(A3),B4)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,groups3827104343326376752nteger(F,A3)),groups3827104343326376752nteger(F,B4)) ) ) ).

% prod_dvd_prod_subset
tff(fact_8117_prod__dvd__prod__subset,axiom,
    ! [B4: set_nat,A3: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,B4)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A3),B4)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_set_nat_nat(groups708209901874060359at_nat(F),A3)),aa_set_nat_nat(groups708209901874060359at_nat(F),B4)) ) ) ).

% prod_dvd_prod_subset
tff(fact_8118_prod__dvd__prod__subset,axiom,
    ! [B4: set_nat,A3: set_nat,F: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,B4)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A3),B4)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,groups705719431365010083at_int(F,A3)),groups705719431365010083at_int(F,B4)) ) ) ).

% prod_dvd_prod_subset
tff(fact_8119_prod__dvd__prod__subset,axiom,
    ! [B4: set_int,A3: set_int,F: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,B4)
     => ( aa_set_int_o(ord_less_eq_set_int(A3),B4)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,groups1705073143266064639nt_int(F,A3)),groups1705073143266064639nt_int(F,B4)) ) ) ).

% prod_dvd_prod_subset
tff(fact_8120_prod__dvd__prod__subset2,axiom,
    ! [B4: set_real,A3: set_real,F: fun_real_nat,G: fun_real_nat] :
      ( finite_finite_real(B4)
     => ( ord_less_eq_set_real(A3,B4)
       => ( ! [A2: real] :
              ( member_real(A2,A3)
             => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_real_nat(F,A2)),aa_real_nat(G,A2)) )
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,groups4696554848551431203al_nat(F,A3)),groups4696554848551431203al_nat(G,B4)) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_8121_prod__dvd__prod__subset2,axiom,
    ! [B4: set_complex,A3: set_complex,F: fun_complex_nat,G: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A3),B4)
       => ( ! [A2: complex] :
              ( aa_set_complex_o(member_complex(A2),A3)
             => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_complex_nat(F,A2)),aa_complex_nat(G,A2)) )
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,groups861055069439313189ex_nat(F,A3)),groups861055069439313189ex_nat(G,B4)) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_8122_prod__dvd__prod__subset2,axiom,
    ! [B4: set_real,A3: set_real,F: fun_real_int,G: fun_real_int] :
      ( finite_finite_real(B4)
     => ( ord_less_eq_set_real(A3,B4)
       => ( ! [A2: real] :
              ( member_real(A2,A3)
             => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_real_int(F,A2)),aa_real_int(G,A2)) )
         => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,groups4694064378042380927al_int(F,A3)),groups4694064378042380927al_int(G,B4)) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_8123_prod__dvd__prod__subset2,axiom,
    ! [B4: set_complex,A3: set_complex,F: fun_complex_int,G: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A3),B4)
       => ( ! [A2: complex] :
              ( aa_set_complex_o(member_complex(A2),A3)
             => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_complex_int(F,A2)),aa_complex_int(G,A2)) )
         => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,groups858564598930262913ex_int(F,A3)),groups858564598930262913ex_int(G,B4)) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_8124_prod__dvd__prod__subset2,axiom,
    ! [B4: set_real,A3: set_real,F: fun_re6978948838062957830nteger,G: fun_re6978948838062957830nteger] :
      ( finite_finite_real(B4)
     => ( ord_less_eq_set_real(A3,B4)
       => ( ! [A2: real] :
              ( member_real(A2,A3)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_real_Code_integer(F,A2)),aa_real_Code_integer(G,A2)) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,groups6225526099057966256nteger(F,A3)),groups6225526099057966256nteger(G,B4)) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_8125_prod__dvd__prod__subset2,axiom,
    ! [B4: set_nat,A3: set_nat,F: fun_nat_Code_integer,G: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,B4)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A3),B4)
       => ( ! [A2: nat] :
              ( aa_set_nat_o(member_nat(A2),A3)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(F,A2)),aa_nat_Code_integer(G,A2)) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,groups3455450783089532116nteger(F,A3)),groups3455450783089532116nteger(G,B4)) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_8126_prod__dvd__prod__subset2,axiom,
    ! [B4: set_complex,A3: set_complex,F: fun_co8153928105989009288nteger,G: fun_co8153928105989009288nteger] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A3),B4)
       => ( ! [A2: complex] :
              ( aa_set_complex_o(member_complex(A2),A3)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_com4217951221296830281nteger(F,A2)),aa_com4217951221296830281nteger(G,A2)) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,groups8682486955453173170nteger(F,A3)),groups8682486955453173170nteger(G,B4)) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_8127_prod__dvd__prod__subset2,axiom,
    ! [B4: set_int,A3: set_int,F: fun_int_nat,G: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,B4)
     => ( aa_set_int_o(ord_less_eq_set_int(A3),B4)
       => ( ! [A2: int] :
              ( aa_set_int_o(member_int(A2),A3)
             => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_int_nat(F,A2)),aa_int_nat(G,A2)) )
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,groups1707563613775114915nt_nat(F,A3)),groups1707563613775114915nt_nat(G,B4)) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_8128_prod__dvd__prod__subset2,axiom,
    ! [B4: set_int,A3: set_int,F: fun_int_Code_integer,G: fun_int_Code_integer] :
      ( aa_set_int_o(finite_finite_int,B4)
     => ( aa_set_int_o(ord_less_eq_set_int(A3),B4)
       => ( ! [A2: int] :
              ( aa_set_int_o(member_int(A2),A3)
             => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_int_Code_integer(F,A2)),aa_int_Code_integer(G,A2)) )
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,groups3827104343326376752nteger(F,A3)),groups3827104343326376752nteger(G,B4)) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_8129_prod__dvd__prod__subset2,axiom,
    ! [B4: set_nat,A3: set_nat,F: fun_nat_nat,G: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,B4)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A3),B4)
       => ( ! [A2: nat] :
              ( aa_set_nat_o(member_nat(A2),A3)
             => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(F,A2)),aa_nat_nat(G,A2)) )
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_set_nat_nat(groups708209901874060359at_nat(F),A3)),aa_set_nat_nat(groups708209901874060359at_nat(G),B4)) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_8130_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_real,T3: set_real,S2: set_real,I: fun_real_real,J: fun_real_real,T: set_real,G: fun_real_complex,H: fun_real_complex] :
      ( finite_finite_real(S4)
     => ( finite_finite_real(T3)
       => ( ! [A2: real] :
              ( member_real(A2,minus_minus_set_real(S2,S4))
             => ( aa_real_real(I,aa_real_real(J,A2)) = A2 ) )
         => ( ! [A2: real] :
                ( member_real(A2,minus_minus_set_real(S2,S4))
               => member_real(aa_real_real(J,A2),minus_minus_set_real(T,T3)) )
           => ( ! [B2: real] :
                  ( member_real(B2,minus_minus_set_real(T,T3))
                 => ( aa_real_real(J,aa_real_real(I,B2)) = B2 ) )
             => ( ! [B2: real] :
                    ( member_real(B2,minus_minus_set_real(T,T3))
                   => member_real(aa_real_real(I,B2),minus_minus_set_real(S2,S4)) )
               => ( ! [A2: real] :
                      ( member_real(A2,S4)
                     => ( aa_real_complex(G,A2) = one_one_complex ) )
                 => ( ! [B2: real] :
                        ( member_real(B2,T3)
                       => ( aa_real_complex(H,B2) = one_one_complex ) )
                   => ( ! [A2: real] :
                          ( member_real(A2,S2)
                         => ( aa_real_complex(H,aa_real_real(J,A2)) = aa_real_complex(G,A2) ) )
                     => ( groups713298508707869441omplex(G,S2) = groups713298508707869441omplex(H,T) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_8131_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_real,T3: set_int,S2: set_real,I: fun_int_real,J: fun_real_int,T: set_int,G: fun_real_complex,H: fun_int_complex] :
      ( finite_finite_real(S4)
     => ( aa_set_int_o(finite_finite_int,T3)
       => ( ! [A2: real] :
              ( member_real(A2,minus_minus_set_real(S2,S4))
             => ( aa_int_real(I,aa_real_int(J,A2)) = A2 ) )
         => ( ! [A2: real] :
                ( member_real(A2,minus_minus_set_real(S2,S4))
               => aa_set_int_o(member_int(aa_real_int(J,A2)),minus_minus_set_int(T,T3)) )
           => ( ! [B2: int] :
                  ( aa_set_int_o(member_int(B2),minus_minus_set_int(T,T3))
                 => ( aa_real_int(J,aa_int_real(I,B2)) = B2 ) )
             => ( ! [B2: int] :
                    ( aa_set_int_o(member_int(B2),minus_minus_set_int(T,T3))
                   => member_real(aa_int_real(I,B2),minus_minus_set_real(S2,S4)) )
               => ( ! [A2: real] :
                      ( member_real(A2,S4)
                     => ( aa_real_complex(G,A2) = one_one_complex ) )
                 => ( ! [B2: int] :
                        ( aa_set_int_o(member_int(B2),T3)
                       => ( aa_int_complex(H,B2) = one_one_complex ) )
                   => ( ! [A2: real] :
                          ( member_real(A2,S2)
                         => ( aa_int_complex(H,aa_real_int(J,A2)) = aa_real_complex(G,A2) ) )
                     => ( groups713298508707869441omplex(G,S2) = groups7440179247065528705omplex(H,T) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_8132_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_real,T3: set_complex,S2: set_real,I: fun_complex_real,J: fun_real_complex,T: set_complex,G: fun_real_complex,H: fun_complex_complex] :
      ( finite_finite_real(S4)
     => ( aa_set_complex_o(finite3207457112153483333omplex,T3)
       => ( ! [A2: real] :
              ( member_real(A2,minus_minus_set_real(S2,S4))
             => ( aa_complex_real(I,aa_real_complex(J,A2)) = A2 ) )
         => ( ! [A2: real] :
                ( member_real(A2,minus_minus_set_real(S2,S4))
               => aa_set_complex_o(member_complex(aa_real_complex(J,A2)),minus_811609699411566653omplex(T,T3)) )
           => ( ! [B2: complex] :
                  ( aa_set_complex_o(member_complex(B2),minus_811609699411566653omplex(T,T3))
                 => ( aa_real_complex(J,aa_complex_real(I,B2)) = B2 ) )
             => ( ! [B2: complex] :
                    ( aa_set_complex_o(member_complex(B2),minus_811609699411566653omplex(T,T3))
                   => member_real(aa_complex_real(I,B2),minus_minus_set_real(S2,S4)) )
               => ( ! [A2: real] :
                      ( member_real(A2,S4)
                     => ( aa_real_complex(G,A2) = one_one_complex ) )
                 => ( ! [B2: complex] :
                        ( aa_set_complex_o(member_complex(B2),T3)
                       => ( aa_complex_complex(H,B2) = one_one_complex ) )
                   => ( ! [A2: real] :
                          ( member_real(A2,S2)
                         => ( aa_complex_complex(H,aa_real_complex(J,A2)) = aa_real_complex(G,A2) ) )
                     => ( groups713298508707869441omplex(G,S2) = groups3708469109370488835omplex(H,T) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_8133_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_int,T3: set_real,S2: set_int,I: fun_real_int,J: fun_int_real,T: set_real,G: fun_int_complex,H: fun_real_complex] :
      ( aa_set_int_o(finite_finite_int,S4)
     => ( finite_finite_real(T3)
       => ( ! [A2: int] :
              ( aa_set_int_o(member_int(A2),minus_minus_set_int(S2,S4))
             => ( aa_real_int(I,aa_int_real(J,A2)) = A2 ) )
         => ( ! [A2: int] :
                ( aa_set_int_o(member_int(A2),minus_minus_set_int(S2,S4))
               => member_real(aa_int_real(J,A2),minus_minus_set_real(T,T3)) )
           => ( ! [B2: real] :
                  ( member_real(B2,minus_minus_set_real(T,T3))
                 => ( aa_int_real(J,aa_real_int(I,B2)) = B2 ) )
             => ( ! [B2: real] :
                    ( member_real(B2,minus_minus_set_real(T,T3))
                   => aa_set_int_o(member_int(aa_real_int(I,B2)),minus_minus_set_int(S2,S4)) )
               => ( ! [A2: int] :
                      ( aa_set_int_o(member_int(A2),S4)
                     => ( aa_int_complex(G,A2) = one_one_complex ) )
                 => ( ! [B2: real] :
                        ( member_real(B2,T3)
                       => ( aa_real_complex(H,B2) = one_one_complex ) )
                   => ( ! [A2: int] :
                          ( aa_set_int_o(member_int(A2),S2)
                         => ( aa_real_complex(H,aa_int_real(J,A2)) = aa_int_complex(G,A2) ) )
                     => ( groups7440179247065528705omplex(G,S2) = groups713298508707869441omplex(H,T) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_8134_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_int,T3: set_int,S2: set_int,I: fun_int_int,J: fun_int_int,T: set_int,G: fun_int_complex,H: fun_int_complex] :
      ( aa_set_int_o(finite_finite_int,S4)
     => ( aa_set_int_o(finite_finite_int,T3)
       => ( ! [A2: int] :
              ( aa_set_int_o(member_int(A2),minus_minus_set_int(S2,S4))
             => ( aa_int_int(I,aa_int_int(J,A2)) = A2 ) )
         => ( ! [A2: int] :
                ( aa_set_int_o(member_int(A2),minus_minus_set_int(S2,S4))
               => aa_set_int_o(member_int(aa_int_int(J,A2)),minus_minus_set_int(T,T3)) )
           => ( ! [B2: int] :
                  ( aa_set_int_o(member_int(B2),minus_minus_set_int(T,T3))
                 => ( aa_int_int(J,aa_int_int(I,B2)) = B2 ) )
             => ( ! [B2: int] :
                    ( aa_set_int_o(member_int(B2),minus_minus_set_int(T,T3))
                   => aa_set_int_o(member_int(aa_int_int(I,B2)),minus_minus_set_int(S2,S4)) )
               => ( ! [A2: int] :
                      ( aa_set_int_o(member_int(A2),S4)
                     => ( aa_int_complex(G,A2) = one_one_complex ) )
                 => ( ! [B2: int] :
                        ( aa_set_int_o(member_int(B2),T3)
                       => ( aa_int_complex(H,B2) = one_one_complex ) )
                   => ( ! [A2: int] :
                          ( aa_set_int_o(member_int(A2),S2)
                         => ( aa_int_complex(H,aa_int_int(J,A2)) = aa_int_complex(G,A2) ) )
                     => ( groups7440179247065528705omplex(G,S2) = groups7440179247065528705omplex(H,T) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_8135_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_int,T3: set_complex,S2: set_int,I: fun_complex_int,J: fun_int_complex,T: set_complex,G: fun_int_complex,H: fun_complex_complex] :
      ( aa_set_int_o(finite_finite_int,S4)
     => ( aa_set_complex_o(finite3207457112153483333omplex,T3)
       => ( ! [A2: int] :
              ( aa_set_int_o(member_int(A2),minus_minus_set_int(S2,S4))
             => ( aa_complex_int(I,aa_int_complex(J,A2)) = A2 ) )
         => ( ! [A2: int] :
                ( aa_set_int_o(member_int(A2),minus_minus_set_int(S2,S4))
               => aa_set_complex_o(member_complex(aa_int_complex(J,A2)),minus_811609699411566653omplex(T,T3)) )
           => ( ! [B2: complex] :
                  ( aa_set_complex_o(member_complex(B2),minus_811609699411566653omplex(T,T3))
                 => ( aa_int_complex(J,aa_complex_int(I,B2)) = B2 ) )
             => ( ! [B2: complex] :
                    ( aa_set_complex_o(member_complex(B2),minus_811609699411566653omplex(T,T3))
                   => aa_set_int_o(member_int(aa_complex_int(I,B2)),minus_minus_set_int(S2,S4)) )
               => ( ! [A2: int] :
                      ( aa_set_int_o(member_int(A2),S4)
                     => ( aa_int_complex(G,A2) = one_one_complex ) )
                 => ( ! [B2: complex] :
                        ( aa_set_complex_o(member_complex(B2),T3)
                       => ( aa_complex_complex(H,B2) = one_one_complex ) )
                   => ( ! [A2: int] :
                          ( aa_set_int_o(member_int(A2),S2)
                         => ( aa_complex_complex(H,aa_int_complex(J,A2)) = aa_int_complex(G,A2) ) )
                     => ( groups7440179247065528705omplex(G,S2) = groups3708469109370488835omplex(H,T) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_8136_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_complex,T3: set_real,S2: set_complex,I: fun_real_complex,J: fun_complex_real,T: set_real,G: fun_complex_complex,H: fun_real_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S4)
     => ( finite_finite_real(T3)
       => ( ! [A2: complex] :
              ( aa_set_complex_o(member_complex(A2),minus_811609699411566653omplex(S2,S4))
             => ( aa_real_complex(I,aa_complex_real(J,A2)) = A2 ) )
         => ( ! [A2: complex] :
                ( aa_set_complex_o(member_complex(A2),minus_811609699411566653omplex(S2,S4))
               => member_real(aa_complex_real(J,A2),minus_minus_set_real(T,T3)) )
           => ( ! [B2: real] :
                  ( member_real(B2,minus_minus_set_real(T,T3))
                 => ( aa_complex_real(J,aa_real_complex(I,B2)) = B2 ) )
             => ( ! [B2: real] :
                    ( member_real(B2,minus_minus_set_real(T,T3))
                   => aa_set_complex_o(member_complex(aa_real_complex(I,B2)),minus_811609699411566653omplex(S2,S4)) )
               => ( ! [A2: complex] :
                      ( aa_set_complex_o(member_complex(A2),S4)
                     => ( aa_complex_complex(G,A2) = one_one_complex ) )
                 => ( ! [B2: real] :
                        ( member_real(B2,T3)
                       => ( aa_real_complex(H,B2) = one_one_complex ) )
                   => ( ! [A2: complex] :
                          ( aa_set_complex_o(member_complex(A2),S2)
                         => ( aa_real_complex(H,aa_complex_real(J,A2)) = aa_complex_complex(G,A2) ) )
                     => ( groups3708469109370488835omplex(G,S2) = groups713298508707869441omplex(H,T) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_8137_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_complex,T3: set_int,S2: set_complex,I: fun_int_complex,J: fun_complex_int,T: set_int,G: fun_complex_complex,H: fun_int_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S4)
     => ( aa_set_int_o(finite_finite_int,T3)
       => ( ! [A2: complex] :
              ( aa_set_complex_o(member_complex(A2),minus_811609699411566653omplex(S2,S4))
             => ( aa_int_complex(I,aa_complex_int(J,A2)) = A2 ) )
         => ( ! [A2: complex] :
                ( aa_set_complex_o(member_complex(A2),minus_811609699411566653omplex(S2,S4))
               => aa_set_int_o(member_int(aa_complex_int(J,A2)),minus_minus_set_int(T,T3)) )
           => ( ! [B2: int] :
                  ( aa_set_int_o(member_int(B2),minus_minus_set_int(T,T3))
                 => ( aa_complex_int(J,aa_int_complex(I,B2)) = B2 ) )
             => ( ! [B2: int] :
                    ( aa_set_int_o(member_int(B2),minus_minus_set_int(T,T3))
                   => aa_set_complex_o(member_complex(aa_int_complex(I,B2)),minus_811609699411566653omplex(S2,S4)) )
               => ( ! [A2: complex] :
                      ( aa_set_complex_o(member_complex(A2),S4)
                     => ( aa_complex_complex(G,A2) = one_one_complex ) )
                 => ( ! [B2: int] :
                        ( aa_set_int_o(member_int(B2),T3)
                       => ( aa_int_complex(H,B2) = one_one_complex ) )
                   => ( ! [A2: complex] :
                          ( aa_set_complex_o(member_complex(A2),S2)
                         => ( aa_int_complex(H,aa_complex_int(J,A2)) = aa_complex_complex(G,A2) ) )
                     => ( groups3708469109370488835omplex(G,S2) = groups7440179247065528705omplex(H,T) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_8138_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_complex,T3: set_complex,S2: set_complex,I: fun_complex_complex,J: fun_complex_complex,T: set_complex,G: fun_complex_complex,H: fun_complex_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S4)
     => ( aa_set_complex_o(finite3207457112153483333omplex,T3)
       => ( ! [A2: complex] :
              ( aa_set_complex_o(member_complex(A2),minus_811609699411566653omplex(S2,S4))
             => ( aa_complex_complex(I,aa_complex_complex(J,A2)) = A2 ) )
         => ( ! [A2: complex] :
                ( aa_set_complex_o(member_complex(A2),minus_811609699411566653omplex(S2,S4))
               => aa_set_complex_o(member_complex(aa_complex_complex(J,A2)),minus_811609699411566653omplex(T,T3)) )
           => ( ! [B2: complex] :
                  ( aa_set_complex_o(member_complex(B2),minus_811609699411566653omplex(T,T3))
                 => ( aa_complex_complex(J,aa_complex_complex(I,B2)) = B2 ) )
             => ( ! [B2: complex] :
                    ( aa_set_complex_o(member_complex(B2),minus_811609699411566653omplex(T,T3))
                   => aa_set_complex_o(member_complex(aa_complex_complex(I,B2)),minus_811609699411566653omplex(S2,S4)) )
               => ( ! [A2: complex] :
                      ( aa_set_complex_o(member_complex(A2),S4)
                     => ( aa_complex_complex(G,A2) = one_one_complex ) )
                 => ( ! [B2: complex] :
                        ( aa_set_complex_o(member_complex(B2),T3)
                       => ( aa_complex_complex(H,B2) = one_one_complex ) )
                   => ( ! [A2: complex] :
                          ( aa_set_complex_o(member_complex(A2),S2)
                         => ( aa_complex_complex(H,aa_complex_complex(J,A2)) = aa_complex_complex(G,A2) ) )
                     => ( groups3708469109370488835omplex(G,S2) = groups3708469109370488835omplex(H,T) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_8139_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_real,T3: set_real,S2: set_real,I: fun_real_real,J: fun_real_real,T: set_real,G: fun_real_real,H: fun_real_real] :
      ( finite_finite_real(S4)
     => ( finite_finite_real(T3)
       => ( ! [A2: real] :
              ( member_real(A2,minus_minus_set_real(S2,S4))
             => ( aa_real_real(I,aa_real_real(J,A2)) = A2 ) )
         => ( ! [A2: real] :
                ( member_real(A2,minus_minus_set_real(S2,S4))
               => member_real(aa_real_real(J,A2),minus_minus_set_real(T,T3)) )
           => ( ! [B2: real] :
                  ( member_real(B2,minus_minus_set_real(T,T3))
                 => ( aa_real_real(J,aa_real_real(I,B2)) = B2 ) )
             => ( ! [B2: real] :
                    ( member_real(B2,minus_minus_set_real(T,T3))
                   => member_real(aa_real_real(I,B2),minus_minus_set_real(S2,S4)) )
               => ( ! [A2: real] :
                      ( member_real(A2,S4)
                     => ( aa_real_real(G,A2) = one_one_real ) )
                 => ( ! [B2: real] :
                        ( member_real(B2,T3)
                       => ( aa_real_real(H,B2) = one_one_real ) )
                   => ( ! [A2: real] :
                          ( member_real(A2,S2)
                         => ( aa_real_real(H,aa_real_real(J,A2)) = aa_real_real(G,A2) ) )
                     => ( groups1681761925125756287l_real(G,S2) = groups1681761925125756287l_real(H,T) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_8140_prod_Osetdiff__irrelevant,axiom,
    ! [A3: set_real,G: fun_real_complex] :
      ( finite_finite_real(A3)
     => ( groups713298508707869441omplex(G,minus_minus_set_real(A3,collect_real(aTP_Lamm_nt(G)))) = groups713298508707869441omplex(G,A3) ) ) ).

% prod.setdiff_irrelevant
tff(fact_8141_prod_Osetdiff__irrelevant,axiom,
    ! [A3: set_int,G: fun_int_complex] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( groups7440179247065528705omplex(G,minus_minus_set_int(A3,collect_int(aTP_Lamm_nu(G)))) = groups7440179247065528705omplex(G,A3) ) ) ).

% prod.setdiff_irrelevant
tff(fact_8142_prod_Osetdiff__irrelevant,axiom,
    ! [A3: set_complex,G: fun_complex_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A3)
     => ( groups3708469109370488835omplex(G,minus_811609699411566653omplex(A3,collect_complex(aTP_Lamm_nv(G)))) = groups3708469109370488835omplex(G,A3) ) ) ).

% prod.setdiff_irrelevant
tff(fact_8143_prod_Osetdiff__irrelevant,axiom,
    ! [A3: set_real,G: fun_real_real] :
      ( finite_finite_real(A3)
     => ( groups1681761925125756287l_real(G,minus_minus_set_real(A3,collect_real(aTP_Lamm_nw(G)))) = groups1681761925125756287l_real(G,A3) ) ) ).

% prod.setdiff_irrelevant
tff(fact_8144_prod_Osetdiff__irrelevant,axiom,
    ! [A3: set_int,G: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( groups2316167850115554303t_real(G,minus_minus_set_int(A3,collect_int(aTP_Lamm_nx(G)))) = groups2316167850115554303t_real(G,A3) ) ) ).

% prod.setdiff_irrelevant
tff(fact_8145_prod_Osetdiff__irrelevant,axiom,
    ! [A3: set_complex,G: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A3)
     => ( groups766887009212190081x_real(G,minus_811609699411566653omplex(A3,collect_complex(aTP_Lamm_ny(G)))) = groups766887009212190081x_real(G,A3) ) ) ).

% prod.setdiff_irrelevant
tff(fact_8146_prod_Osetdiff__irrelevant,axiom,
    ! [A3: set_real,G: fun_real_rat] :
      ( finite_finite_real(A3)
     => ( groups4061424788464935467al_rat(G,minus_minus_set_real(A3,collect_real(aTP_Lamm_nz(G)))) = groups4061424788464935467al_rat(G,A3) ) ) ).

% prod.setdiff_irrelevant
tff(fact_8147_prod_Osetdiff__irrelevant,axiom,
    ! [A3: set_int,G: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( groups1072433553688619179nt_rat(G,minus_minus_set_int(A3,collect_int(aTP_Lamm_oa(G)))) = groups1072433553688619179nt_rat(G,A3) ) ) ).

% prod.setdiff_irrelevant
tff(fact_8148_prod_Osetdiff__irrelevant,axiom,
    ! [A3: set_complex,G: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A3)
     => ( groups225925009352817453ex_rat(G,minus_811609699411566653omplex(A3,collect_complex(aTP_Lamm_ob(G)))) = groups225925009352817453ex_rat(G,A3) ) ) ).

% prod.setdiff_irrelevant
tff(fact_8149_prod_Osetdiff__irrelevant,axiom,
    ! [A3: set_real,G: fun_real_nat] :
      ( finite_finite_real(A3)
     => ( groups4696554848551431203al_nat(G,minus_minus_set_real(A3,collect_real(aTP_Lamm_oc(G)))) = groups4696554848551431203al_nat(G,A3) ) ) ).

% prod.setdiff_irrelevant
tff(fact_8150_exp__sum,axiom,
    ! [I5: set_int,F: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,I5)
     => ( aa_real_real(exp_real,groups8778361861064173332t_real(F,I5)) = groups2316167850115554303t_real(aTP_Lamm_od(F),I5) ) ) ).

% exp_sum
tff(fact_8151_exp__sum,axiom,
    ! [I5: set_complex,F: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,I5)
     => ( aa_real_real(exp_real,groups5808333547571424918x_real(F,I5)) = groups766887009212190081x_real(aTP_Lamm_oe(F),I5) ) ) ).

% exp_sum
tff(fact_8152_exp__sum,axiom,
    ! [I5: set_complex,F: fun_complex_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,I5)
     => ( aa_complex_complex(exp_complex,groups7754918857620584856omplex(F,I5)) = groups3708469109370488835omplex(aTP_Lamm_of(F),I5) ) ) ).

% exp_sum
tff(fact_8153_exp__sum,axiom,
    ! [I5: set_nat,F: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,I5)
     => ( aa_real_real(exp_real,aa_set_nat_real(groups6591440286371151544t_real(F),I5)) = aa_set_nat_real(groups129246275422532515t_real(aTP_Lamm_og(F)),I5) ) ) ).

% exp_sum
tff(fact_8154_norm__exp,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(exp_real,Xb))),aa_real_real(exp_real,aa_real_real(real_V7735802525324610683m_real,Xb))) ).

% norm_exp
tff(fact_8155_norm__exp,axiom,
    ! [Xb: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(exp_complex,Xb))),aa_real_real(exp_real,aa_complex_real(real_V1022390504157884413omplex,Xb))) ).

% norm_exp
tff(fact_8156_prod_OatLeastAtMost__rev,axiom,
    ! [G: fun_nat_nat,Na: nat,M: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),set_or1269000886237332187st_nat(Na,M)) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_jn(G),Na),M)),set_or1269000886237332187st_nat(Na,M)) ).

% prod.atLeastAtMost_rev
tff(fact_8157_prod_OatLeastAtMost__rev,axiom,
    ! [G: fun_nat_int,Na: nat,M: nat] : groups705719431365010083at_int(G,set_or1269000886237332187st_nat(Na,M)) = groups705719431365010083at_int(aa_nat_fun_nat_int(aa_nat1293065384339088724at_int(aTP_Lamm_oh(G),Na),M),set_or1269000886237332187st_nat(Na,M)) ).

% prod.atLeastAtMost_rev
tff(fact_8158_less__1__prod2,axiom,
    ! [I5: set_real,I: real,F: fun_real_real] :
      ( finite_finite_real(I5)
     => ( member_real(I,I5)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(F,I))
         => ( ! [I2: real] :
                ( member_real(I2,I5)
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(F,I2)) )
           => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),groups1681761925125756287l_real(F,I5)) ) ) ) ) ).

% less_1_prod2
tff(fact_8159_less__1__prod2,axiom,
    ! [I5: set_nat,I: nat,F: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,I5)
     => ( aa_set_nat_o(member_nat(I),I5)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_nat_real(F,I))
         => ( ! [I2: nat] :
                ( aa_set_nat_o(member_nat(I2),I5)
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_nat_real(F,I2)) )
           => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_set_nat_real(groups129246275422532515t_real(F),I5)) ) ) ) ) ).

% less_1_prod2
tff(fact_8160_less__1__prod2,axiom,
    ! [I5: set_int,I: int,F: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,I5)
     => ( aa_set_int_o(member_int(I),I5)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_int_real(F,I))
         => ( ! [I2: int] :
                ( aa_set_int_o(member_int(I2),I5)
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_int_real(F,I2)) )
           => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),groups2316167850115554303t_real(F,I5)) ) ) ) ) ).

% less_1_prod2
tff(fact_8161_less__1__prod2,axiom,
    ! [I5: set_complex,I: complex,F: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,I5)
     => ( aa_set_complex_o(member_complex(I),I5)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_complex_real(F,I))
         => ( ! [I2: complex] :
                ( aa_set_complex_o(member_complex(I2),I5)
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_complex_real(F,I2)) )
           => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),groups766887009212190081x_real(F,I5)) ) ) ) ) ).

% less_1_prod2
tff(fact_8162_less__1__prod2,axiom,
    ! [I5: set_real,I: real,F: fun_real_rat] :
      ( finite_finite_real(I5)
     => ( member_real(I,I5)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_real_rat(F,I))
         => ( ! [I2: real] :
                ( member_real(I2,I5)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_real_rat(F,I2)) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),groups4061424788464935467al_rat(F,I5)) ) ) ) ) ).

% less_1_prod2
tff(fact_8163_less__1__prod2,axiom,
    ! [I5: set_nat,I: nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,I5)
     => ( aa_set_nat_o(member_nat(I),I5)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_nat_rat(F,I))
         => ( ! [I2: nat] :
                ( aa_set_nat_o(member_nat(I2),I5)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_nat_rat(F,I2)) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),groups73079841787564623at_rat(F,I5)) ) ) ) ) ).

% less_1_prod2
tff(fact_8164_less__1__prod2,axiom,
    ! [I5: set_int,I: int,F: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,I5)
     => ( aa_set_int_o(member_int(I),I5)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_int_rat(F,I))
         => ( ! [I2: int] :
                ( aa_set_int_o(member_int(I2),I5)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_int_rat(F,I2)) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),groups1072433553688619179nt_rat(F,I5)) ) ) ) ) ).

% less_1_prod2
tff(fact_8165_less__1__prod2,axiom,
    ! [I5: set_complex,I: complex,F: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,I5)
     => ( aa_set_complex_o(member_complex(I),I5)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_complex_rat(F,I))
         => ( ! [I2: complex] :
                ( aa_set_complex_o(member_complex(I2),I5)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_complex_rat(F,I2)) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),groups225925009352817453ex_rat(F,I5)) ) ) ) ) ).

% less_1_prod2
tff(fact_8166_less__1__prod2,axiom,
    ! [I5: set_real,I: real,F: fun_real_int] :
      ( finite_finite_real(I5)
     => ( member_real(I,I5)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_real_int(F,I))
         => ( ! [I2: real] :
                ( member_real(I2,I5)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_real_int(F,I2)) )
           => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),groups4694064378042380927al_int(F,I5)) ) ) ) ) ).

% less_1_prod2
tff(fact_8167_less__1__prod2,axiom,
    ! [I5: set_complex,I: complex,F: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,I5)
     => ( aa_set_complex_o(member_complex(I),I5)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_complex_int(F,I))
         => ( ! [I2: complex] :
                ( aa_set_complex_o(member_complex(I2),I5)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_complex_int(F,I2)) )
           => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),groups858564598930262913ex_int(F,I5)) ) ) ) ) ).

% less_1_prod2
tff(fact_8168_less__1__prod,axiom,
    ! [I5: set_complex,F: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,I5)
     => ( ( I5 != bot_bot_set_complex )
       => ( ! [I2: complex] :
              ( aa_set_complex_o(member_complex(I2),I5)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_complex_real(F,I2)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),groups766887009212190081x_real(F,I5)) ) ) ) ).

% less_1_prod
tff(fact_8169_less__1__prod,axiom,
    ! [I5: set_real,F: fun_real_real] :
      ( finite_finite_real(I5)
     => ( ( I5 != bot_bot_set_real )
       => ( ! [I2: real] :
              ( member_real(I2,I5)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(F,I2)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),groups1681761925125756287l_real(F,I5)) ) ) ) ).

% less_1_prod
tff(fact_8170_less__1__prod,axiom,
    ! [I5: set_nat,F: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,I5)
     => ( ( I5 != bot_bot_set_nat )
       => ( ! [I2: nat] :
              ( aa_set_nat_o(member_nat(I2),I5)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_nat_real(F,I2)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_set_nat_real(groups129246275422532515t_real(F),I5)) ) ) ) ).

% less_1_prod
tff(fact_8171_less__1__prod,axiom,
    ! [I5: set_int,F: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,I5)
     => ( ( I5 != bot_bot_set_int )
       => ( ! [I2: int] :
              ( aa_set_int_o(member_int(I2),I5)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_int_real(F,I2)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),groups2316167850115554303t_real(F,I5)) ) ) ) ).

% less_1_prod
tff(fact_8172_less__1__prod,axiom,
    ! [I5: set_complex,F: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,I5)
     => ( ( I5 != bot_bot_set_complex )
       => ( ! [I2: complex] :
              ( aa_set_complex_o(member_complex(I2),I5)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_complex_rat(F,I2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),groups225925009352817453ex_rat(F,I5)) ) ) ) ).

% less_1_prod
tff(fact_8173_less__1__prod,axiom,
    ! [I5: set_real,F: fun_real_rat] :
      ( finite_finite_real(I5)
     => ( ( I5 != bot_bot_set_real )
       => ( ! [I2: real] :
              ( member_real(I2,I5)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_real_rat(F,I2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),groups4061424788464935467al_rat(F,I5)) ) ) ) ).

% less_1_prod
tff(fact_8174_less__1__prod,axiom,
    ! [I5: set_nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,I5)
     => ( ( I5 != bot_bot_set_nat )
       => ( ! [I2: nat] :
              ( aa_set_nat_o(member_nat(I2),I5)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_nat_rat(F,I2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),groups73079841787564623at_rat(F,I5)) ) ) ) ).

% less_1_prod
tff(fact_8175_less__1__prod,axiom,
    ! [I5: set_int,F: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,I5)
     => ( ( I5 != bot_bot_set_int )
       => ( ! [I2: int] :
              ( aa_set_int_o(member_int(I2),I5)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_int_rat(F,I2)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),groups1072433553688619179nt_rat(F,I5)) ) ) ) ).

% less_1_prod
tff(fact_8176_less__1__prod,axiom,
    ! [I5: set_complex,F: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,I5)
     => ( ( I5 != bot_bot_set_complex )
       => ( ! [I2: complex] :
              ( aa_set_complex_o(member_complex(I2),I5)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_complex_int(F,I2)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),groups858564598930262913ex_int(F,I5)) ) ) ) ).

% less_1_prod
tff(fact_8177_less__1__prod,axiom,
    ! [I5: set_real,F: fun_real_int] :
      ( finite_finite_real(I5)
     => ( ( I5 != bot_bot_set_real )
       => ( ! [I2: real] :
              ( member_real(I2,I5)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_real_int(F,I2)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),groups4694064378042380927al_int(F,I5)) ) ) ) ).

% less_1_prod
tff(fact_8178_prod_Osubset__diff,axiom,
    ! [B4: set_complex,A3: set_complex,G: fun_complex_real] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(B4),A3)
     => ( aa_set_complex_o(finite3207457112153483333omplex,A3)
       => ( groups766887009212190081x_real(G,A3) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups766887009212190081x_real(G,minus_811609699411566653omplex(A3,B4))),groups766887009212190081x_real(G,B4)) ) ) ) ).

% prod.subset_diff
tff(fact_8179_prod_Osubset__diff,axiom,
    ! [B4: set_nat,A3: set_nat,G: fun_nat_real] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),A3)
     => ( aa_set_nat_o(finite_finite_nat,A3)
       => ( aa_set_nat_real(groups129246275422532515t_real(G),A3) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_set_nat_real(groups129246275422532515t_real(G),minus_minus_set_nat(A3,B4))),aa_set_nat_real(groups129246275422532515t_real(G),B4)) ) ) ) ).

% prod.subset_diff
tff(fact_8180_prod_Osubset__diff,axiom,
    ! [B4: set_complex,A3: set_complex,G: fun_complex_rat] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(B4),A3)
     => ( aa_set_complex_o(finite3207457112153483333omplex,A3)
       => ( groups225925009352817453ex_rat(G,A3) = aa_rat_rat(times_times_rat(groups225925009352817453ex_rat(G,minus_811609699411566653omplex(A3,B4))),groups225925009352817453ex_rat(G,B4)) ) ) ) ).

% prod.subset_diff
tff(fact_8181_prod_Osubset__diff,axiom,
    ! [B4: set_nat,A3: set_nat,G: fun_nat_rat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),A3)
     => ( aa_set_nat_o(finite_finite_nat,A3)
       => ( groups73079841787564623at_rat(G,A3) = aa_rat_rat(times_times_rat(groups73079841787564623at_rat(G,minus_minus_set_nat(A3,B4))),groups73079841787564623at_rat(G,B4)) ) ) ) ).

% prod.subset_diff
tff(fact_8182_prod_Osubset__diff,axiom,
    ! [B4: set_complex,A3: set_complex,G: fun_complex_nat] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(B4),A3)
     => ( aa_set_complex_o(finite3207457112153483333omplex,A3)
       => ( groups861055069439313189ex_nat(G,A3) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups861055069439313189ex_nat(G,minus_811609699411566653omplex(A3,B4))),groups861055069439313189ex_nat(G,B4)) ) ) ) ).

% prod.subset_diff
tff(fact_8183_prod_Osubset__diff,axiom,
    ! [B4: set_complex,A3: set_complex,G: fun_complex_int] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(B4),A3)
     => ( aa_set_complex_o(finite3207457112153483333omplex,A3)
       => ( groups858564598930262913ex_int(G,A3) = aa_int_int(aa_int_fun_int_int(times_times_int,groups858564598930262913ex_int(G,minus_811609699411566653omplex(A3,B4))),groups858564598930262913ex_int(G,B4)) ) ) ) ).

% prod.subset_diff
tff(fact_8184_prod_Osubset__diff,axiom,
    ! [B4: set_int,A3: set_int,G: fun_int_real] :
      ( aa_set_int_o(ord_less_eq_set_int(B4),A3)
     => ( aa_set_int_o(finite_finite_int,A3)
       => ( groups2316167850115554303t_real(G,A3) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups2316167850115554303t_real(G,minus_minus_set_int(A3,B4))),groups2316167850115554303t_real(G,B4)) ) ) ) ).

% prod.subset_diff
tff(fact_8185_prod_Osubset__diff,axiom,
    ! [B4: set_int,A3: set_int,G: fun_int_rat] :
      ( aa_set_int_o(ord_less_eq_set_int(B4),A3)
     => ( aa_set_int_o(finite_finite_int,A3)
       => ( groups1072433553688619179nt_rat(G,A3) = aa_rat_rat(times_times_rat(groups1072433553688619179nt_rat(G,minus_minus_set_int(A3,B4))),groups1072433553688619179nt_rat(G,B4)) ) ) ) ).

% prod.subset_diff
tff(fact_8186_prod_Osubset__diff,axiom,
    ! [B4: set_int,A3: set_int,G: fun_int_nat] :
      ( aa_set_int_o(ord_less_eq_set_int(B4),A3)
     => ( aa_set_int_o(finite_finite_int,A3)
       => ( groups1707563613775114915nt_nat(G,A3) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups1707563613775114915nt_nat(G,minus_minus_set_int(A3,B4))),groups1707563613775114915nt_nat(G,B4)) ) ) ) ).

% prod.subset_diff
tff(fact_8187_prod_Osubset__diff,axiom,
    ! [B4: set_nat,A3: set_nat,G: fun_nat_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B4),A3)
     => ( aa_set_nat_o(finite_finite_nat,A3)
       => ( aa_set_nat_nat(groups708209901874060359at_nat(G),A3) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups708209901874060359at_nat(G),minus_minus_set_nat(A3,B4))),aa_set_nat_nat(groups708209901874060359at_nat(G),B4)) ) ) ) ).

% prod.subset_diff
tff(fact_8188_prod_Omono__neutral__cong__right,axiom,
    ! [T: set_real,S2: set_real,G: fun_real_complex,H: fun_real_complex] :
      ( finite_finite_real(T)
     => ( ord_less_eq_set_real(S2,T)
       => ( ! [X: real] :
              ( member_real(X,minus_minus_set_real(T,S2))
             => ( aa_real_complex(G,X) = one_one_complex ) )
         => ( ! [X: real] :
                ( member_real(X,S2)
               => ( aa_real_complex(G,X) = aa_real_complex(H,X) ) )
           => ( groups713298508707869441omplex(G,T) = groups713298508707869441omplex(H,S2) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_8189_prod_Omono__neutral__cong__right,axiom,
    ! [T: set_complex,S2: set_complex,G: fun_complex_complex,H: fun_complex_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T)
       => ( ! [X: complex] :
              ( aa_set_complex_o(member_complex(X),minus_811609699411566653omplex(T,S2))
             => ( aa_complex_complex(G,X) = one_one_complex ) )
         => ( ! [X: complex] :
                ( aa_set_complex_o(member_complex(X),S2)
               => ( aa_complex_complex(G,X) = aa_complex_complex(H,X) ) )
           => ( groups3708469109370488835omplex(G,T) = groups3708469109370488835omplex(H,S2) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_8190_prod_Omono__neutral__cong__right,axiom,
    ! [T: set_real,S2: set_real,G: fun_real_real,H: fun_real_real] :
      ( finite_finite_real(T)
     => ( ord_less_eq_set_real(S2,T)
       => ( ! [X: real] :
              ( member_real(X,minus_minus_set_real(T,S2))
             => ( aa_real_real(G,X) = one_one_real ) )
         => ( ! [X: real] :
                ( member_real(X,S2)
               => ( aa_real_real(G,X) = aa_real_real(H,X) ) )
           => ( groups1681761925125756287l_real(G,T) = groups1681761925125756287l_real(H,S2) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_8191_prod_Omono__neutral__cong__right,axiom,
    ! [T: set_complex,S2: set_complex,G: fun_complex_real,H: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T)
       => ( ! [X: complex] :
              ( aa_set_complex_o(member_complex(X),minus_811609699411566653omplex(T,S2))
             => ( aa_complex_real(G,X) = one_one_real ) )
         => ( ! [X: complex] :
                ( aa_set_complex_o(member_complex(X),S2)
               => ( aa_complex_real(G,X) = aa_complex_real(H,X) ) )
           => ( groups766887009212190081x_real(G,T) = groups766887009212190081x_real(H,S2) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_8192_prod_Omono__neutral__cong__right,axiom,
    ! [T: set_real,S2: set_real,G: fun_real_rat,H: fun_real_rat] :
      ( finite_finite_real(T)
     => ( ord_less_eq_set_real(S2,T)
       => ( ! [X: real] :
              ( member_real(X,minus_minus_set_real(T,S2))
             => ( aa_real_rat(G,X) = one_one_rat ) )
         => ( ! [X: real] :
                ( member_real(X,S2)
               => ( aa_real_rat(G,X) = aa_real_rat(H,X) ) )
           => ( groups4061424788464935467al_rat(G,T) = groups4061424788464935467al_rat(H,S2) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_8193_prod_Omono__neutral__cong__right,axiom,
    ! [T: set_complex,S2: set_complex,G: fun_complex_rat,H: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T)
       => ( ! [X: complex] :
              ( aa_set_complex_o(member_complex(X),minus_811609699411566653omplex(T,S2))
             => ( aa_complex_rat(G,X) = one_one_rat ) )
         => ( ! [X: complex] :
                ( aa_set_complex_o(member_complex(X),S2)
               => ( aa_complex_rat(G,X) = aa_complex_rat(H,X) ) )
           => ( groups225925009352817453ex_rat(G,T) = groups225925009352817453ex_rat(H,S2) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_8194_prod_Omono__neutral__cong__right,axiom,
    ! [T: set_real,S2: set_real,G: fun_real_nat,H: fun_real_nat] :
      ( finite_finite_real(T)
     => ( ord_less_eq_set_real(S2,T)
       => ( ! [X: real] :
              ( member_real(X,minus_minus_set_real(T,S2))
             => ( aa_real_nat(G,X) = one_one_nat ) )
         => ( ! [X: real] :
                ( member_real(X,S2)
               => ( aa_real_nat(G,X) = aa_real_nat(H,X) ) )
           => ( groups4696554848551431203al_nat(G,T) = groups4696554848551431203al_nat(H,S2) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_8195_prod_Omono__neutral__cong__right,axiom,
    ! [T: set_complex,S2: set_complex,G: fun_complex_nat,H: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T)
       => ( ! [X: complex] :
              ( aa_set_complex_o(member_complex(X),minus_811609699411566653omplex(T,S2))
             => ( aa_complex_nat(G,X) = one_one_nat ) )
         => ( ! [X: complex] :
                ( aa_set_complex_o(member_complex(X),S2)
               => ( aa_complex_nat(G,X) = aa_complex_nat(H,X) ) )
           => ( groups861055069439313189ex_nat(G,T) = groups861055069439313189ex_nat(H,S2) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_8196_prod_Omono__neutral__cong__right,axiom,
    ! [T: set_real,S2: set_real,G: fun_real_int,H: fun_real_int] :
      ( finite_finite_real(T)
     => ( ord_less_eq_set_real(S2,T)
       => ( ! [X: real] :
              ( member_real(X,minus_minus_set_real(T,S2))
             => ( aa_real_int(G,X) = one_one_int ) )
         => ( ! [X: real] :
                ( member_real(X,S2)
               => ( aa_real_int(G,X) = aa_real_int(H,X) ) )
           => ( groups4694064378042380927al_int(G,T) = groups4694064378042380927al_int(H,S2) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_8197_prod_Omono__neutral__cong__right,axiom,
    ! [T: set_complex,S2: set_complex,G: fun_complex_int,H: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T)
       => ( ! [X: complex] :
              ( aa_set_complex_o(member_complex(X),minus_811609699411566653omplex(T,S2))
             => ( aa_complex_int(G,X) = one_one_int ) )
         => ( ! [X: complex] :
                ( aa_set_complex_o(member_complex(X),S2)
               => ( aa_complex_int(G,X) = aa_complex_int(H,X) ) )
           => ( groups858564598930262913ex_int(G,T) = groups858564598930262913ex_int(H,S2) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_8198_prod_Omono__neutral__cong__left,axiom,
    ! [T: set_real,S2: set_real,H: fun_real_complex,G: fun_real_complex] :
      ( finite_finite_real(T)
     => ( ord_less_eq_set_real(S2,T)
       => ( ! [X: real] :
              ( member_real(X,minus_minus_set_real(T,S2))
             => ( aa_real_complex(H,X) = one_one_complex ) )
         => ( ! [X: real] :
                ( member_real(X,S2)
               => ( aa_real_complex(G,X) = aa_real_complex(H,X) ) )
           => ( groups713298508707869441omplex(G,S2) = groups713298508707869441omplex(H,T) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_8199_prod_Omono__neutral__cong__left,axiom,
    ! [T: set_complex,S2: set_complex,H: fun_complex_complex,G: fun_complex_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T)
       => ( ! [X: complex] :
              ( aa_set_complex_o(member_complex(X),minus_811609699411566653omplex(T,S2))
             => ( aa_complex_complex(H,X) = one_one_complex ) )
         => ( ! [X: complex] :
                ( aa_set_complex_o(member_complex(X),S2)
               => ( aa_complex_complex(G,X) = aa_complex_complex(H,X) ) )
           => ( groups3708469109370488835omplex(G,S2) = groups3708469109370488835omplex(H,T) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_8200_prod_Omono__neutral__cong__left,axiom,
    ! [T: set_real,S2: set_real,H: fun_real_real,G: fun_real_real] :
      ( finite_finite_real(T)
     => ( ord_less_eq_set_real(S2,T)
       => ( ! [X: real] :
              ( member_real(X,minus_minus_set_real(T,S2))
             => ( aa_real_real(H,X) = one_one_real ) )
         => ( ! [X: real] :
                ( member_real(X,S2)
               => ( aa_real_real(G,X) = aa_real_real(H,X) ) )
           => ( groups1681761925125756287l_real(G,S2) = groups1681761925125756287l_real(H,T) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_8201_prod_Omono__neutral__cong__left,axiom,
    ! [T: set_complex,S2: set_complex,H: fun_complex_real,G: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T)
       => ( ! [X: complex] :
              ( aa_set_complex_o(member_complex(X),minus_811609699411566653omplex(T,S2))
             => ( aa_complex_real(H,X) = one_one_real ) )
         => ( ! [X: complex] :
                ( aa_set_complex_o(member_complex(X),S2)
               => ( aa_complex_real(G,X) = aa_complex_real(H,X) ) )
           => ( groups766887009212190081x_real(G,S2) = groups766887009212190081x_real(H,T) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_8202_prod_Omono__neutral__cong__left,axiom,
    ! [T: set_real,S2: set_real,H: fun_real_rat,G: fun_real_rat] :
      ( finite_finite_real(T)
     => ( ord_less_eq_set_real(S2,T)
       => ( ! [X: real] :
              ( member_real(X,minus_minus_set_real(T,S2))
             => ( aa_real_rat(H,X) = one_one_rat ) )
         => ( ! [X: real] :
                ( member_real(X,S2)
               => ( aa_real_rat(G,X) = aa_real_rat(H,X) ) )
           => ( groups4061424788464935467al_rat(G,S2) = groups4061424788464935467al_rat(H,T) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_8203_prod_Omono__neutral__cong__left,axiom,
    ! [T: set_complex,S2: set_complex,H: fun_complex_rat,G: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T)
       => ( ! [X: complex] :
              ( aa_set_complex_o(member_complex(X),minus_811609699411566653omplex(T,S2))
             => ( aa_complex_rat(H,X) = one_one_rat ) )
         => ( ! [X: complex] :
                ( aa_set_complex_o(member_complex(X),S2)
               => ( aa_complex_rat(G,X) = aa_complex_rat(H,X) ) )
           => ( groups225925009352817453ex_rat(G,S2) = groups225925009352817453ex_rat(H,T) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_8204_prod_Omono__neutral__cong__left,axiom,
    ! [T: set_real,S2: set_real,H: fun_real_nat,G: fun_real_nat] :
      ( finite_finite_real(T)
     => ( ord_less_eq_set_real(S2,T)
       => ( ! [X: real] :
              ( member_real(X,minus_minus_set_real(T,S2))
             => ( aa_real_nat(H,X) = one_one_nat ) )
         => ( ! [X: real] :
                ( member_real(X,S2)
               => ( aa_real_nat(G,X) = aa_real_nat(H,X) ) )
           => ( groups4696554848551431203al_nat(G,S2) = groups4696554848551431203al_nat(H,T) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_8205_prod_Omono__neutral__cong__left,axiom,
    ! [T: set_complex,S2: set_complex,H: fun_complex_nat,G: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T)
       => ( ! [X: complex] :
              ( aa_set_complex_o(member_complex(X),minus_811609699411566653omplex(T,S2))
             => ( aa_complex_nat(H,X) = one_one_nat ) )
         => ( ! [X: complex] :
                ( aa_set_complex_o(member_complex(X),S2)
               => ( aa_complex_nat(G,X) = aa_complex_nat(H,X) ) )
           => ( groups861055069439313189ex_nat(G,S2) = groups861055069439313189ex_nat(H,T) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_8206_prod_Omono__neutral__cong__left,axiom,
    ! [T: set_real,S2: set_real,H: fun_real_int,G: fun_real_int] :
      ( finite_finite_real(T)
     => ( ord_less_eq_set_real(S2,T)
       => ( ! [X: real] :
              ( member_real(X,minus_minus_set_real(T,S2))
             => ( aa_real_int(H,X) = one_one_int ) )
         => ( ! [X: real] :
                ( member_real(X,S2)
               => ( aa_real_int(G,X) = aa_real_int(H,X) ) )
           => ( groups4694064378042380927al_int(G,S2) = groups4694064378042380927al_int(H,T) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_8207_prod_Omono__neutral__cong__left,axiom,
    ! [T: set_complex,S2: set_complex,H: fun_complex_int,G: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T)
       => ( ! [X: complex] :
              ( aa_set_complex_o(member_complex(X),minus_811609699411566653omplex(T,S2))
             => ( aa_complex_int(H,X) = one_one_int ) )
         => ( ! [X: complex] :
                ( aa_set_complex_o(member_complex(X),S2)
               => ( aa_complex_int(G,X) = aa_complex_int(H,X) ) )
           => ( groups858564598930262913ex_int(G,S2) = groups858564598930262913ex_int(H,T) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_8208_prod_Omono__neutral__right,axiom,
    ! [T: set_complex,S2: set_complex,G: fun_complex_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T)
       => ( ! [X: complex] :
              ( aa_set_complex_o(member_complex(X),minus_811609699411566653omplex(T,S2))
             => ( aa_complex_complex(G,X) = one_one_complex ) )
         => ( groups3708469109370488835omplex(G,T) = groups3708469109370488835omplex(G,S2) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_8209_prod_Omono__neutral__right,axiom,
    ! [T: set_complex,S2: set_complex,G: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T)
       => ( ! [X: complex] :
              ( aa_set_complex_o(member_complex(X),minus_811609699411566653omplex(T,S2))
             => ( aa_complex_real(G,X) = one_one_real ) )
         => ( groups766887009212190081x_real(G,T) = groups766887009212190081x_real(G,S2) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_8210_prod_Omono__neutral__right,axiom,
    ! [T: set_complex,S2: set_complex,G: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T)
       => ( ! [X: complex] :
              ( aa_set_complex_o(member_complex(X),minus_811609699411566653omplex(T,S2))
             => ( aa_complex_rat(G,X) = one_one_rat ) )
         => ( groups225925009352817453ex_rat(G,T) = groups225925009352817453ex_rat(G,S2) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_8211_prod_Omono__neutral__right,axiom,
    ! [T: set_complex,S2: set_complex,G: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T)
       => ( ! [X: complex] :
              ( aa_set_complex_o(member_complex(X),minus_811609699411566653omplex(T,S2))
             => ( aa_complex_nat(G,X) = one_one_nat ) )
         => ( groups861055069439313189ex_nat(G,T) = groups861055069439313189ex_nat(G,S2) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_8212_prod_Omono__neutral__right,axiom,
    ! [T: set_complex,S2: set_complex,G: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T)
       => ( ! [X: complex] :
              ( aa_set_complex_o(member_complex(X),minus_811609699411566653omplex(T,S2))
             => ( aa_complex_int(G,X) = one_one_int ) )
         => ( groups858564598930262913ex_int(G,T) = groups858564598930262913ex_int(G,S2) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_8213_prod_Omono__neutral__right,axiom,
    ! [T: set_nat,S2: set_nat,G: fun_nat_complex] :
      ( aa_set_nat_o(finite_finite_nat,T)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S2),T)
       => ( ! [X: nat] :
              ( aa_set_nat_o(member_nat(X),minus_minus_set_nat(T,S2))
             => ( aa_nat_complex(G,X) = one_one_complex ) )
         => ( aa_set_nat_complex(groups6464643781859351333omplex(G),T) = aa_set_nat_complex(groups6464643781859351333omplex(G),S2) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_8214_prod_Omono__neutral__right,axiom,
    ! [T: set_nat,S2: set_nat,G: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,T)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S2),T)
       => ( ! [X: nat] :
              ( aa_set_nat_o(member_nat(X),minus_minus_set_nat(T,S2))
             => ( aa_nat_real(G,X) = one_one_real ) )
         => ( aa_set_nat_real(groups129246275422532515t_real(G),T) = aa_set_nat_real(groups129246275422532515t_real(G),S2) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_8215_prod_Omono__neutral__right,axiom,
    ! [T: set_nat,S2: set_nat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,T)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S2),T)
       => ( ! [X: nat] :
              ( aa_set_nat_o(member_nat(X),minus_minus_set_nat(T,S2))
             => ( aa_nat_rat(G,X) = one_one_rat ) )
         => ( groups73079841787564623at_rat(G,T) = groups73079841787564623at_rat(G,S2) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_8216_prod_Omono__neutral__right,axiom,
    ! [T: set_int,S2: set_int,G: fun_int_complex] :
      ( aa_set_int_o(finite_finite_int,T)
     => ( aa_set_int_o(ord_less_eq_set_int(S2),T)
       => ( ! [X: int] :
              ( aa_set_int_o(member_int(X),minus_minus_set_int(T,S2))
             => ( aa_int_complex(G,X) = one_one_complex ) )
         => ( groups7440179247065528705omplex(G,T) = groups7440179247065528705omplex(G,S2) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_8217_prod_Omono__neutral__right,axiom,
    ! [T: set_int,S2: set_int,G: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,T)
     => ( aa_set_int_o(ord_less_eq_set_int(S2),T)
       => ( ! [X: int] :
              ( aa_set_int_o(member_int(X),minus_minus_set_int(T,S2))
             => ( aa_int_real(G,X) = one_one_real ) )
         => ( groups2316167850115554303t_real(G,T) = groups2316167850115554303t_real(G,S2) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_8218_prod_Omono__neutral__left,axiom,
    ! [T: set_complex,S2: set_complex,G: fun_complex_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T)
       => ( ! [X: complex] :
              ( aa_set_complex_o(member_complex(X),minus_811609699411566653omplex(T,S2))
             => ( aa_complex_complex(G,X) = one_one_complex ) )
         => ( groups3708469109370488835omplex(G,S2) = groups3708469109370488835omplex(G,T) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_8219_prod_Omono__neutral__left,axiom,
    ! [T: set_complex,S2: set_complex,G: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T)
       => ( ! [X: complex] :
              ( aa_set_complex_o(member_complex(X),minus_811609699411566653omplex(T,S2))
             => ( aa_complex_real(G,X) = one_one_real ) )
         => ( groups766887009212190081x_real(G,S2) = groups766887009212190081x_real(G,T) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_8220_prod_Omono__neutral__left,axiom,
    ! [T: set_complex,S2: set_complex,G: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T)
       => ( ! [X: complex] :
              ( aa_set_complex_o(member_complex(X),minus_811609699411566653omplex(T,S2))
             => ( aa_complex_rat(G,X) = one_one_rat ) )
         => ( groups225925009352817453ex_rat(G,S2) = groups225925009352817453ex_rat(G,T) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_8221_prod_Omono__neutral__left,axiom,
    ! [T: set_complex,S2: set_complex,G: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T)
       => ( ! [X: complex] :
              ( aa_set_complex_o(member_complex(X),minus_811609699411566653omplex(T,S2))
             => ( aa_complex_nat(G,X) = one_one_nat ) )
         => ( groups861055069439313189ex_nat(G,S2) = groups861055069439313189ex_nat(G,T) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_8222_prod_Omono__neutral__left,axiom,
    ! [T: set_complex,S2: set_complex,G: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(S2),T)
       => ( ! [X: complex] :
              ( aa_set_complex_o(member_complex(X),minus_811609699411566653omplex(T,S2))
             => ( aa_complex_int(G,X) = one_one_int ) )
         => ( groups858564598930262913ex_int(G,S2) = groups858564598930262913ex_int(G,T) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_8223_prod_Omono__neutral__left,axiom,
    ! [T: set_nat,S2: set_nat,G: fun_nat_complex] :
      ( aa_set_nat_o(finite_finite_nat,T)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S2),T)
       => ( ! [X: nat] :
              ( aa_set_nat_o(member_nat(X),minus_minus_set_nat(T,S2))
             => ( aa_nat_complex(G,X) = one_one_complex ) )
         => ( aa_set_nat_complex(groups6464643781859351333omplex(G),S2) = aa_set_nat_complex(groups6464643781859351333omplex(G),T) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_8224_prod_Omono__neutral__left,axiom,
    ! [T: set_nat,S2: set_nat,G: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,T)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S2),T)
       => ( ! [X: nat] :
              ( aa_set_nat_o(member_nat(X),minus_minus_set_nat(T,S2))
             => ( aa_nat_real(G,X) = one_one_real ) )
         => ( aa_set_nat_real(groups129246275422532515t_real(G),S2) = aa_set_nat_real(groups129246275422532515t_real(G),T) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_8225_prod_Omono__neutral__left,axiom,
    ! [T: set_nat,S2: set_nat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,T)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S2),T)
       => ( ! [X: nat] :
              ( aa_set_nat_o(member_nat(X),minus_minus_set_nat(T,S2))
             => ( aa_nat_rat(G,X) = one_one_rat ) )
         => ( groups73079841787564623at_rat(G,S2) = groups73079841787564623at_rat(G,T) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_8226_prod_Omono__neutral__left,axiom,
    ! [T: set_int,S2: set_int,G: fun_int_complex] :
      ( aa_set_int_o(finite_finite_int,T)
     => ( aa_set_int_o(ord_less_eq_set_int(S2),T)
       => ( ! [X: int] :
              ( aa_set_int_o(member_int(X),minus_minus_set_int(T,S2))
             => ( aa_int_complex(G,X) = one_one_complex ) )
         => ( groups7440179247065528705omplex(G,S2) = groups7440179247065528705omplex(G,T) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_8227_prod_Omono__neutral__left,axiom,
    ! [T: set_int,S2: set_int,G: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,T)
     => ( aa_set_int_o(ord_less_eq_set_int(S2),T)
       => ( ! [X: int] :
              ( aa_set_int_o(member_int(X),minus_minus_set_int(T,S2))
             => ( aa_int_real(G,X) = one_one_real ) )
         => ( groups2316167850115554303t_real(G,S2) = groups2316167850115554303t_real(G,T) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_8228_prod_Osame__carrierI,axiom,
    ! [C2: set_real,A3: set_real,B4: set_real,G: fun_real_complex,H: fun_real_complex] :
      ( finite_finite_real(C2)
     => ( ord_less_eq_set_real(A3,C2)
       => ( ord_less_eq_set_real(B4,C2)
         => ( ! [A2: real] :
                ( member_real(A2,minus_minus_set_real(C2,A3))
               => ( aa_real_complex(G,A2) = one_one_complex ) )
           => ( ! [B2: real] :
                  ( member_real(B2,minus_minus_set_real(C2,B4))
                 => ( aa_real_complex(H,B2) = one_one_complex ) )
             => ( ( groups713298508707869441omplex(G,C2) = groups713298508707869441omplex(H,C2) )
               => ( groups713298508707869441omplex(G,A3) = groups713298508707869441omplex(H,B4) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_8229_prod_Osame__carrierI,axiom,
    ! [C2: set_complex,A3: set_complex,B4: set_complex,G: fun_complex_complex,H: fun_complex_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,C2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A3),C2)
       => ( aa_set_complex_o(ord_le211207098394363844omplex(B4),C2)
         => ( ! [A2: complex] :
                ( aa_set_complex_o(member_complex(A2),minus_811609699411566653omplex(C2,A3))
               => ( aa_complex_complex(G,A2) = one_one_complex ) )
           => ( ! [B2: complex] :
                  ( aa_set_complex_o(member_complex(B2),minus_811609699411566653omplex(C2,B4))
                 => ( aa_complex_complex(H,B2) = one_one_complex ) )
             => ( ( groups3708469109370488835omplex(G,C2) = groups3708469109370488835omplex(H,C2) )
               => ( groups3708469109370488835omplex(G,A3) = groups3708469109370488835omplex(H,B4) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_8230_prod_Osame__carrierI,axiom,
    ! [C2: set_real,A3: set_real,B4: set_real,G: fun_real_real,H: fun_real_real] :
      ( finite_finite_real(C2)
     => ( ord_less_eq_set_real(A3,C2)
       => ( ord_less_eq_set_real(B4,C2)
         => ( ! [A2: real] :
                ( member_real(A2,minus_minus_set_real(C2,A3))
               => ( aa_real_real(G,A2) = one_one_real ) )
           => ( ! [B2: real] :
                  ( member_real(B2,minus_minus_set_real(C2,B4))
                 => ( aa_real_real(H,B2) = one_one_real ) )
             => ( ( groups1681761925125756287l_real(G,C2) = groups1681761925125756287l_real(H,C2) )
               => ( groups1681761925125756287l_real(G,A3) = groups1681761925125756287l_real(H,B4) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_8231_prod_Osame__carrierI,axiom,
    ! [C2: set_complex,A3: set_complex,B4: set_complex,G: fun_complex_real,H: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,C2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A3),C2)
       => ( aa_set_complex_o(ord_le211207098394363844omplex(B4),C2)
         => ( ! [A2: complex] :
                ( aa_set_complex_o(member_complex(A2),minus_811609699411566653omplex(C2,A3))
               => ( aa_complex_real(G,A2) = one_one_real ) )
           => ( ! [B2: complex] :
                  ( aa_set_complex_o(member_complex(B2),minus_811609699411566653omplex(C2,B4))
                 => ( aa_complex_real(H,B2) = one_one_real ) )
             => ( ( groups766887009212190081x_real(G,C2) = groups766887009212190081x_real(H,C2) )
               => ( groups766887009212190081x_real(G,A3) = groups766887009212190081x_real(H,B4) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_8232_prod_Osame__carrierI,axiom,
    ! [C2: set_real,A3: set_real,B4: set_real,G: fun_real_rat,H: fun_real_rat] :
      ( finite_finite_real(C2)
     => ( ord_less_eq_set_real(A3,C2)
       => ( ord_less_eq_set_real(B4,C2)
         => ( ! [A2: real] :
                ( member_real(A2,minus_minus_set_real(C2,A3))
               => ( aa_real_rat(G,A2) = one_one_rat ) )
           => ( ! [B2: real] :
                  ( member_real(B2,minus_minus_set_real(C2,B4))
                 => ( aa_real_rat(H,B2) = one_one_rat ) )
             => ( ( groups4061424788464935467al_rat(G,C2) = groups4061424788464935467al_rat(H,C2) )
               => ( groups4061424788464935467al_rat(G,A3) = groups4061424788464935467al_rat(H,B4) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_8233_prod_Osame__carrierI,axiom,
    ! [C2: set_complex,A3: set_complex,B4: set_complex,G: fun_complex_rat,H: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,C2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A3),C2)
       => ( aa_set_complex_o(ord_le211207098394363844omplex(B4),C2)
         => ( ! [A2: complex] :
                ( aa_set_complex_o(member_complex(A2),minus_811609699411566653omplex(C2,A3))
               => ( aa_complex_rat(G,A2) = one_one_rat ) )
           => ( ! [B2: complex] :
                  ( aa_set_complex_o(member_complex(B2),minus_811609699411566653omplex(C2,B4))
                 => ( aa_complex_rat(H,B2) = one_one_rat ) )
             => ( ( groups225925009352817453ex_rat(G,C2) = groups225925009352817453ex_rat(H,C2) )
               => ( groups225925009352817453ex_rat(G,A3) = groups225925009352817453ex_rat(H,B4) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_8234_prod_Osame__carrierI,axiom,
    ! [C2: set_real,A3: set_real,B4: set_real,G: fun_real_nat,H: fun_real_nat] :
      ( finite_finite_real(C2)
     => ( ord_less_eq_set_real(A3,C2)
       => ( ord_less_eq_set_real(B4,C2)
         => ( ! [A2: real] :
                ( member_real(A2,minus_minus_set_real(C2,A3))
               => ( aa_real_nat(G,A2) = one_one_nat ) )
           => ( ! [B2: real] :
                  ( member_real(B2,minus_minus_set_real(C2,B4))
                 => ( aa_real_nat(H,B2) = one_one_nat ) )
             => ( ( groups4696554848551431203al_nat(G,C2) = groups4696554848551431203al_nat(H,C2) )
               => ( groups4696554848551431203al_nat(G,A3) = groups4696554848551431203al_nat(H,B4) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_8235_prod_Osame__carrierI,axiom,
    ! [C2: set_complex,A3: set_complex,B4: set_complex,G: fun_complex_nat,H: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,C2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A3),C2)
       => ( aa_set_complex_o(ord_le211207098394363844omplex(B4),C2)
         => ( ! [A2: complex] :
                ( aa_set_complex_o(member_complex(A2),minus_811609699411566653omplex(C2,A3))
               => ( aa_complex_nat(G,A2) = one_one_nat ) )
           => ( ! [B2: complex] :
                  ( aa_set_complex_o(member_complex(B2),minus_811609699411566653omplex(C2,B4))
                 => ( aa_complex_nat(H,B2) = one_one_nat ) )
             => ( ( groups861055069439313189ex_nat(G,C2) = groups861055069439313189ex_nat(H,C2) )
               => ( groups861055069439313189ex_nat(G,A3) = groups861055069439313189ex_nat(H,B4) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_8236_prod_Osame__carrierI,axiom,
    ! [C2: set_real,A3: set_real,B4: set_real,G: fun_real_int,H: fun_real_int] :
      ( finite_finite_real(C2)
     => ( ord_less_eq_set_real(A3,C2)
       => ( ord_less_eq_set_real(B4,C2)
         => ( ! [A2: real] :
                ( member_real(A2,minus_minus_set_real(C2,A3))
               => ( aa_real_int(G,A2) = one_one_int ) )
           => ( ! [B2: real] :
                  ( member_real(B2,minus_minus_set_real(C2,B4))
                 => ( aa_real_int(H,B2) = one_one_int ) )
             => ( ( groups4694064378042380927al_int(G,C2) = groups4694064378042380927al_int(H,C2) )
               => ( groups4694064378042380927al_int(G,A3) = groups4694064378042380927al_int(H,B4) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_8237_prod_Osame__carrierI,axiom,
    ! [C2: set_complex,A3: set_complex,B4: set_complex,G: fun_complex_int,H: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,C2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A3),C2)
       => ( aa_set_complex_o(ord_le211207098394363844omplex(B4),C2)
         => ( ! [A2: complex] :
                ( aa_set_complex_o(member_complex(A2),minus_811609699411566653omplex(C2,A3))
               => ( aa_complex_int(G,A2) = one_one_int ) )
           => ( ! [B2: complex] :
                  ( aa_set_complex_o(member_complex(B2),minus_811609699411566653omplex(C2,B4))
                 => ( aa_complex_int(H,B2) = one_one_int ) )
             => ( ( groups858564598930262913ex_int(G,C2) = groups858564598930262913ex_int(H,C2) )
               => ( groups858564598930262913ex_int(G,A3) = groups858564598930262913ex_int(H,B4) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_8238_prod_Osame__carrier,axiom,
    ! [C2: set_real,A3: set_real,B4: set_real,G: fun_real_complex,H: fun_real_complex] :
      ( finite_finite_real(C2)
     => ( ord_less_eq_set_real(A3,C2)
       => ( ord_less_eq_set_real(B4,C2)
         => ( ! [A2: real] :
                ( member_real(A2,minus_minus_set_real(C2,A3))
               => ( aa_real_complex(G,A2) = one_one_complex ) )
           => ( ! [B2: real] :
                  ( member_real(B2,minus_minus_set_real(C2,B4))
                 => ( aa_real_complex(H,B2) = one_one_complex ) )
             => ( ( groups713298508707869441omplex(G,A3) = groups713298508707869441omplex(H,B4) )
              <=> ( groups713298508707869441omplex(G,C2) = groups713298508707869441omplex(H,C2) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_8239_prod_Osame__carrier,axiom,
    ! [C2: set_complex,A3: set_complex,B4: set_complex,G: fun_complex_complex,H: fun_complex_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,C2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A3),C2)
       => ( aa_set_complex_o(ord_le211207098394363844omplex(B4),C2)
         => ( ! [A2: complex] :
                ( aa_set_complex_o(member_complex(A2),minus_811609699411566653omplex(C2,A3))
               => ( aa_complex_complex(G,A2) = one_one_complex ) )
           => ( ! [B2: complex] :
                  ( aa_set_complex_o(member_complex(B2),minus_811609699411566653omplex(C2,B4))
                 => ( aa_complex_complex(H,B2) = one_one_complex ) )
             => ( ( groups3708469109370488835omplex(G,A3) = groups3708469109370488835omplex(H,B4) )
              <=> ( groups3708469109370488835omplex(G,C2) = groups3708469109370488835omplex(H,C2) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_8240_prod_Osame__carrier,axiom,
    ! [C2: set_real,A3: set_real,B4: set_real,G: fun_real_real,H: fun_real_real] :
      ( finite_finite_real(C2)
     => ( ord_less_eq_set_real(A3,C2)
       => ( ord_less_eq_set_real(B4,C2)
         => ( ! [A2: real] :
                ( member_real(A2,minus_minus_set_real(C2,A3))
               => ( aa_real_real(G,A2) = one_one_real ) )
           => ( ! [B2: real] :
                  ( member_real(B2,minus_minus_set_real(C2,B4))
                 => ( aa_real_real(H,B2) = one_one_real ) )
             => ( ( groups1681761925125756287l_real(G,A3) = groups1681761925125756287l_real(H,B4) )
              <=> ( groups1681761925125756287l_real(G,C2) = groups1681761925125756287l_real(H,C2) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_8241_prod_Osame__carrier,axiom,
    ! [C2: set_complex,A3: set_complex,B4: set_complex,G: fun_complex_real,H: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,C2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A3),C2)
       => ( aa_set_complex_o(ord_le211207098394363844omplex(B4),C2)
         => ( ! [A2: complex] :
                ( aa_set_complex_o(member_complex(A2),minus_811609699411566653omplex(C2,A3))
               => ( aa_complex_real(G,A2) = one_one_real ) )
           => ( ! [B2: complex] :
                  ( aa_set_complex_o(member_complex(B2),minus_811609699411566653omplex(C2,B4))
                 => ( aa_complex_real(H,B2) = one_one_real ) )
             => ( ( groups766887009212190081x_real(G,A3) = groups766887009212190081x_real(H,B4) )
              <=> ( groups766887009212190081x_real(G,C2) = groups766887009212190081x_real(H,C2) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_8242_prod_Osame__carrier,axiom,
    ! [C2: set_real,A3: set_real,B4: set_real,G: fun_real_rat,H: fun_real_rat] :
      ( finite_finite_real(C2)
     => ( ord_less_eq_set_real(A3,C2)
       => ( ord_less_eq_set_real(B4,C2)
         => ( ! [A2: real] :
                ( member_real(A2,minus_minus_set_real(C2,A3))
               => ( aa_real_rat(G,A2) = one_one_rat ) )
           => ( ! [B2: real] :
                  ( member_real(B2,minus_minus_set_real(C2,B4))
                 => ( aa_real_rat(H,B2) = one_one_rat ) )
             => ( ( groups4061424788464935467al_rat(G,A3) = groups4061424788464935467al_rat(H,B4) )
              <=> ( groups4061424788464935467al_rat(G,C2) = groups4061424788464935467al_rat(H,C2) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_8243_prod_Osame__carrier,axiom,
    ! [C2: set_complex,A3: set_complex,B4: set_complex,G: fun_complex_rat,H: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,C2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A3),C2)
       => ( aa_set_complex_o(ord_le211207098394363844omplex(B4),C2)
         => ( ! [A2: complex] :
                ( aa_set_complex_o(member_complex(A2),minus_811609699411566653omplex(C2,A3))
               => ( aa_complex_rat(G,A2) = one_one_rat ) )
           => ( ! [B2: complex] :
                  ( aa_set_complex_o(member_complex(B2),minus_811609699411566653omplex(C2,B4))
                 => ( aa_complex_rat(H,B2) = one_one_rat ) )
             => ( ( groups225925009352817453ex_rat(G,A3) = groups225925009352817453ex_rat(H,B4) )
              <=> ( groups225925009352817453ex_rat(G,C2) = groups225925009352817453ex_rat(H,C2) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_8244_prod_Osame__carrier,axiom,
    ! [C2: set_real,A3: set_real,B4: set_real,G: fun_real_nat,H: fun_real_nat] :
      ( finite_finite_real(C2)
     => ( ord_less_eq_set_real(A3,C2)
       => ( ord_less_eq_set_real(B4,C2)
         => ( ! [A2: real] :
                ( member_real(A2,minus_minus_set_real(C2,A3))
               => ( aa_real_nat(G,A2) = one_one_nat ) )
           => ( ! [B2: real] :
                  ( member_real(B2,minus_minus_set_real(C2,B4))
                 => ( aa_real_nat(H,B2) = one_one_nat ) )
             => ( ( groups4696554848551431203al_nat(G,A3) = groups4696554848551431203al_nat(H,B4) )
              <=> ( groups4696554848551431203al_nat(G,C2) = groups4696554848551431203al_nat(H,C2) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_8245_prod_Osame__carrier,axiom,
    ! [C2: set_complex,A3: set_complex,B4: set_complex,G: fun_complex_nat,H: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,C2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A3),C2)
       => ( aa_set_complex_o(ord_le211207098394363844omplex(B4),C2)
         => ( ! [A2: complex] :
                ( aa_set_complex_o(member_complex(A2),minus_811609699411566653omplex(C2,A3))
               => ( aa_complex_nat(G,A2) = one_one_nat ) )
           => ( ! [B2: complex] :
                  ( aa_set_complex_o(member_complex(B2),minus_811609699411566653omplex(C2,B4))
                 => ( aa_complex_nat(H,B2) = one_one_nat ) )
             => ( ( groups861055069439313189ex_nat(G,A3) = groups861055069439313189ex_nat(H,B4) )
              <=> ( groups861055069439313189ex_nat(G,C2) = groups861055069439313189ex_nat(H,C2) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_8246_prod_Osame__carrier,axiom,
    ! [C2: set_real,A3: set_real,B4: set_real,G: fun_real_int,H: fun_real_int] :
      ( finite_finite_real(C2)
     => ( ord_less_eq_set_real(A3,C2)
       => ( ord_less_eq_set_real(B4,C2)
         => ( ! [A2: real] :
                ( member_real(A2,minus_minus_set_real(C2,A3))
               => ( aa_real_int(G,A2) = one_one_int ) )
           => ( ! [B2: real] :
                  ( member_real(B2,minus_minus_set_real(C2,B4))
                 => ( aa_real_int(H,B2) = one_one_int ) )
             => ( ( groups4694064378042380927al_int(G,A3) = groups4694064378042380927al_int(H,B4) )
              <=> ( groups4694064378042380927al_int(G,C2) = groups4694064378042380927al_int(H,C2) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_8247_prod_Osame__carrier,axiom,
    ! [C2: set_complex,A3: set_complex,B4: set_complex,G: fun_complex_int,H: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,C2)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A3),C2)
       => ( aa_set_complex_o(ord_le211207098394363844omplex(B4),C2)
         => ( ! [A2: complex] :
                ( aa_set_complex_o(member_complex(A2),minus_811609699411566653omplex(C2,A3))
               => ( aa_complex_int(G,A2) = one_one_int ) )
           => ( ! [B2: complex] :
                  ( aa_set_complex_o(member_complex(B2),minus_811609699411566653omplex(C2,B4))
                 => ( aa_complex_int(H,B2) = one_one_int ) )
             => ( ( groups858564598930262913ex_int(G,A3) = groups858564598930262913ex_int(H,B4) )
              <=> ( groups858564598930262913ex_int(G,C2) = groups858564598930262913ex_int(H,C2) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_8248_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_real,Na: nat] : aa_set_nat_real(groups129246275422532515t_real(G),set_or1269000886237332187st_nat(zero_zero_nat,aa_nat_nat(suc,Na))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_set_nat_real(groups129246275422532515t_real(G),set_or1269000886237332187st_nat(zero_zero_nat,Na))),aa_nat_real(G,aa_nat_nat(suc,Na))) ).

% prod.atLeast0_atMost_Suc
tff(fact_8249_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_rat,Na: nat] : groups73079841787564623at_rat(G,set_or1269000886237332187st_nat(zero_zero_nat,aa_nat_nat(suc,Na))) = aa_rat_rat(times_times_rat(groups73079841787564623at_rat(G,set_or1269000886237332187st_nat(zero_zero_nat,Na))),aa_nat_rat(G,aa_nat_nat(suc,Na))) ).

% prod.atLeast0_atMost_Suc
tff(fact_8250_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_nat,Na: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),set_or1269000886237332187st_nat(zero_zero_nat,aa_nat_nat(suc,Na))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups708209901874060359at_nat(G),set_or1269000886237332187st_nat(zero_zero_nat,Na))),aa_nat_nat(G,aa_nat_nat(suc,Na))) ).

% prod.atLeast0_atMost_Suc
tff(fact_8251_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_int,Na: nat] : groups705719431365010083at_int(G,set_or1269000886237332187st_nat(zero_zero_nat,aa_nat_nat(suc,Na))) = aa_int_int(aa_int_fun_int_int(times_times_int,groups705719431365010083at_int(G,set_or1269000886237332187st_nat(zero_zero_nat,Na))),aa_nat_int(G,aa_nat_nat(suc,Na))) ).

% prod.atLeast0_atMost_Suc
tff(fact_8252_prod_OatLeast__Suc__atMost,axiom,
    ! [M: nat,Na: nat,G: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( aa_set_nat_real(groups129246275422532515t_real(G),set_or1269000886237332187st_nat(M,Na)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(G,M)),aa_set_nat_real(groups129246275422532515t_real(G),set_or1269000886237332187st_nat(aa_nat_nat(suc,M),Na))) ) ) ).

% prod.atLeast_Suc_atMost
tff(fact_8253_prod_OatLeast__Suc__atMost,axiom,
    ! [M: nat,Na: nat,G: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( groups73079841787564623at_rat(G,set_or1269000886237332187st_nat(M,Na)) = aa_rat_rat(times_times_rat(aa_nat_rat(G,M)),groups73079841787564623at_rat(G,set_or1269000886237332187st_nat(aa_nat_nat(suc,M),Na))) ) ) ).

% prod.atLeast_Suc_atMost
tff(fact_8254_prod_OatLeast__Suc__atMost,axiom,
    ! [M: nat,Na: nat,G: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( aa_set_nat_nat(groups708209901874060359at_nat(G),set_or1269000886237332187st_nat(M,Na)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(G,M)),aa_set_nat_nat(groups708209901874060359at_nat(G),set_or1269000886237332187st_nat(aa_nat_nat(suc,M),Na))) ) ) ).

% prod.atLeast_Suc_atMost
tff(fact_8255_prod_OatLeast__Suc__atMost,axiom,
    ! [M: nat,Na: nat,G: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( groups705719431365010083at_int(G,set_or1269000886237332187st_nat(M,Na)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(G,M)),groups705719431365010083at_int(G,set_or1269000886237332187st_nat(aa_nat_nat(suc,M),Na))) ) ) ).

% prod.atLeast_Suc_atMost
tff(fact_8256_prod_Onat__ivl__Suc_H,axiom,
    ! [M: nat,Na: nat,G: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,Na))
     => ( aa_set_nat_real(groups129246275422532515t_real(G),set_or1269000886237332187st_nat(M,aa_nat_nat(suc,Na))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(G,aa_nat_nat(suc,Na))),aa_set_nat_real(groups129246275422532515t_real(G),set_or1269000886237332187st_nat(M,Na))) ) ) ).

% prod.nat_ivl_Suc'
tff(fact_8257_prod_Onat__ivl__Suc_H,axiom,
    ! [M: nat,Na: nat,G: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,Na))
     => ( groups73079841787564623at_rat(G,set_or1269000886237332187st_nat(M,aa_nat_nat(suc,Na))) = aa_rat_rat(times_times_rat(aa_nat_rat(G,aa_nat_nat(suc,Na))),groups73079841787564623at_rat(G,set_or1269000886237332187st_nat(M,Na))) ) ) ).

% prod.nat_ivl_Suc'
tff(fact_8258_prod_Onat__ivl__Suc_H,axiom,
    ! [M: nat,Na: nat,G: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,Na))
     => ( aa_set_nat_nat(groups708209901874060359at_nat(G),set_or1269000886237332187st_nat(M,aa_nat_nat(suc,Na))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(G,aa_nat_nat(suc,Na))),aa_set_nat_nat(groups708209901874060359at_nat(G),set_or1269000886237332187st_nat(M,Na))) ) ) ).

% prod.nat_ivl_Suc'
tff(fact_8259_prod_Onat__ivl__Suc_H,axiom,
    ! [M: nat,Na: nat,G: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,Na))
     => ( groups705719431365010083at_int(G,set_or1269000886237332187st_nat(M,aa_nat_nat(suc,Na))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(G,aa_nat_nat(suc,Na))),groups705719431365010083at_int(G,set_or1269000886237332187st_nat(M,Na))) ) ) ).

% prod.nat_ivl_Suc'
tff(fact_8260_prod_OSuc__reindex__ivl,axiom,
    ! [M: nat,Na: nat,G: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_set_nat_real(groups129246275422532515t_real(G),set_or1269000886237332187st_nat(M,Na))),aa_nat_real(G,aa_nat_nat(suc,Na))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(G,M)),aa_set_nat_real(groups129246275422532515t_real(aTP_Lamm_jg(G)),set_or1269000886237332187st_nat(M,Na))) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_8261_prod_OSuc__reindex__ivl,axiom,
    ! [M: nat,Na: nat,G: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( aa_rat_rat(times_times_rat(groups73079841787564623at_rat(G,set_or1269000886237332187st_nat(M,Na))),aa_nat_rat(G,aa_nat_nat(suc,Na))) = aa_rat_rat(times_times_rat(aa_nat_rat(G,M)),groups73079841787564623at_rat(aTP_Lamm_jw(G),set_or1269000886237332187st_nat(M,Na))) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_8262_prod_OSuc__reindex__ivl,axiom,
    ! [M: nat,Na: nat,G: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups708209901874060359at_nat(G),set_or1269000886237332187st_nat(M,Na))),aa_nat_nat(G,aa_nat_nat(suc,Na))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(G,M)),aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_jf(G)),set_or1269000886237332187st_nat(M,Na))) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_8263_prod_OSuc__reindex__ivl,axiom,
    ! [M: nat,Na: nat,G: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,groups705719431365010083at_int(G,set_or1269000886237332187st_nat(M,Na))),aa_nat_int(G,aa_nat_nat(suc,Na))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(G,M)),groups705719431365010083at_int(aTP_Lamm_jx(G),set_or1269000886237332187st_nat(M,Na))) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_8264_fact__prod,axiom,
    ! [Na: nat] : aa_nat_int(semiri1406184849735516958ct_int,Na) = aa_nat_int(semiri1314217659103216013at_int,aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_cc),set_or1269000886237332187st_nat(one_one_nat,Na))) ).

% fact_prod
tff(fact_8265_fact__prod,axiom,
    ! [Na: nat] : aa_nat_Code_integer(semiri3624122377584611663nteger,Na) = aa_nat_Code_integer(semiri4939895301339042750nteger,aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_cc),set_or1269000886237332187st_nat(one_one_nat,Na))) ).

% fact_prod
tff(fact_8266_fact__prod,axiom,
    ! [Na: nat] : aa_nat_nat(semiri1408675320244567234ct_nat,Na) = aa_nat_nat(semiri1316708129612266289at_nat,aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_cc),set_or1269000886237332187st_nat(one_one_nat,Na))) ).

% fact_prod
tff(fact_8267_fact__prod,axiom,
    ! [Na: nat] : aa_nat_real(semiri2265585572941072030t_real,Na) = aa_nat_real(semiri5074537144036343181t_real,aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_cc),set_or1269000886237332187st_nat(one_one_nat,Na))) ).

% fact_prod
tff(fact_8268_prod__atLeastAtMost__code,axiom,
    ! [F: fun_nat_complex,Aa: nat,Ba: nat] : aa_set_nat_complex(groups6464643781859351333omplex(F),set_or1269000886237332187st_nat(Aa,Ba)) = aa_complex_complex(aa_nat3829984526574172007omplex(set_fo1517530859248394432omplex(aTP_Lamm_oi(F),Aa),Ba),one_one_complex) ).

% prod_atLeastAtMost_code
tff(fact_8269_prod__atLeastAtMost__code,axiom,
    ! [F: fun_nat_real,Aa: nat,Ba: nat] : aa_set_nat_real(groups129246275422532515t_real(F),set_or1269000886237332187st_nat(Aa,Ba)) = aa_real_real(aa_nat_fun_real_real(set_fo3111899725591712190t_real(aTP_Lamm_oj(F),Aa),Ba),one_one_real) ).

% prod_atLeastAtMost_code
tff(fact_8270_prod__atLeastAtMost__code,axiom,
    ! [F: fun_nat_rat,Aa: nat,Ba: nat] : groups73079841787564623at_rat(F,set_or1269000886237332187st_nat(Aa,Ba)) = aa_rat_rat(set_fo1949268297981939178at_rat(aTP_Lamm_ok(F),Aa,Ba),one_one_rat) ).

% prod_atLeastAtMost_code
tff(fact_8271_prod__atLeastAtMost__code,axiom,
    ! [F: fun_nat_nat,Aa: nat,Ba: nat] : aa_set_nat_nat(groups708209901874060359at_nat(F),set_or1269000886237332187st_nat(Aa,Ba)) = aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(aTP_Lamm_ol(F),Aa),Ba),one_one_nat) ).

% prod_atLeastAtMost_code
tff(fact_8272_prod__atLeastAtMost__code,axiom,
    ! [F: fun_nat_int,Aa: nat,Ba: nat] : groups705719431365010083at_int(F,set_or1269000886237332187st_nat(Aa,Ba)) = aa_int_int(aa_nat_fun_int_int(set_fo2581907887559384638at_int(aTP_Lamm_om(F),Aa),Ba),one_one_int) ).

% prod_atLeastAtMost_code
tff(fact_8273_prod__mono__strict,axiom,
    ! [A3: set_complex,F: fun_complex_real,G: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A3)
     => ( ! [I2: complex] :
            ( aa_set_complex_o(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_real,aa_complex_real(F,I2)),aa_complex_real(G,I2)) ) )
       => ( ( A3 != bot_bot_set_complex )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,groups766887009212190081x_real(F,A3)),groups766887009212190081x_real(G,A3)) ) ) ) ).

% prod_mono_strict
tff(fact_8274_prod__mono__strict,axiom,
    ! [A3: set_real,F: fun_real_real,G: fun_real_real] :
      ( finite_finite_real(A3)
     => ( ! [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_real,aa_real_real(F,I2)),aa_real_real(G,I2)) ) )
       => ( ( A3 != bot_bot_set_real )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,groups1681761925125756287l_real(F,A3)),groups1681761925125756287l_real(G,A3)) ) ) ) ).

% prod_mono_strict
tff(fact_8275_prod__mono__strict,axiom,
    ! [A3: set_nat,F: fun_nat_real,G: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( ! [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_real,aa_nat_real(F,I2)),aa_nat_real(G,I2)) ) )
       => ( ( A3 != bot_bot_set_nat )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_set_nat_real(groups129246275422532515t_real(F),A3)),aa_set_nat_real(groups129246275422532515t_real(G),A3)) ) ) ) ).

% prod_mono_strict
tff(fact_8276_prod__mono__strict,axiom,
    ! [A3: set_int,F: fun_int_real,G: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( ! [I2: int] :
            ( aa_set_int_o(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_real,aa_int_real(F,I2)),aa_int_real(G,I2)) ) )
       => ( ( A3 != bot_bot_set_int )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,groups2316167850115554303t_real(F,A3)),groups2316167850115554303t_real(G,A3)) ) ) ) ).

% prod_mono_strict
tff(fact_8277_prod__mono__strict,axiom,
    ! [A3: set_complex,F: fun_complex_rat,G: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A3)
     => ( ! [I2: complex] :
            ( aa_set_complex_o(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_rat,aa_complex_rat(F,I2)),aa_complex_rat(G,I2)) ) )
       => ( ( A3 != bot_bot_set_complex )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups225925009352817453ex_rat(F,A3)),groups225925009352817453ex_rat(G,A3)) ) ) ) ).

% prod_mono_strict
tff(fact_8278_prod__mono__strict,axiom,
    ! [A3: set_real,F: fun_real_rat,G: fun_real_rat] :
      ( finite_finite_real(A3)
     => ( ! [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_rat,aa_real_rat(F,I2)),aa_real_rat(G,I2)) ) )
       => ( ( A3 != bot_bot_set_real )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups4061424788464935467al_rat(F,A3)),groups4061424788464935467al_rat(G,A3)) ) ) ) ).

% prod_mono_strict
tff(fact_8279_prod__mono__strict,axiom,
    ! [A3: set_nat,F: fun_nat_rat,G: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( ! [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_rat,aa_nat_rat(F,I2)),aa_nat_rat(G,I2)) ) )
       => ( ( A3 != bot_bot_set_nat )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups73079841787564623at_rat(F,A3)),groups73079841787564623at_rat(G,A3)) ) ) ) ).

% prod_mono_strict
tff(fact_8280_prod__mono__strict,axiom,
    ! [A3: set_int,F: fun_int_rat,G: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( ! [I2: int] :
            ( aa_set_int_o(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_rat,aa_int_rat(F,I2)),aa_int_rat(G,I2)) ) )
       => ( ( A3 != bot_bot_set_int )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,groups1072433553688619179nt_rat(F,A3)),groups1072433553688619179nt_rat(G,A3)) ) ) ) ).

% prod_mono_strict
tff(fact_8281_prod__mono__strict,axiom,
    ! [A3: set_complex,F: fun_complex_nat,G: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A3)
     => ( ! [I2: complex] :
            ( aa_set_complex_o(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_nat,aa_complex_nat(F,I2)),aa_complex_nat(G,I2)) ) )
       => ( ( A3 != bot_bot_set_complex )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups861055069439313189ex_nat(F,A3)),groups861055069439313189ex_nat(G,A3)) ) ) ) ).

% prod_mono_strict
tff(fact_8282_prod__mono__strict,axiom,
    ! [A3: set_real,F: fun_real_nat,G: fun_real_nat] :
      ( finite_finite_real(A3)
     => ( ! [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_nat,aa_real_nat(F,I2)),aa_real_nat(G,I2)) ) )
       => ( ( A3 != bot_bot_set_real )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,groups4696554848551431203al_nat(F,A3)),groups4696554848551431203al_nat(G,A3)) ) ) ) ).

% prod_mono_strict
tff(fact_8283_even__prod__iff,axiom,
    ! [A3: set_int,F: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),groups1707563613775114915nt_nat(F,A3))
      <=> ? [X4: int] :
            ( aa_set_int_o(member_int(X4),A3)
            & 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(F,X4)) ) ) ) ).

% even_prod_iff
tff(fact_8284_even__prod__iff,axiom,
    ! [A3: set_complex,F: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A3)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),groups861055069439313189ex_nat(F,A3))
      <=> ? [X4: complex] :
            ( aa_set_complex_o(member_complex(X4),A3)
            & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_complex_nat(F,X4)) ) ) ) ).

% even_prod_iff
tff(fact_8285_even__prod__iff,axiom,
    ! [A3: set_complex,F: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A3)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),groups858564598930262913ex_int(F,A3))
      <=> ? [X4: complex] :
            ( aa_set_complex_o(member_complex(X4),A3)
            & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_complex_int(F,X4)) ) ) ) ).

% even_prod_iff
tff(fact_8286_even__prod__iff,axiom,
    ! [A3: set_nat,F: fun_nat_Code_integer] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),groups3455450783089532116nteger(F,A3))
      <=> ? [X4: nat] :
            ( aa_set_nat_o(member_nat(X4),A3)
            & aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_nat_Code_integer(F,X4)) ) ) ) ).

% even_prod_iff
tff(fact_8287_even__prod__iff,axiom,
    ! [A3: set_int,F: fun_int_Code_integer] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),groups3827104343326376752nteger(F,A3))
      <=> ? [X4: int] :
            ( aa_set_int_o(member_int(X4),A3)
            & aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_int_Code_integer(F,X4)) ) ) ) ).

% even_prod_iff
tff(fact_8288_even__prod__iff,axiom,
    ! [A3: set_complex,F: fun_co8153928105989009288nteger] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A3)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),groups8682486955453173170nteger(F,A3))
      <=> ? [X4: complex] :
            ( aa_set_complex_o(member_complex(X4),A3)
            & aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_com4217951221296830281nteger(F,X4)) ) ) ) ).

% even_prod_iff
tff(fact_8289_even__prod__iff,axiom,
    ! [A3: set_nat,F: fun_nat_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(groups708209901874060359at_nat(F),A3))
      <=> ? [X4: nat] :
            ( aa_set_nat_o(member_nat(X4),A3)
            & 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(F,X4)) ) ) ) ).

% even_prod_iff
tff(fact_8290_even__prod__iff,axiom,
    ! [A3: set_nat,F: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),groups705719431365010083at_int(F,A3))
      <=> ? [X4: nat] :
            ( aa_set_nat_o(member_nat(X4),A3)
            & 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(F,X4)) ) ) ) ).

% even_prod_iff
tff(fact_8291_even__prod__iff,axiom,
    ! [A3: set_int,F: fun_int_int] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),groups1705073143266064639nt_int(F,A3))
      <=> ? [X4: int] :
            ( aa_set_int_o(member_int(X4),A3)
            & 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(F,X4)) ) ) ) ).

% even_prod_iff
tff(fact_8292_prod_Oub__add__nat,axiom,
    ! [M: nat,Na: nat,G: fun_nat_real,P3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),one_one_nat))
     => ( aa_set_nat_real(groups129246275422532515t_real(G),set_or1269000886237332187st_nat(M,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),P3))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_set_nat_real(groups129246275422532515t_real(G),set_or1269000886237332187st_nat(M,Na))),aa_set_nat_real(groups129246275422532515t_real(G),set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),P3)))) ) ) ).

% prod.ub_add_nat
tff(fact_8293_prod_Oub__add__nat,axiom,
    ! [M: nat,Na: nat,G: fun_nat_rat,P3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),one_one_nat))
     => ( groups73079841787564623at_rat(G,set_or1269000886237332187st_nat(M,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),P3))) = aa_rat_rat(times_times_rat(groups73079841787564623at_rat(G,set_or1269000886237332187st_nat(M,Na))),groups73079841787564623at_rat(G,set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),P3)))) ) ) ).

% prod.ub_add_nat
tff(fact_8294_prod_Oub__add__nat,axiom,
    ! [M: nat,Na: nat,G: fun_nat_nat,P3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),one_one_nat))
     => ( aa_set_nat_nat(groups708209901874060359at_nat(G),set_or1269000886237332187st_nat(M,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),P3))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups708209901874060359at_nat(G),set_or1269000886237332187st_nat(M,Na))),aa_set_nat_nat(groups708209901874060359at_nat(G),set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),P3)))) ) ) ).

% prod.ub_add_nat
tff(fact_8295_prod_Oub__add__nat,axiom,
    ! [M: nat,Na: nat,G: fun_nat_int,P3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),one_one_nat))
     => ( groups705719431365010083at_int(G,set_or1269000886237332187st_nat(M,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),P3))) = aa_int_int(aa_int_fun_int_int(times_times_int,groups705719431365010083at_int(G,set_or1269000886237332187st_nat(M,Na))),groups705719431365010083at_int(G,set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),P3)))) ) ) ).

% prod.ub_add_nat
tff(fact_8296_fact__eq__fact__times,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
     => ( aa_nat_nat(semiri1408675320244567234ct_nat,M) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Na)),aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_cc),set_or1269000886237332187st_nat(aa_nat_nat(suc,Na),M))) ) ) ).

% fact_eq_fact_times
tff(fact_8297_prod__mono2,axiom,
    ! [B4: set_real,A3: set_real,F: fun_real_real] :
      ( finite_finite_real(B4)
     => ( ord_less_eq_set_real(A3,B4)
       => ( ! [B2: real] :
              ( member_real(B2,minus_minus_set_real(B4,A3))
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(F,B2)) )
         => ( ! [A2: real] :
                ( member_real(A2,A3)
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,A2)) )
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups1681761925125756287l_real(F,A3)),groups1681761925125756287l_real(F,B4)) ) ) ) ) ).

% prod_mono2
tff(fact_8298_prod__mono2,axiom,
    ! [B4: set_complex,A3: set_complex,F: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A3),B4)
       => ( ! [B2: complex] :
              ( aa_set_complex_o(member_complex(B2),minus_811609699411566653omplex(B4,A3))
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_complex_real(F,B2)) )
         => ( ! [A2: complex] :
                ( aa_set_complex_o(member_complex(A2),A3)
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(F,A2)) )
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups766887009212190081x_real(F,A3)),groups766887009212190081x_real(F,B4)) ) ) ) ) ).

% prod_mono2
tff(fact_8299_prod__mono2,axiom,
    ! [B4: set_nat,A3: set_nat,F: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,B4)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A3),B4)
       => ( ! [B2: nat] :
              ( aa_set_nat_o(member_nat(B2),minus_minus_set_nat(B4,A3))
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_nat_real(F,B2)) )
         => ( ! [A2: nat] :
                ( aa_set_nat_o(member_nat(A2),A3)
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,A2)) )
           => 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(F),B4)) ) ) ) ) ).

% prod_mono2
tff(fact_8300_prod__mono2,axiom,
    ! [B4: set_real,A3: set_real,F: fun_real_rat] :
      ( finite_finite_real(B4)
     => ( ord_less_eq_set_real(A3,B4)
       => ( ! [B2: real] :
              ( member_real(B2,minus_minus_set_real(B4,A3))
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_real_rat(F,B2)) )
         => ( ! [A2: real] :
                ( member_real(A2,A3)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,A2)) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups4061424788464935467al_rat(F,A3)),groups4061424788464935467al_rat(F,B4)) ) ) ) ) ).

% prod_mono2
tff(fact_8301_prod__mono2,axiom,
    ! [B4: set_complex,A3: set_complex,F: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A3),B4)
       => ( ! [B2: complex] :
              ( aa_set_complex_o(member_complex(B2),minus_811609699411566653omplex(B4,A3))
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_complex_rat(F,B2)) )
         => ( ! [A2: complex] :
                ( aa_set_complex_o(member_complex(A2),A3)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,A2)) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups225925009352817453ex_rat(F,A3)),groups225925009352817453ex_rat(F,B4)) ) ) ) ) ).

% prod_mono2
tff(fact_8302_prod__mono2,axiom,
    ! [B4: set_nat,A3: set_nat,F: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,B4)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A3),B4)
       => ( ! [B2: nat] :
              ( aa_set_nat_o(member_nat(B2),minus_minus_set_nat(B4,A3))
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_nat_rat(F,B2)) )
         => ( ! [A2: nat] :
                ( aa_set_nat_o(member_nat(A2),A3)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,A2)) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups73079841787564623at_rat(F,A3)),groups73079841787564623at_rat(F,B4)) ) ) ) ) ).

% prod_mono2
tff(fact_8303_prod__mono2,axiom,
    ! [B4: set_real,A3: set_real,F: fun_real_int] :
      ( finite_finite_real(B4)
     => ( ord_less_eq_set_real(A3,B4)
       => ( ! [B2: real] :
              ( member_real(B2,minus_minus_set_real(B4,A3))
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_real_int(F,B2)) )
         => ( ! [A2: real] :
                ( member_real(A2,A3)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_real_int(F,A2)) )
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups4694064378042380927al_int(F,A3)),groups4694064378042380927al_int(F,B4)) ) ) ) ) ).

% prod_mono2
tff(fact_8304_prod__mono2,axiom,
    ! [B4: set_complex,A3: set_complex,F: fun_complex_int] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B4)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A3),B4)
       => ( ! [B2: complex] :
              ( aa_set_complex_o(member_complex(B2),minus_811609699411566653omplex(B4,A3))
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_complex_int(F,B2)) )
         => ( ! [A2: complex] :
                ( aa_set_complex_o(member_complex(A2),A3)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_complex_int(F,A2)) )
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups858564598930262913ex_int(F,A3)),groups858564598930262913ex_int(F,B4)) ) ) ) ) ).

% prod_mono2
tff(fact_8305_prod__mono2,axiom,
    ! [B4: set_int,A3: set_int,F: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,B4)
     => ( aa_set_int_o(ord_less_eq_set_int(A3),B4)
       => ( ! [B2: int] :
              ( aa_set_int_o(member_int(B2),minus_minus_set_int(B4,A3))
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_int_real(F,B2)) )
         => ( ! [A2: int] :
                ( aa_set_int_o(member_int(A2),A3)
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(F,A2)) )
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups2316167850115554303t_real(F,A3)),groups2316167850115554303t_real(F,B4)) ) ) ) ) ).

% prod_mono2
tff(fact_8306_prod__mono2,axiom,
    ! [B4: set_int,A3: set_int,F: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,B4)
     => ( aa_set_int_o(ord_less_eq_set_int(A3),B4)
       => ( ! [B2: int] :
              ( aa_set_int_o(member_int(B2),minus_minus_set_int(B4,A3))
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_int_rat(F,B2)) )
         => ( ! [A2: int] :
                ( aa_set_int_o(member_int(A2),A3)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,A2)) )
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups1072433553688619179nt_rat(F,A3)),groups1072433553688619179nt_rat(F,B4)) ) ) ) ) ).

% prod_mono2
tff(fact_8307_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_8308_pochhammer__Suc__prod,axiom,
    ! [Aa: rat,Na: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(Aa),aa_nat_nat(suc,Na)) = groups73079841787564623at_rat(aTP_Lamm_on(Aa),set_or1269000886237332187st_nat(zero_zero_nat,Na)) ).

% pochhammer_Suc_prod
tff(fact_8309_pochhammer__Suc__prod,axiom,
    ! [Aa: real,Na: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Aa),aa_nat_nat(suc,Na)) = aa_set_nat_real(groups129246275422532515t_real(aa_real_fun_nat_real(aTP_Lamm_oo,Aa)),set_or1269000886237332187st_nat(zero_zero_nat,Na)) ).

% pochhammer_Suc_prod
tff(fact_8310_pochhammer__Suc__prod,axiom,
    ! [Aa: code_integer,Na: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(Aa),aa_nat_nat(suc,Na)) = groups3455450783089532116nteger(aTP_Lamm_op(Aa),set_or1269000886237332187st_nat(zero_zero_nat,Na)) ).

% pochhammer_Suc_prod
tff(fact_8311_pochhammer__Suc__prod,axiom,
    ! [Aa: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Aa),aa_nat_nat(suc,Na)) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_oq,Aa)),set_or1269000886237332187st_nat(zero_zero_nat,Na)) ).

% pochhammer_Suc_prod
tff(fact_8312_pochhammer__Suc__prod,axiom,
    ! [Aa: int,Na: nat] : aa_nat_int(comm_s4660882817536571857er_int(Aa),aa_nat_nat(suc,Na)) = groups705719431365010083at_int(aTP_Lamm_or(Aa),set_or1269000886237332187st_nat(zero_zero_nat,Na)) ).

% pochhammer_Suc_prod
tff(fact_8313_pochhammer__prod__rev,axiom,
    ! [Aa: rat,Na: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(Aa),Na) = groups73079841787564623at_rat(aa_nat_fun_nat_rat(aTP_Lamm_os(Aa),Na),set_or1269000886237332187st_nat(one_one_nat,Na)) ).

% pochhammer_prod_rev
tff(fact_8314_pochhammer__prod__rev,axiom,
    ! [Aa: real,Na: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Aa),Na) = aa_set_nat_real(groups129246275422532515t_real(aa_nat_fun_nat_real(aTP_Lamm_ot(Aa),Na)),set_or1269000886237332187st_nat(one_one_nat,Na)) ).

% pochhammer_prod_rev
tff(fact_8315_pochhammer__prod__rev,axiom,
    ! [Aa: code_integer,Na: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(Aa),Na) = groups3455450783089532116nteger(aa_nat2010205248571845816nteger(aTP_Lamm_ou(Aa),Na),set_or1269000886237332187st_nat(one_one_nat,Na)) ).

% pochhammer_prod_rev
tff(fact_8316_pochhammer__prod__rev,axiom,
    ! [Aa: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Aa),Na) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ov(Aa),Na)),set_or1269000886237332187st_nat(one_one_nat,Na)) ).

% pochhammer_prod_rev
tff(fact_8317_pochhammer__prod__rev,axiom,
    ! [Aa: int,Na: nat] : aa_nat_int(comm_s4660882817536571857er_int(Aa),Na) = groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_ow(Aa),Na),set_or1269000886237332187st_nat(one_one_nat,Na)) ).

% pochhammer_prod_rev
tff(fact_8318_fact__div__fact,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(semiri1408675320244567234ct_nat,M)),aa_nat_nat(semiri1408675320244567234ct_nat,Na)) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_cc),set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),one_one_nat),M)) ) ) ).

% fact_div_fact
tff(fact_8319_prod_Oin__pairs,axiom,
    ! [G: fun_nat_real,M: nat,Na: nat] : aa_set_nat_real(groups129246275422532515t_real(G),set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)))) = aa_set_nat_real(groups129246275422532515t_real(aTP_Lamm_ox(G)),set_or1269000886237332187st_nat(M,Na)) ).

% prod.in_pairs
tff(fact_8320_prod_Oin__pairs,axiom,
    ! [G: fun_nat_rat,M: nat,Na: nat] : groups73079841787564623at_rat(G,set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)))) = groups73079841787564623at_rat(aTP_Lamm_oy(G),set_or1269000886237332187st_nat(M,Na)) ).

% prod.in_pairs
tff(fact_8321_prod_Oin__pairs,axiom,
    ! [G: fun_nat_nat,M: nat,Na: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)))) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_oz(G)),set_or1269000886237332187st_nat(M,Na)) ).

% prod.in_pairs
tff(fact_8322_prod_Oin__pairs,axiom,
    ! [G: fun_nat_int,M: nat,Na: nat] : groups705719431365010083at_int(G,set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)))) = groups705719431365010083at_int(aTP_Lamm_pa(G),set_or1269000886237332187st_nat(M,Na)) ).

% prod.in_pairs
tff(fact_8323_exp__bound__half,axiom,
    ! [Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,Z)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),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_8324_exp__bound__half,axiom,
    ! [Z: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),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_8325_pochhammer__Suc__prod__rev,axiom,
    ! [Aa: rat,Na: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(Aa),aa_nat_nat(suc,Na)) = groups73079841787564623at_rat(aa_nat_fun_nat_rat(aTP_Lamm_os(Aa),Na),set_or1269000886237332187st_nat(zero_zero_nat,Na)) ).

% pochhammer_Suc_prod_rev
tff(fact_8326_pochhammer__Suc__prod__rev,axiom,
    ! [Aa: real,Na: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Aa),aa_nat_nat(suc,Na)) = aa_set_nat_real(groups129246275422532515t_real(aa_nat_fun_nat_real(aTP_Lamm_ot(Aa),Na)),set_or1269000886237332187st_nat(zero_zero_nat,Na)) ).

% pochhammer_Suc_prod_rev
tff(fact_8327_pochhammer__Suc__prod__rev,axiom,
    ! [Aa: code_integer,Na: nat] : aa_nat_Code_integer(comm_s8582702949713902594nteger(Aa),aa_nat_nat(suc,Na)) = groups3455450783089532116nteger(aa_nat2010205248571845816nteger(aTP_Lamm_ou(Aa),Na),set_or1269000886237332187st_nat(zero_zero_nat,Na)) ).

% pochhammer_Suc_prod_rev
tff(fact_8328_pochhammer__Suc__prod__rev,axiom,
    ! [Aa: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Aa),aa_nat_nat(suc,Na)) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ov(Aa),Na)),set_or1269000886237332187st_nat(zero_zero_nat,Na)) ).

% pochhammer_Suc_prod_rev
tff(fact_8329_pochhammer__Suc__prod__rev,axiom,
    ! [Aa: int,Na: nat] : aa_nat_int(comm_s4660882817536571857er_int(Aa),aa_nat_nat(suc,Na)) = groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_ow(Aa),Na),set_or1269000886237332187st_nat(zero_zero_nat,Na)) ).

% pochhammer_Suc_prod_rev
tff(fact_8330_exp__bound__lemma,axiom,
    ! [Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,Z)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),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(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(real_V7735802525324610683m_real,Z)))) ) ).

% exp_bound_lemma
tff(fact_8331_exp__bound__lemma,axiom,
    ! [Z: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),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(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_complex_real(real_V1022390504157884413omplex,Z)))) ) ).

% exp_bound_lemma
tff(fact_8332_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_8333_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_8334_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_8335_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_8336_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(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),C)) ) ) ).

% suminf_geometric
tff(fact_8337_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_8338_norm__one,axiom,
    aa_real_real(real_V7735802525324610683m_real,one_one_real) = one_one_real ).

% norm_one
tff(fact_8339_norm__one,axiom,
    aa_complex_real(real_V1022390504157884413omplex,one_one_complex) = one_one_real ).

% norm_one
tff(fact_8340_norm__zero,axiom,
    aa_real_real(real_V7735802525324610683m_real,zero_zero_real) = zero_zero_real ).

% norm_zero
tff(fact_8341_norm__zero,axiom,
    aa_complex_real(real_V1022390504157884413omplex,zero_zero_complex) = zero_zero_real ).

% norm_zero
tff(fact_8342_suminf__zero,axiom,
    suminf_real(aTP_Lamm_fc) = zero_zero_real ).

% suminf_zero
tff(fact_8343_suminf__zero,axiom,
    suminf_nat(aTP_Lamm_bx) = zero_zero_nat ).

% suminf_zero
tff(fact_8344_suminf__zero,axiom,
    suminf_int(aTP_Lamm_pb) = zero_zero_int ).

% suminf_zero
tff(fact_8345_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_8346_norm__eq__zero,axiom,
    ! [Xb: complex] :
      ( ( aa_complex_real(real_V1022390504157884413omplex,Xb) = zero_zero_real )
    <=> ( Xb = zero_zero_complex ) ) ).

% norm_eq_zero
tff(fact_8347_prod__eq__1__iff,axiom,
    ! [A3: set_int,F: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( ( groups1707563613775114915nt_nat(F,A3) = one_one_nat )
      <=> ! [X4: int] :
            ( aa_set_int_o(member_int(X4),A3)
           => ( aa_int_nat(F,X4) = one_one_nat ) ) ) ) ).

% prod_eq_1_iff
tff(fact_8348_prod__eq__1__iff,axiom,
    ! [A3: set_complex,F: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A3)
     => ( ( groups861055069439313189ex_nat(F,A3) = one_one_nat )
      <=> ! [X4: complex] :
            ( aa_set_complex_o(member_complex(X4),A3)
           => ( aa_complex_nat(F,X4) = one_one_nat ) ) ) ) ).

% prod_eq_1_iff
tff(fact_8349_prod__eq__1__iff,axiom,
    ! [A3: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( ( aa_set_nat_nat(groups708209901874060359at_nat(F),A3) = one_one_nat )
      <=> ! [X4: nat] :
            ( aa_set_nat_o(member_nat(X4),A3)
           => ( aa_nat_nat(F,X4) = one_one_nat ) ) ) ) ).

% prod_eq_1_iff
tff(fact_8350_prod__pos__nat__iff,axiom,
    ! [A3: set_int,F: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,A3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),groups1707563613775114915nt_nat(F,A3))
      <=> ! [X4: int] :
            ( aa_set_int_o(member_int(X4),A3)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_int_nat(F,X4)) ) ) ) ).

% prod_pos_nat_iff
tff(fact_8351_prod__pos__nat__iff,axiom,
    ! [A3: set_complex,F: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),groups861055069439313189ex_nat(F,A3))
      <=> ! [X4: complex] :
            ( aa_set_complex_o(member_complex(X4),A3)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_complex_nat(F,X4)) ) ) ) ).

% prod_pos_nat_iff
tff(fact_8352_prod__pos__nat__iff,axiom,
    ! [A3: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_set_nat_nat(groups708209901874060359at_nat(F),A3))
      <=> ! [X4: nat] :
            ( aa_set_nat_o(member_nat(X4),A3)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(F,X4)) ) ) ) ).

% prod_pos_nat_iff
tff(fact_8353_prod__int__eq,axiom,
    ! [I: nat,J: nat] : groups705719431365010083at_int(semiri1314217659103216013at_int,set_or1269000886237332187st_nat(I,J)) = groups1705073143266064639nt_int(aTP_Lamm_cd,set_or1266510415728281911st_int(aa_nat_int(semiri1314217659103216013at_int,I),aa_nat_int(semiri1314217659103216013at_int,J))) ).

% prod_int_eq
tff(fact_8354_ln__prod,axiom,
    ! [I5: set_Pr1261947904930325089at_nat,F: fun_Pr672294278902479040t_real] :
      ( finite6177210948735845034at_nat(I5)
     => ( ! [I2: product_prod_nat_nat] :
            ( member8440522571783428010at_nat(I2,I5)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_Pro6168199777849092593t_real(F,I2)) )
       => ( aa_real_real(ln_ln_real,groups6036352826371341000t_real(F,I5)) = groups4567486121110086003t_real(aTP_Lamm_pc(F),I5) ) ) ) ).

% ln_prod
tff(fact_8355_ln__prod,axiom,
    ! [I5: set_real,F: fun_real_real] :
      ( finite_finite_real(I5)
     => ( ! [I2: real] :
            ( member_real(I2,I5)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(F,I2)) )
       => ( aa_real_real(ln_ln_real,groups1681761925125756287l_real(F,I5)) = groups8097168146408367636l_real(aTP_Lamm_pd(F),I5) ) ) ) ).

% ln_prod
tff(fact_8356_ln__prod,axiom,
    ! [I5: set_set_nat,F: fun_set_nat_real] :
      ( finite1152437895449049373et_nat(I5)
     => ( ! [I2: set_nat] :
            ( member_set_nat(I2,I5)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_set_nat_real(F,I2)) )
       => ( aa_real_real(ln_ln_real,groups3619160379726066777t_real(F,I5)) = groups5107569545109728110t_real(aTP_Lamm_pe(F),I5) ) ) ) ).

% ln_prod
tff(fact_8357_ln__prod,axiom,
    ! [I5: set_int,F: fun_int_real] :
      ( aa_set_int_o(finite_finite_int,I5)
     => ( ! [I2: int] :
            ( aa_set_int_o(member_int(I2),I5)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_int_real(F,I2)) )
       => ( aa_real_real(ln_ln_real,groups2316167850115554303t_real(F,I5)) = groups8778361861064173332t_real(aTP_Lamm_pf(F),I5) ) ) ) ).

% ln_prod
tff(fact_8358_ln__prod,axiom,
    ! [I5: set_complex,F: fun_complex_real] :
      ( aa_set_complex_o(finite3207457112153483333omplex,I5)
     => ( ! [I2: complex] :
            ( aa_set_complex_o(member_complex(I2),I5)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(F,I2)) )
       => ( aa_real_real(ln_ln_real,groups766887009212190081x_real(F,I5)) = groups5808333547571424918x_real(aTP_Lamm_pg(F),I5) ) ) ) ).

% ln_prod
tff(fact_8359_ln__prod,axiom,
    ! [I5: set_nat,F: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,I5)
     => ( ! [I2: nat] :
            ( aa_set_nat_o(member_nat(I2),I5)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(F,I2)) )
       => ( aa_real_real(ln_ln_real,aa_set_nat_real(groups129246275422532515t_real(F),I5)) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_ph(F)),I5) ) ) ) ).

% ln_prod
tff(fact_8360_prod__int__plus__eq,axiom,
    ! [I: nat,J: nat] : groups705719431365010083at_int(semiri1314217659103216013at_int,set_or1269000886237332187st_nat(I,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J))) = groups1705073143266064639nt_int(aTP_Lamm_cd,set_or1266510415728281911st_int(aa_nat_int(semiri1314217659103216013at_int,I),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J)))) ).

% prod_int_plus_eq
tff(fact_8361_norm__not__less__zero,axiom,
    ! [Xb: complex] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Xb)),zero_zero_real) ).

% norm_not_less_zero
tff(fact_8362_norm__ge__zero,axiom,
    ! [Xb: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(real_V1022390504157884413omplex,Xb)) ).

% norm_ge_zero
tff(fact_8363_sum__norm__le,axiom,
    ! [S2: set_Pr1261947904930325089at_nat,F: fun_Pr4050773816208733250omplex,G: fun_Pr672294278902479040t_real] :
      ( ! [X: product_prod_nat_nat] :
          ( member8440522571783428010at_nat(X,S2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_Pro82899191565270131omplex(F,X))),aa_Pro6168199777849092593t_real(G,X)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,groups6381953495645901045omplex(F,S2))),groups4567486121110086003t_real(G,S2)) ) ).

% sum_norm_le
tff(fact_8364_sum__norm__le,axiom,
    ! [S2: set_real,F: fun_real_complex,G: fun_real_real] :
      ( ! [X: real] :
          ( member_real(X,S2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_real_complex(F,X))),aa_real_real(G,X)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,groups5754745047067104278omplex(F,S2))),groups8097168146408367636l_real(G,S2)) ) ).

% sum_norm_le
tff(fact_8365_sum__norm__le,axiom,
    ! [S2: set_set_nat,F: fun_set_nat_complex,G: fun_set_nat_real] :
      ( ! [X: set_nat] :
          ( member_set_nat(X,S2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_set_nat_complex(F,X))),aa_set_nat_real(G,X)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,groups8255218700646806128omplex(F,S2))),groups5107569545109728110t_real(G,S2)) ) ).

% sum_norm_le
tff(fact_8366_sum__norm__le,axiom,
    ! [S2: set_int,F: fun_int_complex,G: fun_int_real] :
      ( ! [X: int] :
          ( aa_set_int_o(member_int(X),S2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_int_complex(F,X))),aa_int_real(G,X)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,groups3049146728041665814omplex(F,S2))),groups8778361861064173332t_real(G,S2)) ) ).

% sum_norm_le
tff(fact_8367_sum__norm__le,axiom,
    ! [S2: set_nat,F: fun_nat_complex,G: fun_nat_real] :
      ( ! [X: nat] :
          ( aa_set_nat_o(member_nat(X),S2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(F,X))),aa_nat_real(G,X)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_set_nat_complex(groups2073611262835488442omplex(F),S2))),aa_set_nat_real(groups6591440286371151544t_real(G),S2)) ) ).

% sum_norm_le
tff(fact_8368_sum__norm__le,axiom,
    ! [S2: set_complex,F: fun_complex_complex,G: fun_complex_real] :
      ( ! [X: complex] :
          ( aa_set_complex_o(member_complex(X),S2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(F,X))),aa_complex_real(G,X)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,groups7754918857620584856omplex(F,S2))),groups5808333547571424918x_real(G,S2)) ) ).

% sum_norm_le
tff(fact_8369_sum__norm__le,axiom,
    ! [S2: set_nat,F: fun_nat_real,G: fun_nat_real] :
      ( ! [X: nat] :
          ( aa_set_nat_o(member_nat(X),S2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_nat_real(F,X))),aa_nat_real(G,X)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_set_nat_real(groups6591440286371151544t_real(F),S2))),aa_set_nat_real(groups6591440286371151544t_real(G),S2)) ) ).

% sum_norm_le
tff(fact_8370_norm__sum,axiom,
    ! [F: fun_nat_complex,A3: set_nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_set_nat_complex(groups2073611262835488442omplex(F),A3))),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_pi(F)),A3)) ).

% norm_sum
tff(fact_8371_norm__sum,axiom,
    ! [F: fun_complex_complex,A3: set_complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,groups7754918857620584856omplex(F,A3))),groups5808333547571424918x_real(aTP_Lamm_pj(F),A3)) ).

% norm_sum
tff(fact_8372_norm__sum,axiom,
    ! [F: fun_nat_real,A3: set_nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_set_nat_real(groups6591440286371151544t_real(F),A3))),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_pk(F)),A3)) ).

% norm_sum
tff(fact_8373_nonzero__norm__divide,axiom,
    ! [Ba: real,Aa: real] :
      ( ( Ba != zero_zero_real )
     => ( aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Aa),Ba)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(real_V7735802525324610683m_real,Aa)),aa_real_real(real_V7735802525324610683m_real,Ba)) ) ) ).

% nonzero_norm_divide
tff(fact_8374_nonzero__norm__divide,axiom,
    ! [Ba: complex,Aa: complex] :
      ( ( Ba != zero_zero_complex )
     => ( aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(divide1717551699836669952omplex(Aa),Ba)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(real_V1022390504157884413omplex,Aa)),aa_complex_real(real_V1022390504157884413omplex,Ba)) ) ) ).

% nonzero_norm_divide
tff(fact_8375_power__eq__imp__eq__norm,axiom,
    ! [W2: real,Na: nat,Z: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,W2),Na) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),Na) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => ( aa_real_real(real_V7735802525324610683m_real,W2) = aa_real_real(real_V7735802525324610683m_real,Z) ) ) ) ).

% power_eq_imp_eq_norm
tff(fact_8376_power__eq__imp__eq__norm,axiom,
    ! [W2: complex,Na: nat,Z: complex] :
      ( ( aa_nat_complex(power_power_complex(W2),Na) = aa_nat_complex(power_power_complex(Z),Na) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => ( aa_complex_real(real_V1022390504157884413omplex,W2) = aa_complex_real(real_V1022390504157884413omplex,Z) ) ) ) ).

% power_eq_imp_eq_norm
tff(fact_8377_norm__mult__less,axiom,
    ! [Xb: real,R2: real,Y: real,S3: 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,Y)),S3)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),S3)) ) ) ).

% norm_mult_less
tff(fact_8378_norm__mult__less,axiom,
    ! [Xb: complex,R2: real,Y: complex,S3: 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,Y)),S3)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(times_times_complex(Xb),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),S3)) ) ) ).

% norm_mult_less
tff(fact_8379_norm__mult__ineq,axiom,
    ! [Xb: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(real_V7735802525324610683m_real,Xb)),aa_real_real(real_V7735802525324610683m_real,Y))) ).

% norm_mult_ineq
tff(fact_8380_norm__mult__ineq,axiom,
    ! [Xb: complex,Y: 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),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(real_V1022390504157884413omplex,Xb)),aa_complex_real(real_V1022390504157884413omplex,Y))) ).

% norm_mult_ineq
tff(fact_8381_norm__triangle__lt,axiom,
    ! [Xb: real,Y: real,E2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,Xb)),aa_real_real(real_V7735802525324610683m_real,Y))),E2)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y))),E2) ) ).

% norm_triangle_lt
tff(fact_8382_norm__triangle__lt,axiom,
    ! [Xb: complex,Y: complex,E2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,Xb)),aa_complex_real(real_V1022390504157884413omplex,Y))),E2)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(Xb),Y))),E2) ) ).

% norm_triangle_lt
tff(fact_8383_norm__add__less,axiom,
    ! [Xb: real,R2: real,Y: real,S3: 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,Y)),S3)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,R2),S3)) ) ) ).

% norm_add_less
tff(fact_8384_norm__add__less,axiom,
    ! [Xb: complex,R2: real,Y: complex,S3: 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,Y)),S3)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(Xb),Y))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,R2),S3)) ) ) ).

% norm_add_less
tff(fact_8385_norm__triangle__mono,axiom,
    ! [Aa: real,R2: real,Ba: real,S3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,Aa)),R2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,Ba)),S3)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),Ba))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,R2),S3)) ) ) ).

% norm_triangle_mono
tff(fact_8386_norm__triangle__mono,axiom,
    ! [Aa: complex,R2: real,Ba: complex,S3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Aa)),R2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Ba)),S3)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(Aa),Ba))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,R2),S3)) ) ) ).

% norm_triangle_mono
tff(fact_8387_norm__triangle__ineq,axiom,
    ! [Xb: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,Xb)),aa_real_real(real_V7735802525324610683m_real,Y))) ).

% norm_triangle_ineq
tff(fact_8388_norm__triangle__ineq,axiom,
    ! [Xb: complex,Y: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(Xb),Y))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,Xb)),aa_complex_real(real_V1022390504157884413omplex,Y))) ).

% norm_triangle_ineq
tff(fact_8389_norm__triangle__le,axiom,
    ! [Xb: real,Y: real,E2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,Xb)),aa_real_real(real_V7735802525324610683m_real,Y))),E2)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y))),E2) ) ).

% norm_triangle_le
tff(fact_8390_norm__triangle__le,axiom,
    ! [Xb: complex,Y: complex,E2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,Xb)),aa_complex_real(real_V1022390504157884413omplex,Y))),E2)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(Xb),Y))),E2) ) ).

% norm_triangle_le
tff(fact_8391_norm__add__leD,axiom,
    ! [Aa: real,Ba: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),Ba))),C)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,Ba)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,Aa)),C)) ) ).

% norm_add_leD
tff(fact_8392_norm__add__leD,axiom,
    ! [Aa: complex,Ba: 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(Aa),Ba))),C)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Ba)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,Aa)),C)) ) ).

% norm_add_leD
tff(fact_8393_norm__power__ineq,axiom,
    ! [Xb: real,Na: 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),Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(real_V7735802525324610683m_real,Xb)),Na)) ).

% norm_power_ineq
tff(fact_8394_norm__power__ineq,axiom,
    ! [Xb: complex,Na: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(power_power_complex(Xb),Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(real_V1022390504157884413omplex,Xb)),Na)) ).

% norm_power_ineq
tff(fact_8395_norm__diff__triangle__less,axiom,
    ! [Xb: real,Y: real,E1: real,Z: real,E22: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Y))),E1)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y),Z))),E22)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Z))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,E1),E22)) ) ) ).

% norm_diff_triangle_less
tff(fact_8396_norm__diff__triangle__less,axiom,
    ! [Xb: complex,Y: complex,E1: real,Z: complex,E22: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(Xb),Y))),E1)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(Y),Z))),E22)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(Xb),Z))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,E1),E22)) ) ) ).

% norm_diff_triangle_less
tff(fact_8397_norm__triangle__sub,axiom,
    ! [Xb: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,Xb)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,Y)),aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Y)))) ).

% norm_triangle_sub
tff(fact_8398_norm__triangle__sub,axiom,
    ! [Xb: complex,Y: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Xb)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,Y)),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(Xb),Y)))) ).

% norm_triangle_sub
tff(fact_8399_norm__triangle__ineq4,axiom,
    ! [Aa: real,Ba: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Aa),Ba))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,Aa)),aa_real_real(real_V7735802525324610683m_real,Ba))) ).

% norm_triangle_ineq4
tff(fact_8400_norm__triangle__ineq4,axiom,
    ! [Aa: complex,Ba: 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),Ba))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,Aa)),aa_complex_real(real_V1022390504157884413omplex,Ba))) ).

% norm_triangle_ineq4
tff(fact_8401_norm__diff__triangle__le,axiom,
    ! [Xb: real,Y: real,E1: real,Z: real,E22: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Y))),E1)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y),Z))),E22)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Z))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,E1),E22)) ) ) ).

% norm_diff_triangle_le
tff(fact_8402_norm__diff__triangle__le,axiom,
    ! [Xb: complex,Y: complex,E1: real,Z: complex,E22: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(Xb),Y))),E1)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(Y),Z))),E22)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(Xb),Z))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,E1),E22)) ) ) ).

% norm_diff_triangle_le
tff(fact_8403_norm__triangle__le__diff,axiom,
    ! [Xb: real,Y: real,E2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,Xb)),aa_real_real(real_V7735802525324610683m_real,Y))),E2)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Y))),E2) ) ).

% norm_triangle_le_diff
tff(fact_8404_norm__triangle__le__diff,axiom,
    ! [Xb: complex,Y: complex,E2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,Xb)),aa_complex_real(real_V1022390504157884413omplex,Y))),E2)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(Xb),Y))),E2) ) ).

% norm_triangle_le_diff
tff(fact_8405_norm__diff__ineq,axiom,
    ! [Aa: real,Ba: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(real_V7735802525324610683m_real,Aa)),aa_real_real(real_V7735802525324610683m_real,Ba))),aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),Ba))) ).

% norm_diff_ineq
tff(fact_8406_norm__diff__ineq,axiom,
    ! [Aa: complex,Ba: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_complex_real(real_V1022390504157884413omplex,Aa)),aa_complex_real(real_V1022390504157884413omplex,Ba))),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(Aa),Ba))) ).

% norm_diff_ineq
tff(fact_8407_norm__triangle__ineq2,axiom,
    ! [Aa: real,Ba: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(real_V7735802525324610683m_real,Aa)),aa_real_real(real_V7735802525324610683m_real,Ba))),aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Aa),Ba))) ).

% norm_triangle_ineq2
tff(fact_8408_norm__triangle__ineq2,axiom,
    ! [Aa: complex,Ba: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_complex_real(real_V1022390504157884413omplex,Aa)),aa_complex_real(real_V1022390504157884413omplex,Ba))),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(Aa),Ba))) ).

% norm_triangle_ineq2
tff(fact_8409_suminf__finite,axiom,
    ! [N3: set_nat,F: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,N3)
     => ( ! [N: nat] :
            ( ~ aa_set_nat_o(member_nat(N),N3)
           => ( aa_nat_int(F,N) = zero_zero_int ) )
       => ( suminf_int(F) = groups3539618377306564664at_int(F,N3) ) ) ) ).

% suminf_finite
tff(fact_8410_suminf__finite,axiom,
    ! [N3: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,N3)
     => ( ! [N: nat] :
            ( ~ aa_set_nat_o(member_nat(N),N3)
           => ( aa_nat_nat(F,N) = zero_zero_nat ) )
       => ( suminf_nat(F) = aa_set_nat_nat(groups3542108847815614940at_nat(F),N3) ) ) ) ).

% suminf_finite
tff(fact_8411_suminf__finite,axiom,
    ! [N3: set_nat,F: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,N3)
     => ( ! [N: nat] :
            ( ~ aa_set_nat_o(member_nat(N),N3)
           => ( aa_nat_real(F,N) = zero_zero_real ) )
       => ( suminf_real(F) = aa_set_nat_real(groups6591440286371151544t_real(F),N3) ) ) ) ).

% suminf_finite
tff(fact_8412_power__eq__1__iff,axiom,
    ! [W2: real,Na: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,W2),Na) = one_one_real )
     => ( ( aa_real_real(real_V7735802525324610683m_real,W2) = one_one_real )
        | ( Na = zero_zero_nat ) ) ) ).

% power_eq_1_iff
tff(fact_8413_power__eq__1__iff,axiom,
    ! [W2: complex,Na: nat] :
      ( ( aa_nat_complex(power_power_complex(W2),Na) = one_one_complex )
     => ( ( aa_complex_real(real_V1022390504157884413omplex,W2) = one_one_real )
        | ( Na = zero_zero_nat ) ) ) ).

% power_eq_1_iff
tff(fact_8414_norm__diff__triangle__ineq,axiom,
    ! [Aa: real,Ba: real,C: real,D: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),Ba)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),D)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Aa),C))),aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Ba),D)))) ).

% norm_diff_triangle_ineq
tff(fact_8415_norm__diff__triangle__ineq,axiom,
    ! [Aa: complex,Ba: 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(Aa),Ba)),aa_complex_complex(plus_plus_complex(C),D)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(Aa),C))),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(Ba),D)))) ).

% norm_diff_triangle_ineq
tff(fact_8416_norm__triangle__ineq3,axiom,
    ! [Aa: real,Ba: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(real_V7735802525324610683m_real,Aa)),aa_real_real(real_V7735802525324610683m_real,Ba)))),aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Aa),Ba))) ).

% norm_triangle_ineq3
tff(fact_8417_norm__triangle__ineq3,axiom,
    ! [Aa: complex,Ba: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_complex_real(real_V1022390504157884413omplex,Aa)),aa_complex_real(real_V1022390504157884413omplex,Ba)))),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(Aa),Ba))) ).

% norm_triangle_ineq3
tff(fact_8418_norm__prod__diff,axiom,
    ! [I5: set_Pr1261947904930325089at_nat,Z: fun_Pr672294278902479040t_real,W2: fun_Pr672294278902479040t_real] :
      ( ! [I2: product_prod_nat_nat] :
          ( member8440522571783428010at_nat(I2,I5)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_Pro6168199777849092593t_real(Z,I2))),one_one_real) )
     => ( ! [I2: product_prod_nat_nat] :
            ( member8440522571783428010at_nat(I2,I5)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_Pro6168199777849092593t_real(W2,I2))),one_one_real) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups6036352826371341000t_real(Z,I5)),groups6036352826371341000t_real(W2,I5)))),groups4567486121110086003t_real(aa_fun8712866722335801276t_real(aTP_Lamm_pl(Z),W2),I5)) ) ) ).

% norm_prod_diff
tff(fact_8419_norm__prod__diff,axiom,
    ! [I5: set_real,Z: fun_real_real,W2: fun_real_real] :
      ( ! [I2: real] :
          ( member_real(I2,I5)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(Z,I2))),one_one_real) )
     => ( ! [I2: real] :
            ( member_real(I2,I5)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(W2,I2))),one_one_real) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups1681761925125756287l_real(Z,I5)),groups1681761925125756287l_real(W2,I5)))),groups8097168146408367636l_real(aa_fun6946558556880649704l_real(aTP_Lamm_pm(Z),W2),I5)) ) ) ).

% norm_prod_diff
tff(fact_8420_norm__prod__diff,axiom,
    ! [I5: set_set_nat,Z: fun_set_nat_real,W2: fun_set_nat_real] :
      ( ! [I2: set_nat] :
          ( member_set_nat(I2,I5)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_set_nat_real(Z,I2))),one_one_real) )
     => ( ! [I2: set_nat] :
            ( member_set_nat(I2,I5)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_set_nat_real(W2,I2))),one_one_real) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups3619160379726066777t_real(Z,I5)),groups3619160379726066777t_real(W2,I5)))),groups5107569545109728110t_real(aa_fun3569254276315353500t_real(aTP_Lamm_pn(Z),W2),I5)) ) ) ).

% norm_prod_diff
tff(fact_8421_norm__prod__diff,axiom,
    ! [I5: set_int,Z: fun_int_real,W2: fun_int_real] :
      ( ! [I2: int] :
          ( aa_set_int_o(member_int(I2),I5)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_int_real(Z,I2))),one_one_real) )
     => ( ! [I2: int] :
            ( aa_set_int_o(member_int(I2),I5)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_int_real(W2,I2))),one_one_real) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,groups2316167850115554303t_real(Z,I5)),groups2316167850115554303t_real(W2,I5)))),groups8778361861064173332t_real(aa_fun3252798400008884712t_real(aTP_Lamm_po(Z),W2),I5)) ) ) ).

% norm_prod_diff
tff(fact_8422_norm__prod__diff,axiom,
    ! [I5: set_Pr1261947904930325089at_nat,Z: fun_Pr4050773816208733250omplex,W2: fun_Pr4050773816208733250omplex] :
      ( ! [I2: product_prod_nat_nat] :
          ( member8440522571783428010at_nat(I2,I5)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_Pro82899191565270131omplex(Z,I2))),one_one_real) )
     => ( ! [I2: product_prod_nat_nat] :
            ( member8440522571783428010at_nat(I2,I5)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_Pro82899191565270131omplex(W2,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(groups8110221916422527690omplex(Z,I5)),groups8110221916422527690omplex(W2,I5)))),groups4567486121110086003t_real(aa_fun6819693891783611194t_real(aTP_Lamm_pp(Z),W2),I5)) ) ) ).

% norm_prod_diff
tff(fact_8423_norm__prod__diff,axiom,
    ! [I5: set_real,Z: fun_real_complex,W2: fun_real_complex] :
      ( ! [I2: real] :
          ( member_real(I2,I5)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_real_complex(Z,I2))),one_one_real) )
     => ( ! [I2: real] :
            ( member_real(I2,I5)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_real_complex(W2,I2))),one_one_real) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(groups713298508707869441omplex(Z,I5)),groups713298508707869441omplex(W2,I5)))),groups8097168146408367636l_real(aa_fun2786211375485122666l_real(aTP_Lamm_pq(Z),W2),I5)) ) ) ).

% norm_prod_diff
tff(fact_8424_norm__prod__diff,axiom,
    ! [I5: set_set_nat,Z: fun_set_nat_complex,W2: fun_set_nat_complex] :
      ( ! [I2: set_nat] :
          ( member_set_nat(I2,I5)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_set_nat_complex(Z,I2))),one_one_real) )
     => ( ! [I2: set_nat] :
            ( member_set_nat(I2,I5)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_set_nat_complex(W2,I2))),one_one_real) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(groups1092910753850256091omplex(Z,I5)),groups1092910753850256091omplex(W2,I5)))),groups5107569545109728110t_real(aa_fun5767801665285160478t_real(aTP_Lamm_pr(Z),W2),I5)) ) ) ).

% norm_prod_diff
tff(fact_8425_norm__prod__diff,axiom,
    ! [I5: set_int,Z: fun_int_complex,W2: fun_int_complex] :
      ( ! [I2: int] :
          ( aa_set_int_o(member_int(I2),I5)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_int_complex(Z,I2))),one_one_real) )
     => ( ! [I2: int] :
            ( aa_set_int_o(member_int(I2),I5)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_int_complex(W2,I2))),one_one_real) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(groups7440179247065528705omplex(Z,I5)),groups7440179247065528705omplex(W2,I5)))),groups8778361861064173332t_real(aa_fun1985759105522019690t_real(aTP_Lamm_ps(Z),W2),I5)) ) ) ).

% norm_prod_diff
tff(fact_8426_norm__prod__diff,axiom,
    ! [I5: set_nat,Z: fun_nat_real,W2: fun_nat_real] :
      ( ! [I2: nat] :
          ( aa_set_nat_o(member_nat(I2),I5)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_nat_real(Z,I2))),one_one_real) )
     => ( ! [I2: nat] :
            ( aa_set_nat_o(member_nat(I2),I5)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_nat_real(W2,I2))),one_one_real) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_set_nat_real(groups129246275422532515t_real(Z),I5)),aa_set_nat_real(groups129246275422532515t_real(W2),I5)))),aa_set_nat_real(groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_pt(Z),W2)),I5)) ) ) ).

% norm_prod_diff
tff(fact_8427_norm__prod__diff,axiom,
    ! [I5: set_nat,Z: fun_nat_complex,W2: fun_nat_complex] :
      ( ! [I2: nat] :
          ( aa_set_nat_o(member_nat(I2),I5)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(Z,I2))),one_one_real) )
     => ( ! [I2: nat] :
            ( aa_set_nat_o(member_nat(I2),I5)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(W2,I2))),one_one_real) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(aa_set_nat_complex(groups6464643781859351333omplex(Z),I5)),aa_set_nat_complex(groups6464643781859351333omplex(W2),I5)))),aa_set_nat_real(groups6591440286371151544t_real(aa_fun7054779078796324018t_real(aTP_Lamm_pu(Z),W2)),I5)) ) ) ).

% norm_prod_diff
tff(fact_8428_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_8429_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_8430_norm__power__diff,axiom,
    ! [Z: real,W2: real,M: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,Z)),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,W2)),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,W2),M)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Z),W2)))) ) ) ).

% norm_power_diff
tff(fact_8431_norm__power__diff,axiom,
    ! [Z: complex,W2: complex,M: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Z)),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,W2)),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(Z),M)),aa_nat_complex(power_power_complex(W2),M)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(Z),W2)))) ) ) ).

% norm_power_diff
tff(fact_8432_pi__series,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))) = suminf_real(aTP_Lamm_pv) ).

% pi_series
tff(fact_8433_lemma__termdiff2,axiom,
    ! [H: complex,Z: complex,Na: nat] :
      ( ( H != zero_zero_complex )
     => ( aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(aa_complex_complex(plus_plus_complex(Z),H)),Na)),aa_nat_complex(power_power_complex(Z),Na))),H)),aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,Na)),aa_nat_complex(power_power_complex(Z),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(suc,zero_zero_nat))))) = aa_complex_complex(times_times_complex(H),aa_set_nat_complex(groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_px(H),Z),Na)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(suc,zero_zero_nat))))) ) ) ).

% lemma_termdiff2
tff(fact_8434_lemma__termdiff2,axiom,
    ! [H: rat,Z: rat,Na: 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)),Na)),aa_nat_rat(power_power_rat(Z),Na))),H)),aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,Na)),aa_nat_rat(power_power_rat(Z),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(suc,zero_zero_nat))))) = aa_rat_rat(times_times_rat(H),groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_pz(H),Z),Na),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(suc,zero_zero_nat))))) ) ) ).

% lemma_termdiff2
tff(fact_8435_lemma__termdiff2,axiom,
    ! [H: real,Z: real,Na: nat] :
      ( ( H != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),H)),Na)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),Na))),H)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(suc,zero_zero_nat))))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,H),aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_qb(H),Z),Na)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(suc,zero_zero_nat))))) ) ) ).

% lemma_termdiff2
tff(fact_8436_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_li,Xb)) ) ).

% summable_arctan_series
tff(fact_8437_choose__odd__sum,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_set_nat_complex(groups2073611262835488442omplex(aTP_Lamm_qc(Na)),aa_nat_set_nat(set_ord_atMost_nat,Na))) = aa_nat_complex(power_power_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),Na) ) ) ).

% choose_odd_sum
tff(fact_8438_choose__odd__sum,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),groups2906978787729119204at_rat(aTP_Lamm_qd(Na),aa_nat_set_nat(set_ord_atMost_nat,Na))) = aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),Na) ) ) ).

% choose_odd_sum
tff(fact_8439_choose__odd__sum,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),groups3539618377306564664at_int(aa_nat_fun_nat_int(aTP_Lamm_qe,Na),aa_nat_set_nat(set_ord_atMost_nat,Na))) = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na) ) ) ).

% choose_odd_sum
tff(fact_8440_choose__odd__sum,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),groups7501900531339628137nteger(aTP_Lamm_qf(Na),aa_nat_set_nat(set_ord_atMost_nat,Na))) = aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Na) ) ) ).

% choose_odd_sum
tff(fact_8441_choose__odd__sum,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_qg,Na)),aa_nat_set_nat(set_ord_atMost_nat,Na))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Na) ) ) ).

% choose_odd_sum
tff(fact_8442_choose__even__sum,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_set_nat_complex(groups2073611262835488442omplex(aTP_Lamm_qh(Na)),aa_nat_set_nat(set_ord_atMost_nat,Na))) = aa_nat_complex(power_power_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),Na) ) ) ).

% choose_even_sum
tff(fact_8443_choose__even__sum,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),groups2906978787729119204at_rat(aTP_Lamm_qi(Na),aa_nat_set_nat(set_ord_atMost_nat,Na))) = aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),Na) ) ) ).

% choose_even_sum
tff(fact_8444_choose__even__sum,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),groups3539618377306564664at_int(aa_nat_fun_nat_int(aTP_Lamm_qj,Na),aa_nat_set_nat(set_ord_atMost_nat,Na))) = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na) ) ) ).

% choose_even_sum
tff(fact_8445_choose__even__sum,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),groups7501900531339628137nteger(aTP_Lamm_qk(Na),aa_nat_set_nat(set_ord_atMost_nat,Na))) = aa_nat_Code_integer(power_8256067586552552935nteger(aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Na) ) ) ).

% choose_even_sum
tff(fact_8446_choose__even__sum,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_ql,Na)),aa_nat_set_nat(set_ord_atMost_nat,Na))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Na) ) ) ).

% choose_even_sum
tff(fact_8447_lessThan__eq__iff,axiom,
    ! [Xb: nat,Y: nat] :
      ( ( aa_nat_set_nat(set_ord_lessThan_nat,Xb) = aa_nat_set_nat(set_ord_lessThan_nat,Y) )
    <=> ( Xb = Y ) ) ).

% lessThan_eq_iff
tff(fact_8448_lessThan__eq__iff,axiom,
    ! [Xb: int,Y: int] :
      ( ( set_ord_lessThan_int(Xb) = set_ord_lessThan_int(Y) )
    <=> ( Xb = Y ) ) ).

% lessThan_eq_iff
tff(fact_8449_lessThan__eq__iff,axiom,
    ! [Xb: real,Y: real] :
      ( ( set_or5984915006950818249n_real(Xb) = set_or5984915006950818249n_real(Y) )
    <=> ( Xb = Y ) ) ).

% lessThan_eq_iff
tff(fact_8450_atMost__eq__iff,axiom,
    ! [Xb: nat,Y: nat] :
      ( ( aa_nat_set_nat(set_ord_atMost_nat,Xb) = aa_nat_set_nat(set_ord_atMost_nat,Y) )
    <=> ( Xb = Y ) ) ).

% atMost_eq_iff
tff(fact_8451_atMost__eq__iff,axiom,
    ! [Xb: int,Y: int] :
      ( ( set_ord_atMost_int(Xb) = set_ord_atMost_int(Y) )
    <=> ( Xb = Y ) ) ).

% atMost_eq_iff
tff(fact_8452_lessThan__iff,axiom,
    ! [I: set_nat,K: set_nat] :
      ( member_set_nat(I,set_or890127255671739683et_nat(K))
    <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,I),K) ) ).

% lessThan_iff
tff(fact_8453_lessThan__iff,axiom,
    ! [I: rat,K: rat] :
      ( member_rat(I,set_ord_lessThan_rat(K))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,I),K) ) ).

% lessThan_iff
tff(fact_8454_lessThan__iff,axiom,
    ! [I: num,K: num] :
      ( member_num(I,set_ord_lessThan_num(K))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,I),K) ) ).

% lessThan_iff
tff(fact_8455_lessThan__iff,axiom,
    ! [I: nat,K: nat] :
      ( aa_set_nat_o(member_nat(I),aa_nat_set_nat(set_ord_lessThan_nat,K))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),K) ) ).

% lessThan_iff
tff(fact_8456_lessThan__iff,axiom,
    ! [I: int,K: int] :
      ( aa_set_int_o(member_int(I),set_ord_lessThan_int(K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,I),K) ) ).

% lessThan_iff
tff(fact_8457_lessThan__iff,axiom,
    ! [I: real,K: real] :
      ( member_real(I,set_or5984915006950818249n_real(K))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,I),K) ) ).

% lessThan_iff
tff(fact_8458_atMost__iff,axiom,
    ! [I: real,K: real] :
      ( member_real(I,set_ord_atMost_real(K))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,I),K) ) ).

% atMost_iff
tff(fact_8459_atMost__iff,axiom,
    ! [I: set_nat,K: set_nat] :
      ( member_set_nat(I,set_or4236626031148496127et_nat(K))
    <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,I),K) ) ).

% atMost_iff
tff(fact_8460_atMost__iff,axiom,
    ! [I: set_int,K: set_int] :
      ( member_set_int(I,set_or58775011639299419et_int(K))
    <=> aa_set_int_o(ord_less_eq_set_int(I),K) ) ).

% atMost_iff
tff(fact_8461_atMost__iff,axiom,
    ! [I: rat,K: rat] :
      ( member_rat(I,set_ord_atMost_rat(K))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,I),K) ) ).

% atMost_iff
tff(fact_8462_atMost__iff,axiom,
    ! [I: num,K: num] :
      ( member_num(I,set_ord_atMost_num(K))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,I),K) ) ).

% atMost_iff
tff(fact_8463_atMost__iff,axiom,
    ! [I: nat,K: nat] :
      ( aa_set_nat_o(member_nat(I),aa_nat_set_nat(set_ord_atMost_nat,K))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),K) ) ).

% atMost_iff
tff(fact_8464_atMost__iff,axiom,
    ! [I: int,K: int] :
      ( aa_set_int_o(member_int(I),set_ord_atMost_int(K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),K) ) ).

% atMost_iff
tff(fact_8465_finite__lessThan,axiom,
    ! [K: nat] : aa_set_nat_o(finite_finite_nat,aa_nat_set_nat(set_ord_lessThan_nat,K)) ).

% finite_lessThan
tff(fact_8466_finite__atMost,axiom,
    ! [K: nat] : aa_set_nat_o(finite_finite_nat,aa_nat_set_nat(set_ord_atMost_nat,K)) ).

% finite_atMost
tff(fact_8467_summable__single,axiom,
    ! [I: nat,F: fun_nat_real] : summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_qm(I),F)) ).

% summable_single
tff(fact_8468_summable__single,axiom,
    ! [I: nat,F: fun_nat_nat] : summable_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_qn(I),F)) ).

% summable_single
tff(fact_8469_summable__single,axiom,
    ! [I: nat,F: fun_nat_int] : summable_int(aa_fun5565043122120104496at_int(aTP_Lamm_qo(I),F)) ).

% summable_single
tff(fact_8470_summable__zero,axiom,
    summable_real(aTP_Lamm_fc) ).

% summable_zero
tff(fact_8471_summable__zero,axiom,
    summable_nat(aTP_Lamm_bx) ).

% summable_zero
tff(fact_8472_summable__zero,axiom,
    summable_int(aTP_Lamm_pb) ).

% summable_zero
tff(fact_8473_lessThan__subset__iff,axiom,
    ! [Xb: rat,Y: rat] :
      ( ord_less_eq_set_rat(set_ord_lessThan_rat(Xb),set_ord_lessThan_rat(Y))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y) ) ).

% lessThan_subset_iff
tff(fact_8474_lessThan__subset__iff,axiom,
    ! [Xb: num,Y: num] :
      ( ord_less_eq_set_num(set_ord_lessThan_num(Xb),set_ord_lessThan_num(Y))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Y) ) ).

% lessThan_subset_iff
tff(fact_8475_lessThan__subset__iff,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_nat_set_nat(set_ord_lessThan_nat,Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Y))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y) ) ).

% lessThan_subset_iff
tff(fact_8476_lessThan__subset__iff,axiom,
    ! [Xb: int,Y: int] :
      ( aa_set_int_o(ord_less_eq_set_int(set_ord_lessThan_int(Xb)),set_ord_lessThan_int(Y))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y) ) ).

% lessThan_subset_iff
tff(fact_8477_lessThan__subset__iff,axiom,
    ! [Xb: real,Y: real] :
      ( ord_less_eq_set_real(set_or5984915006950818249n_real(Xb),set_or5984915006950818249n_real(Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y) ) ).

% lessThan_subset_iff
tff(fact_8478_atMost__subset__iff,axiom,
    ! [Xb: set_int,Y: set_int] :
      ( ord_le4403425263959731960et_int(set_or58775011639299419et_int(Xb),set_or58775011639299419et_int(Y))
    <=> aa_set_int_o(ord_less_eq_set_int(Xb),Y) ) ).

% atMost_subset_iff
tff(fact_8479_atMost__subset__iff,axiom,
    ! [Xb: rat,Y: rat] :
      ( ord_less_eq_set_rat(set_ord_atMost_rat(Xb),set_ord_atMost_rat(Y))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y) ) ).

% atMost_subset_iff
tff(fact_8480_atMost__subset__iff,axiom,
    ! [Xb: num,Y: num] :
      ( ord_less_eq_set_num(set_ord_atMost_num(Xb),set_ord_atMost_num(Y))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Y) ) ).

% atMost_subset_iff
tff(fact_8481_atMost__subset__iff,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_nat_set_nat(set_ord_atMost_nat,Xb)),aa_nat_set_nat(set_ord_atMost_nat,Y))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y) ) ).

% atMost_subset_iff
tff(fact_8482_atMost__subset__iff,axiom,
    ! [Xb: int,Y: int] :
      ( aa_set_int_o(ord_less_eq_set_int(set_ord_atMost_int(Xb)),set_ord_atMost_int(Y))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y) ) ).

% atMost_subset_iff
tff(fact_8483_lessThan__0,axiom,
    aa_nat_set_nat(set_ord_lessThan_nat,zero_zero_nat) = bot_bot_set_nat ).

% lessThan_0
tff(fact_8484_summable__cmult__iff,axiom,
    ! [C: real,F: fun_nat_real] :
      ( summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_qp(C),F))
    <=> ( ( C = zero_zero_real )
        | summable_real(F) ) ) ).

% summable_cmult_iff
tff(fact_8485_summable__divide__iff,axiom,
    ! [F: fun_nat_real,C: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_qq(F),C))
    <=> ( ( C = zero_zero_real )
        | summable_real(F) ) ) ).

% summable_divide_iff
tff(fact_8486_summable__If__finite__set,axiom,
    ! [A3: set_nat,F: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_qr(A3),F)) ) ).

% summable_If_finite_set
tff(fact_8487_summable__If__finite__set,axiom,
    ! [A3: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => summable_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_qs(A3),F)) ) ).

% summable_If_finite_set
tff(fact_8488_summable__If__finite__set,axiom,
    ! [A3: set_nat,F: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => summable_int(aa_fun5565043122120104496at_int(aTP_Lamm_qt(A3),F)) ) ).

% summable_If_finite_set
tff(fact_8489_summable__If__finite,axiom,
    ! [P: fun_nat_o,F: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(P))
     => summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_qu(P),F)) ) ).

% summable_If_finite
tff(fact_8490_summable__If__finite,axiom,
    ! [P: fun_nat_o,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(P))
     => summable_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_qv(P),F)) ) ).

% summable_If_finite
tff(fact_8491_summable__If__finite,axiom,
    ! [P: fun_nat_o,F: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(P))
     => summable_int(aa_fun5565043122120104496at_int(aTP_Lamm_qw(P),F)) ) ).

% summable_If_finite
tff(fact_8492_Icc__subset__Iic__iff,axiom,
    ! [L: set_int,H: set_int,H2: set_int] :
      ( ord_le4403425263959731960et_int(set_or370866239135849197et_int(L,H),set_or58775011639299419et_int(H2))
    <=> ( ~ aa_set_int_o(ord_less_eq_set_int(L),H)
        | aa_set_int_o(ord_less_eq_set_int(H),H2) ) ) ).

% Icc_subset_Iic_iff
tff(fact_8493_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_8494_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_8495_Icc__subset__Iic__iff,axiom,
    ! [L: nat,H: nat,H2: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_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_8496_Icc__subset__Iic__iff,axiom,
    ! [L: int,H: int,H2: int] :
      ( aa_set_int_o(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_8497_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_8498_sum_OlessThan__Suc,axiom,
    ! [G: fun_nat_rat,Na: nat] : groups2906978787729119204at_rat(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Na))) = aa_rat_rat(plus_plus_rat(groups2906978787729119204at_rat(G,aa_nat_set_nat(set_ord_lessThan_nat,Na))),aa_nat_rat(G,Na)) ).

% sum.lessThan_Suc
tff(fact_8499_sum_OlessThan__Suc,axiom,
    ! [G: fun_nat_int,Na: nat] : groups3539618377306564664at_int(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Na))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,aa_nat_set_nat(set_ord_lessThan_nat,Na))),aa_nat_int(G,Na)) ).

% sum.lessThan_Suc
tff(fact_8500_sum_OlessThan__Suc,axiom,
    ! [G: fun_nat_nat,Na: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Na))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,Na))),aa_nat_nat(G,Na)) ).

% sum.lessThan_Suc
tff(fact_8501_sum_OlessThan__Suc,axiom,
    ! [G: fun_nat_real,Na: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Na))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_ord_lessThan_nat,Na))),aa_nat_real(G,Na)) ).

% sum.lessThan_Suc
tff(fact_8502_sum_OatMost__Suc,axiom,
    ! [G: fun_nat_rat,Na: nat] : groups2906978787729119204at_rat(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Na))) = aa_rat_rat(plus_plus_rat(groups2906978787729119204at_rat(G,aa_nat_set_nat(set_ord_atMost_nat,Na))),aa_nat_rat(G,aa_nat_nat(suc,Na))) ).

% sum.atMost_Suc
tff(fact_8503_sum_OatMost__Suc,axiom,
    ! [G: fun_nat_int,Na: nat] : groups3539618377306564664at_int(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Na))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(G,aa_nat_set_nat(set_ord_atMost_nat,Na))),aa_nat_int(G,aa_nat_nat(suc,Na))) ).

% sum.atMost_Suc
tff(fact_8504_sum_OatMost__Suc,axiom,
    ! [G: fun_nat_nat,Na: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Na))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,Na))),aa_nat_nat(G,aa_nat_nat(suc,Na))) ).

% sum.atMost_Suc
tff(fact_8505_sum_OatMost__Suc,axiom,
    ! [G: fun_nat_real,Na: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Na))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_ord_atMost_nat,Na))),aa_nat_real(G,aa_nat_nat(suc,Na))) ).

% sum.atMost_Suc
tff(fact_8506_prod_OlessThan__Suc,axiom,
    ! [G: fun_nat_real,Na: nat] : aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Na))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_ord_lessThan_nat,Na))),aa_nat_real(G,Na)) ).

% prod.lessThan_Suc
tff(fact_8507_prod_OlessThan__Suc,axiom,
    ! [G: fun_nat_rat,Na: nat] : groups73079841787564623at_rat(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Na))) = aa_rat_rat(times_times_rat(groups73079841787564623at_rat(G,aa_nat_set_nat(set_ord_lessThan_nat,Na))),aa_nat_rat(G,Na)) ).

% prod.lessThan_Suc
tff(fact_8508_prod_OlessThan__Suc,axiom,
    ! [G: fun_nat_nat,Na: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Na))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,Na))),aa_nat_nat(G,Na)) ).

% prod.lessThan_Suc
tff(fact_8509_prod_OlessThan__Suc,axiom,
    ! [G: fun_nat_int,Na: nat] : groups705719431365010083at_int(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Na))) = aa_int_int(aa_int_fun_int_int(times_times_int,groups705719431365010083at_int(G,aa_nat_set_nat(set_ord_lessThan_nat,Na))),aa_nat_int(G,Na)) ).

% prod.lessThan_Suc
tff(fact_8510_prod_OatMost__Suc,axiom,
    ! [G: fun_nat_real,Na: nat] : aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Na))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_ord_atMost_nat,Na))),aa_nat_real(G,aa_nat_nat(suc,Na))) ).

% prod.atMost_Suc
tff(fact_8511_prod_OatMost__Suc,axiom,
    ! [G: fun_nat_rat,Na: nat] : groups73079841787564623at_rat(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Na))) = aa_rat_rat(times_times_rat(groups73079841787564623at_rat(G,aa_nat_set_nat(set_ord_atMost_nat,Na))),aa_nat_rat(G,aa_nat_nat(suc,Na))) ).

% prod.atMost_Suc
tff(fact_8512_prod_OatMost__Suc,axiom,
    ! [G: fun_nat_nat,Na: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Na))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,Na))),aa_nat_nat(G,aa_nat_nat(suc,Na))) ).

% prod.atMost_Suc
tff(fact_8513_prod_OatMost__Suc,axiom,
    ! [G: fun_nat_int,Na: nat] : groups705719431365010083at_int(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Na))) = aa_int_int(aa_int_fun_int_int(times_times_int,groups705719431365010083at_int(G,aa_nat_set_nat(set_ord_atMost_nat,Na))),aa_nat_int(G,aa_nat_nat(suc,Na))) ).

% prod.atMost_Suc
tff(fact_8514_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_8515_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_8516_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_8517_Iic__subset__Iio__iff,axiom,
    ! [Aa: rat,Ba: rat] :
      ( ord_less_eq_set_rat(set_ord_atMost_rat(Aa),set_ord_lessThan_rat(Ba))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),Ba) ) ).

% Iic_subset_Iio_iff
tff(fact_8518_Iic__subset__Iio__iff,axiom,
    ! [Aa: num,Ba: num] :
      ( ord_less_eq_set_num(set_ord_atMost_num(Aa),set_ord_lessThan_num(Ba))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Aa),Ba) ) ).

% Iic_subset_Iio_iff
tff(fact_8519_Iic__subset__Iio__iff,axiom,
    ! [Aa: nat,Ba: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_nat_set_nat(set_ord_atMost_nat,Aa)),aa_nat_set_nat(set_ord_lessThan_nat,Ba))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Aa),Ba) ) ).

% Iic_subset_Iio_iff
tff(fact_8520_Iic__subset__Iio__iff,axiom,
    ! [Aa: int,Ba: int] :
      ( aa_set_int_o(ord_less_eq_set_int(set_ord_atMost_int(Aa)),set_ord_lessThan_int(Ba))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),Ba) ) ).

% Iic_subset_Iio_iff
tff(fact_8521_Iic__subset__Iio__iff,axiom,
    ! [Aa: real,Ba: real] :
      ( ord_less_eq_set_real(set_ord_atMost_real(Aa),set_or5984915006950818249n_real(Ba))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),Ba) ) ).

% Iic_subset_Iio_iff
tff(fact_8522_infinite__Iio,axiom,
    ! [Aa: int] : ~ aa_set_int_o(finite_finite_int,set_ord_lessThan_int(Aa)) ).

% infinite_Iio
tff(fact_8523_infinite__Iio,axiom,
    ! [Aa: real] : ~ finite_finite_real(set_or5984915006950818249n_real(Aa)) ).

% infinite_Iio
tff(fact_8524_lessThan__non__empty,axiom,
    ! [Xb: int] : set_ord_lessThan_int(Xb) != bot_bot_set_int ).

% lessThan_non_empty
tff(fact_8525_lessThan__non__empty,axiom,
    ! [Xb: real] : set_or5984915006950818249n_real(Xb) != bot_bot_set_real ).

% lessThan_non_empty
tff(fact_8526_not__empty__eq__Iic__eq__empty,axiom,
    ! [H: real] : bot_bot_set_real != set_ord_atMost_real(H) ).

% not_empty_eq_Iic_eq_empty
tff(fact_8527_not__empty__eq__Iic__eq__empty,axiom,
    ! [H: nat] : bot_bot_set_nat != aa_nat_set_nat(set_ord_atMost_nat,H) ).

% not_empty_eq_Iic_eq_empty
tff(fact_8528_not__empty__eq__Iic__eq__empty,axiom,
    ! [H: int] : bot_bot_set_int != set_ord_atMost_int(H) ).

% not_empty_eq_Iic_eq_empty
tff(fact_8529_infinite__Iic,axiom,
    ! [Aa: int] : ~ aa_set_int_o(finite_finite_int,set_ord_atMost_int(Aa)) ).

% infinite_Iic
tff(fact_8530_not__Iic__eq__Icc,axiom,
    ! [H2: int,L: int,H: int] : set_ord_atMost_int(H2) != set_or1266510415728281911st_int(L,H) ).

% not_Iic_eq_Icc
tff(fact_8531_not__Iic__eq__Icc,axiom,
    ! [H2: real,L: real,H: real] : set_ord_atMost_real(H2) != set_or1222579329274155063t_real(L,H) ).

% not_Iic_eq_Icc
tff(fact_8532_summable__comparison__test,axiom,
    ! [F: fun_nat_real,G: fun_nat_real] :
      ( ? [N6: nat] :
        ! [N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N6),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_8533_summable__comparison__test,axiom,
    ! [F: fun_nat_complex,G: fun_nat_real] :
      ( ? [N6: nat] :
        ! [N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N6),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_8534_summable__comparison__test_H,axiom,
    ! [G: fun_nat_real,N3: nat,F: fun_nat_real] :
      ( summable_real(G)
     => ( ! [N: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N3),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_8535_summable__comparison__test_H,axiom,
    ! [G: fun_nat_real,N3: nat,F: fun_nat_complex] :
      ( summable_real(G)
     => ( ! [N: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N3),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_8536_summable__const__iff,axiom,
    ! [C: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_qx,C))
    <=> ( C = zero_zero_real ) ) ).

% summable_const_iff
tff(fact_8537_summable__Suc__iff,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(aTP_Lamm_jg(F))
    <=> summable_real(F) ) ).

% summable_Suc_iff
tff(fact_8538_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_8539_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_8540_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_8541_lessThan__def,axiom,
    ! [U: set_nat] : set_or890127255671739683et_nat(U) = collect_set_nat(aa_set7861427066675100071_nat_o(aTP_Lamm_qy,U)) ).

% lessThan_def
tff(fact_8542_lessThan__def,axiom,
    ! [U: rat] : set_ord_lessThan_rat(U) = collect_rat(aa_rat_fun_rat_o(aTP_Lamm_qz,U)) ).

% lessThan_def
tff(fact_8543_lessThan__def,axiom,
    ! [U: num] : set_ord_lessThan_num(U) = collect_num(aa_num_fun_num_o(aTP_Lamm_ra,U)) ).

% lessThan_def
tff(fact_8544_lessThan__def,axiom,
    ! [U: nat] : aa_nat_set_nat(set_ord_lessThan_nat,U) = collect_nat(aa_nat_fun_nat_o(aTP_Lamm_au,U)) ).

% lessThan_def
tff(fact_8545_lessThan__def,axiom,
    ! [U: int] : set_ord_lessThan_int(U) = collect_int(aa_int_fun_int_o(aTP_Lamm_rb,U)) ).

% lessThan_def
tff(fact_8546_lessThan__def,axiom,
    ! [U: real] : set_or5984915006950818249n_real(U) = collect_real(aa_real_fun_real_o(aTP_Lamm_rc,U)) ).

% lessThan_def
tff(fact_8547_bounded__imp__summable,axiom,
    ! [Aa: 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(Aa,N))
     => ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups3539618377306564664at_int(Aa,aa_nat_set_nat(set_ord_atMost_nat,N))),B4)
       => summable_int(Aa) ) ) ).

% bounded_imp_summable
tff(fact_8548_bounded__imp__summable,axiom,
    ! [Aa: 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(Aa,N))
     => ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(groups3542108847815614940at_nat(Aa),aa_nat_set_nat(set_ord_atMost_nat,N))),B4)
       => summable_nat(Aa) ) ) ).

% bounded_imp_summable
tff(fact_8549_bounded__imp__summable,axiom,
    ! [Aa: 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(Aa,N))
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_set_nat_real(groups6591440286371151544t_real(Aa),aa_nat_set_nat(set_ord_atMost_nat,N))),B4)
       => summable_real(Aa) ) ) ).

% bounded_imp_summable
tff(fact_8550_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_8551_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_8552_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_8553_atMost__def,axiom,
    ! [U: real] : set_ord_atMost_real(U) = collect_real(aa_real_fun_real_o(aTP_Lamm_rd,U)) ).

% atMost_def
tff(fact_8554_atMost__def,axiom,
    ! [U: set_nat] : set_or4236626031148496127et_nat(U) = collect_set_nat(aa_set7861427066675100071_nat_o(aTP_Lamm_ar,U)) ).

% atMost_def
tff(fact_8555_atMost__def,axiom,
    ! [U: set_int] : set_or58775011639299419et_int(U) = collect_set_int(aTP_Lamm_at(U)) ).

% atMost_def
tff(fact_8556_atMost__def,axiom,
    ! [U: rat] : set_ord_atMost_rat(U) = collect_rat(aa_rat_fun_rat_o(aTP_Lamm_re,U)) ).

% atMost_def
tff(fact_8557_atMost__def,axiom,
    ! [U: num] : set_ord_atMost_num(U) = collect_num(aa_num_fun_num_o(aTP_Lamm_rf,U)) ).

% atMost_def
tff(fact_8558_atMost__def,axiom,
    ! [U: nat] : aa_nat_set_nat(set_ord_atMost_nat,U) = collect_nat(aa_nat_fun_nat_o(aTP_Lamm_av,U)) ).

% atMost_def
tff(fact_8559_atMost__def,axiom,
    ! [U: int] : set_ord_atMost_int(U) = collect_int(aa_int_fun_int_o(aTP_Lamm_rg,U)) ).

% atMost_def
tff(fact_8560_sum_Onested__swap_H,axiom,
    ! [Aa: fun_nat_fun_nat_nat,Na: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_rh(Aa)),aa_nat_set_nat(set_ord_atMost_nat,Na)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_rj(Aa),Na)),aa_nat_set_nat(set_ord_lessThan_nat,Na)) ).

% sum.nested_swap'
tff(fact_8561_sum_Onested__swap_H,axiom,
    ! [Aa: fun_nat_fun_nat_real,Na: nat] : aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_rk(Aa)),aa_nat_set_nat(set_ord_atMost_nat,Na)) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_rm(Aa),Na)),aa_nat_set_nat(set_ord_lessThan_nat,Na)) ).

% sum.nested_swap'
tff(fact_8562_prod_Onested__swap_H,axiom,
    ! [Aa: fun_nat_fun_nat_nat,Na: nat] : aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_rn(Aa)),aa_nat_set_nat(set_ord_atMost_nat,Na)) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ro(Aa),Na)),aa_nat_set_nat(set_ord_lessThan_nat,Na)) ).

% prod.nested_swap'
tff(fact_8563_prod_Onested__swap_H,axiom,
    ! [Aa: fun_nat_fun_nat_int,Na: nat] : groups705719431365010083at_int(aTP_Lamm_rp(Aa),aa_nat_set_nat(set_ord_atMost_nat,Na)) = groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_rr(Aa),Na),aa_nat_set_nat(set_ord_lessThan_nat,Na)) ).

% prod.nested_swap'
tff(fact_8564_powser__insidea,axiom,
    ! [F: fun_nat_real,Xb: real,Z: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_rs(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_rt(F),Z)) ) ) ).

% powser_insidea
tff(fact_8565_powser__insidea,axiom,
    ! [F: fun_nat_complex,Xb: complex,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_ru(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_rv(F),Z)) ) ) ).

% powser_insidea
tff(fact_8566_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_8567_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_8568_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_8569_summable__finite,axiom,
    ! [N3: set_nat,F: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,N3)
     => ( ! [N: nat] :
            ( ~ aa_set_nat_o(member_nat(N),N3)
           => ( aa_nat_real(F,N) = zero_zero_real ) )
       => summable_real(F) ) ) ).

% summable_finite
tff(fact_8570_summable__finite,axiom,
    ! [N3: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,N3)
     => ( ! [N: nat] :
            ( ~ aa_set_nat_o(member_nat(N),N3)
           => ( aa_nat_nat(F,N) = zero_zero_nat ) )
       => summable_nat(F) ) ) ).

% summable_finite
tff(fact_8571_summable__finite,axiom,
    ! [N3: set_nat,F: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,N3)
     => ( ! [N: nat] :
            ( ~ aa_set_nat_o(member_nat(N),N3)
           => ( aa_nat_int(F,N) = zero_zero_int ) )
       => summable_int(F) ) ) ).

% summable_finite
tff(fact_8572_sum_OatMost__shift,axiom,
    ! [G: fun_nat_rat,Na: nat] : groups2906978787729119204at_rat(G,aa_nat_set_nat(set_ord_atMost_nat,Na)) = aa_rat_rat(plus_plus_rat(aa_nat_rat(G,zero_zero_nat)),groups2906978787729119204at_rat(aTP_Lamm_jw(G),aa_nat_set_nat(set_ord_lessThan_nat,Na))) ).

% sum.atMost_shift
tff(fact_8573_sum_OatMost__shift,axiom,
    ! [G: fun_nat_int,Na: nat] : groups3539618377306564664at_int(G,aa_nat_set_nat(set_ord_atMost_nat,Na)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,zero_zero_nat)),groups3539618377306564664at_int(aTP_Lamm_jx(G),aa_nat_set_nat(set_ord_lessThan_nat,Na))) ).

% sum.atMost_shift
tff(fact_8574_sum_OatMost__shift,axiom,
    ! [G: fun_nat_nat,Na: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,Na)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G,zero_zero_nat)),aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_jf(G)),aa_nat_set_nat(set_ord_lessThan_nat,Na))) ).

% sum.atMost_shift
tff(fact_8575_sum_OatMost__shift,axiom,
    ! [G: fun_nat_real,Na: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_ord_atMost_nat,Na)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(G,zero_zero_nat)),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_jg(G)),aa_nat_set_nat(set_ord_lessThan_nat,Na))) ).

% sum.atMost_shift
tff(fact_8576_prod_OatMost__shift,axiom,
    ! [G: fun_nat_real,Na: nat] : aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_ord_atMost_nat,Na)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(G,zero_zero_nat)),aa_set_nat_real(groups129246275422532515t_real(aTP_Lamm_jg(G)),aa_nat_set_nat(set_ord_lessThan_nat,Na))) ).

% prod.atMost_shift
tff(fact_8577_prod_OatMost__shift,axiom,
    ! [G: fun_nat_rat,Na: nat] : groups73079841787564623at_rat(G,aa_nat_set_nat(set_ord_atMost_nat,Na)) = aa_rat_rat(times_times_rat(aa_nat_rat(G,zero_zero_nat)),groups73079841787564623at_rat(aTP_Lamm_jw(G),aa_nat_set_nat(set_ord_lessThan_nat,Na))) ).

% prod.atMost_shift
tff(fact_8578_prod_OatMost__shift,axiom,
    ! [G: fun_nat_nat,Na: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,Na)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(G,zero_zero_nat)),aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_jf(G)),aa_nat_set_nat(set_ord_lessThan_nat,Na))) ).

% prod.atMost_shift
tff(fact_8579_prod_OatMost__shift,axiom,
    ! [G: fun_nat_int,Na: nat] : groups705719431365010083at_int(G,aa_nat_set_nat(set_ord_atMost_nat,Na)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(G,zero_zero_nat)),groups705719431365010083at_int(aTP_Lamm_jx(G),aa_nat_set_nat(set_ord_lessThan_nat,Na))) ).

% prod.atMost_shift
tff(fact_8580_Iio__eq__empty__iff,axiom,
    ! [Na: nat] :
      ( ( aa_nat_set_nat(set_ord_lessThan_nat,Na) = bot_bot_set_nat )
    <=> ( Na = bot_bot_nat ) ) ).

% Iio_eq_empty_iff
tff(fact_8581_atMost__atLeast0,axiom,
    ! [Na: nat] : aa_nat_set_nat(set_ord_atMost_nat,Na) = set_or1269000886237332187st_nat(zero_zero_nat,Na) ).

% atMost_atLeast0
tff(fact_8582_lessThan__strict__subset__iff,axiom,
    ! [M: rat,Na: rat] :
      ( ord_less_set_rat(set_ord_lessThan_rat(M),set_ord_lessThan_rat(Na))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,M),Na) ) ).

% lessThan_strict_subset_iff
tff(fact_8583_lessThan__strict__subset__iff,axiom,
    ! [M: num,Na: num] :
      ( ord_less_set_num(set_ord_lessThan_num(M),set_ord_lessThan_num(Na))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,M),Na) ) ).

% lessThan_strict_subset_iff
tff(fact_8584_lessThan__strict__subset__iff,axiom,
    ! [M: nat,Na: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,aa_nat_set_nat(set_ord_lessThan_nat,M)),aa_nat_set_nat(set_ord_lessThan_nat,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ).

% lessThan_strict_subset_iff
tff(fact_8585_lessThan__strict__subset__iff,axiom,
    ! [M: int,Na: int] :
      ( aa_set_int_o(ord_less_set_int(set_ord_lessThan_int(M)),set_ord_lessThan_int(Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,M),Na) ) ).

% lessThan_strict_subset_iff
tff(fact_8586_lessThan__strict__subset__iff,axiom,
    ! [M: real,Na: real] :
      ( ord_less_set_real(set_or5984915006950818249n_real(M),set_or5984915006950818249n_real(Na))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,M),Na) ) ).

% lessThan_strict_subset_iff
tff(fact_8587_summable__mult__D,axiom,
    ! [C: real,F: fun_nat_real] :
      ( summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_qp(C),F))
     => ( ( C != zero_zero_real )
       => summable_real(F) ) ) ).

% summable_mult_D
tff(fact_8588_summable__zero__power,axiom,
    summable_real(aa_real_fun_nat_real(power_power_real,zero_zero_real)) ).

% summable_zero_power
tff(fact_8589_summable__zero__power,axiom,
    summable_int(power_power_int(zero_zero_int)) ).

% summable_zero_power
tff(fact_8590_summable__zero__power,axiom,
    summable_complex(power_power_complex(zero_zero_complex)) ).

% summable_zero_power
tff(fact_8591_pi__gt__zero,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),pi) ).

% pi_gt_zero
tff(fact_8592_pi__not__less__zero,axiom,
    ~ aa_real_o(aa_real_fun_real_o(ord_less_real,pi),zero_zero_real) ).

% pi_not_less_zero
tff(fact_8593_pi__ge__zero,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),pi) ).

% pi_ge_zero
tff(fact_8594_lessThan__empty__iff,axiom,
    ! [Na: nat] :
      ( ( aa_nat_set_nat(set_ord_lessThan_nat,Na) = bot_bot_set_nat )
    <=> ( Na = zero_zero_nat ) ) ).

% lessThan_empty_iff
tff(fact_8595_sum__less__suminf,axiom,
    ! [F: fun_nat_int,Na: nat] :
      ( summable_int(F)
     => ( ! [M3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M3)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(F,M3)) )
       => aa_int_o(aa_int_fun_int_o(ord_less_int,groups3539618377306564664at_int(F,aa_nat_set_nat(set_ord_lessThan_nat,Na))),suminf_int(F)) ) ) ).

% sum_less_suminf
tff(fact_8596_sum__less__suminf,axiom,
    ! [F: fun_nat_nat,Na: nat] :
      ( summable_nat(F)
     => ( ! [M3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M3)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(F,M3)) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_ord_lessThan_nat,Na))),suminf_nat(F)) ) ) ).

% sum_less_suminf
tff(fact_8597_sum__less__suminf,axiom,
    ! [F: fun_nat_real,Na: nat] :
      ( summable_real(F)
     => ( ! [M3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M3)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(F,M3)) )
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_ord_lessThan_nat,Na))),suminf_real(F)) ) ) ).

% sum_less_suminf
tff(fact_8598_not__Iic__le__Icc,axiom,
    ! [H: int,L2: int,H2: int] : ~ aa_set_int_o(ord_less_eq_set_int(set_ord_atMost_int(H)),set_or1266510415728281911st_int(L2,H2)) ).

% not_Iic_le_Icc
tff(fact_8599_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_8600_polyfun__linear__factor__root,axiom,
    ! [C: fun_nat_complex,Aa: complex,Na: nat] :
      ( ( aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_ru(C),Aa)),aa_nat_set_nat(set_ord_atMost_nat,Na)) = zero_zero_complex )
     => ~ ! [B2: fun_nat_complex] :
            ~ ! [Z4: complex] : aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_ru(C),Z4)),aa_nat_set_nat(set_ord_atMost_nat,Na)) = aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(Z4),Aa)),aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_ru(B2),Z4)),aa_nat_set_nat(set_ord_lessThan_nat,Na))) ) ).

% polyfun_linear_factor_root
tff(fact_8601_polyfun__linear__factor__root,axiom,
    ! [C: fun_nat_rat,Aa: rat,Na: nat] :
      ( ( groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_rw(C),Aa),aa_nat_set_nat(set_ord_atMost_nat,Na)) = zero_zero_rat )
     => ~ ! [B2: fun_nat_rat] :
            ~ ! [Z4: rat] : groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_rw(C),Z4),aa_nat_set_nat(set_ord_atMost_nat,Na)) = aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(Z4),Aa)),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_rw(B2),Z4),aa_nat_set_nat(set_ord_lessThan_nat,Na))) ) ).

% polyfun_linear_factor_root
tff(fact_8602_polyfun__linear__factor__root,axiom,
    ! [C: fun_nat_int,Aa: int,Na: nat] :
      ( ( groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_rx(C),Aa),aa_nat_set_nat(set_ord_atMost_nat,Na)) = zero_zero_int )
     => ~ ! [B2: fun_nat_int] :
            ~ ! [Z4: int] : groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_rx(C),Z4),aa_nat_set_nat(set_ord_atMost_nat,Na)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Z4),Aa)),groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_rx(B2),Z4),aa_nat_set_nat(set_ord_lessThan_nat,Na))) ) ).

% polyfun_linear_factor_root
tff(fact_8603_polyfun__linear__factor__root,axiom,
    ! [C: fun_nat_real,Aa: real,Na: nat] :
      ( ( aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_rs(C),Aa)),aa_nat_set_nat(set_ord_atMost_nat,Na)) = zero_zero_real )
     => ~ ! [B2: fun_nat_real] :
            ~ ! [Z4: real] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_rs(C),Z4)),aa_nat_set_nat(set_ord_atMost_nat,Na)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Z4),Aa)),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_rs(B2),Z4)),aa_nat_set_nat(set_ord_lessThan_nat,Na))) ) ).

% polyfun_linear_factor_root
tff(fact_8604_finite__nat__iff__bounded__le,axiom,
    ! [S2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,S2)
    <=> ? [K3: nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S2),aa_nat_set_nat(set_ord_atMost_nat,K3)) ) ).

% finite_nat_iff_bounded_le
tff(fact_8605_sum_Otriangle__reindex,axiom,
    ! [G: fun_nat_fun_nat_nat,Na: nat] : groups977919841031483927at_nat(produc6842872674320459806at_nat(G),collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_ry(Na)))) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_sa(G)),aa_nat_set_nat(set_ord_lessThan_nat,Na)) ).

% sum.triangle_reindex
tff(fact_8606_sum_Otriangle__reindex,axiom,
    ! [G: fun_nat_fun_nat_real,Na: nat] : groups4567486121110086003t_real(produc1703576794950452218t_real(G),collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_ry(Na)))) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_sc(G)),aa_nat_set_nat(set_ord_lessThan_nat,Na)) ).

% sum.triangle_reindex
tff(fact_8607_finite__nat__bounded,axiom,
    ! [S2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ? [K2: nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S2),aa_nat_set_nat(set_ord_lessThan_nat,K2)) ) ).

% finite_nat_bounded
tff(fact_8608_finite__nat__iff__bounded,axiom,
    ! [S2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,S2)
    <=> ? [K3: nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S2),aa_nat_set_nat(set_ord_lessThan_nat,K3)) ) ).

% finite_nat_iff_bounded
tff(fact_8609_prod_Otriangle__reindex,axiom,
    ! [G: fun_nat_fun_nat_nat,Na: nat] : groups4077766827762148844at_nat(produc6842872674320459806at_nat(G),collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_ry(Na)))) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_sd(G)),aa_nat_set_nat(set_ord_lessThan_nat,Na)) ).

% prod.triangle_reindex
tff(fact_8610_prod_Otriangle__reindex,axiom,
    ! [G: fun_nat_fun_nat_int,Na: nat] : groups4075276357253098568at_int(produc6840382203811409530at_int(G),collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_ry(Na)))) = groups705719431365010083at_int(aTP_Lamm_sf(G),aa_nat_set_nat(set_ord_lessThan_nat,Na)) ).

% prod.triangle_reindex
tff(fact_8611_sum__less__suminf2,axiom,
    ! [F: fun_nat_int,Na: nat,I: nat] :
      ( summable_int(F)
     => ( ! [M3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M3)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(F,M3)) )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),I)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(F,I))
           => aa_int_o(aa_int_fun_int_o(ord_less_int,groups3539618377306564664at_int(F,aa_nat_set_nat(set_ord_lessThan_nat,Na))),suminf_int(F)) ) ) ) ) ).

% sum_less_suminf2
tff(fact_8612_sum__less__suminf2,axiom,
    ! [F: fun_nat_nat,Na: nat,I: nat] :
      ( summable_nat(F)
     => ( ! [M3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M3)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,M3)) )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),I)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(F,I))
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_ord_lessThan_nat,Na))),suminf_nat(F)) ) ) ) ) ).

% sum_less_suminf2
tff(fact_8613_sum__less__suminf2,axiom,
    ! [F: fun_nat_real,Na: nat,I: nat] :
      ( summable_real(F)
     => ( ! [M3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M3)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,M3)) )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),I)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(F,I))
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_ord_lessThan_nat,Na))),suminf_real(F)) ) ) ) ) ).

% sum_less_suminf2
tff(fact_8614_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_8615_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_8616_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_8617_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_8618_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_8619_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_8620_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_8621_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_8622_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_8623_summable__0__powser,axiom,
    ! [F: fun_nat_complex] : summable_complex(aTP_Lamm_iu(F)) ).

% summable_0_powser
tff(fact_8624_summable__0__powser,axiom,
    ! [F: fun_nat_real] : summable_real(aTP_Lamm_iw(F)) ).

% summable_0_powser
tff(fact_8625_summable__zero__power_H,axiom,
    ! [F: fun_nat_complex] : summable_complex(aTP_Lamm_iu(F)) ).

% summable_zero_power'
tff(fact_8626_summable__zero__power_H,axiom,
    ! [F: fun_nat_real] : summable_real(aTP_Lamm_iw(F)) ).

% summable_zero_power'
tff(fact_8627_summable__zero__power_H,axiom,
    ! [F: fun_nat_int] : summable_int(aTP_Lamm_sg(F)) ).

% summable_zero_power'
tff(fact_8628_powser__split__head_I3_J,axiom,
    ! [F: fun_nat_complex,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_ru(F),Z))
     => summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_sh(F),Z)) ) ).

% powser_split_head(3)
tff(fact_8629_powser__split__head_I3_J,axiom,
    ! [F: fun_nat_real,Z: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_rs(F),Z))
     => summable_real(aa_real_fun_nat_real(aTP_Lamm_si(F),Z)) ) ).

% powser_split_head(3)
tff(fact_8630_summable__powser__split__head,axiom,
    ! [F: fun_nat_complex,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_sh(F),Z))
    <=> summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_ru(F),Z)) ) ).

% summable_powser_split_head
tff(fact_8631_summable__powser__split__head,axiom,
    ! [F: fun_nat_real,Z: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_si(F),Z))
    <=> summable_real(aa_real_fun_nat_real(aTP_Lamm_rs(F),Z)) ) ).

% summable_powser_split_head
tff(fact_8632_summable__norm__comparison__test,axiom,
    ! [F: fun_nat_complex,G: fun_nat_real] :
      ( ? [N6: nat] :
        ! [N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N6),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_pi(F)) ) ) ).

% summable_norm_comparison_test
tff(fact_8633_summable__rabs__comparison__test,axiom,
    ! [F: fun_nat_real,G: fun_nat_real] :
      ( ? [N6: nat] :
        ! [N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N6),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_fy(F)) ) ) ).

% summable_rabs_comparison_test
tff(fact_8634_sum_Onat__diff__reindex,axiom,
    ! [G: fun_nat_nat,Na: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_sj(G),Na)),aa_nat_set_nat(set_ord_lessThan_nat,Na)) = aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,Na)) ).

% sum.nat_diff_reindex
tff(fact_8635_sum_Onat__diff__reindex,axiom,
    ! [G: fun_nat_real,Na: nat] : aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_sk(G),Na)),aa_nat_set_nat(set_ord_lessThan_nat,Na)) = aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_ord_lessThan_nat,Na)) ).

% sum.nat_diff_reindex
tff(fact_8636_summable__rabs,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(aTP_Lamm_fy(F))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,suminf_real(F))),suminf_real(aTP_Lamm_fy(F))) ) ).

% summable_rabs
tff(fact_8637_prod_Onat__diff__reindex,axiom,
    ! [G: fun_nat_nat,Na: nat] : aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_sj(G),Na)),aa_nat_set_nat(set_ord_lessThan_nat,Na)) = aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,Na)) ).

% prod.nat_diff_reindex
tff(fact_8638_prod_Onat__diff__reindex,axiom,
    ! [G: fun_nat_int,Na: nat] : groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_sl(G),Na),aa_nat_set_nat(set_ord_lessThan_nat,Na)) = groups705719431365010083at_int(G,aa_nat_set_nat(set_ord_lessThan_nat,Na)) ).

% prod.nat_diff_reindex
tff(fact_8639_sum__diff__distrib,axiom,
    ! [Q: fun_int_nat,P: fun_int_nat,Na: int] :
      ( ! [X: int] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(Q,X)),aa_int_nat(P,X))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups4541462559716669496nt_nat(P,set_ord_lessThan_int(Na))),groups4541462559716669496nt_nat(Q,set_ord_lessThan_int(Na))) = groups4541462559716669496nt_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_jd(Q),P),set_ord_lessThan_int(Na)) ) ) ).

% sum_diff_distrib
tff(fact_8640_sum__diff__distrib,axiom,
    ! [Q: fun_real_nat,P: fun_real_nat,Na: real] :
      ( ! [X: real] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_real_nat(Q,X)),aa_real_nat(P,X))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,groups1935376822645274424al_nat(P,set_or5984915006950818249n_real(Na))),groups1935376822645274424al_nat(Q,set_or5984915006950818249n_real(Na))) = groups1935376822645274424al_nat(aa_fun211602392931233840al_nat(aTP_Lamm_jb(Q),P),set_or5984915006950818249n_real(Na)) ) ) ).

% sum_diff_distrib
tff(fact_8641_sum__diff__distrib,axiom,
    ! [Q: fun_nat_nat,P: fun_nat_nat,Na: nat] :
      ( ! [X: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(Q,X)),aa_nat_nat(P,X))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(P),aa_nat_set_nat(set_ord_lessThan_nat,Na))),aa_set_nat_nat(groups3542108847815614940at_nat(Q),aa_nat_set_nat(set_ord_lessThan_nat,Na))) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_je(Q),P)),aa_nat_set_nat(set_ord_lessThan_nat,Na)) ) ) ).

% sum_diff_distrib
tff(fact_8642_suminf__pos2,axiom,
    ! [F: fun_nat_real,I: nat] :
      ( summable_real(F)
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,N))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(F,I))
         => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),suminf_real(F)) ) ) ) ).

% suminf_pos2
tff(fact_8643_suminf__pos2,axiom,
    ! [F: fun_nat_nat,I: nat] :
      ( summable_nat(F)
     => ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,N))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(F,I))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),suminf_nat(F)) ) ) ) ).

% suminf_pos2
tff(fact_8644_suminf__pos2,axiom,
    ! [F: fun_nat_int,I: nat] :
      ( summable_int(F)
     => ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(F,N))
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(F,I))
         => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),suminf_int(F)) ) ) ) ).

% suminf_pos2
tff(fact_8645_suminf__pos__iff,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(F)
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,N))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),suminf_real(F))
        <=> ? [I3: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(F,I3)) ) ) ) ).

% suminf_pos_iff
tff(fact_8646_suminf__pos__iff,axiom,
    ! [F: fun_nat_nat] :
      ( summable_nat(F)
     => ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,N))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),suminf_nat(F))
        <=> ? [I3: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(F,I3)) ) ) ) ).

% suminf_pos_iff
tff(fact_8647_suminf__pos__iff,axiom,
    ! [F: fun_nat_int] :
      ( summable_int(F)
     => ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(F,N))
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),suminf_int(F))
        <=> ? [I3: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(F,I3)) ) ) ) ).

% suminf_pos_iff
tff(fact_8648_sum__choose__upper,axiom,
    ! [M: nat,Na: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_sm,M)),aa_nat_set_nat(set_ord_atMost_nat,Na)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,Na)),aa_nat_nat(suc,M)) ).

% sum_choose_upper
tff(fact_8649_polyfun__diff__alt,axiom,
    ! [Na: nat,Aa: fun_nat_complex,Xb: complex,Y: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Na)
     => ( aa_complex_complex(minus_minus_complex(aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_ru(Aa),Xb)),aa_nat_set_nat(set_ord_atMost_nat,Na))),aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_ru(Aa),Y)),aa_nat_set_nat(set_ord_atMost_nat,Na))) = aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(Xb),Y)),aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_fun5031706661949900810omplex(aTP_Lamm_so(Na),Aa),Xb),Y)),aa_nat_set_nat(set_ord_lessThan_nat,Na))) ) ) ).

% polyfun_diff_alt
tff(fact_8650_polyfun__diff__alt,axiom,
    ! [Na: nat,Aa: fun_nat_rat,Xb: rat,Y: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Na)
     => ( aa_rat_rat(minus_minus_rat(groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_rw(Aa),Xb),aa_nat_set_nat(set_ord_atMost_nat,Na))),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_rw(Aa),Y),aa_nat_set_nat(set_ord_atMost_nat,Na))) = aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(Xb),Y)),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aa_fun8528462509625601970at_rat(aTP_Lamm_sq(Na),Aa),Xb),Y),aa_nat_set_nat(set_ord_lessThan_nat,Na))) ) ) ).

% polyfun_diff_alt
tff(fact_8651_polyfun__diff__alt,axiom,
    ! [Na: nat,Aa: fun_nat_int,Xb: int,Y: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Na)
     => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_rx(Aa),Xb),aa_nat_set_nat(set_ord_atMost_nat,Na))),groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_rx(Aa),Y),aa_nat_set_nat(set_ord_atMost_nat,Na))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),Y)),groups3539618377306564664at_int(aa_int_fun_nat_int(aa_int5842867502692063756at_int(aa_fun7421131748584218370at_int(aTP_Lamm_ss(Na),Aa),Xb),Y),aa_nat_set_nat(set_ord_lessThan_nat,Na))) ) ) ).

% polyfun_diff_alt
tff(fact_8652_polyfun__diff__alt,axiom,
    ! [Na: nat,Aa: fun_nat_real,Xb: real,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Na)
     => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_rs(Aa),Xb)),aa_nat_set_nat(set_ord_atMost_nat,Na))),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_rs(Aa),Y)),aa_nat_set_nat(set_ord_atMost_nat,Na))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Y)),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aa_fun2548873334967081986t_real(aTP_Lamm_su(Na),Aa),Xb),Y)),aa_nat_set_nat(set_ord_lessThan_nat,Na))) ) ) ).

% polyfun_diff_alt
tff(fact_8653_powser__inside,axiom,
    ! [F: fun_nat_real,Xb: real,Z: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_rs(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_rs(F),Z)) ) ) ).

% powser_inside
tff(fact_8654_powser__inside,axiom,
    ! [F: fun_nat_complex,Xb: complex,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_ru(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_ru(F),Z)) ) ) ).

% powser_inside
tff(fact_8655_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_8656_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_8657_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_8658_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_8659_suminf__split__head,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(F)
     => ( suminf_real(aTP_Lamm_jg(F)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,suminf_real(F)),aa_nat_real(F,zero_zero_nat)) ) ) ).

% suminf_split_head
tff(fact_8660_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_8661_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_8662_sum_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_rat,Na: nat] : groups2906978787729119204at_rat(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Na))) = aa_rat_rat(plus_plus_rat(aa_nat_rat(G,zero_zero_nat)),groups2906978787729119204at_rat(aTP_Lamm_jw(G),aa_nat_set_nat(set_ord_atMost_nat,Na))) ).

% sum.atMost_Suc_shift
tff(fact_8663_sum_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_int,Na: nat] : groups3539618377306564664at_int(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Na))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,zero_zero_nat)),groups3539618377306564664at_int(aTP_Lamm_jx(G),aa_nat_set_nat(set_ord_atMost_nat,Na))) ).

% sum.atMost_Suc_shift
tff(fact_8664_sum_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_nat,Na: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Na))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G,zero_zero_nat)),aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_jf(G)),aa_nat_set_nat(set_ord_atMost_nat,Na))) ).

% sum.atMost_Suc_shift
tff(fact_8665_sum_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_real,Na: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Na))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(G,zero_zero_nat)),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_jg(G)),aa_nat_set_nat(set_ord_atMost_nat,Na))) ).

% sum.atMost_Suc_shift
tff(fact_8666_sum__pos__lt__pair,axiom,
    ! [F: fun_nat_real,K: nat] :
      ( summable_real(F)
     => ( ! [D4: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,aa_nat_nat(suc,zero_zero_nat))),D4)))),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))),D4)),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_8667_sum__telescope,axiom,
    ! [F: fun_nat_rat,I: nat] : groups2906978787729119204at_rat(aTP_Lamm_sv(F),aa_nat_set_nat(set_ord_atMost_nat,I)) = aa_rat_rat(minus_minus_rat(aa_nat_rat(F,zero_zero_nat)),aa_nat_rat(F,aa_nat_nat(suc,I))) ).

% sum_telescope
tff(fact_8668_sum__telescope,axiom,
    ! [F: fun_nat_int,I: nat] : groups3539618377306564664at_int(aTP_Lamm_sw(F),aa_nat_set_nat(set_ord_atMost_nat,I)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,zero_zero_nat)),aa_nat_int(F,aa_nat_nat(suc,I))) ).

% sum_telescope
tff(fact_8669_sum__telescope,axiom,
    ! [F: fun_nat_real,I: nat] : aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_sx(F)),aa_nat_set_nat(set_ord_atMost_nat,I)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(F,zero_zero_nat)),aa_nat_real(F,aa_nat_nat(suc,I))) ).

% sum_telescope
tff(fact_8670_polyfun__eq__coeffs,axiom,
    ! [C: fun_nat_complex,Na: nat,D: fun_nat_complex] :
      ( ! [X4: complex] : aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_ru(C),X4)),aa_nat_set_nat(set_ord_atMost_nat,Na)) = aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_ru(D),X4)),aa_nat_set_nat(set_ord_atMost_nat,Na))
    <=> ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I3),Na)
         => ( aa_nat_complex(C,I3) = aa_nat_complex(D,I3) ) ) ) ).

% polyfun_eq_coeffs
tff(fact_8671_polyfun__eq__coeffs,axiom,
    ! [C: fun_nat_real,Na: nat,D: fun_nat_real] :
      ( ! [X4: real] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_rs(C),X4)),aa_nat_set_nat(set_ord_atMost_nat,Na)) = aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_rs(D),X4)),aa_nat_set_nat(set_ord_atMost_nat,Na))
    <=> ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I3),Na)
         => ( aa_nat_real(C,I3) = aa_nat_real(D,I3) ) ) ) ).

% polyfun_eq_coeffs
tff(fact_8672_prod_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_real,Na: nat] : aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Na))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(G,zero_zero_nat)),aa_set_nat_real(groups129246275422532515t_real(aTP_Lamm_jg(G)),aa_nat_set_nat(set_ord_atMost_nat,Na))) ).

% prod.atMost_Suc_shift
tff(fact_8673_prod_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_rat,Na: nat] : groups73079841787564623at_rat(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Na))) = aa_rat_rat(times_times_rat(aa_nat_rat(G,zero_zero_nat)),groups73079841787564623at_rat(aTP_Lamm_jw(G),aa_nat_set_nat(set_ord_atMost_nat,Na))) ).

% prod.atMost_Suc_shift
tff(fact_8674_prod_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_nat,Na: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Na))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(G,zero_zero_nat)),aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_jf(G)),aa_nat_set_nat(set_ord_atMost_nat,Na))) ).

% prod.atMost_Suc_shift
tff(fact_8675_prod_OatMost__Suc__shift,axiom,
    ! [G: fun_nat_int,Na: nat] : groups705719431365010083at_int(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Na))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(G,zero_zero_nat)),groups705719431365010083at_int(aTP_Lamm_jx(G),aa_nat_set_nat(set_ord_atMost_nat,Na))) ).

% prod.atMost_Suc_shift
tff(fact_8676_sum_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_rat,Na: nat] : groups2906978787729119204at_rat(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Na))) = aa_rat_rat(plus_plus_rat(aa_nat_rat(G,zero_zero_nat)),groups2906978787729119204at_rat(aTP_Lamm_jw(G),aa_nat_set_nat(set_ord_lessThan_nat,Na))) ).

% sum.lessThan_Suc_shift
tff(fact_8677_sum_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_int,Na: nat] : groups3539618377306564664at_int(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Na))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(G,zero_zero_nat)),groups3539618377306564664at_int(aTP_Lamm_jx(G),aa_nat_set_nat(set_ord_lessThan_nat,Na))) ).

% sum.lessThan_Suc_shift
tff(fact_8678_sum_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_nat,Na: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Na))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(G,zero_zero_nat)),aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_jf(G)),aa_nat_set_nat(set_ord_lessThan_nat,Na))) ).

% sum.lessThan_Suc_shift
tff(fact_8679_sum_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_real,Na: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Na))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(G,zero_zero_nat)),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_jg(G)),aa_nat_set_nat(set_ord_lessThan_nat,Na))) ).

% sum.lessThan_Suc_shift
tff(fact_8680_sum__lessThan__telescope,axiom,
    ! [F: fun_nat_rat,M: nat] : groups2906978787729119204at_rat(aTP_Lamm_jy(F),aa_nat_set_nat(set_ord_lessThan_nat,M)) = aa_rat_rat(minus_minus_rat(aa_nat_rat(F,M)),aa_nat_rat(F,zero_zero_nat)) ).

% sum_lessThan_telescope
tff(fact_8681_sum__lessThan__telescope,axiom,
    ! [F: fun_nat_int,M: nat] : groups3539618377306564664at_int(aTP_Lamm_jz(F),aa_nat_set_nat(set_ord_lessThan_nat,M)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,M)),aa_nat_int(F,zero_zero_nat)) ).

% sum_lessThan_telescope
tff(fact_8682_sum__lessThan__telescope,axiom,
    ! [F: fun_nat_real,M: nat] : aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_ka(F)),aa_nat_set_nat(set_ord_lessThan_nat,M)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(F,M)),aa_nat_real(F,zero_zero_nat)) ).

% sum_lessThan_telescope
tff(fact_8683_sum__lessThan__telescope_H,axiom,
    ! [F: fun_nat_rat,M: nat] : groups2906978787729119204at_rat(aTP_Lamm_sv(F),aa_nat_set_nat(set_ord_lessThan_nat,M)) = aa_rat_rat(minus_minus_rat(aa_nat_rat(F,zero_zero_nat)),aa_nat_rat(F,M)) ).

% sum_lessThan_telescope'
tff(fact_8684_sum__lessThan__telescope_H,axiom,
    ! [F: fun_nat_int,M: nat] : groups3539618377306564664at_int(aTP_Lamm_sw(F),aa_nat_set_nat(set_ord_lessThan_nat,M)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,zero_zero_nat)),aa_nat_int(F,M)) ).

% sum_lessThan_telescope'
tff(fact_8685_sum__lessThan__telescope_H,axiom,
    ! [F: fun_nat_real,M: nat] : aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_sx(F)),aa_nat_set_nat(set_ord_lessThan_nat,M)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(F,zero_zero_nat)),aa_nat_real(F,M)) ).

% sum_lessThan_telescope'
tff(fact_8686_prod_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_real,Na: nat] : aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Na))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(G,zero_zero_nat)),aa_set_nat_real(groups129246275422532515t_real(aTP_Lamm_jg(G)),aa_nat_set_nat(set_ord_lessThan_nat,Na))) ).

% prod.lessThan_Suc_shift
tff(fact_8687_prod_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_rat,Na: nat] : groups73079841787564623at_rat(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Na))) = aa_rat_rat(times_times_rat(aa_nat_rat(G,zero_zero_nat)),groups73079841787564623at_rat(aTP_Lamm_jw(G),aa_nat_set_nat(set_ord_lessThan_nat,Na))) ).

% prod.lessThan_Suc_shift
tff(fact_8688_prod_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_nat,Na: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Na))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(G,zero_zero_nat)),aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_jf(G)),aa_nat_set_nat(set_ord_lessThan_nat,Na))) ).

% prod.lessThan_Suc_shift
tff(fact_8689_prod_OlessThan__Suc__shift,axiom,
    ! [G: fun_nat_int,Na: nat] : groups705719431365010083at_int(G,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Na))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(G,zero_zero_nat)),groups705719431365010083at_int(aTP_Lamm_jx(G),aa_nat_set_nat(set_ord_lessThan_nat,Na))) ).

% prod.lessThan_Suc_shift
tff(fact_8690_summable__norm,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(aTP_Lamm_pk(F))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,suminf_real(F))),suminf_real(aTP_Lamm_pk(F))) ) ).

% summable_norm
tff(fact_8691_summable__norm,axiom,
    ! [F: fun_nat_complex] :
      ( summable_real(aTP_Lamm_pi(F))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,suminf_complex(F))),suminf_real(aTP_Lamm_pi(F))) ) ).

% summable_norm
tff(fact_8692_sum_OatLeast1__atMost__eq,axiom,
    ! [G: fun_nat_nat,Na: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),Na)) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_jf(G)),aa_nat_set_nat(set_ord_lessThan_nat,Na)) ).

% sum.atLeast1_atMost_eq
tff(fact_8693_sum_OatLeast1__atMost__eq,axiom,
    ! [G: fun_nat_real,Na: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),Na)) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_jg(G)),aa_nat_set_nat(set_ord_lessThan_nat,Na)) ).

% sum.atLeast1_atMost_eq
tff(fact_8694_prod_OatLeast1__atMost__eq,axiom,
    ! [G: fun_nat_nat,Na: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),Na)) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_jf(G)),aa_nat_set_nat(set_ord_lessThan_nat,Na)) ).

% prod.atLeast1_atMost_eq
tff(fact_8695_prod_OatLeast1__atMost__eq,axiom,
    ! [G: fun_nat_int,Na: nat] : groups705719431365010083at_int(G,set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),Na)) = groups705719431365010083at_int(aTP_Lamm_jx(G),aa_nat_set_nat(set_ord_lessThan_nat,Na)) ).

% prod.atLeast1_atMost_eq
tff(fact_8696_sum__bounds__lt__plus1,axiom,
    ! [F: fun_nat_nat,Mm: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_jf(F)),aa_nat_set_nat(set_ord_lessThan_nat,Mm)) = aa_set_nat_nat(groups3542108847815614940at_nat(F),set_or1269000886237332187st_nat(one_one_nat,Mm)) ).

% sum_bounds_lt_plus1
tff(fact_8697_sum__bounds__lt__plus1,axiom,
    ! [F: fun_nat_real,Mm: nat] : aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_jg(F)),aa_nat_set_nat(set_ord_lessThan_nat,Mm)) = aa_set_nat_real(groups6591440286371151544t_real(F),set_or1269000886237332187st_nat(one_one_nat,Mm)) ).

% sum_bounds_lt_plus1
tff(fact_8698_polyfun__diff,axiom,
    ! [Na: nat,Aa: fun_nat_complex,Xb: complex,Y: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Na)
     => ( aa_complex_complex(minus_minus_complex(aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_ru(Aa),Xb)),aa_nat_set_nat(set_ord_atMost_nat,Na))),aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_ru(Aa),Y)),aa_nat_set_nat(set_ord_atMost_nat,Na))) = aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(Xb),Y)),aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_fun5031706661949900810omplex(aTP_Lamm_sz(Na),Aa),Xb),Y)),aa_nat_set_nat(set_ord_lessThan_nat,Na))) ) ) ).

% polyfun_diff
tff(fact_8699_polyfun__diff,axiom,
    ! [Na: nat,Aa: fun_nat_rat,Xb: rat,Y: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Na)
     => ( aa_rat_rat(minus_minus_rat(groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_rw(Aa),Xb),aa_nat_set_nat(set_ord_atMost_nat,Na))),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_rw(Aa),Y),aa_nat_set_nat(set_ord_atMost_nat,Na))) = aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(Xb),Y)),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aa_fun8528462509625601970at_rat(aTP_Lamm_tb(Na),Aa),Xb),Y),aa_nat_set_nat(set_ord_lessThan_nat,Na))) ) ) ).

% polyfun_diff
tff(fact_8700_polyfun__diff,axiom,
    ! [Na: nat,Aa: fun_nat_int,Xb: int,Y: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Na)
     => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_rx(Aa),Xb),aa_nat_set_nat(set_ord_atMost_nat,Na))),groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_rx(Aa),Y),aa_nat_set_nat(set_ord_atMost_nat,Na))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),Y)),groups3539618377306564664at_int(aa_int_fun_nat_int(aa_int5842867502692063756at_int(aa_fun7421131748584218370at_int(aTP_Lamm_td(Na),Aa),Xb),Y),aa_nat_set_nat(set_ord_lessThan_nat,Na))) ) ) ).

% polyfun_diff
tff(fact_8701_polyfun__diff,axiom,
    ! [Na: nat,Aa: fun_nat_real,Xb: real,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Na)
     => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_rs(Aa),Xb)),aa_nat_set_nat(set_ord_atMost_nat,Na))),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_rs(Aa),Y)),aa_nat_set_nat(set_ord_atMost_nat,Na))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Y)),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aa_fun2548873334967081986t_real(aTP_Lamm_tf(Na),Aa),Xb),Y)),aa_nat_set_nat(set_ord_lessThan_nat,Na))) ) ) ).

% polyfun_diff
tff(fact_8702_sum__choose__lower,axiom,
    ! [R2: nat,Na: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_tg,R2)),aa_nat_set_nat(set_ord_atMost_nat,Na)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,R2),Na))),Na) ).

% sum_choose_lower
tff(fact_8703_choose__rising__sum_I2_J,axiom,
    ! [Na: nat,M: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_th,Na)),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),M)),one_one_nat)),M) ).

% choose_rising_sum(2)
tff(fact_8704_choose__rising__sum_I1_J,axiom,
    ! [Na: nat,M: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_th,Na)),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),M)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),one_one_nat)) ).

% choose_rising_sum(1)
tff(fact_8705_sum__le__suminf,axiom,
    ! [F: fun_nat_int,I5: set_nat] :
      ( summable_int(F)
     => ( aa_set_nat_o(finite_finite_nat,I5)
       => ( ! [N: nat] :
              ( aa_set_nat_o(member_nat(N),uminus5710092332889474511et_nat(I5))
             => 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,groups3539618377306564664at_int(F,I5)),suminf_int(F)) ) ) ) ).

% sum_le_suminf
tff(fact_8706_sum__le__suminf,axiom,
    ! [F: fun_nat_nat,I5: set_nat] :
      ( summable_nat(F)
     => ( aa_set_nat_o(finite_finite_nat,I5)
       => ( ! [N: nat] :
              ( aa_set_nat_o(member_nat(N),uminus5710092332889474511et_nat(I5))
             => 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,aa_set_nat_nat(groups3542108847815614940at_nat(F),I5)),suminf_nat(F)) ) ) ) ).

% sum_le_suminf
tff(fact_8707_sum__le__suminf,axiom,
    ! [F: fun_nat_real,I5: set_nat] :
      ( summable_real(F)
     => ( aa_set_nat_o(finite_finite_nat,I5)
       => ( ! [N: nat] :
              ( aa_set_nat_o(member_nat(N),uminus5710092332889474511et_nat(I5))
             => 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,aa_set_nat_real(groups6591440286371151544t_real(F),I5)),suminf_real(F)) ) ) ) ).

% sum_le_suminf
tff(fact_8708_polyfun__eq__0,axiom,
    ! [C: fun_nat_complex,Na: nat] :
      ( ! [X4: complex] : aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_ru(C),X4)),aa_nat_set_nat(set_ord_atMost_nat,Na)) = zero_zero_complex
    <=> ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I3),Na)
         => ( aa_nat_complex(C,I3) = zero_zero_complex ) ) ) ).

% polyfun_eq_0
tff(fact_8709_polyfun__eq__0,axiom,
    ! [C: fun_nat_real,Na: nat] :
      ( ! [X4: real] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_rs(C),X4)),aa_nat_set_nat(set_ord_atMost_nat,Na)) = zero_zero_real
    <=> ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I3),Na)
         => ( aa_nat_real(C,I3) = zero_zero_real ) ) ) ).

% polyfun_eq_0
tff(fact_8710_zero__polynom__imp__zero__coeffs,axiom,
    ! [C: fun_nat_complex,Na: nat,K: nat] :
      ( ! [W: complex] : aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_ru(C),W)),aa_nat_set_nat(set_ord_atMost_nat,Na)) = zero_zero_complex
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
       => ( aa_nat_complex(C,K) = zero_zero_complex ) ) ) ).

% zero_polynom_imp_zero_coeffs
tff(fact_8711_zero__polynom__imp__zero__coeffs,axiom,
    ! [C: fun_nat_real,Na: nat,K: nat] :
      ( ! [W: real] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_rs(C),W)),aa_nat_set_nat(set_ord_atMost_nat,Na)) = zero_zero_real
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
       => ( aa_nat_real(C,K) = zero_zero_real ) ) ) ).

% zero_polynom_imp_zero_coeffs
tff(fact_8712_pi__half__less__two,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),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_8713_pi__half__le__two,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),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_8714_power__diff__1__eq,axiom,
    ! [Xb: complex,Na: nat] : aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(Xb),Na)),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,Na))) ).

% power_diff_1_eq
tff(fact_8715_power__diff__1__eq,axiom,
    ! [Xb: rat,Na: nat] : aa_rat_rat(minus_minus_rat(aa_nat_rat(power_power_rat(Xb),Na)),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,Na))) ).

% power_diff_1_eq
tff(fact_8716_power__diff__1__eq,axiom,
    ! [Xb: int,Na: nat] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(Xb),Na)),one_one_int) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),one_one_int)),groups3539618377306564664at_int(power_power_int(Xb),aa_nat_set_nat(set_ord_lessThan_nat,Na))) ).

% power_diff_1_eq
tff(fact_8717_power__diff__1__eq,axiom,
    ! [Xb: real,Na: nat] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Na)),one_one_real) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_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,Na))) ).

% power_diff_1_eq
tff(fact_8718_one__diff__power__eq,axiom,
    ! [Xb: complex,Na: nat] : aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(power_power_complex(Xb),Na)) = 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,Na))) ).

% one_diff_power_eq
tff(fact_8719_one__diff__power__eq,axiom,
    ! [Xb: rat,Na: nat] : aa_rat_rat(minus_minus_rat(one_one_rat),aa_nat_rat(power_power_rat(Xb),Na)) = 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,Na))) ).

% one_diff_power_eq
tff(fact_8720_one__diff__power__eq,axiom,
    ! [Xb: int,Na: nat] : aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),aa_nat_int(power_power_int(Xb),Na)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),Xb)),groups3539618377306564664at_int(power_power_int(Xb),aa_nat_set_nat(set_ord_lessThan_nat,Na))) ).

% one_diff_power_eq
tff(fact_8721_one__diff__power__eq,axiom,
    ! [Xb: real,Na: nat] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Na)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_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,Na))) ).

% one_diff_power_eq
tff(fact_8722_geometric__sum,axiom,
    ! [Xb: complex,Na: nat] :
      ( ( Xb != one_one_complex )
     => ( aa_set_nat_complex(groups2073611262835488442omplex(power_power_complex(Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Na)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(Xb),Na)),one_one_complex)),aa_complex_complex(minus_minus_complex(Xb),one_one_complex)) ) ) ).

% geometric_sum
tff(fact_8723_geometric__sum,axiom,
    ! [Xb: rat,Na: nat] :
      ( ( Xb != one_one_rat )
     => ( groups2906978787729119204at_rat(power_power_rat(Xb),aa_nat_set_nat(set_ord_lessThan_nat,Na)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_nat_rat(power_power_rat(Xb),Na)),one_one_rat)),aa_rat_rat(minus_minus_rat(Xb),one_one_rat)) ) ) ).

% geometric_sum
tff(fact_8724_geometric__sum,axiom,
    ! [Xb: real,Na: 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,Na)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Na)),one_one_real)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),one_one_real)) ) ) ).

% geometric_sum
tff(fact_8725_sum__up__index__split,axiom,
    ! [F: fun_nat_rat,M: nat,Na: nat] : groups2906978787729119204at_rat(F,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na))) = aa_rat_rat(plus_plus_rat(groups2906978787729119204at_rat(F,aa_nat_set_nat(set_ord_atMost_nat,M))),groups2906978787729119204at_rat(F,set_or1269000886237332187st_nat(aa_nat_nat(suc,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na)))) ).

% sum_up_index_split
tff(fact_8726_sum__up__index__split,axiom,
    ! [F: fun_nat_int,M: nat,Na: nat] : groups3539618377306564664at_int(F,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,groups3539618377306564664at_int(F,aa_nat_set_nat(set_ord_atMost_nat,M))),groups3539618377306564664at_int(F,set_or1269000886237332187st_nat(aa_nat_nat(suc,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na)))) ).

% sum_up_index_split
tff(fact_8727_sum__up__index__split,axiom,
    ! [F: fun_nat_nat,M: nat,Na: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_ord_atMost_nat,M))),aa_set_nat_nat(groups3542108847815614940at_nat(F),set_or1269000886237332187st_nat(aa_nat_nat(suc,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na)))) ).

% sum_up_index_split
tff(fact_8728_sum__up__index__split,axiom,
    ! [F: fun_nat_real,M: nat,Na: nat] : aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_ord_atMost_nat,M))),aa_set_nat_real(groups6591440286371151544t_real(F),set_or1269000886237332187st_nat(aa_nat_nat(suc,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na)))) ).

% sum_up_index_split
tff(fact_8729_sum_Otriangle__reindex__eq,axiom,
    ! [G: fun_nat_fun_nat_nat,Na: nat] : groups977919841031483927at_nat(produc6842872674320459806at_nat(G),collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_ti(Na)))) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_sa(G)),aa_nat_set_nat(set_ord_atMost_nat,Na)) ).

% sum.triangle_reindex_eq
tff(fact_8730_sum_Otriangle__reindex__eq,axiom,
    ! [G: fun_nat_fun_nat_real,Na: nat] : groups4567486121110086003t_real(produc1703576794950452218t_real(G),collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_ti(Na)))) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_sc(G)),aa_nat_set_nat(set_ord_atMost_nat,Na)) ).

% sum.triangle_reindex_eq
tff(fact_8731_prod_Otriangle__reindex__eq,axiom,
    ! [G: fun_nat_fun_nat_nat,Na: nat] : groups4077766827762148844at_nat(produc6842872674320459806at_nat(G),collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_ti(Na)))) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_sd(G)),aa_nat_set_nat(set_ord_atMost_nat,Na)) ).

% prod.triangle_reindex_eq
tff(fact_8732_prod_Otriangle__reindex__eq,axiom,
    ! [G: fun_nat_fun_nat_int,Na: nat] : groups4075276357253098568at_int(produc6840382203811409530at_int(G),collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_ti(Na)))) = groups705719431365010083at_int(aTP_Lamm_sf(G),aa_nat_set_nat(set_ord_atMost_nat,Na)) ).

% prod.triangle_reindex_eq
tff(fact_8733_sum__choose__diagonal,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_tj(M),Na)),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,Na)),M) ) ) ).

% sum_choose_diagonal
tff(fact_8734_powser__split__head_I1_J,axiom,
    ! [F: fun_nat_complex,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_ru(F),Z))
     => ( suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_ru(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_sh(F),Z))),Z)) ) ) ).

% powser_split_head(1)
tff(fact_8735_powser__split__head_I1_J,axiom,
    ! [F: fun_nat_real,Z: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_rs(F),Z))
     => ( suminf_real(aa_real_fun_nat_real(aTP_Lamm_rs(F),Z)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(F,zero_zero_nat)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,suminf_real(aa_real_fun_nat_real(aTP_Lamm_si(F),Z))),Z)) ) ) ).

% powser_split_head(1)
tff(fact_8736_powser__split__head_I2_J,axiom,
    ! [F: fun_nat_complex,Z: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_ru(F),Z))
     => ( aa_complex_complex(times_times_complex(suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_sh(F),Z))),Z) = aa_complex_complex(minus_minus_complex(suminf_complex(aa_com8181001469867462503omplex(aTP_Lamm_ru(F),Z))),aa_nat_complex(F,zero_zero_nat)) ) ) ).

% powser_split_head(2)
tff(fact_8737_powser__split__head_I2_J,axiom,
    ! [F: fun_nat_real,Z: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_rs(F),Z))
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,suminf_real(aa_real_fun_nat_real(aTP_Lamm_si(F),Z))),Z) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,suminf_real(aa_real_fun_nat_real(aTP_Lamm_rs(F),Z))),aa_nat_real(F,zero_zero_nat)) ) ) ).

% powser_split_head(2)
tff(fact_8738_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)
       => ~ ! [N7: nat] :
              ~ ! [M2: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N7),M2)
                 => ! [N5: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_set_nat_complex(groups2073611262835488442omplex(F),set_or1269000886237332187st_nat(M2,N5)))),E2) ) ) ) ).

% summable_partial_sum_bound
tff(fact_8739_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)
       => ~ ! [N7: nat] :
              ~ ! [M2: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N7),M2)
                 => ! [N5: 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),set_or1269000886237332187st_nat(M2,N5)))),E2) ) ) ) ).

% summable_partial_sum_bound
tff(fact_8740_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)
       => ? [N7: nat] :
          ! [N5: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N7),N5)
           => 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_ji(F),N5)))),R2) ) ) ) ).

% suminf_exist_split
tff(fact_8741_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)
       => ? [N7: nat] :
          ! [N5: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N7),N5)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,suminf_complex(aa_nat6454746962644518025omplex(aTP_Lamm_tk(F),N5)))),R2) ) ) ) ).

% suminf_exist_split
tff(fact_8742_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_rs(F),Z)) ) ) ) ) ).

% summable_power_series
tff(fact_8743_Abel__lemma,axiom,
    ! [R2: real,R0: real,Aa: fun_nat_complex,M7: 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(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(Aa,N))),aa_nat_real(aa_real_fun_nat_real(power_power_real,R0),N))),M7)
         => summable_real(aa_fun7054779078796324018t_real(aTP_Lamm_tl(R2),Aa)) ) ) ) ).

% Abel_lemma
tff(fact_8744_sum__gp__basic,axiom,
    ! [Xb: complex,Na: 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,Na))) = aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(power_power_complex(Xb),aa_nat_nat(suc,Na))) ).

% sum_gp_basic
tff(fact_8745_sum__gp__basic,axiom,
    ! [Xb: rat,Na: 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,Na))) = aa_rat_rat(minus_minus_rat(one_one_rat),aa_nat_rat(power_power_rat(Xb),aa_nat_nat(suc,Na))) ).

% sum_gp_basic
tff(fact_8746_sum__gp__basic,axiom,
    ! [Xb: int,Na: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),Xb)),groups3539618377306564664at_int(power_power_int(Xb),aa_nat_set_nat(set_ord_atMost_nat,Na))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),aa_nat_int(power_power_int(Xb),aa_nat_nat(suc,Na))) ).

% sum_gp_basic
tff(fact_8747_sum__gp__basic,axiom,
    ! [Xb: real,Na: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_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,Na))) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_nat_nat(suc,Na))) ).

% sum_gp_basic
tff(fact_8748_polyfun__finite__roots,axiom,
    ! [C: fun_nat_complex,Na: nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_nat_fun_complex_o(aTP_Lamm_tm(C),Na)))
    <=> ? [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I3),Na)
          & ( aa_nat_complex(C,I3) != zero_zero_complex ) ) ) ).

% polyfun_finite_roots
tff(fact_8749_polyfun__finite__roots,axiom,
    ! [C: fun_nat_real,Na: nat] :
      ( finite_finite_real(collect_real(aa_nat_fun_real_o(aTP_Lamm_tn(C),Na)))
    <=> ? [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I3),Na)
          & ( aa_nat_real(C,I3) != zero_zero_real ) ) ) ).

% polyfun_finite_roots
tff(fact_8750_polyfun__roots__finite,axiom,
    ! [C: fun_nat_complex,K: nat,Na: nat] :
      ( ( aa_nat_complex(C,K) != zero_zero_complex )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
       => aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_nat_fun_complex_o(aTP_Lamm_tm(C),Na))) ) ) ).

% polyfun_roots_finite
tff(fact_8751_polyfun__roots__finite,axiom,
    ! [C: fun_nat_real,K: nat,Na: nat] :
      ( ( aa_nat_real(C,K) != zero_zero_real )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
       => finite_finite_real(collect_real(aa_nat_fun_real_o(aTP_Lamm_tn(C),Na))) ) ) ).

% polyfun_roots_finite
tff(fact_8752_sum__gp__strict,axiom,
    ! [Xb: complex,Na: nat] :
      aa_set_nat_complex(groups2073611262835488442omplex(power_power_complex(Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Na)) = $ite(Xb = one_one_complex,aa_nat_complex(semiri8010041392384452111omplex,Na),aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(power_power_complex(Xb),Na))),aa_complex_complex(minus_minus_complex(one_one_complex),Xb))) ).

% sum_gp_strict
tff(fact_8753_sum__gp__strict,axiom,
    ! [Xb: rat,Na: nat] :
      groups2906978787729119204at_rat(power_power_rat(Xb),aa_nat_set_nat(set_ord_lessThan_nat,Na)) = $ite(Xb = one_one_rat,aa_nat_rat(semiri681578069525770553at_rat,Na),aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(one_one_rat),aa_nat_rat(power_power_rat(Xb),Na))),aa_rat_rat(minus_minus_rat(one_one_rat),Xb))) ).

% sum_gp_strict
tff(fact_8754_sum__gp__strict,axiom,
    ! [Xb: real,Na: nat] :
      aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Na)) = $ite(Xb = one_one_real,aa_nat_real(semiri5074537144036343181t_real,Na),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Na))),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),Xb))) ).

% sum_gp_strict
tff(fact_8755_pi__half__gt__zero,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ).

% pi_half_gt_zero
tff(fact_8756_diff__power__eq__sum,axiom,
    ! [Xb: complex,Na: nat,Y: complex] : aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(Xb),aa_nat_nat(suc,Na))),aa_nat_complex(power_power_complex(Y),aa_nat_nat(suc,Na))) = aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(Xb),Y)),aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_to(Xb),Na),Y)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Na)))) ).

% diff_power_eq_sum
tff(fact_8757_diff__power__eq__sum,axiom,
    ! [Xb: rat,Na: nat,Y: rat] : aa_rat_rat(minus_minus_rat(aa_nat_rat(power_power_rat(Xb),aa_nat_nat(suc,Na))),aa_nat_rat(power_power_rat(Y),aa_nat_nat(suc,Na))) = aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(Xb),Y)),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_nat298952458717073160at_rat(aTP_Lamm_tp(Xb),Na),Y),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Na)))) ).

% diff_power_eq_sum
tff(fact_8758_diff__power__eq__sum,axiom,
    ! [Xb: int,Na: nat,Y: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(Xb),aa_nat_nat(suc,Na))),aa_nat_int(power_power_int(Y),aa_nat_nat(suc,Na))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),Y)),groups3539618377306564664at_int(aa_int_fun_nat_int(aa_nat6704147566042551216at_int(aTP_Lamm_tq(Xb),Na),Y),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Na)))) ).

% diff_power_eq_sum
tff(fact_8759_diff__power__eq__sum,axiom,
    ! [Xb: real,Na: nat,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_nat_nat(suc,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_nat_nat(suc,Na))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Y)),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_tr(Xb),Na),Y)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Na)))) ).

% diff_power_eq_sum
tff(fact_8760_power__diff__sumr2,axiom,
    ! [Xb: complex,Na: nat,Y: complex] : aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(Xb),Na)),aa_nat_complex(power_power_complex(Y),Na)) = aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(Xb),Y)),aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_ts(Xb),Na),Y)),aa_nat_set_nat(set_ord_lessThan_nat,Na))) ).

% power_diff_sumr2
tff(fact_8761_power__diff__sumr2,axiom,
    ! [Xb: rat,Na: nat,Y: rat] : aa_rat_rat(minus_minus_rat(aa_nat_rat(power_power_rat(Xb),Na)),aa_nat_rat(power_power_rat(Y),Na)) = aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(Xb),Y)),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_nat298952458717073160at_rat(aTP_Lamm_tt(Xb),Na),Y),aa_nat_set_nat(set_ord_lessThan_nat,Na))) ).

% power_diff_sumr2
tff(fact_8762_power__diff__sumr2,axiom,
    ! [Xb: int,Na: nat,Y: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(Xb),Na)),aa_nat_int(power_power_int(Y),Na)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),Y)),groups3539618377306564664at_int(aa_int_fun_nat_int(aa_nat6704147566042551216at_int(aTP_Lamm_tu(Xb),Na),Y),aa_nat_set_nat(set_ord_lessThan_nat,Na))) ).

% power_diff_sumr2
tff(fact_8763_power__diff__sumr2,axiom,
    ! [Xb: real,Na: nat,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Na)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),Na)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Y)),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_tv(Xb),Na),Y)),aa_nat_set_nat(set_ord_lessThan_nat,Na))) ).

% power_diff_sumr2
tff(fact_8764_pi__half__ge__zero,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ).

% pi_half_ge_zero
tff(fact_8765_sum__power__shift,axiom,
    ! [M: nat,Na: nat,Xb: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( aa_set_nat_complex(groups2073611262835488442omplex(power_power_complex(Xb)),set_or1269000886237332187st_nat(M,Na)) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(Xb),M)),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,Na),M)))) ) ) ).

% sum_power_shift
tff(fact_8766_sum__power__shift,axiom,
    ! [M: nat,Na: nat,Xb: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( groups2906978787729119204at_rat(power_power_rat(Xb),set_or1269000886237332187st_nat(M,Na)) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(Xb),M)),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,Na),M)))) ) ) ).

% sum_power_shift
tff(fact_8767_sum__power__shift,axiom,
    ! [M: nat,Na: nat,Xb: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( groups3539618377306564664at_int(power_power_int(Xb),set_or1269000886237332187st_nat(M,Na)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(Xb),M)),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,Na),M)))) ) ) ).

% sum_power_shift
tff(fact_8768_sum__power__shift,axiom,
    ! [M: nat,Na: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,Xb)),set_or1269000886237332187st_nat(M,Na)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),M)),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,Na),M)))) ) ) ).

% sum_power_shift
tff(fact_8769_m2pi__less__pi,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))),pi) ).

% m2pi_less_pi
tff(fact_8770_summable__ratio__test,axiom,
    ! [C: real,N3: 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,N3),N)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_nat_real(F,aa_nat_nat(suc,N)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),aa_real_real(real_V7735802525324610683m_real,aa_nat_real(F,N)))) )
       => summable_real(F) ) ) ).

% summable_ratio_test
tff(fact_8771_summable__ratio__test,axiom,
    ! [C: real,N3: 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,N3),N)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(F,aa_nat_nat(suc,N)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(F,N)))) )
       => summable_complex(F) ) ) ).

% summable_ratio_test
tff(fact_8772_arctan__ubound,axiom,
    ! [Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arctan,Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ).

% arctan_ubound
tff(fact_8773_real__sum__nat__ivl__bounded2,axiom,
    ! [Na: nat,F: fun_nat_Code_integer,K4: code_integer,K: nat] :
      ( ! [P4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,P4),Na)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(F,P4)),K4) )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),K4)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,groups7501900531339628137nteger(F,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K)))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Na)),K4)) ) ) ).

% real_sum_nat_ivl_bounded2
tff(fact_8774_real__sum__nat__ivl__bounded2,axiom,
    ! [Na: nat,F: fun_nat_rat,K4: rat,K: nat] :
      ( ! [P4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,P4),Na)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,P4)),K4) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),K4)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups2906978787729119204at_rat(F,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K)))),aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,Na)),K4)) ) ) ).

% real_sum_nat_ivl_bounded2
tff(fact_8775_real__sum__nat__ivl__bounded2,axiom,
    ! [Na: nat,F: fun_nat_int,K4: int,K: nat] :
      ( ! [P4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,P4),Na)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,P4)),K4) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K4)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,groups3539618377306564664at_int(F,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Na)),K4)) ) ) ).

% real_sum_nat_ivl_bounded2
tff(fact_8776_real__sum__nat__ivl__bounded2,axiom,
    ! [Na: nat,F: fun_nat_nat,K4: nat,K: nat] :
      ( ! [P4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,P4),Na)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,P4)),K4) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),K4)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(groups3542108847815614940at_nat(F),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Na)),K4)) ) ) ).

% real_sum_nat_ivl_bounded2
tff(fact_8777_real__sum__nat__ivl__bounded2,axiom,
    ! [Na: nat,F: fun_nat_real,K4: real,K: nat] :
      ( ! [P4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,P4),Na)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,P4)),K4) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),K4)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_set_nat_real(groups6591440286371151544t_real(F),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Na)),K4)) ) ) ).

% real_sum_nat_ivl_bounded2
tff(fact_8778_sum_Oin__pairs__0,axiom,
    ! [G: fun_nat_rat,Na: nat] : groups2906978787729119204at_rat(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)))) = groups2906978787729119204at_rat(aTP_Lamm_kl(G),aa_nat_set_nat(set_ord_atMost_nat,Na)) ).

% sum.in_pairs_0
tff(fact_8779_sum_Oin__pairs__0,axiom,
    ! [G: fun_nat_int,Na: nat] : groups3539618377306564664at_int(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)))) = groups3539618377306564664at_int(aTP_Lamm_km(G),aa_nat_set_nat(set_ord_atMost_nat,Na)) ).

% sum.in_pairs_0
tff(fact_8780_sum_Oin__pairs__0,axiom,
    ! [G: fun_nat_nat,Na: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)))) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_kn(G)),aa_nat_set_nat(set_ord_atMost_nat,Na)) ).

% sum.in_pairs_0
tff(fact_8781_sum_Oin__pairs__0,axiom,
    ! [G: fun_nat_real,Na: nat] : aa_set_nat_real(groups6591440286371151544t_real(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)))) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_ko(G)),aa_nat_set_nat(set_ord_atMost_nat,Na)) ).

% sum.in_pairs_0
tff(fact_8782_polynomial__product,axiom,
    ! [M: nat,Aa: fun_nat_complex,Na: nat,Ba: fun_nat_complex,Xb: complex] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),I2)
         => ( aa_nat_complex(Aa,I2) = zero_zero_complex ) )
     => ( ! [J2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),J2)
           => ( aa_nat_complex(Ba,J2) = zero_zero_complex ) )
       => ( aa_complex_complex(times_times_complex(aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_ru(Aa),Xb)),aa_nat_set_nat(set_ord_atMost_nat,M))),aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_ru(Ba),Xb)),aa_nat_set_nat(set_ord_atMost_nat,Na))) = aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_fun3721018923249493343omplex(aTP_Lamm_tx(Aa),Ba),Xb)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na))) ) ) ) ).

% polynomial_product
tff(fact_8783_polynomial__product,axiom,
    ! [M: nat,Aa: fun_nat_rat,Na: nat,Ba: fun_nat_rat,Xb: rat] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),I2)
         => ( aa_nat_rat(Aa,I2) = zero_zero_rat ) )
     => ( ! [J2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),J2)
           => ( aa_nat_rat(Ba,J2) = zero_zero_rat ) )
       => ( aa_rat_rat(times_times_rat(groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_rw(Aa),Xb),aa_nat_set_nat(set_ord_atMost_nat,M))),groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aTP_Lamm_rw(Ba),Xb),aa_nat_set_nat(set_ord_atMost_nat,Na))) = groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_fun5559679822142647773at_rat(aTP_Lamm_tz(Aa),Ba),Xb),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na))) ) ) ) ).

% polynomial_product
tff(fact_8784_polynomial__product,axiom,
    ! [M: nat,Aa: fun_nat_int,Na: nat,Ba: fun_nat_int,Xb: int] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),I2)
         => ( aa_nat_int(Aa,I2) = zero_zero_int ) )
     => ( ! [J2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),J2)
           => ( aa_nat_int(Ba,J2) = zero_zero_int ) )
       => ( aa_int_int(aa_int_fun_int_int(times_times_int,groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_rx(Aa),Xb),aa_nat_set_nat(set_ord_atMost_nat,M))),groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_rx(Ba),Xb),aa_nat_set_nat(set_ord_atMost_nat,Na))) = groups3539618377306564664at_int(aa_int_fun_nat_int(aa_fun2315716387381795033at_int(aTP_Lamm_ub(Aa),Ba),Xb),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na))) ) ) ) ).

% polynomial_product
tff(fact_8785_polynomial__product,axiom,
    ! [M: nat,Aa: fun_nat_real,Na: nat,Ba: fun_nat_real,Xb: real] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),I2)
         => ( aa_nat_real(Aa,I2) = zero_zero_real ) )
     => ( ! [J2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),J2)
           => ( aa_nat_real(Ba,J2) = zero_zero_real ) )
       => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_rs(Aa),Xb)),aa_nat_set_nat(set_ord_atMost_nat,M))),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_rs(Ba),Xb)),aa_nat_set_nat(set_ord_atMost_nat,Na))) = aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_fun4732485501138722137t_real(aTP_Lamm_ud(Aa),Ba),Xb)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na))) ) ) ) ).

% polynomial_product
tff(fact_8786_prod_Oin__pairs__0,axiom,
    ! [G: fun_nat_real,Na: nat] : aa_set_nat_real(groups129246275422532515t_real(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)))) = aa_set_nat_real(groups129246275422532515t_real(aTP_Lamm_ox(G)),aa_nat_set_nat(set_ord_atMost_nat,Na)) ).

% prod.in_pairs_0
tff(fact_8787_prod_Oin__pairs__0,axiom,
    ! [G: fun_nat_rat,Na: nat] : groups73079841787564623at_rat(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)))) = groups73079841787564623at_rat(aTP_Lamm_oy(G),aa_nat_set_nat(set_ord_atMost_nat,Na)) ).

% prod.in_pairs_0
tff(fact_8788_prod_Oin__pairs__0,axiom,
    ! [G: fun_nat_nat,Na: nat] : aa_set_nat_nat(groups708209901874060359at_nat(G),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)))) = aa_set_nat_nat(groups708209901874060359at_nat(aTP_Lamm_oz(G)),aa_nat_set_nat(set_ord_atMost_nat,Na)) ).

% prod.in_pairs_0
tff(fact_8789_prod_Oin__pairs__0,axiom,
    ! [G: fun_nat_int,Na: nat] : groups705719431365010083at_int(G,aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)))) = groups705719431365010083at_int(aTP_Lamm_pa(G),aa_nat_set_nat(set_ord_atMost_nat,Na)) ).

% prod.in_pairs_0
tff(fact_8790_one__diff__power__eq_H,axiom,
    ! [Xb: complex,Na: nat] : aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(power_power_complex(Xb),Na)) = 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_ue(Xb),Na)),aa_nat_set_nat(set_ord_lessThan_nat,Na))) ).

% one_diff_power_eq'
tff(fact_8791_one__diff__power__eq_H,axiom,
    ! [Xb: rat,Na: nat] : aa_rat_rat(minus_minus_rat(one_one_rat),aa_nat_rat(power_power_rat(Xb),Na)) = 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_uf(Xb),Na),aa_nat_set_nat(set_ord_lessThan_nat,Na))) ).

% one_diff_power_eq'
tff(fact_8792_one__diff__power__eq_H,axiom,
    ! [Xb: int,Na: nat] : aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),aa_nat_int(power_power_int(Xb),Na)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),Xb)),groups3539618377306564664at_int(aa_nat_fun_nat_int(aTP_Lamm_ug(Xb),Na),aa_nat_set_nat(set_ord_lessThan_nat,Na))) ).

% one_diff_power_eq'
tff(fact_8793_one__diff__power__eq_H,axiom,
    ! [Xb: real,Na: nat] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Na)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),Xb)),aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_uh(Xb),Na)),aa_nat_set_nat(set_ord_lessThan_nat,Na))) ).

% one_diff_power_eq'
tff(fact_8794_polyfun__eq__const,axiom,
    ! [C: fun_nat_complex,Na: nat,K: complex] :
      ( ! [X4: complex] : aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_ru(C),X4)),aa_nat_set_nat(set_ord_atMost_nat,Na)) = K
    <=> ( ( aa_nat_complex(C,zero_zero_nat) = K )
        & ! [X4: nat] :
            ( aa_set_nat_o(member_nat(X4),set_or1269000886237332187st_nat(one_one_nat,Na))
           => ( aa_nat_complex(C,X4) = zero_zero_complex ) ) ) ) ).

% polyfun_eq_const
tff(fact_8795_polyfun__eq__const,axiom,
    ! [C: fun_nat_real,Na: nat,K: real] :
      ( ! [X4: real] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_rs(C),X4)),aa_nat_set_nat(set_ord_atMost_nat,Na)) = K
    <=> ( ( aa_nat_real(C,zero_zero_nat) = K )
        & ! [X4: nat] :
            ( aa_set_nat_o(member_nat(X4),set_or1269000886237332187st_nat(one_one_nat,Na))
           => ( aa_nat_real(C,X4) = zero_zero_real ) ) ) ) ).

% polyfun_eq_const
tff(fact_8796_minus__pi__half__less__zero,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),zero_zero_real) ).

% minus_pi_half_less_zero
tff(fact_8797_Maclaurin__zero,axiom,
    ! [Xb: real,Na: nat,Diff: fun_na2180513548729900188l_real] :
      ( ( Xb = zero_zero_real )
     => ( ( Na != zero_zero_nat )
       => ( aa_set_nat_real(groups6591440286371151544t_real(aa_fun4674562053777771865t_real(aTP_Lamm_ui(Xb),Diff)),aa_nat_set_nat(set_ord_lessThan_nat,Na)) = aa_literal_real(aa_nat8303615512603269253l_real(Diff,zero_zero_nat),zero_zero_literal) ) ) ) ).

% Maclaurin_zero
tff(fact_8798_Maclaurin__zero,axiom,
    ! [Xb: real,Na: nat,Diff: fun_na6724076748692969250l_real] :
      ( ( Xb = zero_zero_real )
     => ( ( Na != zero_zero_nat )
       => ( aa_set_nat_real(groups6591440286371151544t_real(aa_fun1113820866546176217t_real(aTP_Lamm_uj(Xb),Diff)),aa_nat_set_nat(set_ord_lessThan_nat,Na)) = aa_real_real(aa_nat_fun_real_real(Diff,zero_zero_nat),zero_zero_real) ) ) ) ).

% Maclaurin_zero
tff(fact_8799_Maclaurin__zero,axiom,
    ! [Xb: real,Na: nat,Diff: fun_nat_fun_rat_real] :
      ( ( Xb = zero_zero_real )
     => ( ( Na != zero_zero_nat )
       => ( aa_set_nat_real(groups6591440286371151544t_real(aa_fun507281639146497029t_real(aTP_Lamm_uk(Xb),Diff)),aa_nat_set_nat(set_ord_lessThan_nat,Na)) = aa_rat_real(aa_nat_fun_rat_real(Diff,zero_zero_nat),zero_zero_rat) ) ) ) ).

% Maclaurin_zero
tff(fact_8800_Maclaurin__zero,axiom,
    ! [Xb: real,Na: nat,Diff: fun_nat_fun_nat_real] :
      ( ( Xb = zero_zero_real )
     => ( ( Na != zero_zero_nat )
       => ( aa_set_nat_real(groups6591440286371151544t_real(aa_fun7039663905532832253t_real(aTP_Lamm_ul(Xb),Diff)),aa_nat_set_nat(set_ord_lessThan_nat,Na)) = aa_nat_real(aa_nat_fun_nat_real(Diff,zero_zero_nat),zero_zero_nat) ) ) ) ).

% Maclaurin_zero
tff(fact_8801_Maclaurin__zero,axiom,
    ! [Xb: real,Na: nat,Diff: fun_nat_fun_int_real] :
      ( ( Xb = zero_zero_real )
     => ( ( Na != zero_zero_nat )
       => ( aa_set_nat_real(groups6591440286371151544t_real(aa_fun241848180832411225t_real(aTP_Lamm_um(Xb),Diff)),aa_nat_set_nat(set_ord_lessThan_nat,Na)) = aa_int_real(aa_nat_fun_int_real(Diff,zero_zero_nat),zero_zero_int) ) ) ) ).

% Maclaurin_zero
tff(fact_8802_polynomial__product__nat,axiom,
    ! [M: nat,Aa: fun_nat_nat,Na: nat,Ba: fun_nat_nat,Xb: nat] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),I2)
         => ( aa_nat_nat(Aa,I2) = zero_zero_nat ) )
     => ( ! [J2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),J2)
           => ( aa_nat_nat(Ba,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_un(Aa),Xb)),aa_nat_set_nat(set_ord_atMost_nat,M))),aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_un(Ba),Xb)),aa_nat_set_nat(set_ord_atMost_nat,Na))) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_fun6650658898577385413at_nat(aTP_Lamm_up(Aa),Ba),Xb)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na))) ) ) ) ).

% polynomial_product_nat
tff(fact_8803_arctan__lbound,axiom,
    ! [Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arctan,Y)) ).

% arctan_lbound
tff(fact_8804_arctan__bounded,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arctan,Y))
      & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arctan,Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ).

% arctan_bounded
tff(fact_8805_Maclaurin__lemma,axiom,
    ! [H: real,F: fun_real_real,J: fun_nat_real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H)
     => ? [B6: real] : aa_real_real(F,H) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_uq(H),J)),aa_nat_set_nat(set_ord_lessThan_nat,Na))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B6),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,H),Na)),aa_nat_real(semiri2265585572941072030t_real,Na)))) ) ).

% Maclaurin_lemma
tff(fact_8806_sum_Ozero__middle,axiom,
    ! [P3: nat,K: nat,G: fun_nat_rat,H: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),P3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),P3)
       => ( groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_ur(K),G),H),aa_nat_set_nat(set_ord_atMost_nat,P3)) = groups2906978787729119204at_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_us(K),G),H),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P3),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% sum.zero_middle
tff(fact_8807_sum_Ozero__middle,axiom,
    ! [P3: nat,K: nat,G: fun_nat_int,H: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),P3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),P3)
       => ( groups3539618377306564664at_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_ut(K),G),H),aa_nat_set_nat(set_ord_atMost_nat,P3)) = groups3539618377306564664at_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_uu(K),G),H),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P3),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% sum.zero_middle
tff(fact_8808_sum_Ozero__middle,axiom,
    ! [P3: nat,K: nat,G: fun_nat_nat,H: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),P3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),P3)
       => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_uv(K),G),H)),aa_nat_set_nat(set_ord_atMost_nat,P3)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_uw(K),G),H)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P3),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% sum.zero_middle
tff(fact_8809_sum_Ozero__middle,axiom,
    ! [P3: nat,K: nat,G: fun_nat_real,H: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),P3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),P3)
       => ( aa_set_nat_real(groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_ux(K),G),H)),aa_nat_set_nat(set_ord_atMost_nat,P3)) = aa_set_nat_real(groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_uy(K),G),H)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P3),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% sum.zero_middle
tff(fact_8810_prod_Ozero__middle,axiom,
    ! [P3: nat,K: nat,G: fun_nat_complex,H: fun_nat_complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),P3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),P3)
       => ( aa_set_nat_complex(groups6464643781859351333omplex(aa_fun2955221450537884980omplex(aa_fun7497765718855568684omplex(aTP_Lamm_uz(K),G),H)),aa_nat_set_nat(set_ord_atMost_nat,P3)) = aa_set_nat_complex(groups6464643781859351333omplex(aa_fun2955221450537884980omplex(aa_fun7497765718855568684omplex(aTP_Lamm_va(K),G),H)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P3),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% prod.zero_middle
tff(fact_8811_prod_Ozero__middle,axiom,
    ! [P3: nat,K: nat,G: fun_nat_real,H: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),P3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),P3)
       => ( aa_set_nat_real(groups129246275422532515t_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_vb(K),G),H)),aa_nat_set_nat(set_ord_atMost_nat,P3)) = aa_set_nat_real(groups129246275422532515t_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_uy(K),G),H)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P3),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% prod.zero_middle
tff(fact_8812_prod_Ozero__middle,axiom,
    ! [P3: nat,K: nat,G: fun_nat_rat,H: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),P3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),P3)
       => ( groups73079841787564623at_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_vc(K),G),H),aa_nat_set_nat(set_ord_atMost_nat,P3)) = groups73079841787564623at_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_us(K),G),H),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P3),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% prod.zero_middle
tff(fact_8813_prod_Ozero__middle,axiom,
    ! [P3: nat,K: nat,G: fun_nat_nat,H: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),P3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),P3)
       => ( aa_set_nat_nat(groups708209901874060359at_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_vd(K),G),H)),aa_nat_set_nat(set_ord_atMost_nat,P3)) = aa_set_nat_nat(groups708209901874060359at_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_uw(K),G),H)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P3),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% prod.zero_middle
tff(fact_8814_prod_Ozero__middle,axiom,
    ! [P3: nat,K: nat,G: fun_nat_int,H: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),P3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),P3)
       => ( groups705719431365010083at_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_ve(K),G),H),aa_nat_set_nat(set_ord_atMost_nat,P3)) = groups705719431365010083at_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_uu(K),G),H),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,P3),aa_nat_nat(suc,zero_zero_nat)))) ) ) ) ).

% prod.zero_middle
tff(fact_8815_sum__split__even__odd,axiom,
    ! [F: fun_nat_real,G: fun_nat_real,Na: nat] : aa_set_nat_real(groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_vf(F),G)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_vg(F)),aa_nat_set_nat(set_ord_lessThan_nat,Na))),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_vh(G)),aa_nat_set_nat(set_ord_lessThan_nat,Na))) ).

% sum_split_even_odd
tff(fact_8816_Maclaurin__exp__le,axiom,
    ! [Xb: real,Na: nat] :
    ? [T4: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T4)),aa_real_real(abs_abs_real,Xb))
      & ( aa_real_real(exp_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_vi,Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Na))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(exp_real,T4)),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Na))) ) ) ).

% Maclaurin_exp_le
tff(fact_8817_root__polyfun,axiom,
    ! [Na: nat,Z: complex,Aa: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Na)
     => ( ( aa_nat_complex(power_power_complex(Z),Na) = Aa )
      <=> ( aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aTP_Lamm_vj(Na),Z),Aa)),aa_nat_set_nat(set_ord_atMost_nat,Na)) = zero_zero_complex ) ) ) ).

% root_polyfun
tff(fact_8818_root__polyfun,axiom,
    ! [Na: nat,Z: int,Aa: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Na)
     => ( ( aa_nat_int(power_power_int(Z),Na) = Aa )
      <=> ( groups3539618377306564664at_int(aa_int_fun_nat_int(aa_int5842867502692063756at_int(aTP_Lamm_vk(Na),Z),Aa),aa_nat_set_nat(set_ord_atMost_nat,Na)) = zero_zero_int ) ) ) ).

% root_polyfun
tff(fact_8819_root__polyfun,axiom,
    ! [Na: nat,Z: code_integer,Aa: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Na)
     => ( ( aa_nat_Code_integer(power_8256067586552552935nteger(Z),Na) = Aa )
      <=> ( groups7501900531339628137nteger(aa_Cod1578151519121547077nteger(aa_Cod6397266357770349983nteger(aTP_Lamm_vl(Na),Z),Aa),aa_nat_set_nat(set_ord_atMost_nat,Na)) = zero_z3403309356797280102nteger ) ) ) ).

% root_polyfun
tff(fact_8820_root__polyfun,axiom,
    ! [Na: nat,Z: rat,Aa: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Na)
     => ( ( aa_nat_rat(power_power_rat(Z),Na) = Aa )
      <=> ( groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aTP_Lamm_vm(Na),Z),Aa),aa_nat_set_nat(set_ord_atMost_nat,Na)) = zero_zero_rat ) ) ) ).

% root_polyfun
tff(fact_8821_root__polyfun,axiom,
    ! [Na: nat,Z: real,Aa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Na)
     => ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),Na) = Aa )
      <=> ( aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_vn(Na),Z),Aa)),aa_nat_set_nat(set_ord_atMost_nat,Na)) = zero_zero_real ) ) ) ).

% root_polyfun
tff(fact_8822_sum__gp0,axiom,
    ! [Xb: complex,Na: nat] :
      aa_set_nat_complex(groups2073611262835488442omplex(power_power_complex(Xb)),aa_nat_set_nat(set_ord_atMost_nat,Na)) = $ite(Xb = one_one_complex,aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),one_one_nat)),aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(power_power_complex(Xb),aa_nat_nat(suc,Na)))),aa_complex_complex(minus_minus_complex(one_one_complex),Xb))) ).

% sum_gp0
tff(fact_8823_sum__gp0,axiom,
    ! [Xb: rat,Na: nat] :
      groups2906978787729119204at_rat(power_power_rat(Xb),aa_nat_set_nat(set_ord_atMost_nat,Na)) = $ite(Xb = one_one_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),one_one_nat)),aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(one_one_rat),aa_nat_rat(power_power_rat(Xb),aa_nat_nat(suc,Na)))),aa_rat_rat(minus_minus_rat(one_one_rat),Xb))) ).

% sum_gp0
tff(fact_8824_sum__gp0,axiom,
    ! [Xb: real,Na: nat] :
      aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(power_power_real,Xb)),aa_nat_set_nat(set_ord_atMost_nat,Na)) = $ite(Xb = one_one_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),one_one_nat)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_nat_nat(suc,Na)))),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),Xb))) ).

% sum_gp0
tff(fact_8825_choose__alternating__linear__sum,axiom,
    ! [Na: nat] :
      ( ( Na != one_one_nat )
     => ( aa_set_nat_complex(groups2073611262835488442omplex(aTP_Lamm_vo(Na)),aa_nat_set_nat(set_ord_atMost_nat,Na)) = zero_zero_complex ) ) ).

% choose_alternating_linear_sum
tff(fact_8826_choose__alternating__linear__sum,axiom,
    ! [Na: nat] :
      ( ( Na != one_one_nat )
     => ( groups2906978787729119204at_rat(aTP_Lamm_vp(Na),aa_nat_set_nat(set_ord_atMost_nat,Na)) = zero_zero_rat ) ) ).

% choose_alternating_linear_sum
tff(fact_8827_choose__alternating__linear__sum,axiom,
    ! [Na: nat] :
      ( ( Na != one_one_nat )
     => ( groups3539618377306564664at_int(aa_nat_fun_nat_int(aTP_Lamm_vq,Na),aa_nat_set_nat(set_ord_atMost_nat,Na)) = zero_zero_int ) ) ).

% choose_alternating_linear_sum
tff(fact_8828_choose__alternating__linear__sum,axiom,
    ! [Na: nat] :
      ( ( Na != one_one_nat )
     => ( groups7501900531339628137nteger(aTP_Lamm_vr(Na),aa_nat_set_nat(set_ord_atMost_nat,Na)) = zero_z3403309356797280102nteger ) ) ).

% choose_alternating_linear_sum
tff(fact_8829_choose__alternating__linear__sum,axiom,
    ! [Na: nat] :
      ( ( Na != one_one_nat )
     => ( aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_vs,Na)),aa_nat_set_nat(set_ord_atMost_nat,Na)) = zero_zero_real ) ) ).

% choose_alternating_linear_sum
tff(fact_8830_binomial__r__part__sum,axiom,
    ! [M: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)),one_one_nat))),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) ).

% binomial_r_part_sum
tff(fact_8831_choose__linear__sum,axiom,
    ! [Na: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_vt,Na)),aa_nat_set_nat(set_ord_atMost_nat,Na)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat))) ).

% choose_linear_sum
tff(fact_8832_choose__alternating__sum,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_set_nat_complex(groups2073611262835488442omplex(aTP_Lamm_vu(Na)),aa_nat_set_nat(set_ord_atMost_nat,Na)) = zero_zero_complex ) ) ).

% choose_alternating_sum
tff(fact_8833_choose__alternating__sum,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( groups2906978787729119204at_rat(aTP_Lamm_vv(Na),aa_nat_set_nat(set_ord_atMost_nat,Na)) = zero_zero_rat ) ) ).

% choose_alternating_sum
tff(fact_8834_choose__alternating__sum,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( groups3539618377306564664at_int(aa_nat_fun_nat_int(aTP_Lamm_vw,Na),aa_nat_set_nat(set_ord_atMost_nat,Na)) = zero_zero_int ) ) ).

% choose_alternating_sum
tff(fact_8835_choose__alternating__sum,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( groups7501900531339628137nteger(aTP_Lamm_vx(Na),aa_nat_set_nat(set_ord_atMost_nat,Na)) = zero_z3403309356797280102nteger ) ) ).

% choose_alternating_sum
tff(fact_8836_choose__alternating__sum,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_vy,Na)),aa_nat_set_nat(set_ord_atMost_nat,Na)) = zero_zero_real ) ) ).

% choose_alternating_sum
tff(fact_8837_polyfun__extremal__lemma,axiom,
    ! [E2: real,C: fun_nat_complex,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E2)
     => ? [M8: real] :
        ! [Z4: complex] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,M8),aa_complex_real(real_V1022390504157884413omplex,Z4))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_ru(C),Z4)),aa_nat_set_nat(set_ord_atMost_nat,Na)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,E2),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(real_V1022390504157884413omplex,Z4)),aa_nat_nat(suc,Na)))) ) ) ).

% polyfun_extremal_lemma
tff(fact_8838_polyfun__extremal__lemma,axiom,
    ! [E2: real,C: fun_nat_real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E2)
     => ? [M8: real] :
        ! [Z4: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,M8),aa_real_real(real_V7735802525324610683m_real,Z4))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_rs(C),Z4)),aa_nat_set_nat(set_ord_atMost_nat,Na)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,E2),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(real_V7735802525324610683m_real,Z4)),aa_nat_nat(suc,Na)))) ) ) ).

% polyfun_extremal_lemma
tff(fact_8839_Maclaurin__exp__lt,axiom,
    ! [Xb: real,Na: nat] :
      ( ( Xb != zero_zero_real )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => ? [T4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(abs_abs_real,T4))
            & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,T4)),aa_real_real(abs_abs_real,Xb))
            & ( aa_real_real(exp_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_vi,Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Na))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(exp_real,T4)),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Na))) ) ) ) ) ).

% Maclaurin_exp_lt
tff(fact_8840_Maclaurin__sin__expansion3,axiom,
    ! [Na: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ? [T4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),T4)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,T4),Xb)
            & ( aa_real_real(sin_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_vz,Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Na))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T4),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,Na))),pi)))),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Na))) ) ) ) ) ).

% Maclaurin_sin_expansion3
tff(fact_8841_Maclaurin__sin__expansion4,axiom,
    ! [Xb: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ? [T4: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),T4)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T4),Xb)
          & ( aa_real_real(sin_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_vz,Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Na))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T4),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,Na))),pi)))),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Na))) ) ) ) ).

% Maclaurin_sin_expansion4
tff(fact_8842_Maclaurin__sin__expansion2,axiom,
    ! [Xb: real,Na: nat] :
    ? [T4: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T4)),aa_real_real(abs_abs_real,Xb))
      & ( aa_real_real(sin_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_vz,Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Na))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T4),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,Na))),pi)))),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Na))) ) ) ).

% Maclaurin_sin_expansion2
tff(fact_8843_sin__cos__npi,axiom,
    ! [Na: nat] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)))),pi)),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)),Na) ).

% sin_cos_npi
tff(fact_8844_sumr__cos__zero__one,axiom,
    ! [Na: nat] : aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_wa),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Na))) = one_one_real ).

% sumr_cos_zero_one
tff(fact_8845_sin__zero,axiom,
    aa_real_real(sin_real,zero_zero_real) = zero_zero_real ).

% sin_zero
tff(fact_8846_cos__coeff__0,axiom,
    aa_nat_real(cos_coeff,zero_zero_nat) = one_one_real ).

% cos_coeff_0
tff(fact_8847_sin__x__le__x,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(sin_real,Xb)),Xb) ) ).

% sin_x_le_x
tff(fact_8848_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_8849_abs__sin__x__le__abs__x,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))),aa_real_real(abs_abs_real,Xb)) ).

% abs_sin_x_le_abs_x
tff(fact_8850_sin__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),pi)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(sin_real,Xb)) ) ) ).

% sin_gt_zero
tff(fact_8851_sin__x__ge__neg__x,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(uminus_uminus_real,Xb)),aa_real_real(sin_real,Xb)) ) ).

% sin_x_ge_neg_x
tff(fact_8852_sin__ge__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_eq_real,Xb),pi)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(sin_real,Xb)) ) ) ).

% sin_ge_zero
tff(fact_8853_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_8854_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_8855_sin__eq__0__pi,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,pi)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),pi)
       => ( ( aa_real_real(sin_real,Xb) = zero_zero_real )
         => ( Xb = zero_zero_real ) ) ) ) ).

% sin_eq_0_pi
tff(fact_8856_sin__zero__pi__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xb)),pi)
     => ( ( aa_real_real(sin_real,Xb) = zero_zero_real )
      <=> ( Xb = zero_zero_real ) ) ) ).

% sin_zero_pi_iff
tff(fact_8857_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_8858_sin__pi__divide__n__ge__0,axiom,
    ! [Na: nat] :
      ( ( Na != zero_zero_nat )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_nat_real(semiri5074537144036343181t_real,Na)))) ) ).

% sin_pi_divide_n_ge_0
tff(fact_8859_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(aa_rea7340605660300375871l_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_8860_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(aa_rea7340605660300375871l_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_8861_sin__inj__pi,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),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(aa_rea7340605660300375871l_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(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
           => ( ( aa_real_real(sin_real,Xb) = aa_real_real(sin_real,Y) )
             => ( Xb = Y ) ) ) ) ) ) ).

% sin_inj_pi
tff(fact_8862_sin__mono__le__eq,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),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(aa_rea7340605660300375871l_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(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),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,Y))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y) ) ) ) ) ) ).

% sin_mono_le_eq
tff(fact_8863_sin__monotone__2pi__le,axiom,
    ! [Y: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(aa_rea7340605660300375871l_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,Y)),aa_real_real(sin_real,Xb)) ) ) ) ).

% sin_monotone_2pi_le
tff(fact_8864_sin__coeff__Suc,axiom,
    ! [Na: nat] : aa_nat_real(sin_coeff,aa_nat_nat(suc,Na)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(cos_coeff,Na)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Na))) ).

% sin_coeff_Suc
tff(fact_8865_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(aa_rea7340605660300375871l_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_8866_sin__less__zero,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,pi)),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_8867_sin__monotone__2pi,axiom,
    ! [Y: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(aa_rea7340605660300375871l_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,Y)),aa_real_real(sin_real,Xb)) ) ) ) ).

% sin_monotone_2pi
tff(fact_8868_sin__mono__less__eq,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),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(aa_rea7340605660300375871l_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(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),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,Y))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y) ) ) ) ) ) ).

% sin_mono_less_eq
tff(fact_8869_sin__total,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => ? [X: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
            & ( aa_real_real(sin_real,X) = Y )
            & ! [Y4: real] :
                ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y4)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y4),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
                  & ( aa_real_real(sin_real,Y4) = Y ) )
               => ( Y4 = X ) ) ) ) ) ).

% sin_total
tff(fact_8870_sin__pi__divide__n__gt__0,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_nat_real(semiri5074537144036343181t_real,Na)))) ) ).

% sin_pi_divide_n_gt_0
tff(fact_8871_cos__coeff__Suc,axiom,
    ! [Na: nat] : aa_nat_real(cos_coeff,aa_nat_nat(suc,Na)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,aa_nat_real(sin_coeff,Na))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Na))) ).

% cos_coeff_Suc
tff(fact_8872_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(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ) ) ).

% sin_zero_lemma
tff(fact_8873_Maclaurin__cos__expansion2,axiom,
    ! [Xb: real,Na: 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),Na)
       => ? [T4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),T4)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,T4),Xb)
            & ( aa_real_real(cos_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_wb,Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Na))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T4),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,Na))),pi)))),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Na))) ) ) ) ) ).

% Maclaurin_cos_expansion2
tff(fact_8874_Maclaurin__minus__cos__expansion,axiom,
    ! [Na: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real)
       => ? [T4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),T4)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,T4),zero_zero_real)
            & ( aa_real_real(cos_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_wb,Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Na))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T4),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,Na))),pi)))),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Na))) ) ) ) ) ).

% Maclaurin_minus_cos_expansion
tff(fact_8875_Maclaurin__cos__expansion,axiom,
    ! [Xb: real,Na: nat] :
    ? [T4: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T4)),aa_real_real(abs_abs_real,Xb))
      & ( aa_real_real(cos_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_wb,Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Na))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T4),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,Na))),pi)))),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Na))) ) ) ).

% Maclaurin_cos_expansion
tff(fact_8876_gbinomial__partial__row__sum,axiom,
    ! [Aa: complex,M: nat] : aa_set_nat_complex(groups2073611262835488442omplex(aTP_Lamm_wc(Aa)),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_complex_complex(times_times_complex(aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,M)),one_one_complex)),numera6690914467698888265omplex(aa_num_num(bit0,one)))),aa_nat_complex(gbinomial_complex(Aa),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),one_one_nat))) ).

% gbinomial_partial_row_sum
tff(fact_8877_gbinomial__partial__row__sum,axiom,
    ! [Aa: rat,M: nat] : groups2906978787729119204at_rat(aTP_Lamm_wd(Aa),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_rat_rat(times_times_rat(aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,M)),one_one_rat)),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))),aa_nat_rat(gbinomial_rat(Aa),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),one_one_nat))) ).

% gbinomial_partial_row_sum
tff(fact_8878_gbinomial__partial__row__sum,axiom,
    ! [Aa: real,M: nat] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_we,Aa)),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,M)),one_one_real)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,Aa),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),one_one_nat))) ).

% gbinomial_partial_row_sum
tff(fact_8879_cos__pi__eq__zero,axiom,
    ! [M: nat] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,pi),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = zero_zero_real ).

% cos_pi_eq_zero
tff(fact_8880_cos__zero,axiom,
    aa_complex_complex(cos_complex,zero_zero_complex) = one_one_complex ).

% cos_zero
tff(fact_8881_cos__zero,axiom,
    aa_real_real(cos_real,zero_zero_real) = one_one_real ).

% cos_zero
tff(fact_8882_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_8883_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_8884_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_8885_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_8886_gbinomial__0_I1_J,axiom,
    ! [Aa: complex] : aa_nat_complex(gbinomial_complex(Aa),zero_zero_nat) = one_one_complex ).

% gbinomial_0(1)
tff(fact_8887_gbinomial__0_I1_J,axiom,
    ! [Aa: real] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,Aa),zero_zero_nat) = one_one_real ).

% gbinomial_0(1)
tff(fact_8888_gbinomial__0_I1_J,axiom,
    ! [Aa: rat] : aa_nat_rat(gbinomial_rat(Aa),zero_zero_nat) = one_one_rat ).

% gbinomial_0(1)
tff(fact_8889_gbinomial__0_I1_J,axiom,
    ! [Aa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gbinomial_nat,Aa),zero_zero_nat) = one_one_nat ).

% gbinomial_0(1)
tff(fact_8890_gbinomial__0_I1_J,axiom,
    ! [Aa: int] : aa_nat_int(gbinomial_int(Aa),zero_zero_nat) = one_one_int ).

% gbinomial_0(1)
tff(fact_8891_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_8892_sin__cos__squared__add3,axiom,
    ! [Xb: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,Xb)),aa_real_real(cos_real,Xb))),aa_real_real(aa_rea7340605660300375871l_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_8893_sin__cos__squared__add,axiom,
    ! [Xb: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sin_real,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_8894_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_8895_sin__cos__squared__add2,axiom,
    ! [Xb: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(cos_real,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_8896_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_8897_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_8898_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_8899_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_8900_gbinomial__Suc__Suc,axiom,
    ! [Aa: complex,K: nat] : aa_nat_complex(gbinomial_complex(aa_complex_complex(plus_plus_complex(Aa),one_one_complex)),aa_nat_nat(suc,K)) = aa_complex_complex(plus_plus_complex(aa_nat_complex(gbinomial_complex(Aa),K)),aa_nat_complex(gbinomial_complex(Aa),aa_nat_nat(suc,K))) ).

% gbinomial_Suc_Suc
tff(fact_8901_gbinomial__Suc__Suc,axiom,
    ! [Aa: real,K: nat] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),one_one_real)),aa_nat_nat(suc,K)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(gbinomial_real,Aa),K)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,Aa),aa_nat_nat(suc,K))) ).

% gbinomial_Suc_Suc
tff(fact_8902_gbinomial__Suc__Suc,axiom,
    ! [Aa: rat,K: nat] : aa_nat_rat(gbinomial_rat(aa_rat_rat(plus_plus_rat(Aa),one_one_rat)),aa_nat_nat(suc,K)) = aa_rat_rat(plus_plus_rat(aa_nat_rat(gbinomial_rat(Aa),K)),aa_nat_rat(gbinomial_rat(Aa),aa_nat_nat(suc,K))) ).

% gbinomial_Suc_Suc
tff(fact_8903_cos__monotone__0__pi__le,axiom,
    ! [Y: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),pi)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(cos_real,Xb)),aa_real_real(cos_real,Y)) ) ) ) ).

% cos_monotone_0_pi_le
tff(fact_8904_cos__mono__le__eq,axiom,
    ! [Xb: real,Y: 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),pi)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),pi)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(cos_real,Xb)),aa_real_real(cos_real,Y))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Xb) ) ) ) ) ) ).

% cos_mono_le_eq
tff(fact_8905_cos__inj__pi,axiom,
    ! [Xb: real,Y: 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),pi)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),pi)
           => ( ( aa_real_real(cos_real,Xb) = aa_real_real(cos_real,Y) )
             => ( Xb = Y ) ) ) ) ) ) ).

% cos_inj_pi
tff(fact_8906_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_8907_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_8908_gbinomial__of__nat__symmetric,axiom,
    ! [K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Na)
     => ( aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_nat_real(semiri5074537144036343181t_real,Na)),K) = aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),K)) ) ) ).

% gbinomial_of_nat_symmetric
tff(fact_8909_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_8910_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_8911_cos__monotone__0__pi,axiom,
    ! [Y: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),pi)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cos_real,Xb)),aa_real_real(cos_real,Y)) ) ) ) ).

% cos_monotone_0_pi
tff(fact_8912_cos__mono__less__eq,axiom,
    ! [Xb: real,Y: 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),pi)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),pi)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cos_real,Xb)),aa_real_real(cos_real,Y))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xb) ) ) ) ) ) ).

% cos_mono_less_eq
tff(fact_8913_gbinomial__addition__formula,axiom,
    ! [Aa: complex,K: nat] : aa_nat_complex(gbinomial_complex(Aa),aa_nat_nat(suc,K)) = aa_complex_complex(plus_plus_complex(aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(Aa),one_one_complex)),aa_nat_nat(suc,K))),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(Aa),one_one_complex)),K)) ).

% gbinomial_addition_formula
tff(fact_8914_gbinomial__addition__formula,axiom,
    ! [Aa: real,K: nat] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,Aa),aa_nat_nat(suc,K)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Aa),one_one_real)),aa_nat_nat(suc,K))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Aa),one_one_real)),K)) ).

% gbinomial_addition_formula
tff(fact_8915_gbinomial__addition__formula,axiom,
    ! [Aa: rat,K: nat] : aa_nat_rat(gbinomial_rat(Aa),aa_nat_nat(suc,K)) = aa_rat_rat(plus_plus_rat(aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(Aa),one_one_rat)),aa_nat_nat(suc,K))),aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(Aa),one_one_rat)),K)) ).

% gbinomial_addition_formula
tff(fact_8916_gbinomial__absorb__comp,axiom,
    ! [Aa: complex,K: nat] : aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(Aa),aa_nat_complex(semiri8010041392384452111omplex,K))),aa_nat_complex(gbinomial_complex(Aa),K)) = aa_complex_complex(times_times_complex(Aa),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(Aa),one_one_complex)),K)) ).

% gbinomial_absorb_comp
tff(fact_8917_gbinomial__absorb__comp,axiom,
    ! [Aa: rat,K: nat] : aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(Aa),aa_nat_rat(semiri681578069525770553at_rat,K))),aa_nat_rat(gbinomial_rat(Aa),K)) = aa_rat_rat(times_times_rat(Aa),aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(Aa),one_one_rat)),K)) ).

% gbinomial_absorb_comp
tff(fact_8918_gbinomial__absorb__comp,axiom,
    ! [Aa: real,K: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Aa),aa_nat_real(semiri5074537144036343181t_real,K))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,Aa),K)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Aa),one_one_real)),K)) ).

% gbinomial_absorb_comp
tff(fact_8919_gbinomial__ge__n__over__k__pow__k,axiom,
    ! [K: nat,Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,K)),Aa)
     => 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(aa_rea7340605660300375871l_real(divide_divide_real,Aa),aa_nat_real(semiri5074537144036343181t_real,K))),K)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,Aa),K)) ) ).

% gbinomial_ge_n_over_k_pow_k
tff(fact_8920_gbinomial__ge__n__over__k__pow__k,axiom,
    ! [K: nat,Aa: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,K)),Aa)
     => 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),aa_nat_rat(semiri681578069525770553at_rat,K))),K)),aa_nat_rat(gbinomial_rat(Aa),K)) ) ).

% gbinomial_ge_n_over_k_pow_k
tff(fact_8921_gbinomial__mult__1,axiom,
    ! [Aa: rat,K: nat] : aa_rat_rat(times_times_rat(Aa),aa_nat_rat(gbinomial_rat(Aa),K)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,K)),aa_nat_rat(gbinomial_rat(Aa),K))),aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K))),aa_nat_rat(gbinomial_rat(Aa),aa_nat_nat(suc,K)))) ).

% gbinomial_mult_1
tff(fact_8922_gbinomial__mult__1,axiom,
    ! [Aa: real,K: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,Aa),K)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,K)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,Aa),K))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,K))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,Aa),aa_nat_nat(suc,K)))) ).

% gbinomial_mult_1
tff(fact_8923_gbinomial__mult__1_H,axiom,
    ! [Aa: rat,K: nat] : aa_rat_rat(times_times_rat(aa_nat_rat(gbinomial_rat(Aa),K)),Aa) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,K)),aa_nat_rat(gbinomial_rat(Aa),K))),aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K))),aa_nat_rat(gbinomial_rat(Aa),aa_nat_nat(suc,K)))) ).

% gbinomial_mult_1'
tff(fact_8924_gbinomial__mult__1_H,axiom,
    ! [Aa: real,K: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(gbinomial_real,Aa),K)),Aa) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,K)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,Aa),K))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,K))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,Aa),aa_nat_nat(suc,K)))) ).

% gbinomial_mult_1'
tff(fact_8925_cos__monotone__minus__pi__0_H,axiom,
    ! [Y: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,pi)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),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(cos_real,Y)),aa_real_real(cos_real,Xb)) ) ) ) ).

% cos_monotone_minus_pi_0'
tff(fact_8926_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_8927_cos__is__zero,axiom,
    ? [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
      & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))
      & ( aa_real_real(cos_real,X) = zero_zero_real )
      & ! [Y4: real] :
          ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y4)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y4),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))
            & ( aa_real_real(cos_real,Y4) = zero_zero_real ) )
         => ( Y4 = X ) ) ) ).

% cos_is_zero
tff(fact_8928_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_8929_Suc__times__gbinomial,axiom,
    ! [K: nat,Aa: 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(Aa),one_one_complex)),aa_nat_nat(suc,K))) = aa_complex_complex(times_times_complex(aa_complex_complex(plus_plus_complex(Aa),one_one_complex)),aa_nat_complex(gbinomial_complex(Aa),K)) ).

% Suc_times_gbinomial
tff(fact_8930_Suc__times__gbinomial,axiom,
    ! [K: nat,Aa: 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(Aa),one_one_rat)),aa_nat_nat(suc,K))) = aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(Aa),one_one_rat)),aa_nat_rat(gbinomial_rat(Aa),K)) ).

% Suc_times_gbinomial
tff(fact_8931_Suc__times__gbinomial,axiom,
    ! [K: nat,Aa: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,K))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),one_one_real)),aa_nat_nat(suc,K))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),one_one_real)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,Aa),K)) ).

% Suc_times_gbinomial
tff(fact_8932_gbinomial__absorption,axiom,
    ! [K: nat,Aa: complex] : aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,K))),aa_nat_complex(gbinomial_complex(Aa),aa_nat_nat(suc,K))) = aa_complex_complex(times_times_complex(Aa),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(Aa),one_one_complex)),K)) ).

% gbinomial_absorption
tff(fact_8933_gbinomial__absorption,axiom,
    ! [K: nat,Aa: rat] : aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K))),aa_nat_rat(gbinomial_rat(Aa),aa_nat_nat(suc,K))) = aa_rat_rat(times_times_rat(Aa),aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(Aa),one_one_rat)),K)) ).

% gbinomial_absorption
tff(fact_8934_gbinomial__absorption,axiom,
    ! [K: nat,Aa: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,K))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,Aa),aa_nat_nat(suc,K))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Aa),one_one_real)),K)) ).

% gbinomial_absorption
tff(fact_8935_cos__monotone__minus__pi__0,axiom,
    ! [Y: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,pi)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),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_real,aa_real_real(cos_real,Y)),aa_real_real(cos_real,Xb)) ) ) ) ).

% cos_monotone_minus_pi_0
tff(fact_8936_cos__total,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => ? [X: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),pi)
            & ( aa_real_real(cos_real,X) = Y )
            & ! [Y4: real] :
                ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y4)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y4),pi)
                  & ( aa_real_real(cos_real,Y4) = Y ) )
               => ( Y4 = X ) ) ) ) ) ).

% cos_total
tff(fact_8937_gbinomial__trinomial__revision,axiom,
    ! [K: nat,M: nat,Aa: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),M)
     => ( aa_rat_rat(times_times_rat(aa_nat_rat(gbinomial_rat(Aa),M)),aa_nat_rat(gbinomial_rat(aa_nat_rat(semiri681578069525770553at_rat,M)),K)) = aa_rat_rat(times_times_rat(aa_nat_rat(gbinomial_rat(Aa),K)),aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(Aa),aa_nat_rat(semiri681578069525770553at_rat,K))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K))) ) ) ).

% gbinomial_trinomial_revision
tff(fact_8938_gbinomial__trinomial__revision,axiom,
    ! [K: nat,M: nat,Aa: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),M)
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(gbinomial_real,Aa),M)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_nat_real(semiri5074537144036343181t_real,M)),K)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(gbinomial_real,Aa),K)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Aa),aa_nat_real(semiri5074537144036343181t_real,K))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K))) ) ) ).

% gbinomial_trinomial_revision
tff(fact_8939_sincos__principal__value,axiom,
    ! [Xb: real] :
    ? [Y3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,pi)),Y3)
      & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y3),pi)
      & ( aa_real_real(sin_real,Y3) = aa_real_real(sin_real,Xb) )
      & ( aa_real_real(cos_real,Y3) = aa_real_real(cos_real,Xb) ) ) ).

% sincos_principal_value
tff(fact_8940_gbinomial__parallel__sum,axiom,
    ! [Aa: complex,Na: nat] : aa_set_nat_complex(groups2073611262835488442omplex(aTP_Lamm_wf(Aa)),aa_nat_set_nat(set_ord_atMost_nat,Na)) = aa_nat_complex(gbinomial_complex(aa_complex_complex(plus_plus_complex(aa_complex_complex(plus_plus_complex(Aa),aa_nat_complex(semiri8010041392384452111omplex,Na))),one_one_complex)),Na) ).

% gbinomial_parallel_sum
tff(fact_8941_gbinomial__parallel__sum,axiom,
    ! [Aa: rat,Na: nat] : groups2906978787729119204at_rat(aTP_Lamm_wg(Aa),aa_nat_set_nat(set_ord_atMost_nat,Na)) = aa_nat_rat(gbinomial_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(Aa),aa_nat_rat(semiri681578069525770553at_rat,Na))),one_one_rat)),Na) ).

% gbinomial_parallel_sum
tff(fact_8942_gbinomial__parallel__sum,axiom,
    ! [Aa: real,Na: nat] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_wh,Aa)),aa_nat_set_nat(set_ord_atMost_nat,Na)) = aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),aa_nat_real(semiri5074537144036343181t_real,Na))),one_one_real)),Na) ).

% gbinomial_parallel_sum
tff(fact_8943_gbinomial__rec,axiom,
    ! [Aa: complex,K: nat] : aa_nat_complex(gbinomial_complex(aa_complex_complex(plus_plus_complex(Aa),one_one_complex)),aa_nat_nat(suc,K)) = aa_complex_complex(times_times_complex(aa_nat_complex(gbinomial_complex(Aa),K)),aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(Aa),one_one_complex)),aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,K)))) ).

% gbinomial_rec
tff(fact_8944_gbinomial__rec,axiom,
    ! [Aa: rat,K: nat] : aa_nat_rat(gbinomial_rat(aa_rat_rat(plus_plus_rat(Aa),one_one_rat)),aa_nat_nat(suc,K)) = aa_rat_rat(times_times_rat(aa_nat_rat(gbinomial_rat(Aa),K)),aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(Aa),one_one_rat)),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K)))) ).

% gbinomial_rec
tff(fact_8945_gbinomial__rec,axiom,
    ! [Aa: real,K: nat] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),one_one_real)),aa_nat_nat(suc,K)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(gbinomial_real,Aa),K)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),one_one_real)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,K)))) ).

% gbinomial_rec
tff(fact_8946_gbinomial__factors,axiom,
    ! [Aa: complex,K: nat] : aa_nat_complex(gbinomial_complex(aa_complex_complex(plus_plus_complex(Aa),one_one_complex)),aa_nat_nat(suc,K)) = aa_complex_complex(times_times_complex(aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(Aa),one_one_complex)),aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,K)))),aa_nat_complex(gbinomial_complex(Aa),K)) ).

% gbinomial_factors
tff(fact_8947_gbinomial__factors,axiom,
    ! [Aa: rat,K: nat] : aa_nat_rat(gbinomial_rat(aa_rat_rat(plus_plus_rat(Aa),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(Aa),one_one_rat)),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K)))),aa_nat_rat(gbinomial_rat(Aa),K)) ).

% gbinomial_factors
tff(fact_8948_gbinomial__factors,axiom,
    ! [Aa: real,K: nat] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),one_one_real)),aa_nat_nat(suc,K)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),one_one_real)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,K)))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,Aa),K)) ).

% gbinomial_factors
tff(fact_8949_gbinomial__index__swap,axiom,
    ! [K: nat,Na: nat] : aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),K)),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,Na))),one_one_complex)),K)) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Na)),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,K))),one_one_complex)),Na)) ).

% gbinomial_index_swap
tff(fact_8950_gbinomial__index__swap,axiom,
    ! [K: nat,Na: 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,Na))),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)),Na)),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)),Na)) ).

% gbinomial_index_swap
tff(fact_8951_gbinomial__index__swap,axiom,
    ! [K: nat,Na: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),K)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,Na))),one_one_real)),K)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Na)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,K))),one_one_real)),Na)) ).

% gbinomial_index_swap
tff(fact_8952_gbinomial__negated__upper,axiom,
    ! [Aa: complex,K: nat] : aa_nat_complex(gbinomial_complex(Aa),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)),Aa)),one_one_complex)),K)) ).

% gbinomial_negated_upper
tff(fact_8953_gbinomial__negated__upper,axiom,
    ! [Aa: rat,K: nat] : aa_nat_rat(gbinomial_rat(Aa),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)),Aa)),one_one_rat)),K)) ).

% gbinomial_negated_upper
tff(fact_8954_gbinomial__negated__upper,axiom,
    ! [Aa: real,K: nat] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,Aa),K) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),K)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(semiri5074537144036343181t_real,K)),Aa)),one_one_real)),K)) ).

% gbinomial_negated_upper
tff(fact_8955_sin__cos__le1,axiom,
    ! [Xb: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sin_real,Xb)),aa_real_real(sin_real,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,Xb)),aa_real_real(cos_real,Y))))),one_one_real) ).

% sin_cos_le1
tff(fact_8956_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_8957_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(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(cos_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% sin_squared_eq
tff(fact_8958_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_8959_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(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sin_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% cos_squared_eq
tff(fact_8960_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(aa_rea7340605660300375871l_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_8961_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(aa_rea7340605660300375871l_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_8962_gbinomial__minus,axiom,
    ! [Aa: complex,K: nat] : aa_nat_complex(gbinomial_complex(aa_complex_complex(uminus1482373934393186551omplex,Aa)),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(Aa),aa_nat_complex(semiri8010041392384452111omplex,K))),one_one_complex)),K)) ).

% gbinomial_minus
tff(fact_8963_gbinomial__minus,axiom,
    ! [Aa: rat,K: nat] : aa_nat_rat(gbinomial_rat(aa_rat_rat(uminus_uminus_rat,Aa)),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(Aa),aa_nat_rat(semiri681578069525770553at_rat,K))),one_one_rat)),K)) ).

% gbinomial_minus
tff(fact_8964_gbinomial__minus,axiom,
    ! [Aa: real,K: nat] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(uminus_uminus_real,Aa)),K) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),K)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Aa),aa_nat_real(semiri5074537144036343181t_real,K))),one_one_real)),K)) ).

% gbinomial_minus
tff(fact_8965_gbinomial__reduce__nat,axiom,
    ! [K: nat,Aa: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( aa_nat_complex(gbinomial_complex(Aa),K) = aa_complex_complex(plus_plus_complex(aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(Aa),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(Aa),one_one_complex)),K)) ) ) ).

% gbinomial_reduce_nat
tff(fact_8966_gbinomial__reduce__nat,axiom,
    ! [K: nat,Aa: 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,Aa),K) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Aa),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(aa_rea7340605660300375871l_real(minus_minus_real,Aa),one_one_real)),K)) ) ) ).

% gbinomial_reduce_nat
tff(fact_8967_gbinomial__reduce__nat,axiom,
    ! [K: nat,Aa: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( aa_nat_rat(gbinomial_rat(Aa),K) = aa_rat_rat(plus_plus_rat(aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(Aa),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(Aa),one_one_rat)),K)) ) ) ).

% gbinomial_reduce_nat
tff(fact_8968_gbinomial__pochhammer,axiom,
    ! [Aa: complex,K: nat] : aa_nat_complex(gbinomial_complex(Aa),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,Aa)),K))),aa_nat_complex(semiri5044797733671781792omplex,K)) ).

% gbinomial_pochhammer
tff(fact_8969_gbinomial__pochhammer,axiom,
    ! [Aa: rat,K: nat] : aa_nat_rat(gbinomial_rat(Aa),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,Aa)),K))),aa_nat_rat(semiri773545260158071498ct_rat,K)) ).

% gbinomial_pochhammer
tff(fact_8970_gbinomial__pochhammer,axiom,
    ! [Aa: real,K: nat] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,Aa),K) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),K)),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(uminus_uminus_real,Aa)),K))),aa_nat_real(semiri2265585572941072030t_real,K)) ).

% gbinomial_pochhammer
tff(fact_8971_gbinomial__pochhammer_H,axiom,
    ! [Aa: complex,K: nat] : aa_nat_complex(gbinomial_complex(Aa),K) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(minus_minus_complex(Aa),aa_nat_complex(semiri8010041392384452111omplex,K))),one_one_complex)),K)),aa_nat_complex(semiri5044797733671781792omplex,K)) ).

% gbinomial_pochhammer'
tff(fact_8972_gbinomial__pochhammer_H,axiom,
    ! [Aa: rat,K: nat] : aa_nat_rat(gbinomial_rat(Aa),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(Aa),aa_nat_rat(semiri681578069525770553at_rat,K))),one_one_rat)),K)),aa_nat_rat(semiri773545260158071498ct_rat,K)) ).

% gbinomial_pochhammer'
tff(fact_8973_gbinomial__pochhammer_H,axiom,
    ! [Aa: real,K: nat] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,Aa),K) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Aa),aa_nat_real(semiri5074537144036343181t_real,K))),one_one_real)),K)),aa_nat_real(semiri2265585572941072030t_real,K)) ).

% gbinomial_pochhammer'
tff(fact_8974_cos__double__cos,axiom,
    ! [W2: complex] : aa_complex_complex(cos_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),W2)) = 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,W2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),one_one_complex) ).

% cos_double_cos
tff(fact_8975_cos__double__cos,axiom,
    ! [W2: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),W2)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(cos_real,W2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),one_one_real) ).

% cos_double_cos
tff(fact_8976_gbinomial__sum__lower__neg,axiom,
    ! [Aa: complex,M: nat] : aa_set_nat_complex(groups2073611262835488442omplex(aTP_Lamm_wi(Aa)),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),M)),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(Aa),one_one_complex)),M)) ).

% gbinomial_sum_lower_neg
tff(fact_8977_gbinomial__sum__lower__neg,axiom,
    ! [Aa: rat,M: nat] : groups2906978787729119204at_rat(aTP_Lamm_wj(Aa),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),M)),aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(Aa),one_one_rat)),M)) ).

% gbinomial_sum_lower_neg
tff(fact_8978_gbinomial__sum__lower__neg,axiom,
    ! [Aa: real,M: nat] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_wk,Aa)),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),M)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Aa),one_one_real)),M)) ).

% gbinomial_sum_lower_neg
tff(fact_8979_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(aa_rea7340605660300375871l_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(aa_rea7340605660300375871l_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_8980_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(aa_rea7340605660300375871l_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(aa_rea7340605660300375871l_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_8981_cos__double__sin,axiom,
    ! [W2: complex] : aa_complex_complex(cos_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),W2)) = 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,W2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% cos_double_sin
tff(fact_8982_cos__double__sin,axiom,
    ! [W2: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),W2)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sin_real,W2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% cos_double_sin
tff(fact_8983_gbinomial__sum__up__index,axiom,
    ! [K: nat,Na: nat] : aa_set_nat_complex(groups2073611262835488442omplex(aTP_Lamm_wl(K)),set_or1269000886237332187st_nat(zero_zero_nat,Na)) = aa_nat_complex(gbinomial_complex(aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,Na)),one_one_complex)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),one_one_nat)) ).

% gbinomial_sum_up_index
tff(fact_8984_gbinomial__sum__up__index,axiom,
    ! [K: nat,Na: nat] : groups2906978787729119204at_rat(aTP_Lamm_wm(K),set_or1269000886237332187st_nat(zero_zero_nat,Na)) = aa_nat_rat(gbinomial_rat(aa_rat_rat(plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,Na)),one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),one_one_nat)) ).

% gbinomial_sum_up_index
tff(fact_8985_gbinomial__sum__up__index,axiom,
    ! [K: nat,Na: nat] : aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_wn,K)),set_or1269000886237332187st_nat(zero_zero_nat,Na)) = aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,Na)),one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),one_one_nat)) ).

% gbinomial_sum_up_index
tff(fact_8986_gbinomial__Suc,axiom,
    ! [Aa: rat,K: nat] : aa_nat_rat(gbinomial_rat(Aa),aa_nat_nat(suc,K)) = aa_rat_rat(divide_divide_rat(groups73079841787564623at_rat(aTP_Lamm_wo(Aa),set_or1269000886237332187st_nat(zero_zero_nat,K))),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(suc,K))) ).

% gbinomial_Suc
tff(fact_8987_gbinomial__Suc,axiom,
    ! [Aa: code_integer,K: nat] : aa_nat_Code_integer(gbinom8545251970709558553nteger(Aa),aa_nat_nat(suc,K)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,groups3455450783089532116nteger(aTP_Lamm_wp(Aa),set_or1269000886237332187st_nat(zero_zero_nat,K))),aa_nat_Code_integer(semiri3624122377584611663nteger,aa_nat_nat(suc,K))) ).

% gbinomial_Suc
tff(fact_8988_gbinomial__Suc,axiom,
    ! [Aa: real,K: nat] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,Aa),aa_nat_nat(suc,K)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_set_nat_real(groups129246275422532515t_real(aa_real_fun_nat_real(aTP_Lamm_wq,Aa)),set_or1269000886237332187st_nat(zero_zero_nat,K))),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(suc,K))) ).

% gbinomial_Suc
tff(fact_8989_gbinomial__Suc,axiom,
    ! [Aa: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gbinomial_nat,Aa),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_wr,Aa)),set_or1269000886237332187st_nat(zero_zero_nat,K))),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(suc,K))) ).

% gbinomial_Suc
tff(fact_8990_gbinomial__Suc,axiom,
    ! [Aa: int,K: nat] : aa_nat_int(gbinomial_int(Aa),aa_nat_nat(suc,K)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,groups705719431365010083at_int(aTP_Lamm_ws(Aa),set_or1269000886237332187st_nat(zero_zero_nat,K))),aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(suc,K))) ).

% gbinomial_Suc
tff(fact_8991_gbinomial__absorption_H,axiom,
    ! [K: nat,Aa: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( aa_nat_complex(gbinomial_complex(Aa),K) = aa_complex_complex(times_times_complex(aa_complex_complex(divide1717551699836669952omplex(Aa),aa_nat_complex(semiri8010041392384452111omplex,K))),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(Aa),one_one_complex)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))) ) ) ).

% gbinomial_absorption'
tff(fact_8992_gbinomial__absorption_H,axiom,
    ! [K: nat,Aa: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( aa_nat_rat(gbinomial_rat(Aa),K) = aa_rat_rat(times_times_rat(aa_rat_rat(divide_divide_rat(Aa),aa_nat_rat(semiri681578069525770553at_rat,K))),aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(Aa),one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))) ) ) ).

% gbinomial_absorption'
tff(fact_8993_gbinomial__absorption_H,axiom,
    ! [K: nat,Aa: 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,Aa),K) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Aa),aa_nat_real(semiri5074537144036343181t_real,K))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Aa),one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))) ) ) ).

% gbinomial_absorption'
tff(fact_8994_gbinomial__partial__sum__poly__xpos,axiom,
    ! [M: nat,Aa: complex,Xb: complex,Y: complex] : aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_com177695874307951165omplex(aTP_Lamm_wt(M),Aa),Xb),Y)),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_com177695874307951165omplex(aTP_Lamm_wu(M),Aa),Xb),Y)),aa_nat_set_nat(set_ord_atMost_nat,M)) ).

% gbinomial_partial_sum_poly_xpos
tff(fact_8995_gbinomial__partial__sum__poly__xpos,axiom,
    ! [M: nat,Aa: rat,Xb: rat,Y: rat] : groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aa_rat7467449789483521253at_rat(aTP_Lamm_wv(M),Aa),Xb),Y),aa_nat_set_nat(set_ord_atMost_nat,M)) = groups2906978787729119204at_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aa_rat7467449789483521253at_rat(aTP_Lamm_ww(M),Aa),Xb),Y),aa_nat_set_nat(set_ord_atMost_nat,M)) ).

% gbinomial_partial_sum_poly_xpos
tff(fact_8996_gbinomial__partial__sum__poly__xpos,axiom,
    ! [M: nat,Aa: real,Xb: real,Y: real] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aa_rea1729780184207679413t_real(aTP_Lamm_wx(M),Aa),Xb),Y)),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aa_rea1729780184207679413t_real(aTP_Lamm_wy(M),Aa),Xb),Y)),aa_nat_set_nat(set_ord_atMost_nat,M)) ).

% gbinomial_partial_sum_poly_xpos
tff(fact_8997_sincos__total__pi,axiom,
    ! [Y: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(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,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real )
       => ? [T4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T4)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T4),pi)
            & ( Xb = aa_real_real(cos_real,T4) )
            & ( Y = aa_real_real(sin_real,T4) ) ) ) ) ).

% sincos_total_pi
tff(fact_8998_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(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(cos_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ).

% sin_cos_sqrt
tff(fact_8999_sin__expansion__lemma,axiom,
    ! [Xb: real,M: nat] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,M))),pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) = aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,M)),pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% sin_expansion_lemma
tff(fact_9000_gbinomial__code,axiom,
    ! [Aa: complex,K: nat] :
      aa_nat_complex(gbinomial_complex(Aa),K) = $ite(K = zero_zero_nat,one_one_complex,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(aa_nat3829984526574172007omplex(set_fo1517530859248394432omplex(aTP_Lamm_wz(Aa),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_9001_gbinomial__code,axiom,
    ! [Aa: rat,K: nat] :
      aa_nat_rat(gbinomial_rat(Aa),K) = $ite(K = zero_zero_nat,one_one_rat,aa_rat_rat(divide_divide_rat(aa_rat_rat(set_fo1949268297981939178at_rat(aTP_Lamm_xa(Aa),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_9002_gbinomial__code,axiom,
    ! [Aa: real,K: nat] :
      aa_nat_real(aa_real_fun_nat_real(gbinomial_real,Aa),K) = $ite(K = zero_zero_nat,one_one_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(set_fo3111899725591712190t_real(aTP_Lamm_xb(Aa),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_9003_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(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ) ) ).

% cos_zero_lemma
tff(fact_9004_cos__expansion__lemma,axiom,
    ! [Xb: real,M: nat] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,M))),pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) = aa_real_real(uminus_uminus_real,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,M)),pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))) ).

% cos_expansion_lemma
tff(fact_9005_gchoose__row__sum__weighted,axiom,
    ! [R2: rat,M: nat] : groups2906978787729119204at_rat(aTP_Lamm_wd(R2),set_or1269000886237332187st_nat(zero_zero_nat,M)) = aa_rat_rat(times_times_rat(aa_rat_rat(divide_divide_rat(aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,M))),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))),aa_nat_rat(gbinomial_rat(R2),aa_nat_nat(suc,M))) ).

% gchoose_row_sum_weighted
tff(fact_9006_gchoose__row__sum__weighted,axiom,
    ! [R2: real,M: nat] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_we,R2)),set_or1269000886237332187st_nat(zero_zero_nat,M)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,M))),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,M))) ).

% gchoose_row_sum_weighted
tff(fact_9007_sincos__total__pi__half,axiom,
    ! [Xb: real,Y: 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),Y)
       => ( ( aa_real_real(aa_rea7340605660300375871l_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,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real )
         => ? [T4: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T4)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T4),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
              & ( Xb = aa_real_real(cos_real,T4) )
              & ( Y = aa_real_real(sin_real,T4) ) ) ) ) ) ).

% sincos_total_pi_half
tff(fact_9008_sincos__total__2pi__le,axiom,
    ! [Xb: real,Y: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_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,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real )
     => ? [T4: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T4)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T4),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))
          & ( Xb = aa_real_real(cos_real,T4) )
          & ( Y = aa_real_real(sin_real,T4) ) ) ) ).

% sincos_total_2pi_le
tff(fact_9009_gbinomial__r__part__sum,axiom,
    ! [M: nat] : aa_set_nat_complex(groups2073611262835488442omplex(gbinomial_complex(aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_nat_complex(semiri8010041392384452111omplex,M))),one_one_complex))),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_nat_complex(power_power_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) ).

% gbinomial_r_part_sum
tff(fact_9010_gbinomial__r__part__sum,axiom,
    ! [M: nat] : groups2906978787729119204at_rat(gbinomial_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),aa_nat_rat(semiri681578069525770553at_rat,M))),one_one_rat)),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) ).

% gbinomial_r_part_sum
tff(fact_9011_gbinomial__r__part__sum,axiom,
    ! [M: nat] : aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,M))),one_one_real))),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) ).

% gbinomial_r_part_sum
tff(fact_9012_sincos__total__2pi,axiom,
    ! [Xb: real,Y: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_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,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real )
     => ~ ! [T4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T4)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,T4),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))
             => ( ( Xb = aa_real_real(cos_real,T4) )
               => ( Y != aa_real_real(sin_real,T4) ) ) ) ) ) ).

% sincos_total_2pi
tff(fact_9013_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_9014_tan__double,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(cos_real,Xb) != zero_zero_real )
     => ( ( aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Xb)) != zero_zero_real )
       => ( aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Xb)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(tan_real,Xb))),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(tan_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ) ).

% tan_double
tff(fact_9015_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(aa_rea7340605660300375871l_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(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(tan_real,Xb)),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(tan_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ).

% sin_tan
tff(fact_9016_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(aa_rea7340605660300375871l_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(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(tan_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ).

% cos_tan
tff(fact_9017_ceiling__log__nat__eq__powr__iff,axiom,
    ! [Ba: nat,K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Ba)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => ( ( aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_nat_real(semiri5074537144036343181t_real,Ba)),aa_nat_real(semiri5074537144036343181t_real,K))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Na)),one_one_int) )
        <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Ba),Na)),K)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Ba),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),one_one_nat))) ) ) ) ) ).

% ceiling_log_nat_eq_powr_iff
tff(fact_9018_complex__unimodular__polar,axiom,
    ! [Z: complex] :
      ( ( aa_complex_real(real_V1022390504157884413omplex,Z) = one_one_real )
     => ~ ! [T4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T4)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,T4),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))
             => ( Z != aa_real_complex(complex2(aa_real_real(cos_real,T4)),aa_real_real(sin_real,T4)) ) ) ) ) ).

% complex_unimodular_polar
tff(fact_9019_tan__zero,axiom,
    aa_real_real(tan_real,zero_zero_real) = zero_zero_real ).

% tan_zero
tff(fact_9020_zero__less__log__cancel__iff,axiom,
    ! [Aa: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Aa)
     => ( 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_rea7340605660300375871l_real(log,Aa),Xb))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb) ) ) ) ).

% zero_less_log_cancel_iff
tff(fact_9021_log__less__zero__cancel__iff,axiom,
    ! [Aa: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Aa)
     => ( 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(aa_rea7340605660300375871l_real(log,Aa),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_9022_one__less__log__cancel__iff,axiom,
    ! [Aa: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Aa)
     => ( 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(aa_rea7340605660300375871l_real(log,Aa),Xb))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),Xb) ) ) ) ).

% one_less_log_cancel_iff
tff(fact_9023_log__less__one__cancel__iff,axiom,
    ! [Aa: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Aa)
     => ( 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(aa_rea7340605660300375871l_real(log,Aa),Xb)),one_one_real)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Aa) ) ) ) ).

% log_less_one_cancel_iff
tff(fact_9024_log__less__cancel__iff,axiom,
    ! [Aa: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Aa)
     => ( 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),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,Aa),Xb)),aa_real_real(aa_rea7340605660300375871l_real(log,Aa),Y))
          <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y) ) ) ) ) ).

% log_less_cancel_iff
tff(fact_9025_log__eq__one,axiom,
    ! [Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Aa)
     => ( ( Aa != one_one_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(log,Aa),Aa) = one_one_real ) ) ) ).

% log_eq_one
tff(fact_9026_log__le__cancel__iff,axiom,
    ! [Aa: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Aa)
     => ( 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),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,Aa),Xb)),aa_real_real(aa_rea7340605660300375871l_real(log,Aa),Y))
          <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y) ) ) ) ) ).

% log_le_cancel_iff
tff(fact_9027_log__le__one__cancel__iff,axiom,
    ! [Aa: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Aa)
     => ( 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(aa_rea7340605660300375871l_real(log,Aa),Xb)),one_one_real)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Aa) ) ) ) ).

% log_le_one_cancel_iff
tff(fact_9028_one__le__log__cancel__iff,axiom,
    ! [Aa: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Aa)
     => ( 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(aa_rea7340605660300375871l_real(log,Aa),Xb))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),Xb) ) ) ) ).

% one_le_log_cancel_iff
tff(fact_9029_log__le__zero__cancel__iff,axiom,
    ! [Aa: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Aa)
     => ( 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(aa_rea7340605660300375871l_real(log,Aa),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_9030_zero__le__log__cancel__iff,axiom,
    ! [Aa: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Aa)
     => ( 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_rea7340605660300375871l_real(log,Aa),Xb))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xb) ) ) ) ).

% zero_le_log_cancel_iff
tff(fact_9031_log__pow__cancel,axiom,
    ! [Aa: real,Ba: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Aa)
     => ( ( Aa != one_one_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(log,Aa),aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),Ba)) = aa_nat_real(semiri5074537144036343181t_real,Ba) ) ) ) ).

% log_pow_cancel
tff(fact_9032_log__base__change,axiom,
    ! [Aa: real,Ba: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Aa)
     => ( ( Aa != one_one_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(log,Ba),Xb) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(log,Aa),Xb)),aa_real_real(aa_rea7340605660300375871l_real(log,Aa),Ba)) ) ) ) ).

% log_base_change
tff(fact_9033_less__log__of__power,axiom,
    ! [Ba: real,Na: nat,M: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Ba),Na)),M)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Ba)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_real_real(aa_rea7340605660300375871l_real(log,Ba),M)) ) ) ).

% less_log_of_power
tff(fact_9034_log__of__power__eq,axiom,
    ! [M: nat,Ba: real,Na: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,M) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Ba),Na) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Ba)
       => ( aa_nat_real(semiri5074537144036343181t_real,Na) = aa_real_real(aa_rea7340605660300375871l_real(log,Ba),aa_nat_real(semiri5074537144036343181t_real,M)) ) ) ) ).

% log_of_power_eq
tff(fact_9035_log__mult,axiom,
    ! [Aa: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Aa)
     => ( ( Aa != 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),Y)
           => ( aa_real_real(aa_rea7340605660300375871l_real(log,Aa),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(log,Aa),Xb)),aa_real_real(aa_rea7340605660300375871l_real(log,Aa),Y)) ) ) ) ) ) ).

% log_mult
tff(fact_9036_log__divide,axiom,
    ! [Aa: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Aa)
     => ( ( Aa != 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),Y)
           => ( aa_real_real(aa_rea7340605660300375871l_real(log,Aa),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(log,Aa),Xb)),aa_real_real(aa_rea7340605660300375871l_real(log,Aa),Y)) ) ) ) ) ) ).

% log_divide
tff(fact_9037_le__log__of__power,axiom,
    ! [Ba: real,Na: nat,M: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Ba),Na)),M)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Ba)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_real_real(aa_rea7340605660300375871l_real(log,Ba),M)) ) ) ).

% le_log_of_power
tff(fact_9038_log__base__pow,axiom,
    ! [Aa: real,Na: nat,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Aa)
     => ( aa_real_real(aa_rea7340605660300375871l_real(log,aa_nat_real(aa_real_fun_nat_real(power_power_real,Aa),Na)),Xb) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(log,Aa),Xb)),aa_nat_real(semiri5074537144036343181t_real,Na)) ) ) ).

% log_base_pow
tff(fact_9039_log__nat__power,axiom,
    ! [Xb: real,Ba: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_real(aa_rea7340605660300375871l_real(log,Ba),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Na)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_real_real(aa_rea7340605660300375871l_real(log,Ba),Xb)) ) ) ).

% log_nat_power
tff(fact_9040_log__of__power__less,axiom,
    ! [M: nat,Ba: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ba),Na))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Ba)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,Ba),aa_nat_real(semiri5074537144036343181t_real,M))),aa_nat_real(semiri5074537144036343181t_real,Na)) ) ) ) ).

% log_of_power_less
tff(fact_9041_log__eq__div__ln__mult__log,axiom,
    ! [Aa: real,Ba: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Aa)
     => ( ( Aa != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Ba)
         => ( ( Ba != one_one_real )
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
             => ( aa_real_real(aa_rea7340605660300375871l_real(log,Aa),Xb) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,Ba)),aa_real_real(ln_ln_real,Aa))),aa_real_real(aa_rea7340605660300375871l_real(log,Ba),Xb)) ) ) ) ) ) ) ).

% log_eq_div_ln_mult_log
tff(fact_9042_log__of__power__le,axiom,
    ! [M: nat,Ba: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ba),Na))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Ba)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,Ba),aa_nat_real(semiri5074537144036343181t_real,M))),aa_nat_real(semiri5074537144036343181t_real,Na)) ) ) ) ).

% log_of_power_le
tff(fact_9043_lemma__tan__total,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
     => ? [X: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
          & aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(tan_real,X)) ) ) ).

% lemma_tan_total
tff(fact_9044_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(aa_rea7340605660300375871l_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_9045_lemma__tan__total1,axiom,
    ! [Y: real] :
    ? [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X)
      & aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
      & ( aa_real_real(tan_real,X) = Y ) ) ).

% lemma_tan_total1
tff(fact_9046_tan__mono__lt__eq,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),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(aa_rea7340605660300375871l_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(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),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,Y))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y) ) ) ) ) ) ).

% tan_mono_lt_eq
tff(fact_9047_tan__monotone_H,axiom,
    ! [Y: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),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(aa_rea7340605660300375871l_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(aa_rea7340605660300375871l_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,Y),Xb)
            <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(tan_real,Y)),aa_real_real(tan_real,Xb)) ) ) ) ) ) ).

% tan_monotone'
tff(fact_9048_tan__monotone,axiom,
    ! [Y: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_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,Y)),aa_real_real(tan_real,Xb)) ) ) ) ).

% tan_monotone
tff(fact_9049_tan__total,axiom,
    ! [Y: real] :
    ? [X: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X)
      & aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
      & ( aa_real_real(tan_real,X) = Y )
      & ! [Y4: real] :
          ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y4)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,Y4),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
            & ( aa_real_real(tan_real,Y4) = Y ) )
         => ( Y4 = X ) ) ) ).

% tan_total
tff(fact_9050_add__tan__eq,axiom,
    ! [Xb: real,Y: real] :
      ( ( aa_real_real(cos_real,Xb) != zero_zero_real )
     => ( ( aa_real_real(cos_real,Y) != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(tan_real,Xb)),aa_real_real(tan_real,Y)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,Xb)),aa_real_real(cos_real,Y))) ) ) ) ).

% add_tan_eq
tff(fact_9051_less__log2__of__power,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)),M)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,M))) ) ).

% less_log2_of_power
tff(fact_9052_le__log2__of__power,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)),M)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,M))) ) ).

% le_log2_of_power
tff(fact_9053_tan__total__pos,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
     => ? [X: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,X),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
          & ( aa_real_real(tan_real,X) = Y ) ) ) ).

% tan_total_pos
tff(fact_9054_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(aa_rea7340605660300375871l_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_9055_tan__less__zero,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,pi)),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_9056_tan__mono__le__eq,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),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(aa_rea7340605660300375871l_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(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),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,Y))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y) ) ) ) ) ) ).

% tan_mono_le_eq
tff(fact_9057_tan__mono__le,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),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),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(tan_real,Xb)),aa_real_real(tan_real,Y)) ) ) ) ).

% tan_mono_le
tff(fact_9058_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(aa_rea7340605660300375871l_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_9059_arctan__unique,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),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(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => ( ( aa_real_real(tan_real,Xb) = Y )
         => ( aa_real_real(arctan,Y) = Xb ) ) ) ) ).

% arctan_unique
tff(fact_9060_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(aa_rea7340605660300375871l_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(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => ( aa_real_real(arctan,aa_real_real(tan_real,Xb)) = Xb ) ) ) ).

% arctan_tan
tff(fact_9061_arctan,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arctan,Y))
      & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arctan,Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
      & ( aa_real_real(tan_real,aa_real_real(arctan,Y)) = Y ) ) ).

% arctan
tff(fact_9062_tan__add,axiom,
    ! [Xb: complex,Y: complex] :
      ( ( aa_complex_complex(cos_complex,Xb) != zero_zero_complex )
     => ( ( aa_complex_complex(cos_complex,Y) != zero_zero_complex )
       => ( ( aa_complex_complex(cos_complex,aa_complex_complex(plus_plus_complex(Xb),Y)) != zero_zero_complex )
         => ( aa_complex_complex(tan_complex,aa_complex_complex(plus_plus_complex(Xb),Y)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(tan_complex,Xb)),aa_complex_complex(tan_complex,Y))),aa_complex_complex(minus_minus_complex(one_one_complex),aa_complex_complex(times_times_complex(aa_complex_complex(tan_complex,Xb)),aa_complex_complex(tan_complex,Y)))) ) ) ) ) ).

% tan_add
tff(fact_9063_tan__add,axiom,
    ! [Xb: real,Y: real] :
      ( ( aa_real_real(cos_real,Xb) != zero_zero_real )
     => ( ( aa_real_real(cos_real,Y) != zero_zero_real )
       => ( ( aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y)) != zero_zero_real )
         => ( aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(tan_real,Xb)),aa_real_real(tan_real,Y))),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(tan_real,Xb)),aa_real_real(tan_real,Y)))) ) ) ) ) ).

% tan_add
tff(fact_9064_tan__diff,axiom,
    ! [Xb: complex,Y: complex] :
      ( ( aa_complex_complex(cos_complex,Xb) != zero_zero_complex )
     => ( ( aa_complex_complex(cos_complex,Y) != zero_zero_complex )
       => ( ( aa_complex_complex(cos_complex,aa_complex_complex(minus_minus_complex(Xb),Y)) != zero_zero_complex )
         => ( aa_complex_complex(tan_complex,aa_complex_complex(minus_minus_complex(Xb),Y)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(tan_complex,Xb)),aa_complex_complex(tan_complex,Y))),aa_complex_complex(plus_plus_complex(one_one_complex),aa_complex_complex(times_times_complex(aa_complex_complex(tan_complex,Xb)),aa_complex_complex(tan_complex,Y)))) ) ) ) ) ).

% tan_diff
tff(fact_9065_tan__diff,axiom,
    ! [Xb: real,Y: real] :
      ( ( aa_real_real(cos_real,Xb) != zero_zero_real )
     => ( ( aa_real_real(cos_real,Y) != zero_zero_real )
       => ( ( aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Y)) != zero_zero_real )
         => ( aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(tan_real,Xb)),aa_real_real(tan_real,Y))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(tan_real,Xb)),aa_real_real(tan_real,Y)))) ) ) ) ) ).

% tan_diff
tff(fact_9066_lemma__tan__add1,axiom,
    ! [Xb: complex,Y: complex] :
      ( ( aa_complex_complex(cos_complex,Xb) != zero_zero_complex )
     => ( ( aa_complex_complex(cos_complex,Y) != zero_zero_complex )
       => ( aa_complex_complex(minus_minus_complex(one_one_complex),aa_complex_complex(times_times_complex(aa_complex_complex(tan_complex,Xb)),aa_complex_complex(tan_complex,Y))) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(cos_complex,aa_complex_complex(plus_plus_complex(Xb),Y))),aa_complex_complex(times_times_complex(aa_complex_complex(cos_complex,Xb)),aa_complex_complex(cos_complex,Y))) ) ) ) ).

% lemma_tan_add1
tff(fact_9067_lemma__tan__add1,axiom,
    ! [Xb: real,Y: real] :
      ( ( aa_real_real(cos_real,Xb) != zero_zero_real )
     => ( ( aa_real_real(cos_real,Y) != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(tan_real,Xb)),aa_real_real(tan_real,Y))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,Xb)),aa_real_real(cos_real,Y))) ) ) ) ).

% lemma_tan_add1
tff(fact_9068_log2__of__power__less,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,M))),aa_nat_real(semiri5074537144036343181t_real,Na)) ) ) ).

% log2_of_power_less
tff(fact_9069_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)
     => ? [Z3: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))))),Z3)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))))
          & ( aa_real_real(tan_real,Z3) = Xb ) ) ) ).

% tan_total_pi4
tff(fact_9070_log2__of__power__le,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,M))),aa_nat_real(semiri5074537144036343181t_real,Na)) ) ) ).

% log2_of_power_le
tff(fact_9071_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(aa_rea7340605660300375871l_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(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),aa_real_real(exp_real,one_one_real))),aa_real_real(ln_ln_real,Xb)) ) ) ).

% log_base_10_eq2
tff(fact_9072_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(aa_rea7340605660300375871l_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(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,aa_real_real(exp_real,one_one_real))),aa_real_real(ln_ln_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))))),aa_real_real(ln_ln_real,Xb)) ) ) ).

% log_base_10_eq1
tff(fact_9073_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_9074_tan__half,axiom,
    ! [Xb: real] : aa_real_real(tan_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Xb))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Xb))),one_one_real)) ).

% tan_half
tff(fact_9075_ceiling__log2__div2,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Na))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,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,Na),one_one_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat))))),one_one_int) ) ) ).

% ceiling_log2_div2
tff(fact_9076_ceiling__log__nat__eq__if,axiom,
    ! [Ba: nat,Na: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Ba),Na)),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Ba),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),one_one_nat)))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Ba)
         => ( aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_nat_real(semiri5074537144036343181t_real,Ba)),aa_nat_real(semiri5074537144036343181t_real,K))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Na)),one_one_int) ) ) ) ) ).

% ceiling_log_nat_eq_if
tff(fact_9077_sin__paired,axiom,
    ! [Xb: real] : aa_real_o(sums_real(aa_real_fun_nat_real(aTP_Lamm_xc,Xb)),aa_real_real(sin_real,Xb)) ).

% sin_paired
tff(fact_9078_ceiling__log__eq__powr__iff,axiom,
    ! [Xb: real,Ba: 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),Ba)
       => ( ( aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,Ba),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(aa_rea7340605660300375871l_real(powr_real,Ba),aa_nat_real(semiri5074537144036343181t_real,K))),Xb)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Ba),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_9079_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(aa_rea7340605660300375871l_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_9080_sin__arccos__abs,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Y)),one_one_real)
     => ( aa_real_real(sin_real,aa_real_real(arccos,Y)) = aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ).

% sin_arccos_abs
tff(fact_9081_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(aa_rea7340605660300375871l_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_9082_powr__0,axiom,
    ! [Z: real] : aa_real_real(aa_rea7340605660300375871l_real(powr_real,zero_zero_real),Z) = zero_zero_real ).

% powr_0
tff(fact_9083_powr__eq__0__iff,axiom,
    ! [W2: real,Z: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,W2),Z) = zero_zero_real )
    <=> ( W2 = zero_zero_real ) ) ).

% powr_eq_0_iff
tff(fact_9084_powr__one__eq__one,axiom,
    ! [Aa: real] : aa_real_real(aa_rea7340605660300375871l_real(powr_real,one_one_real),Aa) = one_one_real ).

% powr_one_eq_one
tff(fact_9085_sums__zero,axiom,
    aa_real_o(sums_real(aTP_Lamm_fc),zero_zero_real) ).

% sums_zero
tff(fact_9086_sums__zero,axiom,
    aa_nat_o(sums_nat(aTP_Lamm_bx),zero_zero_nat) ).

% sums_zero
tff(fact_9087_sums__zero,axiom,
    aa_int_o(sums_int(aTP_Lamm_pb),zero_zero_int) ).

% sums_zero
tff(fact_9088_powr__zero__eq__one,axiom,
    ! [Xb: real] :
      aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),zero_zero_real) = $ite(Xb = zero_zero_real,zero_zero_real,one_one_real) ).

% powr_zero_eq_one
tff(fact_9089_powr__gt__zero,axiom,
    ! [Xb: real,Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),Aa))
    <=> ( Xb != zero_zero_real ) ) ).

% powr_gt_zero
tff(fact_9090_powr__nonneg__iff,axiom,
    ! [Aa: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Aa),Xb)),zero_zero_real)
    <=> ( Aa = zero_zero_real ) ) ).

% powr_nonneg_iff
tff(fact_9091_powr__less__cancel__iff,axiom,
    ! [Xb: real,Aa: real,Ba: 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(aa_rea7340605660300375871l_real(powr_real,Xb),Aa)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),Ba))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),Ba) ) ) ).

% powr_less_cancel_iff
tff(fact_9092_powr__eq__one__iff,axiom,
    ! [Aa: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Aa)
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,Aa),Xb) = one_one_real )
      <=> ( Xb = zero_zero_real ) ) ) ).

% powr_eq_one_iff
tff(fact_9093_powr__one,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),one_one_real) = Xb ) ) ).

% powr_one
tff(fact_9094_powr__one__gt__zero__iff,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_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_9095_powr__le__cancel__iff,axiom,
    ! [Xb: real,Aa: real,Ba: 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(aa_rea7340605660300375871l_real(powr_real,Xb),Aa)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),Ba))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),Ba) ) ) ).

% powr_le_cancel_iff
tff(fact_9096_powr__log__cancel,axiom,
    ! [Aa: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Aa)
     => ( ( Aa != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
         => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,Aa),aa_real_real(aa_rea7340605660300375871l_real(log,Aa),Xb)) = Xb ) ) ) ) ).

% powr_log_cancel
tff(fact_9097_log__powr__cancel,axiom,
    ! [Aa: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Aa)
     => ( ( Aa != one_one_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(log,Aa),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Aa),Y)) = Y ) ) ) ).

% log_powr_cancel
tff(fact_9098_cos__arccos,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => ( aa_real_real(cos_real,aa_real_real(arccos,Y)) = Y ) ) ) ).

% cos_arccos
tff(fact_9099_sin__arcsin,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => ( aa_real_real(sin_real,aa_real_real(arcsin,Y)) = Y ) ) ) ).

% sin_arcsin
tff(fact_9100_powser__sums__zero__iff,axiom,
    ! [Aa: fun_nat_complex,Xb: complex] :
      ( aa_complex_o(sums_complex(aTP_Lamm_iu(Aa)),Xb)
    <=> ( aa_nat_complex(Aa,zero_zero_nat) = Xb ) ) ).

% powser_sums_zero_iff
tff(fact_9101_powser__sums__zero__iff,axiom,
    ! [Aa: fun_nat_real,Xb: real] :
      ( aa_real_o(sums_real(aTP_Lamm_iw(Aa)),Xb)
    <=> ( aa_nat_real(Aa,zero_zero_nat) = Xb ) ) ).

% powser_sums_zero_iff
tff(fact_9102_powr__numeral,axiom,
    ! [Xb: real,Na: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),aa_num_real(numeral_numeral_real,Na)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,Na)) ) ) ).

% powr_numeral
tff(fact_9103_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_9104_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_9105_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_9106_sums__le,axiom,
    ! [F: fun_nat_real,G: fun_nat_real,S3: 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),S3)
       => ( aa_real_o(sums_real(G),Ta)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,S3),Ta) ) ) ) ).

% sums_le
tff(fact_9107_sums__le,axiom,
    ! [F: fun_nat_nat,G: fun_nat_nat,S3: 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),S3)
       => ( aa_nat_o(sums_nat(G),Ta)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,S3),Ta) ) ) ) ).

% sums_le
tff(fact_9108_sums__le,axiom,
    ! [F: fun_nat_int,G: fun_nat_int,S3: 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),S3)
       => ( aa_int_o(sums_int(G),Ta)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,S3),Ta) ) ) ) ).

% sums_le
tff(fact_9109_sums__single,axiom,
    ! [I: nat,F: fun_nat_real] : aa_real_o(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_qm(I),F)),aa_nat_real(F,I)) ).

% sums_single
tff(fact_9110_sums__single,axiom,
    ! [I: nat,F: fun_nat_nat] : aa_nat_o(sums_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_qn(I),F)),aa_nat_nat(F,I)) ).

% sums_single
tff(fact_9111_sums__single,axiom,
    ! [I: nat,F: fun_nat_int] : aa_int_o(sums_int(aa_fun5565043122120104496at_int(aTP_Lamm_qo(I),F)),aa_nat_int(F,I)) ).

% sums_single
tff(fact_9112_powr__non__neg,axiom,
    ! [Aa: real,Xb: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Aa),Xb)),zero_zero_real) ).

% powr_non_neg
tff(fact_9113_powr__less__mono2__neg,axiom,
    ! [Aa: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),zero_zero_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),Y)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Y),Aa)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),Aa)) ) ) ) ).

% powr_less_mono2_neg
tff(fact_9114_powr__mono2,axiom,
    ! [Aa: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Aa)
     => ( 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),Y)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),Aa)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Y),Aa)) ) ) ) ).

% powr_mono2
tff(fact_9115_powr__ge__pzero,axiom,
    ! [Xb: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),Y)) ).

% powr_ge_pzero
tff(fact_9116_powr__less__mono,axiom,
    ! [Aa: real,Ba: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),Ba)
     => ( 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_rea7340605660300375871l_real(powr_real,Xb),Aa)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),Ba)) ) ) ).

% powr_less_mono
tff(fact_9117_powr__less__cancel,axiom,
    ! [Xb: real,Aa: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),Aa)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),Ba))
     => ( 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),Ba) ) ) ).

% powr_less_cancel
tff(fact_9118_powr__mono,axiom,
    ! [Aa: real,Ba: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),Ba)
     => ( 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_rea7340605660300375871l_real(powr_real,Xb),Aa)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),Ba)) ) ) ).

% powr_mono
tff(fact_9119_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_qp(C),F)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),D))
      <=> aa_real_o(sums_real(F),D) ) ) ).

% sums_mult_iff
tff(fact_9120_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_xd(C),F)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,D),C))
      <=> aa_real_o(sums_real(F),D) ) ) ).

% sums_mult2_iff
tff(fact_9121_powr__less__mono2,axiom,
    ! [Aa: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Aa)
     => ( 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),Y)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),Aa)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Y),Aa)) ) ) ) ).

% powr_less_mono2
tff(fact_9122_powr__mono2_H,axiom,
    ! [Aa: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),zero_zero_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),Y)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Y),Aa)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),Aa)) ) ) ) ).

% powr_mono2'
tff(fact_9123_gr__one__powr,axiom,
    ! [Xb: real,Y: 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),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),Y)) ) ) ).

% gr_one_powr
tff(fact_9124_powr__inj,axiom,
    ! [Aa: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Aa)
     => ( ( Aa != one_one_real )
       => ( ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,Aa),Xb) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,Aa),Y) )
        <=> ( Xb = Y ) ) ) ) ).

% powr_inj
tff(fact_9125_powr__le1,axiom,
    ! [Aa: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Aa)
     => ( 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_rea7340605660300375871l_real(powr_real,Xb),Aa)),one_one_real) ) ) ) ).

% powr_le1
tff(fact_9126_powr__mono__both,axiom,
    ! [Aa: real,Ba: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Aa)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),Ba)
       => ( 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),Y)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),Aa)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Y),Ba)) ) ) ) ) ).

% powr_mono_both
tff(fact_9127_ge__one__powr__ge__zero,axiom,
    ! [Xb: real,Aa: 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)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),Aa)) ) ) ).

% ge_one_powr_ge_zero
tff(fact_9128_powr__divide,axiom,
    ! [Xb: real,Y: real,Aa: 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),Y)
       => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)),Aa) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),Aa)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Y),Aa)) ) ) ) ).

% powr_divide
tff(fact_9129_powr__mult,axiom,
    ! [Xb: real,Y: real,Aa: 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),Y)
       => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Y)),Aa) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),Aa)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Y),Aa)) ) ) ) ).

% powr_mult
tff(fact_9130_sums__mult__D,axiom,
    ! [C: real,F: fun_nat_real,Aa: real] :
      ( aa_real_o(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_qp(C),F)),Aa)
     => ( ( C != zero_zero_real )
       => aa_real_o(sums_real(F),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Aa),C)) ) ) ).

% sums_mult_D
tff(fact_9131_sums__Suc__imp,axiom,
    ! [F: fun_nat_real,S3: real] :
      ( ( aa_nat_real(F,zero_zero_nat) = zero_zero_real )
     => ( aa_real_o(sums_real(aTP_Lamm_jg(F)),S3)
       => aa_real_o(sums_real(F),S3) ) ) ).

% sums_Suc_imp
tff(fact_9132_sums__Suc,axiom,
    ! [F: fun_nat_real,L: real] :
      ( aa_real_o(sums_real(aTP_Lamm_jg(F)),L)
     => aa_real_o(sums_real(F),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,L),aa_nat_real(F,zero_zero_nat))) ) ).

% sums_Suc
tff(fact_9133_sums__Suc,axiom,
    ! [F: fun_nat_nat,L: nat] :
      ( aa_nat_o(sums_nat(aTP_Lamm_jf(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_9134_sums__Suc,axiom,
    ! [F: fun_nat_int,L: int] :
      ( aa_int_o(sums_int(aTP_Lamm_jx(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_9135_sums__Suc__iff,axiom,
    ! [F: fun_nat_real,S3: real] :
      ( aa_real_o(sums_real(aTP_Lamm_jg(F)),S3)
    <=> aa_real_o(sums_real(F),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,S3),aa_nat_real(F,zero_zero_nat))) ) ).

% sums_Suc_iff
tff(fact_9136_sums__zero__iff__shift,axiom,
    ! [Na: nat,F: fun_nat_real,S3: real] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),Na)
         => ( aa_nat_real(F,I2) = zero_zero_real ) )
     => ( aa_real_o(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_xe(Na),F)),S3)
      <=> aa_real_o(sums_real(F),S3) ) ) ).

% sums_zero_iff_shift
tff(fact_9137_sums__If__finite__set,axiom,
    ! [A3: set_nat,F: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => aa_int_o(sums_int(aa_fun5565043122120104496at_int(aTP_Lamm_qt(A3),F)),groups3539618377306564664at_int(F,A3)) ) ).

% sums_If_finite_set
tff(fact_9138_sums__If__finite__set,axiom,
    ! [A3: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => aa_nat_o(sums_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_qs(A3),F)),aa_set_nat_nat(groups3542108847815614940at_nat(F),A3)) ) ).

% sums_If_finite_set
tff(fact_9139_sums__If__finite__set,axiom,
    ! [A3: set_nat,F: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => aa_real_o(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_qr(A3),F)),aa_set_nat_real(groups6591440286371151544t_real(F),A3)) ) ).

% sums_If_finite_set
tff(fact_9140_sums__If__finite,axiom,
    ! [P: fun_nat_o,F: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(P))
     => aa_int_o(sums_int(aa_fun5565043122120104496at_int(aTP_Lamm_qw(P),F)),groups3539618377306564664at_int(F,collect_nat(P))) ) ).

% sums_If_finite
tff(fact_9141_sums__If__finite,axiom,
    ! [P: fun_nat_o,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(P))
     => aa_nat_o(sums_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_qv(P),F)),aa_set_nat_nat(groups3542108847815614940at_nat(F),collect_nat(P))) ) ).

% sums_If_finite
tff(fact_9142_sums__If__finite,axiom,
    ! [P: fun_nat_o,F: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(P))
     => aa_real_o(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_qu(P),F)),aa_set_nat_real(groups6591440286371151544t_real(F),collect_nat(P))) ) ).

% sums_If_finite
tff(fact_9143_sums__finite,axiom,
    ! [N3: set_nat,F: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,N3)
     => ( ! [N: nat] :
            ( ~ aa_set_nat_o(member_nat(N),N3)
           => ( aa_nat_int(F,N) = zero_zero_int ) )
       => aa_int_o(sums_int(F),groups3539618377306564664at_int(F,N3)) ) ) ).

% sums_finite
tff(fact_9144_sums__finite,axiom,
    ! [N3: set_nat,F: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,N3)
     => ( ! [N: nat] :
            ( ~ aa_set_nat_o(member_nat(N),N3)
           => ( aa_nat_nat(F,N) = zero_zero_nat ) )
       => aa_nat_o(sums_nat(F),aa_set_nat_nat(groups3542108847815614940at_nat(F),N3)) ) ) ).

% sums_finite
tff(fact_9145_sums__finite,axiom,
    ! [N3: set_nat,F: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,N3)
     => ( ! [N: nat] :
            ( ~ aa_set_nat_o(member_nat(N),N3)
           => ( aa_nat_real(F,N) = zero_zero_real ) )
       => aa_real_o(sums_real(F),aa_set_nat_real(groups6591440286371151544t_real(F),N3)) ) ) ).

% sums_finite
tff(fact_9146_arccos__le__arccos,axiom,
    ! [Xb: real,Y: 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),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arccos,Y)),aa_real_real(arccos,Xb)) ) ) ) ).

% arccos_le_arccos
tff(fact_9147_arccos__eq__iff,axiom,
    ! [Xb: real,Y: 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,Y)),one_one_real) )
     => ( ( aa_real_real(arccos,Xb) = aa_real_real(arccos,Y) )
      <=> ( Xb = Y ) ) ) ).

% arccos_eq_iff
tff(fact_9148_arccos__le__mono,axiom,
    ! [Xb: real,Y: 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,Y)),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arccos,Xb)),aa_real_real(arccos,Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Xb) ) ) ) ).

% arccos_le_mono
tff(fact_9149_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_9150_arcsin__le__arcsin,axiom,
    ! [Xb: real,Y: 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),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arcsin,Xb)),aa_real_real(arcsin,Y)) ) ) ) ).

% arcsin_le_arcsin
tff(fact_9151_arcsin__eq__iff,axiom,
    ! [Xb: real,Y: 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,Y)),one_one_real)
       => ( ( aa_real_real(arcsin,Xb) = aa_real_real(arcsin,Y) )
        <=> ( Xb = Y ) ) ) ) ).

% arcsin_eq_iff
tff(fact_9152_arcsin__le__mono,axiom,
    ! [Xb: real,Y: 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,Y)),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arcsin,Xb)),aa_real_real(arcsin,Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y) ) ) ) ).

% arcsin_le_mono
tff(fact_9153_powser__sums__if,axiom,
    ! [M: nat,Z: complex] : aa_complex_o(sums_complex(aa_com8181001469867462503omplex(aTP_Lamm_xf(M),Z)),aa_nat_complex(power_power_complex(Z),M)) ).

% powser_sums_if
tff(fact_9154_powser__sums__if,axiom,
    ! [M: nat,Z: real] : aa_real_o(sums_real(aa_real_fun_nat_real(aTP_Lamm_xg(M),Z)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),M)) ).

% powser_sums_if
tff(fact_9155_powser__sums__if,axiom,
    ! [M: nat,Z: int] : aa_int_o(sums_int(aa_int_fun_nat_int(aTP_Lamm_xh(M),Z)),aa_nat_int(power_power_int(Z),M)) ).

% powser_sums_if
tff(fact_9156_powser__sums__zero,axiom,
    ! [Aa: fun_nat_complex] : aa_complex_o(sums_complex(aTP_Lamm_iu(Aa)),aa_nat_complex(Aa,zero_zero_nat)) ).

% powser_sums_zero
tff(fact_9157_powser__sums__zero,axiom,
    ! [Aa: fun_nat_real] : aa_real_o(sums_real(aTP_Lamm_iw(Aa)),aa_nat_real(Aa,zero_zero_nat)) ).

% powser_sums_zero
tff(fact_9158_powr__realpow,axiom,
    ! [Xb: real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),aa_nat_real(semiri5074537144036343181t_real,Na)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Na) ) ) ).

% powr_realpow
tff(fact_9159_powr__less__iff,axiom,
    ! [Ba: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Ba)
     => ( 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(aa_rea7340605660300375871l_real(powr_real,Ba),Y)),Xb)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(log,Ba),Xb)) ) ) ) ).

% powr_less_iff
tff(fact_9160_less__powr__iff,axiom,
    ! [Ba: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Ba)
     => ( 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(aa_rea7340605660300375871l_real(powr_real,Ba),Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,Ba),Xb)),Y) ) ) ) ).

% less_powr_iff
tff(fact_9161_log__less__iff,axiom,
    ! [Ba: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Ba)
     => ( 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(aa_rea7340605660300375871l_real(log,Ba),Xb)),Y)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Ba),Y)) ) ) ) ).

% log_less_iff
tff(fact_9162_less__log__iff,axiom,
    ! [Ba: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Ba)
     => ( 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,Y),aa_real_real(aa_rea7340605660300375871l_real(log,Ba),Xb))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Ba),Y)),Xb) ) ) ) ).

% less_log_iff
tff(fact_9163_sums__If__finite__set_H,axiom,
    ! [G: fun_nat_real,S2: real,A3: set_nat,S4: real,F: fun_nat_real] :
      ( aa_real_o(sums_real(G),S2)
     => ( aa_set_nat_o(finite_finite_nat,A3)
       => ( ( S4 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,S2),aa_set_nat_real(groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_xi(G),F)),A3)) )
         => aa_real_o(sums_real(aa_fun1839206346889086768t_real(aa_set1316517696749040435t_real(aTP_Lamm_xj(G),A3),F)),S4) ) ) ) ).

% sums_If_finite_set'
tff(fact_9164_powr__minus__divide,axiom,
    ! [Xb: real,Aa: real] : aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),aa_real_real(uminus_uminus_real,Aa)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),Aa)) ).

% powr_minus_divide
tff(fact_9165_arccos__lbound,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(arccos,Y)) ) ) ).

% arccos_lbound
tff(fact_9166_arccos__less__arccos,axiom,
    ! [Xb: real,Y: 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),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arccos,Y)),aa_real_real(arccos,Xb)) ) ) ) ).

% arccos_less_arccos
tff(fact_9167_arccos__less__mono,axiom,
    ! [Xb: real,Y: 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,Y)),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arccos,Xb)),aa_real_real(arccos,Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xb) ) ) ) ).

% arccos_less_mono
tff(fact_9168_arccos__ubound,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arccos,Y)),pi) ) ) ).

% arccos_ubound
tff(fact_9169_arccos__cos,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),pi)
       => ( aa_real_real(arccos,aa_real_real(cos_real,Xb)) = Xb ) ) ) ).

% arccos_cos
tff(fact_9170_powr__neg__one,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),Xb) ) ) ).

% powr_neg_one
tff(fact_9171_arcsin__less__arcsin,axiom,
    ! [Xb: real,Y: 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),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arcsin,Xb)),aa_real_real(arcsin,Y)) ) ) ) ).

% arcsin_less_arcsin
tff(fact_9172_powr__mult__base,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Y)) ) ) ).

% powr_mult_base
tff(fact_9173_arcsin__less__mono,axiom,
    ! [Xb: real,Y: 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,Y)),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arcsin,Xb)),aa_real_real(arcsin,Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y) ) ) ) ).

% arcsin_less_mono
tff(fact_9174_le__log__iff,axiom,
    ! [Ba: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Ba)
     => ( 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,Y),aa_real_real(aa_rea7340605660300375871l_real(log,Ba),Xb))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Ba),Y)),Xb) ) ) ) ).

% le_log_iff
tff(fact_9175_log__le__iff,axiom,
    ! [Ba: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Ba)
     => ( 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(aa_rea7340605660300375871l_real(log,Ba),Xb)),Y)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Ba),Y)) ) ) ) ).

% log_le_iff
tff(fact_9176_le__powr__iff,axiom,
    ! [Ba: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Ba)
     => ( 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(aa_rea7340605660300375871l_real(powr_real,Ba),Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,Ba),Xb)),Y) ) ) ) ).

% le_powr_iff
tff(fact_9177_powr__le__iff,axiom,
    ! [Ba: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Ba)
     => ( 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(aa_rea7340605660300375871l_real(powr_real,Ba),Y)),Xb)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(aa_rea7340605660300375871l_real(log,Ba),Xb)) ) ) ) ).

% powr_le_iff
tff(fact_9178_cos__arccos__abs,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Y)),one_one_real)
     => ( aa_real_real(cos_real,aa_real_real(arccos,Y)) = Y ) ) ).

% cos_arccos_abs
tff(fact_9179_arccos__cos__eq__abs,axiom,
    ! [Theta: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Theta)),pi)
     => ( aa_real_real(arccos,aa_real_real(cos_real,Theta)) = aa_real_real(abs_abs_real,Theta) ) ) ).

% arccos_cos_eq_abs
tff(fact_9180_ln__powr__bound,axiom,
    ! [Xb: real,Aa: 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),Aa)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,Xb)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),Aa)),Aa)) ) ) ).

% ln_powr_bound
tff(fact_9181_ln__powr__bound2,axiom,
    ! [Xb: real,Aa: 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)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(ln_ln_real,Xb)),Aa)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Aa),Aa)),Xb)) ) ) ).

% ln_powr_bound2
tff(fact_9182_log__add__eq__powr,axiom,
    ! [Ba: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Ba)
     => ( ( Ba != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
         => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(log,Ba),Xb)),Y) = aa_real_real(aa_rea7340605660300375871l_real(log,Ba),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Ba),Y))) ) ) ) ) ).

% log_add_eq_powr
tff(fact_9183_add__log__eq__powr,axiom,
    ! [Ba: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Ba)
     => ( ( Ba != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
         => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Y),aa_real_real(aa_rea7340605660300375871l_real(log,Ba),Xb)) = aa_real_real(aa_rea7340605660300375871l_real(log,Ba),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Ba),Y)),Xb)) ) ) ) ) ).

% add_log_eq_powr
tff(fact_9184_minus__log__eq__powr,axiom,
    ! [Ba: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Ba)
     => ( ( Ba != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
         => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y),aa_real_real(aa_rea7340605660300375871l_real(log,Ba),Xb)) = aa_real_real(aa_rea7340605660300375871l_real(log,Ba),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Ba),Y)),Xb)) ) ) ) ) ).

% minus_log_eq_powr
tff(fact_9185_arccos__lt__bounded,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(arccos,Y))
          & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arccos,Y)),pi) ) ) ) ).

% arccos_lt_bounded
tff(fact_9186_arccos__bounded,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(arccos,Y))
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arccos,Y)),pi) ) ) ) ).

% arccos_bounded
tff(fact_9187_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_9188_arccos__cos2,axiom,
    ! [Xb: 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,aa_real_real(uminus_uminus_real,pi)),Xb)
       => ( aa_real_real(arccos,aa_real_real(cos_real,Xb)) = aa_real_real(uminus_uminus_real,Xb) ) ) ) ).

% arccos_cos2
tff(fact_9189_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(aa_rea7340605660300375871l_real(minus_minus_real,pi),aa_real_real(arccos,Xb)) ) ) ) ).

% arccos_minus
tff(fact_9190_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_9191_power__half__series,axiom,
    aa_real_o(sums_real(aTP_Lamm_xk),one_one_real) ).

% power_half_series
tff(fact_9192_log__minus__eq__powr,axiom,
    ! [Ba: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Ba)
     => ( ( Ba != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
         => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(log,Ba),Xb)),Y) = aa_real_real(aa_rea7340605660300375871l_real(log,Ba),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Ba),aa_real_real(uminus_uminus_real,Y)))) ) ) ) ) ).

% log_minus_eq_powr
tff(fact_9193_arccos,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(arccos,Y))
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arccos,Y)),pi)
          & ( aa_real_real(cos_real,aa_real_real(arccos,Y)) = Y ) ) ) ) ).

% arccos
tff(fact_9194_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(aa_rea7340605660300375871l_real(minus_minus_real,pi),aa_real_real(arccos,Xb)) ) ) ).

% arccos_minus_abs
tff(fact_9195_sums__if_H,axiom,
    ! [G: fun_nat_real,Xb: real] :
      ( aa_real_o(sums_real(G),Xb)
     => aa_real_o(sums_real(aTP_Lamm_xl(G)),Xb) ) ).

% sums_if'
tff(fact_9196_sums__if,axiom,
    ! [G: fun_nat_real,Xb: real,F: fun_nat_real,Y: real] :
      ( aa_real_o(sums_real(G),Xb)
     => ( aa_real_o(sums_real(F),Y)
       => aa_real_o(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_xm(G),F)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y)) ) ) ).

% sums_if
tff(fact_9197_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(aa_rea7340605660300375871l_real(powr_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = aa_real_real(sqrt,Xb) ) ) ).

% powr_half_sqrt
tff(fact_9198_powr__neg__numeral,axiom,
    ! [Xb: real,Na: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Na))) = aa_real_real(aa_rea7340605660300375871l_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,Na))) ) ) ).

% powr_neg_numeral
tff(fact_9199_arccos__le__pi2,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arccos,Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ).

% arccos_le_pi2
tff(fact_9200_arcsin__lt__bounded,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arcsin,Y))
          & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arcsin,Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ) ).

% arcsin_lt_bounded
tff(fact_9201_arcsin__lbound,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arcsin,Y)) ) ) ).

% arcsin_lbound
tff(fact_9202_arcsin__ubound,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arcsin,Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ).

% arcsin_ubound
tff(fact_9203_arcsin__bounded,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arcsin,Y))
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arcsin,Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ) ).

% arcsin_bounded
tff(fact_9204_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(aa_rea7340605660300375871l_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(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => ( aa_real_real(arcsin,aa_real_real(sin_real,Xb)) = Xb ) ) ) ).

% arcsin_sin
tff(fact_9205_arcsin,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arcsin,Y))
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arcsin,Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
          & ( aa_real_real(sin_real,aa_real_real(arcsin,Y)) = Y ) ) ) ) ).

% arcsin
tff(fact_9206_arcsin__pi,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arcsin,Y))
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arcsin,Y)),pi)
          & ( aa_real_real(sin_real,aa_real_real(arcsin,Y)) = Y ) ) ) ) ).

% arcsin_pi
tff(fact_9207_arcsin__le__iff,axiom,
    ! [Xb: real,Y: 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(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arcsin,Xb)),Y)
            <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(sin_real,Y)) ) ) ) ) ) ).

% arcsin_le_iff
tff(fact_9208_le__arcsin__iff,axiom,
    ! [Xb: real,Y: 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(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(arcsin,Xb))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sin_real,Y)),Xb) ) ) ) ) ) ).

% le_arcsin_iff
tff(fact_9209_floor__log__nat__eq__powr__iff,axiom,
    ! [Ba: nat,K: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Ba)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => ( ( aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_nat_real(semiri5074537144036343181t_real,Ba)),aa_nat_real(semiri5074537144036343181t_real,K))) = aa_nat_int(semiri1314217659103216013at_int,Na) )
        <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Ba),Na)),K)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Ba),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),one_one_nat))) ) ) ) ) ).

% floor_log_nat_eq_powr_iff
tff(fact_9210_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(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,R2))),one_one_real)) ).

% real_of_int_floor_add_one_gt
tff(fact_9211_floor__eq,axiom,
    ! [Na: int,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Na)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,Na)),one_one_real))
       => ( aa_real_int(archim6058952711729229775r_real,Xb) = Na ) ) ) ).

% floor_eq
tff(fact_9212_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(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,R2))),one_one_real)) ).

% real_of_int_floor_add_one_ge
tff(fact_9213_real__of__int__floor__gt__diff__one,axiom,
    ! [R2: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,R2),one_one_real)),aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,R2))) ).

% real_of_int_floor_gt_diff_one
tff(fact_9214_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(aa_rea7340605660300375871l_real(minus_minus_real,R2),one_one_real)),aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,R2))) ).

% real_of_int_floor_ge_diff_one
tff(fact_9215_floor__eq2,axiom,
    ! [Na: int,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Na)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,Na)),one_one_real))
       => ( aa_real_int(archim6058952711729229775r_real,Xb) = Na ) ) ) ).

% floor_eq2
tff(fact_9216_floor__divide__real__eq__div,axiom,
    ! [Ba: int,Aa: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Ba)
     => ( aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Aa),aa_int_real(ring_1_of_int_real,Ba))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_real_int(archim6058952711729229775r_real,Aa)),Ba) ) ) ).

% floor_divide_real_eq_div
tff(fact_9217_floor__log__eq__powr__iff,axiom,
    ! [Xb: real,Ba: 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),Ba)
       => ( ( aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(log,Ba),Xb)) = K )
        <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Ba),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(aa_rea7340605660300375871l_real(powr_real,Ba),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_9218_floor__log2__div2,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
     => ( aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Na))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(log,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,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),one_one_int) ) ) ).

% floor_log2_div2
tff(fact_9219_floor__log__nat__eq__if,axiom,
    ! [Ba: nat,Na: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Ba),Na)),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Ba),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),one_one_nat)))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Ba)
         => ( aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_nat_real(semiri5074537144036343181t_real,Ba)),aa_nat_real(semiri5074537144036343181t_real,K))) = aa_nat_int(semiri1314217659103216013at_int,Na) ) ) ) ) ).

% floor_log_nat_eq_if
tff(fact_9220_Maclaurin__sin__bound,axiom,
    ! [Xb: real,Na: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(sin_real,Xb)),aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_vz,Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Na))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,Xb)),Na))) ).

% Maclaurin_sin_bound
tff(fact_9221_cot__less__zero,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,pi)),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_9222_inverse__powr,axiom,
    ! [Y: real,Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(inverse_inverse_real,Y)),Aa) = aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Y),Aa)) ) ) ).

% inverse_powr
tff(fact_9223_forall__pos__mono__1,axiom,
    ! [P: fun_real_o,E2: real] :
      ( ! [D4: real,E: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,D4),E)
         => ( aa_real_o(P,D4)
           => 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_9224_forall__pos__mono,axiom,
    ! [P: fun_real_o,E2: real] :
      ( ! [D4: real,E: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,D4),E)
         => ( aa_real_o(P,D4)
           => 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_9225_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_9226_sqrt__divide__self__eq,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sqrt,Xb)),Xb) = aa_real_real(inverse_inverse_real,aa_real_real(sqrt,Xb)) ) ) ).

% sqrt_divide_self_eq
tff(fact_9227_ln__inverse,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(inverse_inverse_real,Xb)) = aa_real_real(uminus_uminus_real,aa_real_real(ln_ln_real,Xb)) ) ) ).

% ln_inverse
tff(fact_9228_log__inverse,axiom,
    ! [Aa: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Aa)
     => ( ( Aa != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
         => ( aa_real_real(aa_rea7340605660300375871l_real(log,Aa),aa_real_real(inverse_inverse_real,Xb)) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(log,Aa),Xb)) ) ) ) ) ).

% log_inverse
tff(fact_9229_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(aa_rea7340605660300375871l_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_9230_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(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(inverse_inverse_real,Xb))) ) ).

% plus_inverse_ge_2
tff(fact_9231_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_9232_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(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_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_9233_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(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_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_9234_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(aa_rea7340605660300375871l_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_9235_int__ge__less__than__def,axiom,
    ! [D: int] : int_ge_less_than(D) = collec213857154873943460nt_int(produc4947309494688390418_int_o(aTP_Lamm_xn(D))) ).

% int_ge_less_than_def
tff(fact_9236_int__ge__less__than2__def,axiom,
    ! [D: int] : int_ge_less_than2(D) = collec213857154873943460nt_int(produc4947309494688390418_int_o(aTP_Lamm_xo(D))) ).

% int_ge_less_than2_def
tff(fact_9237_sinh__real__less__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sinh_real,Xb)),aa_real_real(sinh_real,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y) ) ).

% sinh_real_less_iff
tff(fact_9238_sinh__real__le__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sinh_real,Xb)),aa_real_real(sinh_real,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y) ) ).

% sinh_real_le_iff
tff(fact_9239_sinh__real__neg__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sinh_real,Xb)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real) ) ).

% sinh_real_neg_iff
tff(fact_9240_sinh__real__pos__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(sinh_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb) ) ).

% sinh_real_pos_iff
tff(fact_9241_sinh__real__nonpos__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sinh_real,Xb)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real) ) ).

% sinh_real_nonpos_iff
tff(fact_9242_sinh__real__nonneg__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(sinh_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb) ) ).

% sinh_real_nonneg_iff
tff(fact_9243_sinh__le__cosh__real,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sinh_real,Xb)),aa_real_real(cosh_real,Xb)) ).

% sinh_le_cosh_real
tff(fact_9244_sinh__less__cosh__real,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sinh_real,Xb)),aa_real_real(cosh_real,Xb)) ).

% sinh_less_cosh_real
tff(fact_9245_cosh__real__pos,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(cosh_real,Xb)) ).

% cosh_real_pos
tff(fact_9246_cosh__real__nonneg,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(cosh_real,Xb)) ).

% cosh_real_nonneg
tff(fact_9247_cosh__real__nonneg__le__iff,axiom,
    ! [Xb: real,Y: 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),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(cosh_real,Xb)),aa_real_real(cosh_real,Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y) ) ) ) ).

% cosh_real_nonneg_le_iff
tff(fact_9248_cosh__real__nonpos__le__iff,axiom,
    ! [Xb: real,Y: 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,Y),zero_zero_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(cosh_real,Xb)),aa_real_real(cosh_real,Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Xb) ) ) ) ).

% cosh_real_nonpos_le_iff
tff(fact_9249_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_9250_cosh__real__nonpos__less__iff,axiom,
    ! [Xb: real,Y: 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,Y),zero_zero_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cosh_real,Xb)),aa_real_real(cosh_real,Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xb) ) ) ) ).

% cosh_real_nonpos_less_iff
tff(fact_9251_cosh__real__nonneg__less__iff,axiom,
    ! [Xb: real,Y: 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),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cosh_real,Xb)),aa_real_real(cosh_real,Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y) ) ) ) ).

% cosh_real_nonneg_less_iff
tff(fact_9252_cosh__real__strict__mono,axiom,
    ! [Xb: real,Y: 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),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cosh_real,Xb)),aa_real_real(cosh_real,Y)) ) ) ).

% cosh_real_strict_mono
tff(fact_9253_arcosh__cosh__real,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_real(arcosh_real,aa_real_real(cosh_real,Xb)) = Xb ) ) ).

% arcosh_cosh_real
tff(fact_9254_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(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_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_9255_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(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_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_9256_arctan__def,axiom,
    ! [Y: real] : aa_real_real(arctan,Y) = the_real(aa_real_fun_real_o(aTP_Lamm_xp,Y)) ).

% arctan_def
tff(fact_9257_arcsin__def,axiom,
    ! [Y: real] : aa_real_real(arcsin,Y) = the_real(aa_real_fun_real_o(aTP_Lamm_xq,Y)) ).

% arcsin_def
tff(fact_9258_modulo__int__unfold,axiom,
    ! [K: int,M: nat,L: int,Na: 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,M))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,Na))) = $ite(
        ( ( aa_int_int(sgn_sgn_int,L) = zero_zero_int )
        | ( aa_int_int(sgn_sgn_int,K) = zero_zero_int )
        | ( Na = zero_zero_nat ) ),
        aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K)),aa_nat_int(semiri1314217659103216013at_int,M)),
        $ite(aa_int_int(sgn_sgn_int,K) = aa_int_int(sgn_sgn_int,L),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Na))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),zero_n2687167440665602831ol_nat(~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Na),M))))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Na))))) ) ).

% modulo_int_unfold
tff(fact_9259_mask__nat__positive__iff,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(bit_se2002935070580805687sk_nat,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na) ) ).

% mask_nat_positive_iff
tff(fact_9260_less__eq__mask,axiom,
    ! [Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),aa_nat_nat(bit_se2002935070580805687sk_nat,Na)) ).

% less_eq_mask
tff(fact_9261_mask__integer_Oabs__eq,axiom,
    ! [Xb: nat] : aa_nat_Code_integer(bit_se2119862282449309892nteger,Xb) = aa_int_Code_integer(code_integer_of_int,aa_nat_int(bit_se2000444600071755411sk_int,Xb)) ).

% mask_integer.abs_eq
tff(fact_9262_mask__nonnegative__int,axiom,
    ! [Na: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(bit_se2000444600071755411sk_int,Na)) ).

% mask_nonnegative_int
tff(fact_9263_not__mask__negative__int,axiom,
    ! [Na: nat] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(bit_se2000444600071755411sk_int,Na)),zero_zero_int) ).

% not_mask_negative_int
tff(fact_9264_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_9265_less__mask,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Na)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(bit_se2002935070580805687sk_nat,Na)) ) ).

% less_mask
tff(fact_9266_ln__neg__is__const,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real)
     => ( aa_real_real(ln_ln_real,Xb) = the_real(aTP_Lamm_bs) ) ) ).

% ln_neg_is_const
tff(fact_9267_zsgn__def,axiom,
    ! [I: int] :
      aa_int_int(sgn_sgn_int,I) = $ite(
        I = zero_zero_int,
        zero_zero_int,
        $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),I),one_one_int,aa_int_int(uminus_uminus_int,one_one_int)) ) ).

% zsgn_def
tff(fact_9268_Suc__mask__eq__exp,axiom,
    ! [Na: nat] : aa_nat_nat(suc,aa_nat_nat(bit_se2002935070580805687sk_nat,Na)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na) ).

% Suc_mask_eq_exp
tff(fact_9269_mask__nat__less__exp,axiom,
    ! [Na: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(bit_se2002935070580805687sk_nat,Na)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)) ).

% mask_nat_less_exp
tff(fact_9270_arccos__def,axiom,
    ! [Y: real] : aa_real_real(arccos,Y) = the_real(aa_real_fun_real_o(aTP_Lamm_xr,Y)) ).

% arccos_def
tff(fact_9271_eucl__rel__int__remainderI,axiom,
    ! [R2: int,L: int,K: int,Q4: 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,Q4),L)),R2) )
         => aa_Pro4592068669816069595_int_o(eucl_rel_int(K,L),product_Pair_int_int(Q4,R2)) ) ) ) ).

% eucl_rel_int_remainderI
tff(fact_9272_mask__nat__def,axiom,
    ! [Na: nat] : aa_nat_nat(bit_se2002935070580805687sk_nat,Na) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)),one_one_nat) ).

% mask_nat_def
tff(fact_9273_mask__half__int,axiom,
    ! [Na: nat] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(bit_se2000444600071755411sk_int,Na)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_nat_int(bit_se2000444600071755411sk_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat)) ).

% mask_half_int
tff(fact_9274_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 = product_Pair_int_int(zero_zero_int,K3) ) )
        | ? [L3: int,K3: int,Q5: int] :
            ( ( A1 = K3 )
            & ( A22 = L3 )
            & ( A32 = product_Pair_int_int(Q5,zero_zero_int) )
            & ( L3 != zero_zero_int )
            & ( K3 = aa_int_int(aa_int_fun_int_int(times_times_int,Q5),L3) ) )
        | ? [R5: int,L3: int,K3: int,Q5: int] :
            ( ( A1 = K3 )
            & ( A22 = L3 )
            & ( A32 = product_Pair_int_int(Q5,R5) )
            & ( aa_int_int(sgn_sgn_int,R5) = aa_int_int(sgn_sgn_int,L3) )
            & aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,R5)),aa_int_int(abs_abs_int,L3))
            & ( K3 = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Q5),L3)),R5) ) ) ) ) ).

% eucl_rel_int.simps
tff(fact_9275_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 != product_Pair_int_int(zero_zero_int,A1) ) )
       => ( ! [Q3: int] :
              ( ( A32 = product_Pair_int_int(Q3,zero_zero_int) )
             => ( ( A22 != zero_zero_int )
               => ( A1 != aa_int_int(aa_int_fun_int_int(times_times_int,Q3),A22) ) ) )
         => ~ ! [R3: int,Q3: int] :
                ( ( A32 = 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_9276_pi__half,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) = the_real(aTP_Lamm_xs) ).

% pi_half
tff(fact_9277_pi__def,axiom,
    pi = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),the_real(aTP_Lamm_xs)) ).

% pi_def
tff(fact_9278_divide__int__unfold,axiom,
    ! [K: int,M: nat,L: int,Na: nat] :
      aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K)),aa_nat_int(semiri1314217659103216013at_int,M))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,Na))) = $ite(
        ( ( aa_int_int(sgn_sgn_int,L) = zero_zero_int )
        | ( aa_int_int(sgn_sgn_int,K) = zero_zero_int )
        | ( Na = zero_zero_nat ) ),
        zero_zero_int,
        $ite(aa_int_int(sgn_sgn_int,K) = aa_int_int(sgn_sgn_int,L),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na)),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na)),zero_n2687167440665602831ol_nat(~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Na),M)))))) ) ).

% divide_int_unfold
tff(fact_9279_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_9280_powr__int,axiom,
    ! [Xb: real,I: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),aa_int_real(ring_1_of_int_real,I)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),I),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_int_nat(nat2,I)),aa_real_real(aa_rea7340605660300375871l_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,I))))) ) ) ).

% powr_int
tff(fact_9281_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_9282_zero__le__sgn__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(sgn_sgn_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb) ) ).

% zero_le_sgn_iff
tff(fact_9283_sgn__le__0__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sgn_sgn_real,Xb)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real) ) ).

% sgn_le_0_iff
tff(fact_9284_abs__division__segment,axiom,
    ! [K: int] : aa_int_int(abs_abs_int,aa_int_int(euclid3395696857347342551nt_int,K)) = one_one_int ).

% abs_division_segment
tff(fact_9285_nat__1,axiom,
    aa_int_nat(nat2,one_one_int) = aa_nat_nat(suc,zero_zero_nat) ).

% nat_1
tff(fact_9286_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_9287_nat__0__iff,axiom,
    ! [I: int] :
      ( ( aa_int_nat(nat2,I) = zero_zero_nat )
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),zero_zero_int) ) ).

% nat_0_iff
tff(fact_9288_zless__nat__conj,axiom,
    ! [W2: int,Z: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(nat2,W2)),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,W2),Z) ) ) ).

% zless_nat_conj
tff(fact_9289_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_9290_nat__zminus__int,axiom,
    ! [Na: nat] : aa_int_nat(nat2,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,Na))) = zero_zero_nat ).

% nat_zminus_int
tff(fact_9291_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_9292_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_9293_nat__ceiling__le__eq,axiom,
    ! [Xb: real,Aa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,aa_real_int(archim7802044766580827645g_real,Xb))),Aa)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_nat_real(semiri5074537144036343181t_real,Aa)) ) ).

% nat_ceiling_le_eq
tff(fact_9294_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_9295_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_9296_nat__less__numeral__power__cancel__iff,axiom,
    ! [Aa: int,Xb: num,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(nat2,Aa)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Na)) ) ).

% nat_less_numeral_power_cancel_iff
tff(fact_9297_numeral__power__less__nat__cancel__iff,axiom,
    ! [Xb: num,Na: nat,Aa: 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)),Na)),aa_int_nat(nat2,Aa))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Na)),Aa) ) ).

% numeral_power_less_nat_cancel_iff
tff(fact_9298_numeral__power__le__nat__cancel__iff,axiom,
    ! [Xb: num,Na: nat,Aa: 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)),Na)),aa_int_nat(nat2,Aa))
    <=> 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)),Na)),Aa) ) ).

% numeral_power_le_nat_cancel_iff
tff(fact_9299_nat__le__numeral__power__cancel__iff,axiom,
    ! [Aa: int,Xb: num,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,Aa)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Na))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Na)) ) ).

% nat_le_numeral_power_cancel_iff
tff(fact_9300_division__segment__integer_Oabs__eq,axiom,
    ! [Xb: int] : aa_Cod1783974652533396728nteger(euclid6289375185220004616nteger,aa_int_Code_integer(code_integer_of_int,Xb)) = aa_int_Code_integer(code_integer_of_int,aa_int_int(euclid3395696857347342551nt_int,Xb)) ).

% division_segment_integer.abs_eq
tff(fact_9301_division__segment__nat__def,axiom,
    ! [Na: nat] : aa_nat_nat(euclid3398187327856392827nt_nat,Na) = one_one_nat ).

% division_segment_nat_def
tff(fact_9302_nat__zero__as__int,axiom,
    zero_zero_nat = aa_int_nat(nat2,zero_zero_int) ).

% nat_zero_as_int
tff(fact_9303_nat__mono,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,Xb)),aa_int_nat(nat2,Y)) ) ).

% nat_mono
tff(fact_9304_eq__nat__nat__iff,axiom,
    ! [Z: int,Z6: 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),Z6)
       => ( ( aa_int_nat(nat2,Z) = aa_int_nat(nat2,Z6) )
        <=> ( Z = Z6 ) ) ) ) ).

% eq_nat_nat_iff
tff(fact_9305_all__nat,axiom,
    ! [P: fun_nat_o] :
      ( ! [X_12: nat] : aa_nat_o(P,X_12)
    <=> ! [X4: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X4)
         => aa_nat_o(P,aa_int_nat(nat2,X4)) ) ) ).

% all_nat
tff(fact_9306_ex__nat,axiom,
    ! [P: fun_nat_o] :
      ( ? [X_12: nat] : aa_nat_o(P,X_12)
    <=> ? [X4: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X4)
          & aa_nat_o(P,aa_int_nat(nat2,X4)) ) ) ).

% ex_nat
tff(fact_9307_nat__one__as__int,axiom,
    one_one_nat = aa_int_nat(nat2,one_one_int) ).

% nat_one_as_int
tff(fact_9308_sgn__integer_Oabs__eq,axiom,
    ! [Xb: int] : aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,aa_int_Code_integer(code_integer_of_int,Xb)) = aa_int_Code_integer(code_integer_of_int,aa_int_int(sgn_sgn_int,Xb)) ).

% sgn_integer.abs_eq
tff(fact_9309_division__segment__eq__sgn,axiom,
    ! [K: int] :
      ( ( K != zero_zero_int )
     => ( aa_int_int(euclid3395696857347342551nt_int,K) = aa_int_int(sgn_sgn_int,K) ) ) ).

% division_segment_eq_sgn
tff(fact_9310_nat__mono__iff,axiom,
    ! [Z: int,W2: 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,W2)),aa_int_nat(nat2,Z))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W2),Z) ) ) ).

% nat_mono_iff
tff(fact_9311_zless__nat__eq__int__zless,axiom,
    ! [M: nat,Z: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_int_nat(nat2,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M)),Z) ) ).

% zless_nat_eq_int_zless
tff(fact_9312_nat__le__iff,axiom,
    ! [Xb: int,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,Xb)),Na)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),aa_nat_int(semiri1314217659103216013at_int,Na)) ) ).

% nat_le_iff
tff(fact_9313_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_9314_int__eq__iff,axiom,
    ! [M: nat,Z: int] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,M) = Z )
    <=> ( ( M = aa_int_nat(nat2,Z) )
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z) ) ) ).

% int_eq_iff
tff(fact_9315_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_9316_sgn__real__def,axiom,
    ! [Aa: real] :
      aa_real_real(sgn_sgn_real,Aa) = $ite(
        Aa = zero_zero_real,
        zero_zero_real,
        $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Aa),one_one_real,aa_real_real(uminus_uminus_real,one_one_real)) ) ).

% sgn_real_def
tff(fact_9317_nat__less__eq__zless,axiom,
    ! [W2: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),W2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(nat2,W2)),aa_int_nat(nat2,Z))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W2),Z) ) ) ).

% nat_less_eq_zless
tff(fact_9318_nat__le__eq__zle,axiom,
    ! [W2: int,Z: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),W2)
        | 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,W2)),aa_int_nat(nat2,Z))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W2),Z) ) ) ).

% nat_le_eq_zle
tff(fact_9319_nat__eq__iff2,axiom,
    ! [M: nat,W2: int] :
      ( ( M = aa_int_nat(nat2,W2) )
    <=> $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),W2),W2 = aa_nat_int(semiri1314217659103216013at_int,M),M = zero_zero_nat) ) ).

% nat_eq_iff2
tff(fact_9320_nat__eq__iff,axiom,
    ! [W2: int,M: nat] :
      ( ( aa_int_nat(nat2,W2) = M )
    <=> $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),W2),W2 = aa_nat_int(semiri1314217659103216013at_int,M),M = zero_zero_nat) ) ).

% nat_eq_iff
tff(fact_9321_split__nat,axiom,
    ! [P: fun_nat_o,I: int] :
      ( aa_nat_o(P,aa_int_nat(nat2,I))
    <=> ( ! [N4: nat] :
            ( ( I = aa_nat_int(semiri1314217659103216013at_int,N4) )
           => aa_nat_o(P,N4) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,I),zero_zero_int)
         => aa_nat_o(P,zero_zero_nat) ) ) ) ).

% split_nat
tff(fact_9322_le__nat__iff,axiom,
    ! [K: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),aa_int_nat(nat2,K))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,Na)),K) ) ) ).

% le_nat_iff
tff(fact_9323_nat__add__distrib,axiom,
    ! [Z: int,Z6: 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),Z6)
       => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),Z6)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z6)) ) ) ) ).

% nat_add_distrib
tff(fact_9324_nat__mult__distrib,axiom,
    ! [Z: int,Z6: 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),Z6)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z6)) ) ) ).

% nat_mult_distrib
tff(fact_9325_Suc__as__int,axiom,
    ! [X3: nat] : aa_nat_nat(suc,X3) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,X3)),one_one_int)) ).

% Suc_as_int
tff(fact_9326_nat__diff__distrib_H,axiom,
    ! [Xb: int,Y: 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),Y)
       => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(nat2,Xb)),aa_int_nat(nat2,Y)) ) ) ) ).

% nat_diff_distrib'
tff(fact_9327_nat__diff__distrib,axiom,
    ! [Z6: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z6)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z6),Z)
       => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,Z),Z6)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z6)) ) ) ) ).

% nat_diff_distrib
tff(fact_9328_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_9329_nat__div__distrib,axiom,
    ! [Xb: int,Y: 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),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(nat2,Xb)),aa_int_nat(nat2,Y)) ) ) ).

% nat_div_distrib
tff(fact_9330_nat__div__distrib_H,axiom,
    ! [Y: int,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
     => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(divide_divide_int,Xb),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(nat2,Xb)),aa_int_nat(nat2,Y)) ) ) ).

% nat_div_distrib'
tff(fact_9331_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_9332_nat__power__eq,axiom,
    ! [Z: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => ( aa_int_nat(nat2,aa_nat_int(power_power_int(Z),Na)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_int_nat(nat2,Z)),Na) ) ) ).

% nat_power_eq
tff(fact_9333_nat__mod__distrib,axiom,
    ! [Xb: int,Y: 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),Y)
       => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Xb),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_int_nat(nat2,Xb)),aa_int_nat(nat2,Y)) ) ) ) ).

% nat_mod_distrib
tff(fact_9334_floor__eq3,axiom,
    ! [Na: nat,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Na)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Na)))
       => ( aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,Xb)) = Na ) ) ) ).

% floor_eq3
tff(fact_9335_le__nat__floor,axiom,
    ! [Xb: nat,Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Xb)),Aa)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,Aa))) ) ).

% le_nat_floor
tff(fact_9336_take__bit__nat__eq,axiom,
    ! [K: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),aa_int_nat(nat2,K)) = aa_int_nat(nat2,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Na),K)) ) ) ).

% take_bit_nat_eq
tff(fact_9337_nat__take__bit__eq,axiom,
    ! [K: int,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
     => ( aa_int_nat(nat2,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Na),K)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),aa_int_nat(nat2,K)) ) ) ).

% nat_take_bit_eq
tff(fact_9338_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_9339_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_9340_sgn__power__injE,axiom,
    ! [Aa: real,Na: nat,Xb: real,Ba: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,Aa)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,Aa)),Na)) = Xb )
     => ( ( Xb = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,Ba)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,Ba)),Na)) )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
         => ( Aa = Ba ) ) ) ) ).

% sgn_power_injE
tff(fact_9341_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_9342_nat__less__iff,axiom,
    ! [W2: int,M: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),W2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(nat2,W2)),M)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W2),aa_nat_int(semiri1314217659103216013at_int,M)) ) ) ).

% nat_less_iff
tff(fact_9343_nat__mult__distrib__neg,axiom,
    ! [Z: int,Z6: 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),Z6)) = 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,Z6))) ) ) ).

% nat_mult_distrib_neg
tff(fact_9344_nat__abs__int__diff,axiom,
    ! [Aa: nat,Ba: 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,Aa)),aa_nat_int(semiri1314217659103216013at_int,Ba)))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Aa),Ba),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ba),Aa),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Aa),Ba)) ).

% nat_abs_int_diff
tff(fact_9345_division__segment__int__def,axiom,
    ! [K: int] :
      aa_int_int(euclid3395696857347342551nt_int,K) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K),one_one_int,aa_int_int(uminus_uminus_int,one_one_int)) ).

% division_segment_int_def
tff(fact_9346_floor__eq4,axiom,
    ! [Na: nat,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Na)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Na)))
       => ( aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,Xb)) = Na ) ) ) ).

% floor_eq4
tff(fact_9347_diff__nat__eq__if,axiom,
    ! [Z: int,Z6: int] :
      aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z6)) = $ite(
        aa_int_o(aa_int_fun_int_o(ord_less_int,Z6),zero_zero_int),
        aa_int_nat(nat2,Z),
        $let(
          d: int,
          d:= aa_int_int(aa_int_fun_int_int(minus_minus_int,Z),Z6),
          $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_9348_nat__dvd__iff,axiom,
    ! [Z: int,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_int_nat(nat2,Z)),M)
    <=> $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z),aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Z),aa_nat_int(semiri1314217659103216013at_int,M)),M = zero_zero_nat) ) ).

% nat_dvd_iff
tff(fact_9349_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_9350_powr__real__of__int,axiom,
    ! [Xb: real,Na: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),aa_int_real(ring_1_of_int_real,Na)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Na),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_int_nat(nat2,Na)),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,Na))))) ) ) ).

% powr_real_of_int
tff(fact_9351_signed__take__bit__eq__take__bit__minus,axiom,
    ! [Na: nat,K: int] : aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,Na),K) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_nat_nat(suc,Na)),K)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,Na))),aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),Na)))) ).

% signed_take_bit_eq_take_bit_minus
tff(fact_9352_set__encode__insert,axiom,
    ! [A3: set_nat,Na: nat] :
      ( aa_set_nat_o(finite_finite_nat,A3)
     => ( ~ aa_set_nat_o(member_nat(Na),A3)
       => ( aa_set_nat_nat(nat_set_encode,insert_nat(Na,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))),Na)),aa_set_nat_nat(nat_set_encode,A3)) ) ) ) ).

% set_encode_insert
tff(fact_9353_take__bit__Suc__from__most,axiom,
    ! [Na: nat,K: int] : aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,aa_nat_nat(suc,Na)),K) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)),aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),Na)))),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Na),K)) ).

% take_bit_Suc_from_most
tff(fact_9354_xor__Suc__0__eq,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,Na),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),zero_n2687167440665602831ol_nat(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)))),zero_n2687167440665602831ol_nat(~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))) ).

% xor_Suc_0_eq
tff(fact_9355_signed__take__bit__nonnegative__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,Na),K))
    <=> ~ aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),Na) ) ).

% signed_take_bit_nonnegative_iff
tff(fact_9356_signed__take__bit__negative__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_nat_fun_int_int(bit_ri631733984087533419it_int,Na),K)),zero_zero_int)
    <=> aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),Na) ) ).

% signed_take_bit_negative_iff
tff(fact_9357_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_9358_bit__minus__numeral__Bit0__Suc__iff,axiom,
    ! [W2: num,Na: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,W2)))),aa_nat_nat(suc,Na))
    <=> aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W2))),Na) ) ).

% bit_minus_numeral_Bit0_Suc_iff
tff(fact_9359_bit__minus__numeral__Bit1__Suc__iff,axiom,
    ! [W2: num,Na: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,W2)))),aa_nat_nat(suc,Na))
    <=> ~ aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_num_int(numeral_numeral_int,W2)),Na) ) ).

% bit_minus_numeral_Bit1_Suc_iff
tff(fact_9360_xor__nat__numerals_I1_J,axiom,
    ! [Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Y))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y)) ).

% xor_nat_numerals(1)
tff(fact_9361_xor__nat__numerals_I2_J,axiom,
    ! [Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Y)) ).

% xor_nat_numerals(2)
tff(fact_9362_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_9363_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_9364_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_9365_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_9366_atLeast0__atMost__Suc,axiom,
    ! [Na: nat] : set_or1269000886237332187st_nat(zero_zero_nat,aa_nat_nat(suc,Na)) = insert_nat(aa_nat_nat(suc,Na),set_or1269000886237332187st_nat(zero_zero_nat,Na)) ).

% atLeast0_atMost_Suc
tff(fact_9367_atLeastAtMost__insertL,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( insert_nat(M,set_or1269000886237332187st_nat(aa_nat_nat(suc,M),Na)) = set_or1269000886237332187st_nat(M,Na) ) ) ).

% atLeastAtMost_insertL
tff(fact_9368_atLeastAtMostSuc__conv,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,Na))
     => ( set_or1269000886237332187st_nat(M,aa_nat_nat(suc,Na)) = insert_nat(aa_nat_nat(suc,Na),set_or1269000886237332187st_nat(M,Na)) ) ) ).

% atLeastAtMostSuc_conv
tff(fact_9369_Icc__eq__insert__lb__nat,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( set_or1269000886237332187st_nat(M,Na) = insert_nat(M,set_or1269000886237332187st_nat(aa_nat_nat(suc,M),Na)) ) ) ).

% Icc_eq_insert_lb_nat
tff(fact_9370_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_9371_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_9372_bit__imp__take__bit__positive,axiom,
    ! [Na: nat,M: nat,K: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M)
     => ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),Na)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,M),K)) ) ) ).

% bit_imp_take_bit_positive
tff(fact_9373_bit__concat__bit__iff,axiom,
    ! [M: nat,K: int,L: int,Na: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_int_int(bit_concat_bit(M,K),L)),Na)
    <=> ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),M)
          & aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),Na) )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
          & aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,L),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),M)) ) ) ) ).

% bit_concat_bit_iff
tff(fact_9374_int__bit__bound,axiom,
    ! [K: int] :
      ~ ! [N: nat] :
          ( ! [M2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M2)
             => ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),M2)
              <=> aa_nat_o(aa_int_fun_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(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))
              <=> ~ aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),N) ) ) ) ).

% int_bit_bound
tff(fact_9375_atLeast1__atMost__eq__remove0,axiom,
    ! [Na: nat] : set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat),Na) = minus_minus_set_nat(aa_nat_set_nat(set_ord_atMost_nat,Na),insert_nat(zero_zero_nat,bot_bot_set_nat)) ).

% atLeast1_atMost_eq_remove0
tff(fact_9376_xor__nat__unfold,axiom,
    ! [M: nat,Na: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,M),Na) = $ite(
        M = zero_zero_nat,
        Na,
        $ite(Na = zero_zero_nat,M,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ) ).

% xor_nat_unfold
tff(fact_9377_Suc__0__xor__eq,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(suc,zero_zero_nat)),Na) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),zero_n2687167440665602831ol_nat(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)))),zero_n2687167440665602831ol_nat(~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))) ).

% Suc_0_xor_eq
tff(fact_9378_upto__aux__rec,axiom,
    ! [I: int,J: int,Js: list_int] :
      upto_aux(I,J,Js) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,J),I),Js,upto_aux(I,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int),cons_int(J,Js))) ).

% upto_aux_rec
tff(fact_9379_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_9380_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_9381_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_9382_xor__integer_Oabs__eq,axiom,
    ! [Xaa: int,Xb: int] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3222712562003087583nteger,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(bit_se6526347334894502574or_int,Xaa),Xb)) ).

% xor_integer.abs_eq
tff(fact_9383_not__bit__Suc__0__Suc,axiom,
    ! [Na: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(suc,Na)) ).

% not_bit_Suc_0_Suc
tff(fact_9384_bit__Suc__0__iff,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(suc,zero_zero_nat)),Na)
    <=> ( Na = zero_zero_nat ) ) ).

% bit_Suc_0_iff
tff(fact_9385_XOR__lower,axiom,
    ! [Xb: int,Y: 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),Y)
       => 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),Y)) ) ) ).

% XOR_lower
tff(fact_9386_bit__integer_Oabs__eq,axiom,
    ! [Xb: int] : aa_Cod1075288278386115654_nat_o(bit_se9216721137139052372nteger,aa_int_Code_integer(code_integer_of_int,Xb)) = aa_int_fun_nat_o(bit_se1146084159140164899it_int,Xb) ).

% bit_integer.abs_eq
tff(fact_9387_not__bit__Suc__0__numeral,axiom,
    ! [Na: num] : ~ aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,Na)) ).

% not_bit_Suc_0_numeral
tff(fact_9388_bit__nat__iff,axiom,
    ! [K: int,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_int_nat(nat2,K)),Na)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
        & aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),Na) ) ) ).

% bit_nat_iff
tff(fact_9389_atLeastAtMostPlus1__int__conv,axiom,
    ! [M: int,Na: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,M),aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),Na))
     => ( set_or1266510415728281911st_int(M,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),Na)) = insert_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),Na),set_or1266510415728281911st_int(M,Na)) ) ) ).

% atLeastAtMostPlus1_int_conv
tff(fact_9390_simp__from__to,axiom,
    ! [I: int,J: int] :
      set_or1266510415728281911st_int(I,J) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,J),I),bot_bot_set_int,insert_int(I,set_or1266510415728281911st_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,I),one_one_int),J))) ).

% simp_from_to
tff(fact_9391_XOR__upper,axiom,
    ! [Xb: int,Na: nat,Y: 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))),Na))
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na))
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,Xb),Y)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)) ) ) ) ).

% XOR_upper
tff(fact_9392_floor__real__def,axiom,
    ! [Xb: real] : aa_real_int(archim6058952711729229775r_real,Xb) = the_int(aa_real_fun_int_o(aTP_Lamm_xt,Xb)) ).

% floor_real_def
tff(fact_9393_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),product_Pair_int_int(A0,A1))
     => ( ! [K2: int,L4: int] :
            ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(bit_and_int_rel),product_Pair_int_int(K2,L4))
           => ( ( ~ ( aa_set_int_o(member_int(K2),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
                    & aa_set_int_o(member_int(L4),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,L4),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) )
             => aa_int_o(aa_int_fun_int_o(P,K2),L4) ) )
       => aa_int_o(aa_int_fun_int_o(P,A0),A1) ) ) ).

% and_int.pinduct
tff(fact_9394_and__int_Osimps,axiom,
    ! [K: int,L: int] :
      aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K),L) = $ite(
        ( aa_set_int_o(member_int(K),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
        & aa_set_int_o(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_9395_and__int_Oelims,axiom,
    ! [Xb: int,Xaa: int,Y: int] :
      ( ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Xb),Xaa) = Y )
     => ( Y = $ite(
            ( aa_set_int_o(member_int(Xb),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
            & aa_set_int_o(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_9396_and__int_Opsimps,axiom,
    ! [K: int,L: int] :
      ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(bit_and_int_rel),product_Pair_int_int(K,L))
     => ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K),L) = $ite(
            ( aa_set_int_o(member_int(K),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
            & aa_set_int_o(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_9397_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_9398_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_9399_AND__upper2_H,axiom,
    ! [Y: int,Z: int,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Z)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Xb),Y)),Z) ) ) ).

% AND_upper2'
tff(fact_9400_AND__upper1_H,axiom,
    ! [Y: int,Z: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Z)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Y),Ya)),Z) ) ) ).

% AND_upper1'
tff(fact_9401_AND__upper2,axiom,
    ! [Y: int,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Xb),Y)),Y) ) ).

% AND_upper2
tff(fact_9402_AND__upper1,axiom,
    ! [Xb: int,Y: 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),Y)),Xb) ) ).

% AND_upper1
tff(fact_9403_AND__lower,axiom,
    ! [Xb: int,Y: 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),Y)) ) ).

% AND_lower
tff(fact_9404_AND__upper2_H_H,axiom,
    ! [Y: int,Z: int,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Z)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Xb),Y)),Z) ) ) ).

% AND_upper2''
tff(fact_9405_AND__upper1_H_H,axiom,
    ! [Y: int,Z: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Z)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Y),Ya)),Z) ) ) ).

% AND_upper1''
tff(fact_9406_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_9407_and__int_Opelims,axiom,
    ! [Xb: int,Xaa: int,Y: int] :
      ( ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,Xb),Xaa) = Y )
     => ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(bit_and_int_rel),product_Pair_int_int(Xb,Xaa))
       => ~ ( ( Y = $ite(
                  ( aa_set_int_o(member_int(Xb),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
                  & aa_set_int_o(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),product_Pair_int_int(Xb,Xaa)) ) ) ) ).

% and_int.pelims
tff(fact_9408_floor__rat__def,axiom,
    ! [Xb: rat] : aa_rat_int(archim3151403230148437115or_rat,Xb) = the_int(aTP_Lamm_xu(Xb)) ).

% floor_rat_def
tff(fact_9409_or__nat__unfold,axiom,
    ! [M: nat,Na: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,M),Na) = $ite(
        M = zero_zero_nat,
        Na,
        $ite(Na = zero_zero_nat,M,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ) ).

% or_nat_unfold
tff(fact_9410_push__bit__nonnegative__int__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,Na),K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K) ) ).

% push_bit_nonnegative_int_iff
tff(fact_9411_push__bit__negative__int__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,Na),K)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int) ) ).

% push_bit_negative_int_iff
tff(fact_9412_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_9413_and__nat__numerals_I1_J,axiom,
    ! [Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Y))) = zero_zero_nat ).

% and_nat_numerals(1)
tff(fact_9414_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_9415_and__nat__numerals_I2_J,axiom,
    ! [Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y))) = one_one_nat ).

% and_nat_numerals(2)
tff(fact_9416_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_9417_or__nat__numerals_I2_J,axiom,
    ! [Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y)) ).

% or_nat_numerals(2)
tff(fact_9418_push__bit__of__Suc__0,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Na),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na) ).

% push_bit_of_Suc_0
tff(fact_9419_and__Suc__0__eq,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,Na),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% and_Suc_0_eq
tff(fact_9420_Suc__0__and__eq,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(suc,zero_zero_nat)),Na) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% Suc_0_and_eq
tff(fact_9421_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_9422_or__nat__numerals_I1_J,axiom,
    ! [Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Y))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y)) ).

% or_nat_numerals(1)
tff(fact_9423_sgn__rat__def,axiom,
    ! [Aa: rat] :
      aa_rat_rat(sgn_sgn_rat,Aa) = $ite(
        Aa = zero_zero_rat,
        zero_zero_rat,
        $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Aa),one_one_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)) ) ).

% sgn_rat_def
tff(fact_9424_abs__rat__def,axiom,
    ! [Aa: rat] :
      aa_rat_rat(abs_abs_rat,Aa) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Aa),zero_zero_rat),aa_rat_rat(uminus_uminus_rat,Aa),Aa) ).

% abs_rat_def
tff(fact_9425_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)
           => ! [T4: rat] :
                ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),T4)
               => ( R2 != aa_rat_rat(plus_plus_rat(S),T4) ) ) ) ) ).

% obtain_pos_sum
tff(fact_9426_push__bit__integer_Oabs__eq,axiom,
    ! [Xaa: nat,Xb: int] : aa_Cod1783974652533396728nteger(aa_nat6383147911832638789nteger(bit_se7788150548672797655nteger,Xaa),aa_int_Code_integer(code_integer_of_int,Xb)) = aa_int_Code_integer(code_integer_of_int,aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,Xaa),Xb)) ).

% push_bit_integer.abs_eq
tff(fact_9427_less__eq__rat__def,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
        | ( Xb = Y ) ) ) ).

% less_eq_rat_def
tff(fact_9428_set__bit__nat__def,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,M),Na) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,M),one_one_nat)) ).

% set_bit_nat_def
tff(fact_9429_and__integer_Oabs__eq,axiom,
    ! [Xaa: int,Xb: int] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3949692690581998587nteger,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(bit_se725231765392027082nd_int,Xaa),Xb)) ).

% and_integer.abs_eq
tff(fact_9430_flip__bit__nat__def,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,M),Na) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,M),one_one_nat)) ).

% flip_bit_nat_def
tff(fact_9431_bit__push__bit__iff__int,axiom,
    ! [M: nat,K: int,Na: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,M),K)),Na)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
        & aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),M)) ) ) ).

% bit_push_bit_iff_int
tff(fact_9432_bit__push__bit__iff__nat,axiom,
    ! [M: nat,Q4: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,M),Q4)),Na)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
        & aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,Q4),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),M)) ) ) ).

% bit_push_bit_iff_nat
tff(fact_9433_and__nat__unfold,axiom,
    ! [M: nat,Na: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,M),Na) = $ite(
        ( ( M = zero_zero_nat )
        | ( Na = zero_zero_nat ) ),
        zero_zero_nat,
        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ).

% and_nat_unfold
tff(fact_9434_or__Suc__0__eq,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,Na),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),zero_n2687167440665602831ol_nat(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))) ).

% or_Suc_0_eq
tff(fact_9435_Suc__0__or__eq,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(suc,zero_zero_nat)),Na) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),zero_n2687167440665602831ol_nat(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na))) ).

% Suc_0_or_eq
tff(fact_9436_normalize__negative,axiom,
    ! [Q4: int,P3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Q4),zero_zero_int)
     => ( normalize(product_Pair_int_int(P3,Q4)) = normalize(product_Pair_int_int(aa_int_int(uminus_uminus_int,P3),aa_int_int(uminus_uminus_int,Q4))) ) ) ).

% normalize_negative
tff(fact_9437_Sum__Ico__nat,axiom,
    ! [M: nat,Na: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_cc),set_or4665077453230672383an_nat(M,Na)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% Sum_Ico_nat
tff(fact_9438_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_9439_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_9440_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_9441_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_9442_finite__atLeastLessThan,axiom,
    ! [L: nat,U: nat] : aa_set_nat_o(finite_finite_nat,set_or4665077453230672383an_nat(L,U)) ).

% finite_atLeastLessThan
tff(fact_9443_atLeastLessThan__singleton,axiom,
    ! [M: nat] : set_or4665077453230672383an_nat(M,aa_nat_nat(suc,M)) = insert_nat(M,bot_bot_set_nat) ).

% atLeastLessThan_singleton
tff(fact_9444_or__integer_Oabs__eq,axiom,
    ! [Xaa: int,Xb: int] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se1080825931792720795nteger,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(bit_se1409905431419307370or_int,Xaa),Xb)) ).

% or_integer.abs_eq
tff(fact_9445_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_9446_OR__lower,axiom,
    ! [Xb: int,Y: 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),Y)
       => 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),Y)) ) ) ).

% OR_lower
tff(fact_9447_ex__nat__less__eq,axiom,
    ! [Na: nat,P: fun_nat_o] :
      ( ? [M6: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M6),Na)
          & aa_nat_o(P,M6) )
    <=> ? [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),set_or4665077453230672383an_nat(zero_zero_nat,Na))
          & aa_nat_o(P,X4) ) ) ).

% ex_nat_less_eq
tff(fact_9448_all__nat__less__eq,axiom,
    ! [Na: nat,P: fun_nat_o] :
      ( ! [M6: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M6),Na)
         => aa_nat_o(P,M6) )
    <=> ! [X4: nat] :
          ( aa_set_nat_o(member_nat(X4),set_or4665077453230672383an_nat(zero_zero_nat,Na))
         => aa_nat_o(P,X4) ) ) ).

% all_nat_less_eq
tff(fact_9449_atLeastLessThanSuc__atLeastAtMost,axiom,
    ! [L: nat,U: nat] : set_or4665077453230672383an_nat(L,aa_nat_nat(suc,U)) = set_or1269000886237332187st_nat(L,U) ).

% atLeastLessThanSuc_atLeastAtMost
tff(fact_9450_lessThan__atLeast0,axiom,
    ! [Na: nat] : aa_nat_set_nat(set_ord_lessThan_nat,Na) = set_or4665077453230672383an_nat(zero_zero_nat,Na) ).

% lessThan_atLeast0
tff(fact_9451_atLeastLessThan0,axiom,
    ! [M: nat] : set_or4665077453230672383an_nat(M,zero_zero_nat) = bot_bot_set_nat ).

% atLeastLessThan0
tff(fact_9452_atLeast0__lessThan__Suc,axiom,
    ! [Na: nat] : set_or4665077453230672383an_nat(zero_zero_nat,aa_nat_nat(suc,Na)) = insert_nat(Na,set_or4665077453230672383an_nat(zero_zero_nat,Na)) ).

% atLeast0_lessThan_Suc
tff(fact_9453_subset__eq__atLeast0__lessThan__finite,axiom,
    ! [N3: set_nat,Na: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,N3),set_or4665077453230672383an_nat(zero_zero_nat,Na))
     => aa_set_nat_o(finite_finite_nat,N3) ) ).

% subset_eq_atLeast0_lessThan_finite
tff(fact_9454_atLeastLessThanSuc,axiom,
    ! [M: nat,Na: nat] :
      set_or4665077453230672383an_nat(M,aa_nat_nat(suc,Na)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na),insert_nat(Na,set_or4665077453230672383an_nat(M,Na)),bot_bot_set_nat) ).

% atLeastLessThanSuc
tff(fact_9455_prod__Suc__Suc__fact,axiom,
    ! [Na: nat] : aa_set_nat_nat(groups708209901874060359at_nat(suc),set_or4665077453230672383an_nat(aa_nat_nat(suc,zero_zero_nat),Na)) = aa_nat_nat(semiri1408675320244567234ct_nat,Na) ).

% prod_Suc_Suc_fact
tff(fact_9456_prod__Suc__fact,axiom,
    ! [Na: nat] : aa_set_nat_nat(groups708209901874060359at_nat(suc),set_or4665077453230672383an_nat(zero_zero_nat,Na)) = aa_nat_nat(semiri1408675320244567234ct_nat,Na) ).

% prod_Suc_fact
tff(fact_9457_normalize__denom__pos,axiom,
    ! [R2: product_prod_int_int,P3: int,Q4: int] :
      ( ( normalize(R2) = product_Pair_int_int(P3,Q4) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Q4) ) ).

% normalize_denom_pos
tff(fact_9458_atLeastLessThan__nat__numeral,axiom,
    ! [M: nat,K: num] :
      set_or4665077453230672383an_nat(M,aa_num_nat(numeral_numeral_nat,K)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_num_nat(pred_numeral,K)),insert_nat(aa_num_nat(pred_numeral,K),set_or4665077453230672383an_nat(M,aa_num_nat(pred_numeral,K))),bot_bot_set_nat) ).

% atLeastLessThan_nat_numeral
tff(fact_9459_OR__upper,axiom,
    ! [Xb: int,Na: nat,Y: 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))),Na))
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na))
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,Xb),Y)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Na)) ) ) ) ).

% OR_upper
tff(fact_9460_atLeast1__lessThan__eq__remove0,axiom,
    ! [Na: nat] : set_or4665077453230672383an_nat(aa_nat_nat(suc,zero_zero_nat),Na) = minus_minus_set_nat(aa_nat_set_nat(set_ord_lessThan_nat,Na),insert_nat(zero_zero_nat,bot_bot_set_nat)) ).

% atLeast1_lessThan_eq_remove0
tff(fact_9461_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)))),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_9462_Chebyshev__sum__upper__nat,axiom,
    ! [Na: nat,Aa: fun_nat_nat,Ba: fun_nat_nat] :
      ( ! [I2: nat,J2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J2)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J2),Na)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(Aa,I2)),aa_nat_nat(Aa,J2)) ) )
     => ( ! [I2: nat,J2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J2)
           => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J2),Na)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(Ba,J2)),aa_nat_nat(Ba,I2)) ) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_xv(Aa),Ba)),set_or4665077453230672383an_nat(zero_zero_nat,Na)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups3542108847815614940at_nat(Aa),set_or4665077453230672383an_nat(zero_zero_nat,Na))),aa_set_nat_nat(groups3542108847815614940at_nat(Ba),set_or4665077453230672383an_nat(zero_zero_nat,Na)))) ) ) ).

% Chebyshev_sum_upper_nat
tff(fact_9463_Cauchy__iff2,axiom,
    ! [X5: fun_nat_real] :
      ( topolo4055970368930404560y_real(X5)
    <=> ! [J3: nat] :
        ? [M9: nat] :
        ! [M6: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M9),M6)
         => ! [N4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M9),N4)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(X5,M6)),aa_nat_real(X5,N4)))),aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,J3)))) ) ) ) ).

% Cauchy_iff2
tff(fact_9464_finite__atLeastLessThan__int,axiom,
    ! [L: int,U: int] : aa_set_int_o(finite_finite_int,set_or4662586982721622107an_int(L,U)) ).

% finite_atLeastLessThan_int
tff(fact_9465_finite__atLeastZeroLessThan__int,axiom,
    ! [U: int] : aa_set_int_o(finite_finite_int,set_or4662586982721622107an_int(zero_zero_int,U)) ).

% finite_atLeastZeroLessThan_int
tff(fact_9466_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_9467_integer__of__num_I3_J,axiom,
    ! [Na: num] :
      aa_num_Code_integer(code_integer_of_num,aa_num_num(bit1,Na)) = $let(
        k: code_integer,
        k:= aa_num_Code_integer(code_integer_of_num,Na),
        aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,k),k)),one_one_Code_integer) ) ).

% integer_of_num(3)
tff(fact_9468_bit__cut__integer__def,axiom,
    ! [K: code_integer] : code_bit_cut_integer(K) = produc6677183202524767010eger_o(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,K),aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),K)) ).

% bit_cut_integer_def
tff(fact_9469_divmod__integer__def,axiom,
    ! [K: code_integer,L: code_integer] : code_divmod_integer(K,L) = produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,K),L),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,K),L)) ).

% divmod_integer_def
tff(fact_9470_integer__of__num__def,axiom,
    code_integer_of_num = numera6620942414471956472nteger ).

% integer_of_num_def
tff(fact_9471_integer__of__num__triv_I1_J,axiom,
    aa_num_Code_integer(code_integer_of_num,one) = one_one_Code_integer ).

% integer_of_num_triv(1)
tff(fact_9472_integer__of__num_I2_J,axiom,
    ! [Na: num] :
      aa_num_Code_integer(code_integer_of_num,aa_num_num(bit0,Na)) = $let(
        k: code_integer,
        k:= aa_num_Code_integer(code_integer_of_num,Na),
        aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,k),k) ) ).

% integer_of_num(2)
tff(fact_9473_integer__of__num__triv_I2_J,axiom,
    aa_num_Code_integer(code_integer_of_num,aa_num_num(bit0,one)) = aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one)) ).

% integer_of_num_triv(2)
tff(fact_9474_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_xw(K),code_divmod_abs(K,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))))) ).

% bit_cut_integer_code
tff(fact_9475_divmod__abs__def,axiom,
    ! [K: code_integer,L: code_integer] : code_divmod_abs(K,L) = produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,K)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,L)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,K)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,L))) ).

% divmod_abs_def
tff(fact_9476_card__lessThan,axiom,
    ! [U: nat] : aa_set_nat_nat(finite_card_nat,aa_nat_set_nat(set_ord_lessThan_nat,U)) = U ).

% card_lessThan
tff(fact_9477_card__Collect__less__nat,axiom,
    ! [Na: nat] : aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_au,Na))) = Na ).

% card_Collect_less_nat
tff(fact_9478_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_9479_card__atLeastLessThan,axiom,
    ! [L: nat,U: nat] : aa_set_nat_nat(finite_card_nat,set_or4665077453230672383an_nat(L,U)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,U),L) ).

% card_atLeastLessThan
tff(fact_9480_card__Collect__le__nat,axiom,
    ! [Na: nat] : aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_av,Na))) = aa_nat_nat(suc,Na) ).

% card_Collect_le_nat
tff(fact_9481_card__atLeastAtMost,axiom,
    ! [L: nat,U: nat] : aa_set_nat_nat(finite_card_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_9482_card__atLeastLessThan__int,axiom,
    ! [L: int,U: int] : finite_card_int(set_or4662586982721622107an_int(L,U)) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,U),L)) ).

% card_atLeastLessThan_int
tff(fact_9483_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_9484_nat_Odisc__eq__case_I1_J,axiom,
    ! [Nata: nat] :
      ( ( Nata = zero_zero_nat )
    <=> aa_nat_o(case_nat_o($true,aTP_Lamm_bt),Nata) ) ).

% nat.disc_eq_case(1)
tff(fact_9485_nat_Odisc__eq__case_I2_J,axiom,
    ! [Nata: nat] :
      ( ( Nata != zero_zero_nat )
    <=> aa_nat_o(case_nat_o($false,aTP_Lamm_xx),Nata) ) ).

% nat.disc_eq_case(2)
tff(fact_9486_card__less__Suc2,axiom,
    ! [M7: set_nat,I: nat] :
      ( ~ aa_set_nat_o(member_nat(zero_zero_nat),M7)
     => ( aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_xy(M7),I))) = aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_xz(M7),I))) ) ) ).

% card_less_Suc2
tff(fact_9487_card__less__Suc,axiom,
    ! [M7: set_nat,I: nat] :
      ( aa_set_nat_o(member_nat(zero_zero_nat),M7)
     => ( aa_nat_nat(suc,aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_xy(M7),I)))) = aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_xz(M7),I))) ) ) ).

% card_less_Suc
tff(fact_9488_card__less,axiom,
    ! [M7: set_nat,I: nat] :
      ( aa_set_nat_o(member_nat(zero_zero_nat),M7)
     => ( aa_set_nat_nat(finite_card_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_xz(M7),I))) != zero_zero_nat ) ) ).

% card_less
tff(fact_9489_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_9490_subset__card__intvl__is__intvl,axiom,
    ! [A3: set_nat,K: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A3),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 = 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_9491_less__eq__nat_Osimps_I2_J,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,M)),Na)
    <=> aa_nat_o(case_nat_o($false,aa_nat_fun_nat_o(ord_less_eq_nat,M)),Na) ) ).

% less_eq_nat.simps(2)
tff(fact_9492_max__Suc2,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,M),aa_nat_nat(suc,Na)) = aa_nat_nat(case_nat_nat(aa_nat_nat(suc,Na),aa_nat_fun_nat_nat(aTP_Lamm_ya,Na)),M) ).

% max_Suc2
tff(fact_9493_max__Suc1,axiom,
    ! [Na: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(suc,Na)),M) = aa_nat_nat(case_nat_nat(aa_nat_nat(suc,Na),aa_nat_fun_nat_nat(aTP_Lamm_yb,Na)),M) ).

% max_Suc1
tff(fact_9494_divmod__abs__code_I6_J,axiom,
    ! [J: code_integer] : code_divmod_abs(zero_z3403309356797280102nteger,J) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,zero_z3403309356797280102nteger) ).

% divmod_abs_code(6)
tff(fact_9495_subset__eq__atLeast0__lessThan__card,axiom,
    ! [N3: set_nat,Na: nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,N3),set_or4665077453230672383an_nat(zero_zero_nat,Na))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,N3)),Na) ) ).

% subset_eq_atLeast0_lessThan_card
tff(fact_9496_card__sum__le__nat__sum,axiom,
    ! [S2: set_nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_cc),set_or4665077453230672383an_nat(zero_zero_nat,aa_set_nat_nat(finite_card_nat,S2)))),aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_cc),S2)) ).

% card_sum_le_nat_sum
tff(fact_9497_card__nth__roots,axiom,
    ! [C: complex,Na: nat] :
      ( ( C != zero_zero_complex )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => ( finite_card_complex(collect_complex(aa_nat_fun_complex_o(aTP_Lamm_yc(C),Na))) = Na ) ) ) ).

% card_nth_roots
tff(fact_9498_card__roots__unity__eq,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( finite_card_complex(collect_complex(aa_nat_fun_complex_o(aTP_Lamm_cs,Na))) = Na ) ) ).

% card_roots_unity_eq
tff(fact_9499_diff__Suc,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(suc,Na)) = aa_nat_nat(case_nat_nat(zero_zero_nat,aTP_Lamm_cc),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na)) ).

% diff_Suc
tff(fact_9500_divmod__abs__code_I5_J,axiom,
    ! [J: code_integer] : code_divmod_abs(J,zero_z3403309356797280102nteger) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,J)) ).

% divmod_abs_code(5)
tff(fact_9501_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_yd(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_ye(L),code_divmod_abs(K,L)))) ) ) ) ).

% divmod_integer_code
tff(fact_9502_pred__def,axiom,
    ! [Nata: nat] : aa_nat_nat(pred,Nata) = aa_nat_nat(case_nat_nat(zero_zero_nat,aTP_Lamm_cc),Nata) ).

% pred_def
tff(fact_9503_bezw__0,axiom,
    ! [Xb: nat] : bezw(Xb,zero_zero_nat) = product_Pair_int_int(one_one_int,zero_zero_int) ).

% bezw_0
tff(fact_9504_finite__enumerate,axiom,
    ! [S2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ? [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,S2)))
          & ! [N5: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N5),aa_set_nat_nat(finite_card_nat,S2))
             => aa_set_nat_o(member_nat(aa_nat_nat(R3,N5)),S2) ) ) ) ).

% finite_enumerate
tff(fact_9505_binomial__def,axiom,
    ! [Na: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,Na),K) = finite_card_set_nat(collect_set_nat(aa_nat_fun_set_nat_o(aTP_Lamm_yf(Na),K))) ).

% binomial_def
tff(fact_9506_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_yg(L),code_divmod_abs(K,L)))) ) ) ).

% divmod_integer_eq_cases
tff(fact_9507_drop__bit__nonnegative__int__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,Na),K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K) ) ).

% drop_bit_nonnegative_int_iff
tff(fact_9508_drop__bit__negative__int__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,Na),K)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int) ) ).

% drop_bit_negative_int_iff
tff(fact_9509_drop__bit__of__Suc__0,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se8570568707652914677it_nat,Na),aa_nat_nat(suc,zero_zero_nat)) = zero_n2687167440665602831ol_nat(Na = zero_zero_nat) ).

% drop_bit_of_Suc_0
tff(fact_9510_drop__bit__Suc__minus__bit0,axiom,
    ! [Na: nat,K: num] : aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,aa_nat_nat(suc,Na)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,K)))) = aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,Na),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) ).

% drop_bit_Suc_minus_bit0
tff(fact_9511_drop__bit__Suc__minus__bit1,axiom,
    ! [Na: nat,K: num] : aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,aa_nat_nat(suc,Na)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K)))) = aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,Na),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(inc,K)))) ).

% drop_bit_Suc_minus_bit1
tff(fact_9512_drop__bit__integer_Oabs__eq,axiom,
    ! [Xaa: nat,Xb: int] : aa_Cod1783974652533396728nteger(aa_nat6383147911832638789nteger(bit_se3928097537394005634nteger,Xaa),aa_int_Code_integer(code_integer_of_int,Xb)) = aa_int_Code_integer(code_integer_of_int,aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,Xaa),Xb)) ).

% drop_bit_integer.abs_eq
tff(fact_9513_Sup__nat__empty,axiom,
    aa_set_nat_nat(complete_Sup_Sup_nat,bot_bot_set_nat) = zero_zero_nat ).

% Sup_nat_empty
tff(fact_9514_Inf__nat__def1,axiom,
    ! [K4: set_nat] :
      ( ( K4 != bot_bot_set_nat )
     => aa_set_nat_o(member_nat(aa_set_nat_nat(complete_Inf_Inf_nat,K4)),K4) ) ).

% Inf_nat_def1
tff(fact_9515_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_9516_euclidean__size__int__def,axiom,
    euclid4774559944035922753ze_int = comp_int_nat_int(nat2,abs_abs_int) ).

% euclidean_size_int_def
tff(fact_9517_Code__Numeral_Onegative__def,axiom,
    code_negative = comp_C3531382070062128313er_num(uminus1351360451143612070nteger,numera6620942414471956472nteger) ).

% Code_Numeral.negative_def
tff(fact_9518_root__powr__inverse,axiom,
    ! [Na: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_real_real(aa_nat_fun_real_real(root,Na),Xb) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,Na))) ) ) ) ).

% root_powr_inverse
tff(fact_9519_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_9520_finite__greaterThanLessThan__int,axiom,
    ! [L: int,U: int] : aa_set_int_o(finite_finite_int,set_or5832277885323065728an_int(L,U)) ).

% finite_greaterThanLessThan_int
tff(fact_9521_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_9522_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_9523_real__root__eq__iff,axiom,
    ! [Na: nat,Xb: real,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( ( aa_real_real(aa_nat_fun_real_real(root,Na),Xb) = aa_real_real(aa_nat_fun_real_real(root,Na),Y) )
      <=> ( Xb = Y ) ) ) ).

% real_root_eq_iff
tff(fact_9524_real__root__eq__0__iff,axiom,
    ! [Na: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( ( aa_real_real(aa_nat_fun_real_real(root,Na),Xb) = zero_zero_real )
      <=> ( Xb = zero_zero_real ) ) ) ).

% real_root_eq_0_iff
tff(fact_9525_real__root__less__iff,axiom,
    ! [Na: nat,Xb: real,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,Na),Xb)),aa_real_real(aa_nat_fun_real_real(root,Na),Y))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y) ) ) ).

% real_root_less_iff
tff(fact_9526_real__root__le__iff,axiom,
    ! [Na: nat,Xb: real,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,Na),Xb)),aa_real_real(aa_nat_fun_real_real(root,Na),Y))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y) ) ) ).

% real_root_le_iff
tff(fact_9527_real__root__one,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_real(aa_nat_fun_real_real(root,Na),one_one_real) = one_one_real ) ) ).

% real_root_one
tff(fact_9528_real__root__eq__1__iff,axiom,
    ! [Na: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( ( aa_real_real(aa_nat_fun_real_real(root,Na),Xb) = one_one_real )
      <=> ( Xb = one_one_real ) ) ) ).

% real_root_eq_1_iff
tff(fact_9529_real__root__lt__0__iff,axiom,
    ! [Na: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,Na),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_9530_real__root__gt__0__iff,axiom,
    ! [Na: nat,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,Na),Y))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y) ) ) ).

% real_root_gt_0_iff
tff(fact_9531_real__root__le__0__iff,axiom,
    ! [Na: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,Na),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_9532_real__root__ge__0__iff,axiom,
    ! [Na: nat,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,Na),Y))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y) ) ) ).

% real_root_ge_0_iff
tff(fact_9533_real__root__gt__1__iff,axiom,
    ! [Na: nat,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(aa_nat_fun_real_real(root,Na),Y))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Y) ) ) ).

% real_root_gt_1_iff
tff(fact_9534_real__root__lt__1__iff,axiom,
    ! [Na: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,Na),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_9535_real__root__ge__1__iff,axiom,
    ! [Na: nat,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(aa_nat_fun_real_real(root,Na),Y))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Y) ) ) ).

% real_root_ge_1_iff
tff(fact_9536_real__root__le__1__iff,axiom,
    ! [Na: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,Na),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_9537_real__root__pow__pos2,axiom,
    ! [Na: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Na),Xb)),Na) = Xb ) ) ) ).

% real_root_pow_pos2
tff(fact_9538_real__root__pos__pos__le,axiom,
    ! [Xb: real,Na: 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,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,Na),Xb)) ) ).

% real_root_pos_pos_le
tff(fact_9539_real__root__less__mono,axiom,
    ! [Na: nat,Xb: real,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,Na),Xb)),aa_real_real(aa_nat_fun_real_real(root,Na),Y)) ) ) ).

% real_root_less_mono
tff(fact_9540_real__root__le__mono,axiom,
    ! [Na: nat,Xb: real,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,Na),Xb)),aa_real_real(aa_nat_fun_real_real(root,Na),Y)) ) ) ).

% real_root_le_mono
tff(fact_9541_real__root__power,axiom,
    ! [Na: nat,Xb: real,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_real(aa_nat_fun_real_real(root,Na),aa_nat_real(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,Na),Xb)),K) ) ) ).

% real_root_power
tff(fact_9542_real__root__abs,axiom,
    ! [Na: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_real(aa_nat_fun_real_real(root,Na),aa_real_real(abs_abs_real,Xb)) = aa_real_real(abs_abs_real,aa_real_real(aa_nat_fun_real_real(root,Na),Xb)) ) ) ).

% real_root_abs
tff(fact_9543_sgn__root,axiom,
    ! [Na: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_real(sgn_sgn_real,aa_real_real(aa_nat_fun_real_real(root,Na),Xb)) = aa_real_real(sgn_sgn_real,Xb) ) ) ).

% sgn_root
tff(fact_9544_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_9545_real__root__gt__zero,axiom,
    ! [Na: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,Na),Xb)) ) ) ).

% real_root_gt_zero
tff(fact_9546_real__root__strict__decreasing,axiom,
    ! [Na: nat,N3: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),N3)
       => ( 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,N3),Xb)),aa_real_real(aa_nat_fun_real_real(root,Na),Xb)) ) ) ) ).

% real_root_strict_decreasing
tff(fact_9547_root__abs__power,axiom,
    ! [Na: nat,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_real(abs_abs_real,aa_real_real(aa_nat_fun_real_real(root,Na),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),Na))) = aa_real_real(abs_abs_real,Y) ) ) ).

% root_abs_power
tff(fact_9548_real__root__pos__pos,axiom,
    ! [Na: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),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,Na),Xb)) ) ) ).

% real_root_pos_pos
tff(fact_9549_real__root__strict__increasing,axiom,
    ! [Na: nat,N3: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),N3)
       => ( 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,Na),Xb)),aa_real_real(aa_nat_fun_real_real(root,N3),Xb)) ) ) ) ) ).

% real_root_strict_increasing
tff(fact_9550_real__root__decreasing,axiom,
    ! [Na: nat,N3: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),N3)
       => ( 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,N3),Xb)),aa_real_real(aa_nat_fun_real_real(root,Na),Xb)) ) ) ) ).

% real_root_decreasing
tff(fact_9551_real__root__pow__pos,axiom,
    ! [Na: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Na),Xb)),Na) = Xb ) ) ) ).

% real_root_pow_pos
tff(fact_9552_real__root__power__cancel,axiom,
    ! [Na: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
       => ( aa_real_real(aa_nat_fun_real_real(root,Na),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Na)) = Xb ) ) ) ).

% real_root_power_cancel
tff(fact_9553_real__root__pos__unique,axiom,
    ! [Na: nat,Y: real,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),Na) = Xb )
         => ( aa_real_real(aa_nat_fun_real_real(root,Na),Xb) = Y ) ) ) ) ).

% real_root_pos_unique
tff(fact_9554_real__root__increasing,axiom,
    ! [Na: nat,N3: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),N3)
       => ( 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,Na),Xb)),aa_real_real(aa_nat_fun_real_real(root,N3),Xb)) ) ) ) ) ).

% real_root_increasing
tff(fact_9555_sgn__power__root,axiom,
    ! [Na: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,aa_real_real(aa_nat_fun_real_real(root,Na),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,Na),Xb))),Na)) = Xb ) ) ).

% sgn_power_root
tff(fact_9556_root__sgn__power,axiom,
    ! [Na: nat,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_real(aa_nat_fun_real_real(root,Na),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,Y)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,Y)),Na))) = Y ) ) ).

% root_sgn_power
tff(fact_9557_ln__root,axiom,
    ! [Na: nat,Ba: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Ba)
       => ( aa_real_real(ln_ln_real,aa_real_real(aa_nat_fun_real_real(root,Na),Ba)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,Ba)),aa_nat_real(semiri5074537144036343181t_real,Na)) ) ) ) ).

% ln_root
tff(fact_9558_log__root,axiom,
    ! [Na: nat,Aa: real,Ba: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Aa)
       => ( aa_real_real(aa_rea7340605660300375871l_real(log,Ba),aa_real_real(aa_nat_fun_real_real(root,Na),Aa)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(log,Ba),Aa)),aa_nat_real(semiri5074537144036343181t_real,Na)) ) ) ) ).

% log_root
tff(fact_9559_log__base__root,axiom,
    ! [Na: nat,Ba: real,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Ba)
       => ( aa_real_real(aa_rea7340605660300375871l_real(log,aa_real_real(aa_nat_fun_real_real(root,Na),Ba)),Xb) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_real_real(aa_rea7340605660300375871l_real(log,Ba),Xb)) ) ) ) ).

% log_base_root
tff(fact_9560_split__root,axiom,
    ! [P: fun_real_o,Na: nat,Xb: real] :
      ( aa_real_o(P,aa_real_real(aa_nat_fun_real_real(root,Na),Xb))
    <=> ( ( ( Na = zero_zero_nat )
         => aa_real_o(P,zero_zero_real) )
        & ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
         => ! [Y5: real] :
              ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,Y5)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,Y5)),Na)) = Xb )
             => aa_real_o(P,Y5) ) ) ) ) ).

% split_root
tff(fact_9561_finite__greaterThanLessThan,axiom,
    ! [L: nat,U: nat] : aa_set_nat_o(finite_finite_nat,set_or5834768355832116004an_nat(L,U)) ).

% finite_greaterThanLessThan
tff(fact_9562_card__greaterThanLessThan,axiom,
    ! [L: nat,U: nat] : aa_set_nat_nat(finite_card_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_9563_atLeastSucLessThan__greaterThanLessThan,axiom,
    ! [L: nat,U: nat] : set_or4665077453230672383an_nat(aa_nat_nat(suc,L),U) = set_or5834768355832116004an_nat(L,U) ).

% atLeastSucLessThan_greaterThanLessThan
tff(fact_9564_nat__of__integer__non__positive,axiom,
    ! [K: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,K),zero_z3403309356797280102nteger)
     => ( aa_Code_integer_nat(code_nat_of_integer,K) = zero_zero_nat ) ) ).

% nat_of_integer_non_positive
tff(fact_9565_nat__of__integer__of__nat,axiom,
    ! [Na: nat] : aa_Code_integer_nat(code_nat_of_integer,aa_nat_Code_integer(semiri4939895301339042750nteger,Na)) = Na ).

% nat_of_integer_of_nat
tff(fact_9566_Suc__funpow,axiom,
    ! [Na: nat] : compow_fun_nat_nat(Na,suc) = aa_nat_fun_nat_nat(plus_plus_nat,Na) ).

% Suc_funpow
tff(fact_9567_of__nat__of__integer,axiom,
    ! [K: code_integer] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_Code_integer_nat(code_nat_of_integer,K)) = ord_max_Code_integer(zero_z3403309356797280102nteger,K) ).

% of_nat_of_integer
tff(fact_9568_nat__of__integer__code__post_I1_J,axiom,
    aa_Code_integer_nat(code_nat_of_integer,zero_z3403309356797280102nteger) = zero_zero_nat ).

% nat_of_integer_code_post(1)
tff(fact_9569_nat__of__integer_Oabs__eq,axiom,
    ! [Xb: int] : aa_Code_integer_nat(code_nat_of_integer,aa_int_Code_integer(code_integer_of_int,Xb)) = aa_int_nat(nat2,Xb) ).

% nat_of_integer.abs_eq
tff(fact_9570_nat__of__integer__code__post_I2_J,axiom,
    aa_Code_integer_nat(code_nat_of_integer,one_one_Code_integer) = one_one_nat ).

% nat_of_integer_code_post(2)
tff(fact_9571_nat__of__integer__code,axiom,
    ! [K: code_integer] :
      aa_Code_integer_nat(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_yh,code_divmod_integer(K,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))))) ).

% nat_of_integer_code
tff(fact_9572_max__nat_Osemilattice__neutr__order__axioms,axiom,
    semila1623282765462674594er_nat(ord_max_nat,zero_zero_nat,aTP_Lamm_av,aTP_Lamm_au) ).

% max_nat.semilattice_neutr_order_axioms
tff(fact_9573_int__of__integer__code,axiom,
    ! [K: code_integer] :
      aa_Code_integer_int(code_int_of_integer,K) = $ite(
        aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,K),zero_z3403309356797280102nteger),
        aa_int_int(uminus_uminus_int,aa_Code_integer_int(code_int_of_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,K))),
        $ite(K = zero_z3403309356797280102nteger,zero_zero_int,produc1553301316500091796er_int(aTP_Lamm_yi,code_divmod_integer(K,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))))) ) ).

% int_of_integer_code
tff(fact_9574_Gcd__remove0__nat,axiom,
    ! [M7: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,M7)
     => ( aa_set_nat_nat(gcd_Gcd_nat,M7) = aa_set_nat_nat(gcd_Gcd_nat,minus_minus_set_nat(M7,insert_nat(zero_zero_nat,bot_bot_set_nat))) ) ) ).

% Gcd_remove0_nat
tff(fact_9575_int__of__integer__sub,axiom,
    ! [K: num,L: num] : aa_Code_integer_int(code_int_of_integer,neg_nu5755505904847501662nteger(K,L)) = aa_num_int(neg_numeral_sub_int(K),L) ).

% int_of_integer_sub
tff(fact_9576_integer__of__int__int__of__integer,axiom,
    ! [K: code_integer] : aa_int_Code_integer(code_integer_of_int,aa_Code_integer_int(code_int_of_integer,K)) = K ).

% integer_of_int_int_of_integer
tff(fact_9577_int__of__integer__integer__of__int,axiom,
    ! [K: int] : aa_Code_integer_int(code_int_of_integer,aa_int_Code_integer(code_integer_of_int,K)) = K ).

% int_of_integer_integer_of_int
tff(fact_9578_int__of__integer__inverse,axiom,
    ! [Xb: code_integer] : aa_int_Code_integer(code_integer_of_int,aa_Code_integer_int(code_int_of_integer,Xb)) = Xb ).

% int_of_integer_inverse
tff(fact_9579_int__of__integer__max,axiom,
    ! [K: code_integer,L: code_integer] : aa_Code_integer_int(code_int_of_integer,ord_max_Code_integer(K,L)) = aa_int_int(ord_max_int(aa_Code_integer_int(code_int_of_integer,K)),aa_Code_integer_int(code_int_of_integer,L)) ).

% int_of_integer_max
tff(fact_9580_of__int__integer__of,axiom,
    ! [K: code_integer] : aa_int_Code_integer(ring_18347121197199848620nteger,aa_Code_integer_int(code_int_of_integer,K)) = K ).

% of_int_integer_of
tff(fact_9581_int__of__integer__of__int,axiom,
    ! [K: int] : aa_Code_integer_int(code_int_of_integer,aa_int_Code_integer(ring_18347121197199848620nteger,K)) = K ).

% int_of_integer_of_int
tff(fact_9582_zero__integer_Orep__eq,axiom,
    aa_Code_integer_int(code_int_of_integer,zero_z3403309356797280102nteger) = zero_zero_int ).

% zero_integer.rep_eq
tff(fact_9583_int__of__integer__numeral,axiom,
    ! [K: num] : aa_Code_integer_int(code_int_of_integer,aa_num_Code_integer(numera6620942414471956472nteger,K)) = aa_num_int(numeral_numeral_int,K) ).

% int_of_integer_numeral
tff(fact_9584_plus__integer_Orep__eq,axiom,
    ! [Xb: code_integer,Xaa: code_integer] : aa_Code_integer_int(code_int_of_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Xb),Xaa)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_Code_integer_int(code_int_of_integer,Xb)),aa_Code_integer_int(code_int_of_integer,Xaa)) ).

% plus_integer.rep_eq
tff(fact_9585_uminus__integer_Orep__eq,axiom,
    ! [Xb: code_integer] : aa_Code_integer_int(code_int_of_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Xb)) = aa_int_int(uminus_uminus_int,aa_Code_integer_int(code_int_of_integer,Xb)) ).

% uminus_integer.rep_eq
tff(fact_9586_times__integer_Orep__eq,axiom,
    ! [Xb: code_integer,Xaa: code_integer] : aa_Code_integer_int(code_int_of_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Xb),Xaa)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_Code_integer_int(code_int_of_integer,Xb)),aa_Code_integer_int(code_int_of_integer,Xaa)) ).

% times_integer.rep_eq
tff(fact_9587_one__integer_Orep__eq,axiom,
    aa_Code_integer_int(code_int_of_integer,one_one_Code_integer) = one_one_int ).

% one_integer.rep_eq
tff(fact_9588_minus__integer_Orep__eq,axiom,
    ! [Xb: code_integer,Xaa: code_integer] : aa_Code_integer_int(code_int_of_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,Xb),Xaa)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_Code_integer_int(code_int_of_integer,Xb)),aa_Code_integer_int(code_int_of_integer,Xaa)) ).

% minus_integer.rep_eq
tff(fact_9589_abs__integer_Orep__eq,axiom,
    ! [Xb: code_integer] : aa_Code_integer_int(code_int_of_integer,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,Xb)) = aa_int_int(abs_abs_int,aa_Code_integer_int(code_int_of_integer,Xb)) ).

% abs_integer.rep_eq
tff(fact_9590_divide__integer_Orep__eq,axiom,
    ! [Xb: code_integer,Xaa: code_integer] : aa_Code_integer_int(code_int_of_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,Xb),Xaa)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_Code_integer_int(code_int_of_integer,Xb)),aa_Code_integer_int(code_int_of_integer,Xaa)) ).

% divide_integer.rep_eq
tff(fact_9591_modulo__integer_Orep__eq,axiom,
    ! [Xb: code_integer,Xaa: code_integer] : aa_Code_integer_int(code_int_of_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,Xb),Xaa)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_Code_integer_int(code_int_of_integer,Xb)),aa_Code_integer_int(code_int_of_integer,Xaa)) ).

% modulo_integer.rep_eq
tff(fact_9592_int__of__integer__of__nat,axiom,
    ! [Na: nat] : aa_Code_integer_int(code_int_of_integer,aa_nat_Code_integer(semiri4939895301339042750nteger,Na)) = aa_nat_int(semiri1314217659103216013at_int,Na) ).

% int_of_integer_of_nat
tff(fact_9593_sgn__integer_Orep__eq,axiom,
    ! [Xb: code_integer] : aa_Code_integer_int(code_int_of_integer,aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,Xb)) = aa_int_int(sgn_sgn_int,aa_Code_integer_int(code_int_of_integer,Xb)) ).

% sgn_integer.rep_eq
tff(fact_9594_division__segment__integer_Orep__eq,axiom,
    ! [Xb: code_integer] : aa_Code_integer_int(code_int_of_integer,aa_Cod1783974652533396728nteger(euclid6289375185220004616nteger,Xb)) = aa_int_int(euclid3395696857347342551nt_int,aa_Code_integer_int(code_int_of_integer,Xb)) ).

% division_segment_integer.rep_eq
tff(fact_9595_euclidean__size__integer_Orep__eq,axiom,
    ! [Xb: code_integer] : aa_Code_integer_nat(euclid6377331345833325938nteger,Xb) = aa_int_nat(euclid4774559944035922753ze_int,aa_Code_integer_int(code_int_of_integer,Xb)) ).

% euclidean_size_integer.rep_eq
tff(fact_9596_int__of__integer__inject,axiom,
    ! [Xb: code_integer,Y: code_integer] :
      ( ( aa_Code_integer_int(code_int_of_integer,Xb) = aa_Code_integer_int(code_int_of_integer,Y) )
    <=> ( Xb = Y ) ) ).

% int_of_integer_inject
tff(fact_9597_integer__eqI,axiom,
    ! [K: code_integer,L: code_integer] :
      ( ( aa_Code_integer_int(code_int_of_integer,K) = aa_Code_integer_int(code_int_of_integer,L) )
     => ( K = L ) ) ).

% integer_eqI
tff(fact_9598_integer__eq__iff,axiom,
    ! [K: code_integer,L: code_integer] :
      ( ( K = L )
    <=> ( aa_Code_integer_int(code_int_of_integer,K) = aa_Code_integer_int(code_int_of_integer,L) ) ) ).

% integer_eq_iff
tff(fact_9599_Gcd__nat__eq__one,axiom,
    ! [N3: set_nat] :
      ( aa_set_nat_o(member_nat(one_one_nat),N3)
     => ( aa_set_nat_nat(gcd_Gcd_nat,N3) = one_one_nat ) ) ).

% Gcd_nat_eq_one
tff(fact_9600_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,aa_Code_integer_int(code_int_of_integer,Xb)),aa_Code_integer_int(code_int_of_integer,Xaa)) ) ).

% less_integer.rep_eq
tff(fact_9601_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,aa_Code_integer_int(code_int_of_integer,K)),aa_Code_integer_int(code_int_of_integer,L)) ) ).

% integer_less_iff
tff(fact_9602_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,aa_Code_integer_int(code_int_of_integer,Xb)),aa_Code_integer_int(code_int_of_integer,Xaa)) ) ).

% less_eq_integer.rep_eq
tff(fact_9603_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,aa_Code_integer_int(code_int_of_integer,K)),aa_Code_integer_int(code_int_of_integer,L)) ) ).

% integer_less_eq_iff
tff(fact_9604_nat__of__integer_Orep__eq,axiom,
    ! [Xb: code_integer] : aa_Code_integer_nat(code_nat_of_integer,Xb) = aa_int_nat(nat2,aa_Code_integer_int(code_int_of_integer,Xb)) ).

% nat_of_integer.rep_eq
tff(fact_9605_take__bit__integer_Orep__eq,axiom,
    ! [Xb: nat,Xaa: code_integer] : aa_Code_integer_int(code_int_of_integer,aa_Cod1783974652533396728nteger(aa_nat6383147911832638789nteger(bit_se1745604003318907178nteger,Xb),Xaa)) = aa_int_int(aa_nat_fun_int_int(bit_se2923211474154528505it_int,Xb),aa_Code_integer_int(code_int_of_integer,Xaa)) ).

% take_bit_integer.rep_eq
tff(fact_9606_and__integer_Orep__eq,axiom,
    ! [Xb: code_integer,Xaa: code_integer] : aa_Code_integer_int(code_int_of_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3949692690581998587nteger,Xb),Xaa)) = aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_Code_integer_int(code_int_of_integer,Xb)),aa_Code_integer_int(code_int_of_integer,Xaa)) ).

% and_integer.rep_eq
tff(fact_9607_bit__integer_Orep__eq,axiom,
    ! [Xb: code_integer] : aa_Cod1075288278386115654_nat_o(bit_se9216721137139052372nteger,Xb) = aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_Code_integer_int(code_int_of_integer,Xb)) ).

% bit_integer.rep_eq
tff(fact_9608_or__integer_Orep__eq,axiom,
    ! [Xb: code_integer,Xaa: code_integer] : aa_Code_integer_int(code_int_of_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se1080825931792720795nteger,Xb),Xaa)) = aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_Code_integer_int(code_int_of_integer,Xb)),aa_Code_integer_int(code_int_of_integer,Xaa)) ).

% or_integer.rep_eq
tff(fact_9609_xor__integer_Orep__eq,axiom,
    ! [Xb: code_integer,Xaa: code_integer] : aa_Code_integer_int(code_int_of_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3222712562003087583nteger,Xb),Xaa)) = aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,aa_Code_integer_int(code_int_of_integer,Xb)),aa_Code_integer_int(code_int_of_integer,Xaa)) ).

% xor_integer.rep_eq
tff(fact_9610_drop__bit__integer_Orep__eq,axiom,
    ! [Xb: nat,Xaa: code_integer] : aa_Code_integer_int(code_int_of_integer,aa_Cod1783974652533396728nteger(aa_nat6383147911832638789nteger(bit_se3928097537394005634nteger,Xb),Xaa)) = aa_int_int(aa_nat_fun_int_int(bit_se8568078237143864401it_int,Xb),aa_Code_integer_int(code_int_of_integer,Xaa)) ).

% drop_bit_integer.rep_eq
tff(fact_9611_push__bit__integer_Orep__eq,axiom,
    ! [Xb: nat,Xaa: code_integer] : aa_Code_integer_int(code_int_of_integer,aa_Cod1783974652533396728nteger(aa_nat6383147911832638789nteger(bit_se7788150548672797655nteger,Xb),Xaa)) = aa_int_int(aa_nat_fun_int_int(bit_se545348938243370406it_int,Xb),aa_Code_integer_int(code_int_of_integer,Xaa)) ).

% push_bit_integer.rep_eq
tff(fact_9612_mask__integer_Orep__eq,axiom,
    ! [Xb: nat] : aa_Code_integer_int(code_int_of_integer,aa_nat_Code_integer(bit_se2119862282449309892nteger,Xb)) = aa_nat_int(bit_se2000444600071755411sk_int,Xb) ).

% mask_integer.rep_eq
tff(fact_9613_unset__bit__integer_Orep__eq,axiom,
    ! [Xb: nat,Xaa: code_integer] : aa_Code_integer_int(code_int_of_integer,aa_Cod1783974652533396728nteger(aa_nat6383147911832638789nteger(bit_se8260200283734997820nteger,Xb),Xaa)) = aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,Xb),aa_Code_integer_int(code_int_of_integer,Xaa)) ).

% unset_bit_integer.rep_eq
tff(fact_9614_set__bit__integer_Orep__eq,axiom,
    ! [Xb: nat,Xaa: code_integer] : aa_Code_integer_int(code_int_of_integer,aa_Cod1783974652533396728nteger(aa_nat6383147911832638789nteger(bit_se2793503036327961859nteger,Xb),Xaa)) = aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,Xb),aa_Code_integer_int(code_int_of_integer,Xaa)) ).

% set_bit_integer.rep_eq
tff(fact_9615_flip__bit__integer_Orep__eq,axiom,
    ! [Xb: nat,Xaa: code_integer] : aa_Code_integer_int(code_int_of_integer,aa_Cod1783974652533396728nteger(aa_nat6383147911832638789nteger(bit_se1345352211410354436nteger,Xb),Xaa)) = aa_int_int(aa_nat_fun_int_int(bit_se2159334234014336723it_int,Xb),aa_Code_integer_int(code_int_of_integer,Xaa)) ).

% flip_bit_integer.rep_eq
tff(fact_9616_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_9617_Gcd__int__greater__eq__0,axiom,
    ! [K4: set_int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),gcd_Gcd_int(K4)) ).

% Gcd_int_greater_eq_0
tff(fact_9618_inc__BitM__eq,axiom,
    ! [Na: num] : aa_num_num(inc,aa_num_num(bitM,Na)) = aa_num_num(bit0,Na) ).

% inc_BitM_eq
tff(fact_9619_BitM__inc__eq,axiom,
    ! [Na: num] : aa_num_num(bitM,aa_num_num(inc,Na)) = aa_num_num(bit1,Na) ).

% BitM_inc_eq
tff(fact_9620_eval__nat__numeral_I2_J,axiom,
    ! [Na: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Na)) = aa_nat_nat(suc,aa_num_nat(numeral_numeral_nat,aa_num_num(bitM,Na))) ).

% eval_nat_numeral(2)
tff(fact_9621_one__plus__BitM,axiom,
    ! [Na: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,one),aa_num_num(bitM,Na)) = aa_num_num(bit0,Na) ).

% one_plus_BitM
tff(fact_9622_BitM__plus__one,axiom,
    ! [Na: num] : aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(bitM,Na)),one) = aa_num_num(bit0,Na) ).

% BitM_plus_one
tff(fact_9623_num__of__nat_Osimps_I2_J,axiom,
    ! [Na: nat] :
      aa_nat_num(num_of_nat,aa_nat_nat(suc,Na)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na),aa_num_num(inc,aa_nat_num(num_of_nat,Na)),one) ).

% num_of_nat.simps(2)
tff(fact_9624_pred__nat__def,axiom,
    pred_nat = collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_yj)) ).

% pred_nat_def
tff(fact_9625_num__of__nat__numeral__eq,axiom,
    ! [Q4: num] : aa_nat_num(num_of_nat,aa_num_nat(numeral_numeral_nat,Q4)) = Q4 ).

% num_of_nat_numeral_eq
tff(fact_9626_num__of__nat_Osimps_I1_J,axiom,
    aa_nat_num(num_of_nat,zero_zero_nat) = one ).

% num_of_nat.simps(1)
tff(fact_9627_numeral__num__of__nat,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_num_nat(numeral_numeral_nat,aa_nat_num(num_of_nat,Na)) = Na ) ) ).

% numeral_num_of_nat
tff(fact_9628_num__of__nat__One,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),one_one_nat)
     => ( aa_nat_num(num_of_nat,Na) = one ) ) ).

% num_of_nat_One
tff(fact_9629_num__of__nat__double,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_num(num_of_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Na),Na)) = aa_num_num(bit0,aa_nat_num(num_of_nat,Na)) ) ) ).

% num_of_nat_double
tff(fact_9630_num__of__nat__plus__distrib,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => ( aa_nat_num(num_of_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na)) = aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_nat_num(num_of_nat,M)),aa_nat_num(num_of_nat,Na)) ) ) ) ).

% num_of_nat_plus_distrib
tff(fact_9631_upto_Opsimps,axiom,
    ! [I: int,J: int] :
      ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(upto_rel),product_Pair_int_int(I,J))
     => ( upto(I,J) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J),cons_int(I,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,I),one_one_int),J)),nil_int) ) ) ).

% upto.psimps
tff(fact_9632_upto_Opelims,axiom,
    ! [Xb: int,Xaa: int,Y: list_int] :
      ( ( upto(Xb,Xaa) = Y )
     => ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(upto_rel),product_Pair_int_int(Xb,Xaa))
       => ~ ( ( Y = $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),product_Pair_int_int(Xb,Xaa)) ) ) ) ).

% upto.pelims
tff(fact_9633_upto__Nil,axiom,
    ! [I: int,J: int] :
      ( ( upto(I,J) = nil_int )
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,J),I) ) ).

% upto_Nil
tff(fact_9634_upto__Nil2,axiom,
    ! [I: int,J: int] :
      ( ( nil_int = upto(I,J) )
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,J),I) ) ).

% upto_Nil2
tff(fact_9635_upto__empty,axiom,
    ! [J: int,I: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,J),I)
     => ( upto(I,J) = nil_int ) ) ).

% upto_empty
tff(fact_9636_nth__upto,axiom,
    ! [I: int,K: nat,J: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),aa_nat_int(semiri1314217659103216013at_int,K))),J)
     => ( aa_nat_int(nth_int(upto(I,J)),K) = aa_int_int(aa_int_fun_int_int(plus_plus_int,I),aa_nat_int(semiri1314217659103216013at_int,K)) ) ) ).

% nth_upto
tff(fact_9637_upto__rec__numeral_I1_J,axiom,
    ! [M: num,Na: num] :
      upto(aa_num_int(numeral_numeral_int,M),aa_num_int(numeral_numeral_int,Na)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,Na)),cons_int(aa_num_int(numeral_numeral_int,M),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,M)),one_one_int),aa_num_int(numeral_numeral_int,Na))),nil_int) ).

% upto_rec_numeral(1)
tff(fact_9638_upto__rec__numeral_I2_J,axiom,
    ! [M: num,Na: num] :
      upto(aa_num_int(numeral_numeral_int,M),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Na))) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Na))),cons_int(aa_num_int(numeral_numeral_int,M),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,M)),one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Na)))),nil_int) ).

% upto_rec_numeral(2)
tff(fact_9639_upto__rec__numeral_I3_J,axiom,
    ! [M: num,Na: num] :
      upto(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,Na)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,Na)),cons_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),one_one_int),aa_num_int(numeral_numeral_int,Na))),nil_int) ).

% upto_rec_numeral(3)
tff(fact_9640_upto__rec__numeral_I4_J,axiom,
    ! [M: num,Na: num] :
      upto(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Na))) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Na))),cons_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Na)))),nil_int) ).

% upto_rec_numeral(4)
tff(fact_9641_upto__split1,axiom,
    ! [I: int,J: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J),K)
       => ( upto(I,K) = append_int(upto(I,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int)),upto(J,K)) ) ) ) ).

% upto_split1
tff(fact_9642_upto__split2,axiom,
    ! [I: int,J: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J),K)
       => ( upto(I,K) = append_int(upto(I,J),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,J),one_one_int),K)) ) ) ) ).

% upto_split2
tff(fact_9643_upto__rec2,axiom,
    ! [I: int,J: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J)
     => ( upto(I,J) = append_int(upto(I,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int)),cons_int(J,nil_int)) ) ) ).

% upto_rec2
tff(fact_9644_upto__split3,axiom,
    ! [I: int,J: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J),K)
       => ( upto(I,K) = append_int(upto(I,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int)),cons_int(J,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,J),one_one_int),K))) ) ) ) ).

% upto_split3
tff(fact_9645_upto__rec1,axiom,
    ! [I: int,J: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J)
     => ( upto(I,J) = cons_int(I,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,I),one_one_int),J)) ) ) ).

% upto_rec1
tff(fact_9646_upto_Oelims,axiom,
    ! [Xb: int,Xaa: int,Y: list_int] :
      ( ( upto(Xb,Xaa) = Y )
     => ( Y = $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_9647_upto_Osimps,axiom,
    ! [I: int,J: int] :
      upto(I,J) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J),cons_int(I,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,I),one_one_int),J)),nil_int) ).

% upto.simps
tff(fact_9648_num__of__integer_Oabs__eq,axiom,
    ! [Xb: int] : aa_Code_integer_num(code_num_of_integer,aa_int_Code_integer(code_integer_of_int,Xb)) = aa_nat_num(num_of_nat,aa_int_nat(nat2,Xb)) ).

% num_of_integer.abs_eq
tff(fact_9649_num__of__integer_Orep__eq,axiom,
    ! [Xb: code_integer] : aa_Code_integer_num(code_num_of_integer,Xb) = aa_nat_num(num_of_nat,aa_int_nat(nat2,aa_Code_integer_int(code_int_of_integer,Xb))) ).

% num_of_integer.rep_eq
tff(fact_9650_num__of__integer__code,axiom,
    ! [K: code_integer] :
      aa_Code_integer_num(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_yk,code_divmod_integer(K,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))))) ).

% num_of_integer_code
tff(fact_9651_rat__floor__lemma,axiom,
    ! [Aa: int,Ba: 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,Aa),Ba))),aa_int_rat(fract(Aa),Ba))
      & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(fract(Aa),Ba)),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,Aa),Ba)),one_one_int))) ) ).

% rat_floor_lemma
tff(fact_9652_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_9653_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_9654_less__rat,axiom,
    ! [Ba: int,D: int,Aa: int,C: int] :
      ( ( Ba != zero_zero_int )
     => ( ( D != zero_zero_int )
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(fract(Aa),Ba)),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,Aa),D)),aa_int_int(aa_int_fun_int_int(times_times_int,Ba),D))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),Ba)),aa_int_int(aa_int_fun_int_int(times_times_int,Ba),D))) ) ) ) ).

% less_rat
tff(fact_9655_le__rat,axiom,
    ! [Ba: int,D: int,Aa: int,C: int] :
      ( ( Ba != zero_zero_int )
     => ( ( D != zero_zero_int )
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(fract(Aa),Ba)),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,Aa),D)),aa_int_int(aa_int_fun_int_int(times_times_int,Ba),D))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),Ba)),aa_int_int(aa_int_fun_int_int(times_times_int,Ba),D))) ) ) ) ).

% le_rat
tff(fact_9656_Rat__induct__pos,axiom,
    ! [P: fun_rat_o,Q4: rat] :
      ( ! [A2: int,B2: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
         => aa_rat_o(P,aa_int_rat(fract(A2),B2)) )
     => aa_rat_o(P,Q4) ) ).

% Rat_induct_pos
tff(fact_9657_sorted__list__of__set__greaterThanLessThan,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,I)),J)
     => ( linord2614967742042102400et_nat(set_or5834768355832116004an_nat(I,J)) = cons_nat(aa_nat_nat(suc,I),linord2614967742042102400et_nat(set_or5834768355832116004an_nat(aa_nat_nat(suc,I),J))) ) ) ).

% sorted_list_of_set_greaterThanLessThan
tff(fact_9658_zero__less__Fract__iff,axiom,
    ! [Ba: int,Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Ba)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_int_rat(fract(Aa),Ba))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Aa) ) ) ).

% zero_less_Fract_iff
tff(fact_9659_Fract__less__zero__iff,axiom,
    ! [Ba: int,Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Ba)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(fract(Aa),Ba)),zero_zero_rat)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),zero_zero_int) ) ) ).

% Fract_less_zero_iff
tff(fact_9660_Fract__less__one__iff,axiom,
    ! [Ba: int,Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Ba)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(fract(Aa),Ba)),one_one_rat)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Aa),Ba) ) ) ).

% Fract_less_one_iff
tff(fact_9661_one__less__Fract__iff,axiom,
    ! [Ba: int,Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Ba)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_int_rat(fract(Aa),Ba))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Ba),Aa) ) ) ).

% one_less_Fract_iff
tff(fact_9662_Fract__le__zero__iff,axiom,
    ! [Ba: int,Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Ba)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(fract(Aa),Ba)),zero_zero_rat)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),zero_zero_int) ) ) ).

% Fract_le_zero_iff
tff(fact_9663_zero__le__Fract__iff,axiom,
    ! [Ba: int,Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Ba)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(fract(Aa),Ba))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Aa) ) ) ).

% zero_le_Fract_iff
tff(fact_9664_one__le__Fract__iff,axiom,
    ! [Ba: int,Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Ba)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_int_rat(fract(Aa),Ba))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ba),Aa) ) ) ).

% one_le_Fract_iff
tff(fact_9665_Fract__le__one__iff,axiom,
    ! [Ba: int,Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Ba)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(fract(Aa),Ba)),one_one_rat)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Aa),Ba) ) ) ).

% Fract_le_one_iff
tff(fact_9666_nth__sorted__list__of__set__greaterThanLessThan,axiom,
    ! [Na: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),aa_nat_nat(suc,I)))
     => ( aa_nat_nat(nth_nat(linord2614967742042102400et_nat(set_or5834768355832116004an_nat(I,J))),Na) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),Na)) ) ) ).

% nth_sorted_list_of_set_greaterThanLessThan
tff(fact_9667_image__minus__const__atLeastLessThan__nat,axiom,
    ! [C: nat,Xb: nat,Y: nat] :
      image_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_yl,C),set_or4665077453230672383an_nat(Xb,Y)) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),Y),
        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,Y),C)),
        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y),insert_nat(zero_zero_nat,bot_bot_set_nat),bot_bot_set_nat) ) ).

% image_minus_const_atLeastLessThan_nat
tff(fact_9668_image__Suc__atLeastAtMost,axiom,
    ! [I: nat,J: nat] : image_nat_nat(suc,set_or1269000886237332187st_nat(I,J)) = set_or1269000886237332187st_nat(aa_nat_nat(suc,I),aa_nat_nat(suc,J)) ).

% image_Suc_atLeastAtMost
tff(fact_9669_image__Suc__atLeastLessThan,axiom,
    ! [I: nat,J: nat] : image_nat_nat(suc,set_or4665077453230672383an_nat(I,J)) = set_or4665077453230672383an_nat(aa_nat_nat(suc,I),aa_nat_nat(suc,J)) ).

% image_Suc_atLeastLessThan
tff(fact_9670_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_9671_image__Suc__lessThan,axiom,
    ! [Na: nat] : image_nat_nat(suc,aa_nat_set_nat(set_ord_lessThan_nat,Na)) = set_or1269000886237332187st_nat(one_one_nat,Na) ).

% image_Suc_lessThan
tff(fact_9672_image__Suc__atMost,axiom,
    ! [Na: nat] : image_nat_nat(suc,aa_nat_set_nat(set_ord_atMost_nat,Na)) = set_or1269000886237332187st_nat(one_one_nat,aa_nat_nat(suc,Na)) ).

% image_Suc_atMost
tff(fact_9673_atLeast0__atMost__Suc__eq__insert__0,axiom,
    ! [Na: nat] : set_or1269000886237332187st_nat(zero_zero_nat,aa_nat_nat(suc,Na)) = insert_nat(zero_zero_nat,image_nat_nat(suc,set_or1269000886237332187st_nat(zero_zero_nat,Na))) ).

% atLeast0_atMost_Suc_eq_insert_0
tff(fact_9674_atLeast0__lessThan__Suc__eq__insert__0,axiom,
    ! [Na: nat] : set_or4665077453230672383an_nat(zero_zero_nat,aa_nat_nat(suc,Na)) = insert_nat(zero_zero_nat,image_nat_nat(suc,set_or4665077453230672383an_nat(zero_zero_nat,Na))) ).

% atLeast0_lessThan_Suc_eq_insert_0
tff(fact_9675_lessThan__Suc__eq__insert__0,axiom,
    ! [Na: nat] : aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Na)) = insert_nat(zero_zero_nat,image_nat_nat(suc,aa_nat_set_nat(set_ord_lessThan_nat,Na))) ).

% lessThan_Suc_eq_insert_0
tff(fact_9676_atMost__Suc__eq__insert__0,axiom,
    ! [Na: nat] : aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Na)) = insert_nat(zero_zero_nat,image_nat_nat(suc,aa_nat_set_nat(set_ord_atMost_nat,Na))) ).

% atMost_Suc_eq_insert_0
tff(fact_9677_nth__sorted__list__of__set__greaterThanAtMost,axiom,
    ! [Na: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I))
     => ( aa_nat_nat(nth_nat(linord2614967742042102400et_nat(set_or6659071591806873216st_nat(I,J))),Na) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),Na)) ) ) ).

% nth_sorted_list_of_set_greaterThanAtMost
tff(fact_9678_finite__greaterThanAtMost,axiom,
    ! [L: nat,U: nat] : aa_set_nat_o(finite_finite_nat,set_or6659071591806873216st_nat(L,U)) ).

% finite_greaterThanAtMost
tff(fact_9679_card__greaterThanAtMost,axiom,
    ! [L: nat,U: nat] : aa_set_nat_nat(finite_card_nat,set_or6659071591806873216st_nat(L,U)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,U),L) ).

% card_greaterThanAtMost
tff(fact_9680_atLeastSucAtMost__greaterThanAtMost,axiom,
    ! [L: nat,U: nat] : set_or1269000886237332187st_nat(aa_nat_nat(suc,L),U) = set_or6659071591806873216st_nat(L,U) ).

% atLeastSucAtMost_greaterThanAtMost
tff(fact_9681_finite__int__iff__bounded__le,axiom,
    ! [S2: set_int] :
      ( aa_set_int_o(finite_finite_int,S2)
    <=> ? [K3: int] : aa_set_int_o(ord_less_eq_set_int(image_int_int(abs_abs_int,S2)),set_ord_atMost_int(K3)) ) ).

% finite_int_iff_bounded_le
tff(fact_9682_finite__int__iff__bounded,axiom,
    ! [S2: set_int] :
      ( aa_set_int_o(finite_finite_int,S2)
    <=> ? [K3: int] : aa_set_int_o(ord_less_eq_set_int(image_int_int(abs_abs_int,S2)),set_ord_lessThan_int(K3)) ) ).

% finite_int_iff_bounded
tff(fact_9683_image__int__atLeastAtMost,axiom,
    ! [Aa: nat,Ba: nat] : image_nat_int(semiri1314217659103216013at_int,set_or1269000886237332187st_nat(Aa,Ba)) = set_or1266510415728281911st_int(aa_nat_int(semiri1314217659103216013at_int,Aa),aa_nat_int(semiri1314217659103216013at_int,Ba)) ).

% image_int_atLeastAtMost
tff(fact_9684_image__int__atLeastLessThan,axiom,
    ! [Aa: nat,Ba: nat] : image_nat_int(semiri1314217659103216013at_int,set_or4665077453230672383an_nat(Aa,Ba)) = set_or4662586982721622107an_int(aa_nat_int(semiri1314217659103216013at_int,Aa),aa_nat_int(semiri1314217659103216013at_int,Ba)) ).

% image_int_atLeastLessThan
tff(fact_9685_sorted__list__of__set__greaterThanAtMost,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,I)),J)
     => ( linord2614967742042102400et_nat(set_or6659071591806873216st_nat(I,J)) = cons_nat(aa_nat_nat(suc,I),linord2614967742042102400et_nat(set_or6659071591806873216st_nat(aa_nat_nat(suc,I),J))) ) ) ).

% sorted_list_of_set_greaterThanAtMost
tff(fact_9686_image__add__int__atLeastLessThan,axiom,
    ! [L: int,U: int] : image_int_int(aTP_Lamm_ym(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_9687_infinite__int__iff__infinite__nat__abs,axiom,
    ! [S2: set_int] :
      ( ~ aa_set_int_o(finite_finite_int,S2)
    <=> ~ aa_set_nat_o(finite_finite_nat,image_int_nat(comp_int_nat_int(nat2,abs_abs_int),S2)) ) ).

% infinite_int_iff_infinite_nat_abs
tff(fact_9688_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_9689_finite__greaterThanAtMost__int,axiom,
    ! [L: int,U: int] : aa_set_int_o(finite_finite_int,set_or6656581121297822940st_int(L,U)) ).

% finite_greaterThanAtMost_int
tff(fact_9690_card__greaterThanAtMost__int,axiom,
    ! [L: int,U: int] : finite_card_int(set_or6656581121297822940st_int(L,U)) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,U),L)) ).

% card_greaterThanAtMost_int
tff(fact_9691_UN__lessThan__UNIV,axiom,
    comple7399068483239264473et_nat(image_nat_set_nat(set_ord_lessThan_nat,top_top_set_nat)) = top_top_set_nat ).

% UN_lessThan_UNIV
tff(fact_9692_UN__atMost__UNIV,axiom,
    comple7399068483239264473et_nat(image_nat_set_nat(set_ord_atMost_nat,top_top_set_nat)) = top_top_set_nat ).

% UN_atMost_UNIV
tff(fact_9693_nat__not__finite,axiom,
    ~ aa_set_nat_o(finite_finite_nat,top_top_set_nat) ).

% nat_not_finite
tff(fact_9694_infinite__UNIV__nat,axiom,
    ~ aa_set_nat_o(finite_finite_nat,top_top_set_nat) ).

% infinite_UNIV_nat
tff(fact_9695_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_9696_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_9697_range__mod,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( image_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_yn,Na),top_top_set_nat) = set_or4665077453230672383an_nat(zero_zero_nat,Na) ) ) ).

% range_mod
tff(fact_9698_suminf__eq__SUP__real,axiom,
    ! [X5: fun_nat_real] :
      ( summable_real(X5)
     => ( ! [I2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(X5,I2))
       => ( suminf_real(X5) = comple1385675409528146559p_real(image_nat_real(aTP_Lamm_yo(X5),top_top_set_nat)) ) ) ) ).

% suminf_eq_SUP_real
tff(fact_9699_card__UNIV__unit,axiom,
    finite410649719033368117t_unit(top_to1996260823553986621t_unit) = one_one_nat ).

% card_UNIV_unit
tff(fact_9700_range__mult,axiom,
    ! [Aa: real] :
      image_real_real(aa_rea7340605660300375871l_real(times_times_real,Aa),top_top_set_real) = $ite(Aa = zero_zero_real,insert_real(zero_zero_real,bot_bot_set_real),top_top_set_real) ).

% range_mult
tff(fact_9701_sup__nat__def,axiom,
    sup_sup_nat = ord_max_nat ).

% sup_nat_def
tff(fact_9702_integer__of__int__inject,axiom,
    ! [Xb: int,Y: int] :
      ( aa_set_int_o(member_int(Xb),top_top_set_int)
     => ( aa_set_int_o(member_int(Y),top_top_set_int)
       => ( ( aa_int_Code_integer(code_integer_of_int,Xb) = aa_int_Code_integer(code_integer_of_int,Y) )
        <=> ( Xb = Y ) ) ) ) ).

% integer_of_int_inject
tff(fact_9703_integer__of__int__induct,axiom,
    ! [P: fun_Code_integer_o,Xb: code_integer] :
      ( ! [Y3: int] :
          ( aa_set_int_o(member_int(Y3),top_top_set_int)
         => aa_Code_integer_o(P,aa_int_Code_integer(code_integer_of_int,Y3)) )
     => aa_Code_integer_o(P,Xb) ) ).

% integer_of_int_induct
tff(fact_9704_integer__of__int__cases,axiom,
    ! [Xb: code_integer] :
      ~ ! [Y3: int] :
          ( ( Xb = aa_int_Code_integer(code_integer_of_int,Y3) )
         => ~ aa_set_int_o(member_int(Y3),top_top_set_int) ) ).

% integer_of_int_cases
tff(fact_9705_int__of__integer__induct,axiom,
    ! [Y: int,P: fun_int_o] :
      ( aa_set_int_o(member_int(Y),top_top_set_int)
     => ( ! [X: code_integer] : aa_int_o(P,aa_Code_integer_int(code_int_of_integer,X))
       => aa_int_o(P,Y) ) ) ).

% int_of_integer_induct
tff(fact_9706_int__of__integer__cases,axiom,
    ! [Y: int] :
      ( aa_set_int_o(member_int(Y),top_top_set_int)
     => ~ ! [X: code_integer] : Y != aa_Code_integer_int(code_int_of_integer,X) ) ).

% int_of_integer_cases
tff(fact_9707_int__of__integer,axiom,
    ! [Xb: code_integer] : aa_set_int_o(member_int(aa_Code_integer_int(code_int_of_integer,Xb)),top_top_set_int) ).

% int_of_integer
tff(fact_9708_integer__of__int__inverse,axiom,
    ! [Y: int] :
      ( aa_set_int_o(member_int(Y),top_top_set_int)
     => ( aa_Code_integer_int(code_int_of_integer,aa_int_Code_integer(code_integer_of_int,Y)) = Y ) ) ).

% integer_of_int_inverse
tff(fact_9709_atLeastLessThan__add__Un,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( set_or4665077453230672383an_nat(I,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K)) = sup_sup_set_nat(set_or4665077453230672383an_nat(I,J),set_or4665077453230672383an_nat(J,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K))) ) ) ).

% atLeastLessThan_add_Un
tff(fact_9710_root__def,axiom,
    ! [Na: nat,Xb: real] :
      aa_real_real(aa_nat_fun_real_real(root,Na),Xb) = $ite(Na = zero_zero_nat,zero_zero_real,aa_real_real(the_in5290026491893676941l_real(top_top_set_real,aa_nat_fun_real_real(aTP_Lamm_yp,Na)),Xb)) ).

% root_def
tff(fact_9711_DERIV__even__real__root,axiom,
    ! [Na: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real)
         => has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,Na),aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Na),Xb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(suc,zero_zero_nat))))),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ) ) ).

% DERIV_even_real_root
tff(fact_9712_DERIV__isconst3,axiom,
    ! [Aa: real,Ba: real,Xb: real,Y: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),Ba)
     => ( member_real(Xb,set_or1633881224788618240n_real(Aa,Ba))
       => ( member_real(Y,set_or1633881224788618240n_real(Aa,Ba))
         => ( ! [X: real] :
                ( member_real(X,set_or1633881224788618240n_real(Aa,Ba))
               => has_fi5821293074295781190e_real(F,zero_zero_real,topolo2177554685111907308n_real(X,top_top_set_real)) )
           => ( aa_real_real(F,Xb) = aa_real_real(F,Y) ) ) ) ) ) ).

% DERIV_isconst3
tff(fact_9713_DERIV__neg__dec__left,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)
       => ? [D4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D4)
            & ! [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),D4)
                 => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,Xb)),aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),H3))) ) ) ) ) ) ).

% DERIV_neg_dec_left
tff(fact_9714_DERIV__pos__inc__left,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)
       => ? [D4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D4)
            & ! [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),D4)
                 => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),H3))),aa_real_real(F,Xb)) ) ) ) ) ) ).

% DERIV_pos_inc_left
tff(fact_9715_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)
       => ? [D4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D4)
            & ! [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),D4)
                 => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),H3))),aa_real_real(F,Xb)) ) ) ) ) ) ).

% DERIV_neg_dec_right
tff(fact_9716_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)
       => ? [D4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D4)
            & ! [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),D4)
                 => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,Xb)),aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),H3))) ) ) ) ) ) ).

% DERIV_pos_inc_right
tff(fact_9717_DERIV__pos__imp__increasing,axiom,
    ! [Aa: real,Ba: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),Ba)
     => ( ! [X: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),X)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Ba)
             => ? [Y4: real] :
                  ( has_fi5821293074295781190e_real(F,Y4,topolo2177554685111907308n_real(X,top_top_set_real))
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y4) ) ) )
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,Aa)),aa_real_real(F,Ba)) ) ) ).

% DERIV_pos_imp_increasing
tff(fact_9718_DERIV__neg__imp__decreasing,axiom,
    ! [Aa: real,Ba: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),Ba)
     => ( ! [X: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),X)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Ba)
             => ? [Y4: real] :
                  ( has_fi5821293074295781190e_real(F,Y4,topolo2177554685111907308n_real(X,top_top_set_real))
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,Y4),zero_zero_real) ) ) )
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,Ba)),aa_real_real(F,Aa)) ) ) ).

% DERIV_neg_imp_decreasing
tff(fact_9719_DERIV__nonneg__imp__nondecreasing,axiom,
    ! [Aa: real,Ba: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),Ba)
     => ( ! [X: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),X)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Ba)
             => ? [Y4: real] :
                  ( has_fi5821293074295781190e_real(F,Y4,topolo2177554685111907308n_real(X,top_top_set_real))
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y4) ) ) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,Aa)),aa_real_real(F,Ba)) ) ) ).

% DERIV_nonneg_imp_nondecreasing
tff(fact_9720_DERIV__nonpos__imp__nonincreasing,axiom,
    ! [Aa: real,Ba: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),Ba)
     => ( ! [X: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),X)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Ba)
             => ? [Y4: real] :
                  ( has_fi5821293074295781190e_real(F,Y4,topolo2177554685111907308n_real(X,top_top_set_real))
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y4),zero_zero_real) ) ) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,Ba)),aa_real_real(F,Aa)) ) ) ).

% DERIV_nonpos_imp_nonincreasing
tff(fact_9721_deriv__nonneg__imp__mono,axiom,
    ! [Aa: real,Ba: real,G: fun_real_real,G2: fun_real_real] :
      ( ! [X: real] :
          ( member_real(X,set_or1222579329274155063t_real(Aa,Ba))
         => has_fi5821293074295781190e_real(G,aa_real_real(G2,X),topolo2177554685111907308n_real(X,top_top_set_real)) )
     => ( ! [X: real] :
            ( member_real(X,set_or1222579329274155063t_real(Aa,Ba))
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(G2,X)) )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),Ba)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(G,Aa)),aa_real_real(G,Ba)) ) ) ) ).

% deriv_nonneg_imp_mono
tff(fact_9722_has__real__derivative__neg__dec__left,axiom,
    ! [F: fun_real_real,L: real,Xb: real,S2: set_real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xb,S2))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,L),zero_zero_real)
       => ? [D4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D4)
            & ! [H3: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H3)
               => ( member_real(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),H3),S2)
                 => ( aa_real_o(aa_real_fun_real_o(ord_less_real,H3),D4)
                   => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,Xb)),aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),H3))) ) ) ) ) ) ) ).

% has_real_derivative_neg_dec_left
tff(fact_9723_has__real__derivative__pos__inc__left,axiom,
    ! [F: fun_real_real,L: real,Xb: real,S2: set_real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xb,S2))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),L)
       => ? [D4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D4)
            & ! [H3: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H3)
               => ( member_real(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),H3),S2)
                 => ( aa_real_o(aa_real_fun_real_o(ord_less_real,H3),D4)
                   => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),H3))),aa_real_real(F,Xb)) ) ) ) ) ) ) ).

% has_real_derivative_pos_inc_left
tff(fact_9724_has__real__derivative__neg__dec__right,axiom,
    ! [F: fun_real_real,L: real,Xb: real,S2: set_real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xb,S2))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,L),zero_zero_real)
       => ? [D4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D4)
            & ! [H3: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H3)
               => ( member_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),H3),S2)
                 => ( aa_real_o(aa_real_fun_real_o(ord_less_real,H3),D4)
                   => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),H3))),aa_real_real(F,Xb)) ) ) ) ) ) ) ).

% has_real_derivative_neg_dec_right
tff(fact_9725_has__real__derivative__pos__inc__right,axiom,
    ! [F: fun_real_real,L: real,Xb: real,S2: set_real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xb,S2))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),L)
       => ? [D4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D4)
            & ! [H3: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H3)
               => ( member_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),H3),S2)
                 => ( aa_real_o(aa_real_fun_real_o(ord_less_real,H3),D4)
                   => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,Xb)),aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),H3))) ) ) ) ) ) ) ).

% has_real_derivative_pos_inc_right
tff(fact_9726_MVT2,axiom,
    ! [Aa: real,Ba: real,F: fun_real_real,F4: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),Ba)
     => ( ! [X: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),X)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Ba)
             => has_fi5821293074295781190e_real(F,aa_real_real(F4,X),topolo2177554685111907308n_real(X,top_top_set_real)) ) )
       => ? [Z3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),Z3)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),Ba)
            & ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(F,Ba)),aa_real_real(F,Aa)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Ba),Aa)),aa_real_real(F4,Z3)) ) ) ) ) ).

% MVT2
tff(fact_9727_DERIV__local__const,axiom,
    ! [F: fun_real_real,L: real,Xb: real,D: 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),D)
       => ( ! [Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Y3))),D)
             => ( aa_real_real(F,Xb) = aa_real_real(F,Y3) ) )
         => ( L = zero_zero_real ) ) ) ) ).

% DERIV_local_const
tff(fact_9728_DERIV__ln,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(inverse_inverse_real,Xb),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ).

% DERIV_ln
tff(fact_9729_DERIV__local__min,axiom,
    ! [F: fun_real_real,L: real,Xb: real,D: 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),D)
       => ( ! [Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Y3))),D)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,Xb)),aa_real_real(F,Y3)) )
         => ( L = zero_zero_real ) ) ) ) ).

% DERIV_local_min
tff(fact_9730_DERIV__local__max,axiom,
    ! [F: fun_real_real,L: real,Xb: real,D: 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),D)
       => ( ! [Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Y3))),D)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,Y3)),aa_real_real(F,Xb)) )
         => ( L = zero_zero_real ) ) ) ) ).

% DERIV_local_max
tff(fact_9731_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(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),Xb),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ).

% DERIV_ln_divide
tff(fact_9732_DERIV__pow,axiom,
    ! [Na: nat,Xb: real,S3: set_real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(aTP_Lamm_yq,Na),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(suc,zero_zero_nat)))),topolo2177554685111907308n_real(Xb,S3)) ).

% DERIV_pow
tff(fact_9733_DERIV__fun__pow,axiom,
    ! [G: fun_real_real,M: real,Xb: real,Na: nat] :
      ( has_fi5821293074295781190e_real(G,M,topolo2177554685111907308n_real(Xb,top_top_set_real))
     => has_fi5821293074295781190e_real(aa_nat_fun_real_real(aTP_Lamm_yr(G),Na),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(G,Xb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat)))),M),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ).

% DERIV_fun_pow
tff(fact_9734_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(aa_rea7340605660300375871l_real(aTP_Lamm_ys,R2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Z),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,R2),one_one_real))),topolo2177554685111907308n_real(Z,top_top_set_real)) ) ).

% has_real_derivative_powr
tff(fact_9735_DERIV__series_H,axiom,
    ! [F: fun_re4238438383274954274t_real,F4: fun_re4238438383274954274t_real,X0: real,Aa: real,Ba: real,L5: fun_nat_real] :
      ( ! [N: nat] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(aTP_Lamm_hl(F),N),aa_nat_real(aa_real_fun_nat_real(F4,X0),N),topolo2177554685111907308n_real(X0,top_top_set_real))
     => ( ! [X: real] :
            ( member_real(X,set_or1633881224788618240n_real(Aa,Ba))
           => summable_real(aa_real_fun_nat_real(F,X)) )
       => ( member_real(X0,set_or1633881224788618240n_real(Aa,Ba))
         => ( summable_real(aa_real_fun_nat_real(F4,X0))
           => ( summable_real(L5)
             => ( ! [N: nat,X: real,Y3: real] :
                    ( member_real(X,set_or1633881224788618240n_real(Aa,Ba))
                   => ( member_real(Y3,set_or1633881224788618240n_real(Aa,Ba))
                     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(F,X),N)),aa_nat_real(aa_real_fun_nat_real(F,Y3),N)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(L5,N)),aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X),Y3)))) ) )
               => has_fi5821293074295781190e_real(aTP_Lamm_yt(F),suminf_real(aa_real_fun_nat_real(F4,X0)),topolo2177554685111907308n_real(X0,top_top_set_real)) ) ) ) ) ) ) ).

% DERIV_series'
tff(fact_9736_DERIV__log,axiom,
    ! [Xb: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => has_fi5821293074295781190e_real(aa_rea7340605660300375871l_real(log,Ba),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(ln_ln_real,Ba)),Xb)),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ).

% DERIV_log
tff(fact_9737_DERIV__fun__powr,axiom,
    ! [G: fun_real_real,M: real,Xb: real,R2: real] :
      ( has_fi5821293074295781190e_real(G,M,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_yu(G),R2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(G,Xb)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,R2),aa_nat_real(semiri5074537144036343181t_real,one_one_nat))))),M),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ) ).

% DERIV_fun_powr
tff(fact_9738_DERIV__powr,axiom,
    ! [G: fun_real_real,M: real,Xb: real,F: fun_real_real,R2: real] :
      ( has_fi5821293074295781190e_real(G,M,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_yv(G),F),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(G,Xb)),aa_real_real(F,Xb))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),aa_real_real(ln_ln_real,aa_real_real(G,Xb)))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,M),aa_real_real(F,Xb))),aa_real_real(G,Xb)))),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ) ) ).

% DERIV_powr
tff(fact_9739_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(aa_rea7340605660300375871l_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_9740_DERIV__real__sqrt__generic,axiom,
    ! [Xb: real,D6: real] :
      ( ( Xb != zero_zero_real )
     => ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
         => ( D6 = aa_real_real(aa_rea7340605660300375871l_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)
           => ( D6 = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,aa_real_real(inverse_inverse_real,aa_real_real(sqrt,Xb)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) )
         => has_fi5821293074295781190e_real(sqrt,D6,topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ) ) ).

% DERIV_real_sqrt_generic
tff(fact_9741_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(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,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_9742_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(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),topolo2177554685111907308n_real(Xb,A3)) ) ).

% artanh_real_has_field_derivative
tff(fact_9743_DERIV__power__series_H,axiom,
    ! [R: real,F: fun_nat_real,X0: real] :
      ( ! [X: real] :
          ( member_real(X,set_or1633881224788618240n_real(aa_real_real(uminus_uminus_real,R),R))
         => summable_real(aa_real_fun_nat_real(aTP_Lamm_yw(F),X)) )
     => ( 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_yy(F),suminf_real(aa_real_fun_nat_real(aTP_Lamm_yw(F),X0)),topolo2177554685111907308n_real(X0,top_top_set_real)) ) ) ) ).

% DERIV_power_series'
tff(fact_9744_DERIV__real__root,axiom,
    ! [Na: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,Na),aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Na),Xb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(suc,zero_zero_nat))))),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ) ).

% DERIV_real_root
tff(fact_9745_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(aa_rea7340605660300375871l_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_9746_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(aa_rea7340605660300375871l_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_9747_Maclaurin__all__le,axiom,
    ! [Diff: fun_na6724076748692969250l_real,F: fun_real_real,Xb: real,Na: nat] :
      ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
     => ( ! [M3: nat,X: real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M3),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M3)),X),topolo2177554685111907308n_real(X,top_top_set_real))
       => ? [T4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T4)),aa_real_real(abs_abs_real,Xb))
            & ( aa_real_real(F,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_yz(Diff),Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Na))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Diff,Na),T4)),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Na))) ) ) ) ) ).

% Maclaurin_all_le
tff(fact_9748_Maclaurin__all__le__objl,axiom,
    ! [Diff: fun_na6724076748692969250l_real,F: fun_real_real,Xb: real,Na: nat] :
      ( ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
        & ! [M3: nat,X: real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M3),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M3)),X),topolo2177554685111907308n_real(X,top_top_set_real)) )
     => ? [T4: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T4)),aa_real_real(abs_abs_real,Xb))
          & ( aa_real_real(F,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_yz(Diff),Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Na))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Diff,Na),T4)),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Na))) ) ) ) ).

% Maclaurin_all_le_objl
tff(fact_9749_DERIV__odd__real__root,axiom,
    ! [Na: 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))),Na)
     => ( ( Xb != zero_zero_real )
       => has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,Na),aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Na),Xb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(suc,zero_zero_nat))))),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ) ).

% DERIV_odd_real_root
tff(fact_9750_Maclaurin,axiom,
    ! [H: real,Na: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
         => ( ! [M3: nat,T4: real] :
                ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M3),Na)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T4)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T4),H) )
               => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M3),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M3)),T4),topolo2177554685111907308n_real(T4,top_top_set_real)) )
           => ? [T4: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),T4)
                & aa_real_o(aa_real_fun_real_o(ord_less_real,T4),H)
                & ( aa_real_real(F,H) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_fun1113820866546176217t_real(aTP_Lamm_uj(H),Diff)),aa_nat_set_nat(set_ord_lessThan_nat,Na))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Diff,Na),T4)),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,H),Na))) ) ) ) ) ) ) ).

% Maclaurin
tff(fact_9751_Maclaurin2,axiom,
    ! [H: real,Diff: fun_na6724076748692969250l_real,F: fun_real_real,Na: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H)
     => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
       => ( ! [M3: nat,T4: real] :
              ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M3),Na)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T4)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T4),H) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M3),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M3)),T4),topolo2177554685111907308n_real(T4,top_top_set_real)) )
         => ? [T4: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),T4)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T4),H)
              & ( aa_real_real(F,H) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_fun1113820866546176217t_real(aTP_Lamm_uj(H),Diff)),aa_nat_set_nat(set_ord_lessThan_nat,Na))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Diff,Na),T4)),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,H),Na))) ) ) ) ) ) ).

% Maclaurin2
tff(fact_9752_Maclaurin__minus,axiom,
    ! [H: real,Na: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,H),zero_zero_real)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
         => ( ! [M3: nat,T4: real] :
                ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M3),Na)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,H),T4)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T4),zero_zero_real) )
               => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M3),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M3)),T4),topolo2177554685111907308n_real(T4,top_top_set_real)) )
           => ? [T4: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,H),T4)
                & aa_real_o(aa_real_fun_real_o(ord_less_real,T4),zero_zero_real)
                & ( aa_real_real(F,H) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_fun1113820866546176217t_real(aTP_Lamm_uj(H),Diff)),aa_nat_set_nat(set_ord_lessThan_nat,Na))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Diff,Na),T4)),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,H),Na))) ) ) ) ) ) ) ).

% Maclaurin_minus
tff(fact_9753_Maclaurin__all__lt,axiom,
    ! [Diff: fun_na6724076748692969250l_real,F: fun_real_real,Na: 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),Na)
       => ( ( Xb != zero_zero_real )
         => ( ! [M3: nat,X: real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M3),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M3)),X),topolo2177554685111907308n_real(X,top_top_set_real))
           => ? [T4: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(abs_abs_real,T4))
                & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,T4)),aa_real_real(abs_abs_real,Xb))
                & ( aa_real_real(F,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_yz(Diff),Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Na))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Diff,Na),T4)),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Na))) ) ) ) ) ) ) ).

% Maclaurin_all_lt
tff(fact_9754_Maclaurin__bi__le,axiom,
    ! [Diff: fun_na6724076748692969250l_real,F: fun_real_real,Na: nat,Xb: real] :
      ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
     => ( ! [M3: nat,T4: real] :
            ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M3),Na)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T4)),aa_real_real(abs_abs_real,Xb)) )
           => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M3),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M3)),T4),topolo2177554685111907308n_real(T4,top_top_set_real)) )
       => ? [T4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T4)),aa_real_real(abs_abs_real,Xb))
            & ( aa_real_real(F,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_yz(Diff),Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Na))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Diff,Na),T4)),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Na))) ) ) ) ) ).

% Maclaurin_bi_le
tff(fact_9755_Taylor__down,axiom,
    ! [Na: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real,Aa: real,Ba: real,C: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
       => ( ! [M3: nat,T4: real] :
              ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M3),Na)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),T4)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T4),Ba) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M3),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M3)),T4),topolo2177554685111907308n_real(T4,top_top_set_real)) )
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),C)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),Ba)
             => ? [T4: real] :
                  ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),T4)
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,T4),C)
                  & ( aa_real_real(F,Aa) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_za(Diff),Aa),C)),aa_nat_set_nat(set_ord_lessThan_nat,Na))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Diff,Na),T4)),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Aa),C)),Na))) ) ) ) ) ) ) ) ).

% Taylor_down
tff(fact_9756_Taylor__up,axiom,
    ! [Na: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real,Aa: real,Ba: real,C: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
       => ( ! [M3: nat,T4: real] :
              ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M3),Na)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),T4)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T4),Ba) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M3),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M3)),T4),topolo2177554685111907308n_real(T4,top_top_set_real)) )
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),C)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),Ba)
             => ? [T4: real] :
                  ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),T4)
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,T4),Ba)
                  & ( aa_real_real(F,Ba) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_za(Diff),Ba),C)),aa_nat_set_nat(set_ord_lessThan_nat,Na))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Diff,Na),T4)),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Ba),C)),Na))) ) ) ) ) ) ) ) ).

% Taylor_up
tff(fact_9757_Taylor,axiom,
    ! [Na: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real,Aa: real,Ba: real,C: real,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
       => ( ! [M3: nat,T4: real] :
              ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M3),Na)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),T4)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T4),Ba) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M3),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M3)),T4),topolo2177554685111907308n_real(T4,top_top_set_real)) )
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),C)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),Ba)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),Xb)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Ba)
                 => ( ( Xb != C )
                   => ? [T4: 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),T4)
                            & aa_real_o(aa_real_fun_real_o(ord_less_real,T4),C) ),
                            ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),T4)
                            & aa_real_o(aa_real_fun_real_o(ord_less_real,T4),Xb) ) )
                        & ( aa_real_real(F,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_zb(Diff),C),Xb)),aa_nat_set_nat(set_ord_lessThan_nat,Na))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Diff,Na),T4)),aa_nat_real(semiri2265585572941072030t_real,Na))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),C)),Na))) ) ) ) ) ) ) ) ) ) ) ).

% Taylor
tff(fact_9758_Maclaurin__lemma2,axiom,
    ! [Na: nat,H: real,Diff: fun_na6724076748692969250l_real,K: nat,B4: real] :
      ( ! [M3: nat,T4: real] :
          ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M3),Na)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T4)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T4),H) )
         => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M3),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M3)),T4),topolo2177554685111907308n_real(T4,top_top_set_real)) )
     => ( ( Na = aa_nat_nat(suc,K) )
       => ! [M2: nat,T5: real] :
            ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Na)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T5)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T5),H) )
           => has_fi5821293074295781190e_real(aa_nat_fun_real_real(aa_rea1103652080130114956l_real(aa_fun2567993205388704171l_real(aTP_Lamm_zd(Na),Diff),B4),M2),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M2)),T5)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_ze(Diff),M2),T5)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(suc,M2))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B4),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,T5),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(suc,M2)))),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(suc,M2))))))),topolo2177554685111907308n_real(T5,top_top_set_real)) ) ) ) ).

% Maclaurin_lemma2
tff(fact_9759_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_zf,suminf_real(aa_real_fun_nat_real(aTP_Lamm_zg,Xb)),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ).

% DERIV_arctan_series
tff(fact_9760_DERIV__real__root__generic,axiom,
    ! [Na: nat,Xb: real,D6: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( ( 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))),Na)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
             => ( D6 = aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Na),Xb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(suc,zero_zero_nat))))) ) ) )
         => ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real)
               => ( D6 = aa_real_real(uminus_uminus_real,aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Na),Xb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(suc,zero_zero_nat)))))) ) ) )
           => ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
               => ( D6 = aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Na)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Na),Xb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),aa_nat_nat(suc,zero_zero_nat))))) ) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,Na),D6,topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ) ) ) ) ).

% DERIV_real_root_generic
tff(fact_9761_LIM__fun__less__zero,axiom,
    ! [F: fun_real_real,L: real,C: real] :
      ( filterlim_real_real(F,topolo2815343760600316023s_real(L),topolo2177554685111907308n_real(C,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,L),zero_zero_real)
       => ? [R3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R3)
            & ! [X3: real] :
                ( ( ( X3 != C )
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),X3))),R3) )
               => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,X3)),zero_zero_real) ) ) ) ) ).

% LIM_fun_less_zero
tff(fact_9762_LIM__fun__not__zero,axiom,
    ! [F: fun_real_real,L: real,C: real] :
      ( filterlim_real_real(F,topolo2815343760600316023s_real(L),topolo2177554685111907308n_real(C,top_top_set_real))
     => ( ( L != zero_zero_real )
       => ? [R3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R3)
            & ! [X3: real] :
                ( ( ( X3 != C )
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),X3))),R3) )
               => ( aa_real_real(F,X3) != zero_zero_real ) ) ) ) ) ).

% LIM_fun_not_zero
tff(fact_9763_LIM__fun__gt__zero,axiom,
    ! [F: fun_real_real,L: real,C: real] :
      ( filterlim_real_real(F,topolo2815343760600316023s_real(L),topolo2177554685111907308n_real(C,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),L)
       => ? [R3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R3)
            & ! [X3: real] :
                ( ( ( X3 != C )
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),X3))),R3) )
               => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(F,X3)) ) ) ) ) ).

% LIM_fun_gt_zero
tff(fact_9764_summable__Leibniz_I2_J,axiom,
    ! [Aa: fun_nat_real] :
      ( filterlim_nat_real(Aa,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( topolo6980174941875973593q_real(Aa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(Aa,zero_zero_nat))
         => ! [N5: nat] : member_real(suminf_real(aTP_Lamm_zh(Aa)),set_or1222579329274155063t_real(aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_zh(Aa)),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))),N5))),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_zh(Aa)),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))),N5)),one_one_nat))))) ) ) ) ).

% summable_Leibniz(2)
tff(fact_9765_summable__Leibniz_I3_J,axiom,
    ! [Aa: fun_nat_real] :
      ( filterlim_nat_real(Aa,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( topolo6980174941875973593q_real(Aa)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(Aa,zero_zero_nat)),zero_zero_real)
         => ! [N5: nat] : member_real(suminf_real(aTP_Lamm_zh(Aa)),set_or1222579329274155063t_real(aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_zh(Aa)),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))),N5)),one_one_nat))),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_zh(Aa)),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))),N5))))) ) ) ) ).

% summable_Leibniz(3)
tff(fact_9766_summable__Leibniz_H_I4_J,axiom,
    ! [Aa: fun_nat_real,Na: nat] :
      ( filterlim_nat_real(Aa,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(Aa,N))
       => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(Aa,aa_nat_nat(suc,N))),aa_nat_real(Aa,N))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,suminf_real(aTP_Lamm_zh(Aa))),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_zh(Aa)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)),one_one_nat)))) ) ) ) ).

% summable_Leibniz'(4)
tff(fact_9767_trivial__limit__sequentially,axiom,
    at_top_nat != bot_bot_filter_nat ).

% trivial_limit_sequentially
tff(fact_9768_filterlim__Suc,axiom,
    filterlim_nat_nat(suc,at_top_nat,at_top_nat) ).

% filterlim_Suc
tff(fact_9769_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_9770_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_zi,C),at_top_nat,at_top_nat) ) ).

% mult_nat_right_at_top
tff(fact_9771_monoseq__convergent,axiom,
    ! [X5: fun_nat_real,B4: real] :
      ( topolo6980174941875973593q_real(X5)
     => ( ! [I2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_nat_real(X5,I2))),B4)
       => ~ ! [L6: real] : ~ filterlim_nat_real(X5,topolo2815343760600316023s_real(L6),at_top_nat) ) ) ).

% monoseq_convergent
tff(fact_9772_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_zj(F),G),topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
           => ? [L4: real] :
                ( ! [N5: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,N5)),L4)
                & filterlim_nat_real(F,topolo2815343760600316023s_real(L4),at_top_nat)
                & ! [N5: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L4),aa_nat_real(G,N5))
                & filterlim_nat_real(G,topolo2815343760600316023s_real(L4),at_top_nat) ) ) ) ) ) ).

% nested_sequence_unique
tff(fact_9773_LIMSEQ__inverse__zero,axiom,
    ! [X5: fun_nat_real] :
      ( ! [R3: real] :
        ? [N6: nat] :
        ! [N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N6),N)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,R3),aa_nat_real(X5,N)) )
     => filterlim_nat_real(aTP_Lamm_zk(X5),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_inverse_zero
tff(fact_9774_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_zl,C),topolo2815343760600316023s_real(one_one_real),at_top_nat) ) ).

% LIMSEQ_root_const
tff(fact_9775_LIMSEQ__inverse__real__of__nat,axiom,
    filterlim_nat_real(aTP_Lamm_zm,topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ).

% LIMSEQ_inverse_real_of_nat
tff(fact_9776_LIMSEQ__inverse__real__of__nat__add,axiom,
    ! [R2: real] : filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_zn,R2),topolo2815343760600316023s_real(R2),at_top_nat) ).

% LIMSEQ_inverse_real_of_nat_add
tff(fact_9777_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)
             => ? [N5: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(F,N5)),E)) )
         => filterlim_nat_real(F,topolo2815343760600316023s_real(L),at_top_nat) ) ) ) ).

% increasing_LIMSEQ
tff(fact_9778_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_9779_LIMSEQ__divide__realpow__zero,axiom,
    ! [Xb: real,Aa: 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_zo(Xb),Aa),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_divide_realpow_zero
tff(fact_9780_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_9781_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_9782_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_zp,Xb),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_inverse_realpow_zero
tff(fact_9783_LIMSEQ__inverse__real__of__nat__add__minus,axiom,
    ! [R2: real] : filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_zq,R2),topolo2815343760600316023s_real(R2),at_top_nat) ).

% LIMSEQ_inverse_real_of_nat_add_minus
tff(fact_9784_LIMSEQ__inverse__real__of__nat__add__minus__mult,axiom,
    ! [R2: real] : filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_zr,R2),topolo2815343760600316023s_real(R2),at_top_nat) ).

% LIMSEQ_inverse_real_of_nat_add_minus_mult
tff(fact_9785_summable,axiom,
    ! [Aa: fun_nat_real] :
      ( filterlim_nat_real(Aa,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(Aa,N))
       => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(Aa,aa_nat_nat(suc,N))),aa_nat_real(Aa,N))
         => summable_real(aTP_Lamm_zh(Aa)) ) ) ) ).

% summable
tff(fact_9786_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_ld,Xb),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% zeroseq_arctan_series
tff(fact_9787_summable__Leibniz_H_I2_J,axiom,
    ! [Aa: fun_nat_real,Na: nat] :
      ( filterlim_nat_real(Aa,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(Aa,N))
       => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(Aa,aa_nat_nat(suc,N))),aa_nat_real(Aa,N))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_zh(Aa)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)))),suminf_real(aTP_Lamm_zh(Aa))) ) ) ) ).

% summable_Leibniz'(2)
tff(fact_9788_summable__Leibniz_H_I3_J,axiom,
    ! [Aa: fun_nat_real] :
      ( filterlim_nat_real(Aa,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(Aa,N))
       => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(Aa,aa_nat_nat(suc,N))),aa_nat_real(Aa,N))
         => filterlim_nat_real(aTP_Lamm_zs(Aa),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_zh(Aa))),at_top_nat) ) ) ) ).

% summable_Leibniz'(3)
tff(fact_9789_sums__alternating__upper__lower,axiom,
    ! [Aa: fun_nat_real] :
      ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(Aa,aa_nat_nat(suc,N))),aa_nat_real(Aa,N))
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(Aa,N))
       => ( filterlim_nat_real(Aa,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
         => ? [L4: real] :
              ( ! [N5: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_zh(Aa)),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))),N5)))),L4)
              & filterlim_nat_real(aTP_Lamm_zs(Aa),topolo2815343760600316023s_real(L4),at_top_nat)
              & ! [N5: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L4),aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_zh(Aa)),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))),N5)),one_one_nat))))
              & filterlim_nat_real(aTP_Lamm_zt(Aa),topolo2815343760600316023s_real(L4),at_top_nat) ) ) ) ) ).

% sums_alternating_upper_lower
tff(fact_9790_summable__Leibniz_I5_J,axiom,
    ! [Aa: fun_nat_real] :
      ( filterlim_nat_real(Aa,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( topolo6980174941875973593q_real(Aa)
       => filterlim_nat_real(aTP_Lamm_zt(Aa),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_zh(Aa))),at_top_nat) ) ) ).

% summable_Leibniz(5)
tff(fact_9791_summable__Leibniz_H_I5_J,axiom,
    ! [Aa: fun_nat_real] :
      ( filterlim_nat_real(Aa,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(Aa,N))
       => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(Aa,aa_nat_nat(suc,N))),aa_nat_real(Aa,N))
         => filterlim_nat_real(aTP_Lamm_zt(Aa),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_zh(Aa))),at_top_nat) ) ) ) ).

% summable_Leibniz'(5)
tff(fact_9792_DERIV__neg__imp__decreasing__at__top,axiom,
    ! [Ba: real,F: fun_real_real,Flim: real] :
      ( ! [X: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ba),X)
         => ? [Y4: real] :
              ( has_fi5821293074295781190e_real(F,Y4,topolo2177554685111907308n_real(X,top_top_set_real))
              & aa_real_o(aa_real_fun_real_o(ord_less_real,Y4),zero_zero_real) ) )
     => ( filterlim_real_real(F,topolo2815343760600316023s_real(Flim),at_top_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Flim),aa_real_real(F,Ba)) ) ) ).

% DERIV_neg_imp_decreasing_at_top
tff(fact_9793_filterlim__pow__at__bot__even,axiom,
    ! [Na: nat,F: fun_real_real,F3: filter_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( filterlim_real_real(F,at_bot_real,F3)
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
         => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_zu(Na),F),at_top_real,F3) ) ) ) ).

% filterlim_pow_at_bot_even
tff(fact_9794_at__top__le__at__infinity,axiom,
    ord_le4104064031414453916r_real(at_top_real,at_infinity_real) ).

% at_top_le_at_infinity
tff(fact_9795_at__bot__le__at__infinity,axiom,
    ord_le4104064031414453916r_real(at_bot_real,at_infinity_real) ).

% at_bot_le_at_infinity
tff(fact_9796_DERIV__pos__imp__increasing__at__bot,axiom,
    ! [Ba: real,F: fun_real_real,Flim: real] :
      ( ! [X: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Ba)
         => ? [Y4: real] :
              ( has_fi5821293074295781190e_real(F,Y4,topolo2177554685111907308n_real(X,top_top_set_real))
              & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y4) ) )
     => ( filterlim_real_real(F,topolo2815343760600316023s_real(Flim),at_bot_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Flim),aa_real_real(F,Ba)) ) ) ).

% DERIV_pos_imp_increasing_at_bot
tff(fact_9797_filterlim__pow__at__bot__odd,axiom,
    ! [Na: nat,F: fun_real_real,F3: filter_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( filterlim_real_real(F,at_bot_real,F3)
       => ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Na)
         => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_zu(Na),F),at_bot_real,F3) ) ) ) ).

% filterlim_pow_at_bot_odd
tff(fact_9798_eventually__sequentially__Suc,axiom,
    ! [P: fun_nat_o] :
      ( eventually_nat(aTP_Lamm_zv(P),at_top_nat)
    <=> eventually_nat(P,at_top_nat) ) ).

% eventually_sequentially_Suc
tff(fact_9799_eventually__sequentiallyI,axiom,
    ! [C: nat,P: fun_nat_o] :
      ( ! [X: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),X)
         => aa_nat_o(P,X) )
     => eventually_nat(P,at_top_nat) ) ).

% eventually_sequentiallyI
tff(fact_9800_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_9801_le__sequentially,axiom,
    ! [F3: filter_nat] :
      ( ord_le2510731241096832064er_nat(F3,at_top_nat)
    <=> ! [N8: nat] : eventually_nat(aa_nat_fun_nat_o(ord_less_eq_nat,N8),F3) ) ).

% le_sequentially
tff(fact_9802_eventually__at__left__real,axiom,
    ! [Ba: real,Aa: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Ba),Aa)
     => eventually_real(aa_real_fun_real_o(aTP_Lamm_zw(Ba),Aa),topolo2177554685111907308n_real(Aa,set_or5984915006950818249n_real(Aa))) ) ).

% eventually_at_left_real
tff(fact_9803_Bseq__eq__bounded,axiom,
    ! [F: fun_nat_real,Aa: real,Ba: real] :
      ( ord_less_eq_set_real(image_nat_real(F,top_top_set_nat),set_or1222579329274155063t_real(Aa,Ba))
     => bfun_nat_real(F,at_top_nat) ) ).

% Bseq_eq_bounded
tff(fact_9804_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_9805_eventually__at__right__real,axiom,
    ! [Aa: real,Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),Ba)
     => eventually_real(aa_real_fun_real_o(aTP_Lamm_zw(Aa),Ba),topolo2177554685111907308n_real(Aa,set_or5849166863359141190n_real(Aa))) ) ).

% eventually_at_right_real
tff(fact_9806_INT__greaterThan__UNIV,axiom,
    comple7806235888213564991et_nat(image_nat_set_nat(set_or1210151606488870762an_nat,top_top_set_nat)) = bot_bot_set_nat ).

% INT_greaterThan_UNIV
tff(fact_9807_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_9808_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_9809_atLeast__0,axiom,
    aa_nat_set_nat(set_ord_atLeast_nat,zero_zero_nat) = top_top_set_nat ).

% atLeast_0
tff(fact_9810_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_9811_GreatestI__ex__nat,axiom,
    ! [P: fun_nat_o,Ba: nat] :
      ( ? [X_13: nat] : aa_nat_o(P,X_13)
     => ( ! [Y3: nat] :
            ( aa_nat_o(P,Y3)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y3),Ba) )
       => aa_nat_o(P,order_Greatest_nat(P)) ) ) ).

% GreatestI_ex_nat
tff(fact_9812_Greatest__le__nat,axiom,
    ! [P: fun_nat_o,K: nat,Ba: nat] :
      ( aa_nat_o(P,K)
     => ( ! [Y3: nat] :
            ( aa_nat_o(P,Y3)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y3),Ba) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),order_Greatest_nat(P)) ) ) ).

% Greatest_le_nat
tff(fact_9813_GreatestI__nat,axiom,
    ! [P: fun_nat_o,K: nat,Ba: nat] :
      ( aa_nat_o(P,K)
     => ( ! [Y3: nat] :
            ( aa_nat_o(P,Y3)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y3),Ba) )
       => aa_nat_o(P,order_Greatest_nat(P)) ) ) ).

% GreatestI_nat
tff(fact_9814_decseq__bounded,axiom,
    ! [X5: fun_nat_real,B4: real] :
      ( order_9091379641038594480t_real(X5)
     => ( ! [I2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B4),aa_nat_real(X5,I2))
       => bfun_nat_real(X5,at_top_nat) ) ) ).

% decseq_bounded
tff(fact_9815_decseq__convergent,axiom,
    ! [X5: fun_nat_real,B4: real] :
      ( order_9091379641038594480t_real(X5)
     => ( ! [I2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B4),aa_nat_real(X5,I2))
       => ~ ! [L6: real] :
              ( filterlim_nat_real(X5,topolo2815343760600316023s_real(L6),at_top_nat)
             => ~ ! [I4: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L6),aa_nat_real(X5,I4)) ) ) ) ).

% decseq_convergent
tff(fact_9816_UN__atLeast__UNIV,axiom,
    comple7399068483239264473et_nat(image_nat_set_nat(set_ord_atLeast_nat,top_top_set_nat)) = top_top_set_nat ).

% UN_atLeast_UNIV
tff(fact_9817_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_9818_Gcd__eq__Max,axiom,
    ! [M7: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,M7)
     => ( ( M7 != bot_bot_set_nat )
       => ( ~ aa_set_nat_o(member_nat(zero_zero_nat),M7)
         => ( aa_set_nat_nat(gcd_Gcd_nat,M7) = aa_set_nat_nat(lattic8265883725875713057ax_nat,comple7806235888213564991et_nat(image_nat_set_nat(aTP_Lamm_zx,M7))) ) ) ) ) ).

% Gcd_eq_Max
tff(fact_9819_Max__divisors__self__nat,axiom,
    ! [Na: nat] :
      ( ( Na != zero_zero_nat )
     => ( aa_set_nat_nat(lattic8265883725875713057ax_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_cf,Na))) = Na ) ) ).

% Max_divisors_self_nat
tff(fact_9820_card__le__Suc__Max,axiom,
    ! [S2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(finite_card_nat,S2)),aa_nat_nat(suc,aa_set_nat_nat(lattic8265883725875713057ax_nat,S2))) ) ).

% card_le_Suc_Max
tff(fact_9821_Sup__nat__def,axiom,
    ! [X5: set_nat] :
      aa_set_nat_nat(complete_Sup_Sup_nat,X5) = $ite(X5 = bot_bot_set_nat,zero_zero_nat,aa_set_nat_nat(lattic8265883725875713057ax_nat,X5)) ).

% Sup_nat_def
tff(fact_9822_divide__nat__def,axiom,
    ! [M: nat,Na: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Na) = $ite(Na = zero_zero_nat,zero_zero_nat,aa_set_nat_nat(lattic8265883725875713057ax_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_zy(M),Na)))) ).

% divide_nat_def
tff(fact_9823_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,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( Xb = vEBT_Node(Mima,Deg2,TreeList2,Summary2) )
             => ( ( Deg2 = Xaa )
                & $let(
                    n: nat,
                    n:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                    $let(
                      m: nat,
                      m:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),n),
                      ( ! [X4: vEBT_VEBT] :
                          ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList2))
                         => aa_nat_o(vEBT_VEBT_valid(X4),n) )
                      & aa_nat_o(vEBT_VEBT_valid(Summary2),m)
                      & ( 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))),m) )
                      & aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat(( ~ ? [X7: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary2),X7)
                            & ! [X4: vEBT_VEBT] :
                                ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList2))
                               => ~ ? [X7: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X4),X7) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_zz(Deg2),TreeList2),Summary2),n),m))),Mima) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(3)
tff(fact_9824_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,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( Xb = vEBT_Node(Mima,Deg2,TreeList2,Summary2) )
             => ~ ( ( Deg2 = Xaa )
                  & $let(
                      n: nat,
                      n:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                      $let(
                        m: nat,
                        m:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),n),
                        ( ! [X4: vEBT_VEBT] :
                            ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList2))
                           => aa_nat_o(vEBT_VEBT_valid(X4),n) )
                        & aa_nat_o(vEBT_VEBT_valid(Summary2),m)
                        & ( 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))),m) )
                        & aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat(( ~ ? [X7: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary2),X7)
                              & ! [X4: vEBT_VEBT] :
                                  ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList2))
                                 => ~ ? [X7: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X4),X7) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_zz(Deg2),TreeList2),Summary2),n),m))),Mima) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(2)
tff(fact_9825_VEBT__internal_Ovalid_H_Osimps_I2_J,axiom,
    ! [Mima2: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Deg4: nat] :
      ( aa_nat_o(vEBT_VEBT_valid(vEBT_Node(Mima2,Deg,TreeList,Summary)),Deg4)
    <=> ( ( Deg = Deg4 )
        & $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(
              m: nat,
              m:= 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),m)
              & ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m) )
              & aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat(( ~ ? [X7: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary),X7)
                    & ! [X4: vEBT_VEBT] :
                        ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList))
                       => ~ ? [X7: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X4),X7) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_zz(Deg),TreeList),Summary),n),m))),Mima2) ) ) ) ) ) ).

% VEBT_internal.valid'.simps(2)
tff(fact_9826_VEBT__internal_Ovalid_H_Oelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( aa_nat_o(vEBT_VEBT_valid(Xb),Xaa)
      <=> (Y) )
     => ( ( ? [Uu2: $o,Uv2: $o] : Xb = vEBT_Leaf((Uu2),(Uv2))
         => ( (Y)
          <=> ( Xaa != one_one_nat ) ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( Xb = vEBT_Node(Mima,Deg2,TreeList2,Summary2) )
             => ( (Y)
              <=> ~ ( ( Deg2 = Xaa )
                    & $let(
                        n: nat,
                        n:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                        $let(
                          m: nat,
                          m:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),n),
                          ( ! [X4: vEBT_VEBT] :
                              ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList2))
                             => aa_nat_o(vEBT_VEBT_valid(X4),n) )
                          & aa_nat_o(vEBT_VEBT_valid(Summary2),m)
                          & ( 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))),m) )
                          & aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat(( ~ ? [X7: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary2),X7)
                                & ! [X4: vEBT_VEBT] :
                                    ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList2))
                                   => ~ ? [X7: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X4),X7) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_zz(Deg2),TreeList2),Summary2),n),m))),Mima) ) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(1)
tff(fact_9827_VEBT__internal_Ovalid_H_Opelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( aa_nat_o(vEBT_VEBT_valid(Xb),Xaa)
      <=> (Y) )
     => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel),produc738532404422230701BT_nat(Xb,Xaa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( Xb = vEBT_Leaf((Uu2),(Uv2)) )
             => ( ( (Y)
                <=> ( Xaa = one_one_nat ) )
               => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel),produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(Uv2)),Xaa)) ) )
         => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Mima,Deg2,TreeList2,Summary2) )
               => ( ( (Y)
                  <=> ( ( Deg2 = Xaa )
                      & $let(
                          n: nat,
                          n:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                          $let(
                            m: nat,
                            m:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),n),
                            ( ! [X4: vEBT_VEBT] :
                                ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList2))
                               => aa_nat_o(vEBT_VEBT_valid(X4),n) )
                            & aa_nat_o(vEBT_VEBT_valid(Summary2),m)
                            & ( 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))),m) )
                            & aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat(( ~ ? [X7: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary2),X7)
                                  & ! [X4: vEBT_VEBT] :
                                      ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList2))
                                     => ~ ? [X7: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X4),X7) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_zz(Deg2),TreeList2),Summary2),n),m))),Mima) ) ) ) ) )
                 => ~ aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel),produc738532404422230701BT_nat(vEBT_Node(Mima,Deg2,TreeList2,Summary2),Xaa)) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(1)
tff(fact_9828_VEBT__internal_Ovalid_H_Opelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(vEBT_VEBT_valid(Xb),Xaa)
     => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel),produc738532404422230701BT_nat(Xb,Xaa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( Xb = vEBT_Leaf((Uu2),(Uv2)) )
             => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel),produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(Uv2)),Xaa))
               => ( Xaa != one_one_nat ) ) )
         => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Mima,Deg2,TreeList2,Summary2) )
               => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel),produc738532404422230701BT_nat(vEBT_Node(Mima,Deg2,TreeList2,Summary2),Xaa))
                 => ~ ( ( Deg2 = Xaa )
                      & $let(
                          n: nat,
                          n:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                          $let(
                            m: nat,
                            m:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),n),
                            ( ! [X4: vEBT_VEBT] :
                                ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList2))
                               => aa_nat_o(vEBT_VEBT_valid(X4),n) )
                            & aa_nat_o(vEBT_VEBT_valid(Summary2),m)
                            & ( 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))),m) )
                            & aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat(( ~ ? [X7: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary2),X7)
                                  & ! [X4: vEBT_VEBT] :
                                      ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList2))
                                     => ~ ? [X7: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X4),X7) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_zz(Deg2),TreeList2),Summary2),n),m))),Mima) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(2)
tff(fact_9829_Sup__int__def,axiom,
    ! [X5: set_int] : complete_Sup_Sup_int(X5) = the_int(aTP_Lamm_aaa(X5)) ).

% Sup_int_def
tff(fact_9830_VEBT__internal_Ovalid_H_Opelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(vEBT_VEBT_valid(Xb),Xaa)
     => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel),produc738532404422230701BT_nat(Xb,Xaa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( Xb = vEBT_Leaf((Uu2),(Uv2)) )
             => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel),produc738532404422230701BT_nat(vEBT_Leaf((Uu2),(Uv2)),Xaa))
               => ( Xaa = one_one_nat ) ) )
         => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Mima,Deg2,TreeList2,Summary2) )
               => ( aa_Pro632845113475883111_nat_o(accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel),produc738532404422230701BT_nat(vEBT_Node(Mima,Deg2,TreeList2,Summary2),Xaa))
                 => ( ( Deg2 = Xaa )
                    & $let(
                        n: nat,
                        n:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                        $let(
                          m: nat,
                          m:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg2),n),
                          ( ! [X4: vEBT_VEBT] :
                              ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList2))
                             => aa_nat_o(vEBT_VEBT_valid(X4),n) )
                          & aa_nat_o(vEBT_VEBT_valid(Summary2),m)
                          & ( 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))),m) )
                          & aa_opt7112393969400857165_nat_o(case_o184042715313410164at_nat(( ~ ? [X7: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary2),X7)
                                & ! [X4: vEBT_VEBT] :
                                    ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList2))
                                   => ~ ? [X7: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X4),X7) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_zz(Deg2),TreeList2),Summary2),n),m))),Mima) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(3)
tff(fact_9831_uniformity__real__def,axiom,
    topolo1511823702728130853y_real = comple2936214249959783750l_real(image_2178119161166701260l_real(aTP_Lamm_aac,set_or5849166863359141190n_real(zero_zero_real))) ).

% uniformity_real_def
tff(fact_9832_uniformity__complex__def,axiom,
    topolo896644834953643431omplex = comple8358262395181532106omplex(image_5971271580939081552omplex(aTP_Lamm_aae,set_or5849166863359141190n_real(zero_zero_real))) ).

% uniformity_complex_def
tff(fact_9833_isCont__Lb__Ub,axiom,
    ! [Aa: real,Ba: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),Ba)
     => ( ! [X: real] :
            ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),X)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Ba) )
           => topolo4422821103128117721l_real(topolo2177554685111907308n_real(X,top_top_set_real),F) )
       => ? [L6: real,M8: real] :
            ( ! [X3: real] :
                ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),X3)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),Ba) )
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L6),aa_real_real(F,X3))
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,X3)),M8) ) )
            & ! [Y4: real] :
                ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L6),Y4)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y4),M8) )
               => ? [X: real] :
                    ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),X)
                    & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Ba)
                    & ( aa_real_real(F,X) = Y4 ) ) ) ) ) ) ).

% isCont_Lb_Ub
tff(fact_9834_isCont__inverse__function2,axiom,
    ! [Aa: real,Xb: real,Ba: real,G: fun_real_real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Ba)
       => ( ! [Z3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),Z3)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Z3),Ba)
               => ( aa_real_real(G,aa_real_real(F,Z3)) = Z3 ) ) )
         => ( ! [Z3: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),Z3)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Z3),Ba)
                 => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Z3,top_top_set_real),F) ) )
           => topolo4422821103128117721l_real(topolo2177554685111907308n_real(aa_real_real(F,Xb),top_top_set_real),G) ) ) ) ) ).

% isCont_inverse_function2
tff(fact_9835_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_9836_DERIV__inverse__function,axiom,
    ! [F: fun_real_real,D6: real,G: fun_real_real,Xb: real,Aa: real,Ba: real] :
      ( has_fi5821293074295781190e_real(F,D6,topolo2177554685111907308n_real(aa_real_real(G,Xb),top_top_set_real))
     => ( ( D6 != zero_zero_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),Xb)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Ba)
           => ( ! [Y3: real] :
                  ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),Y3)
                 => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y3),Ba)
                   => ( aa_real_real(F,aa_real_real(G,Y3)) = Y3 ) ) )
             => ( topolo4422821103128117721l_real(topolo2177554685111907308n_real(Xb,top_top_set_real),G)
               => has_fi5821293074295781190e_real(G,aa_real_real(inverse_inverse_real,D6),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ) ) ) ) ) ).

% DERIV_inverse_function
tff(fact_9837_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_9838_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_9839_LIM__less__bound,axiom,
    ! [Ba: real,Xb: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Ba),Xb)
     => ( ! [X: real] :
            ( member_real(X,set_or1633881224788618240n_real(Ba,Xb))
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,X)) )
       => ( topolo4422821103128117721l_real(topolo2177554685111907308n_real(Xb,top_top_set_real),F)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,Xb)) ) ) ) ).

% LIM_less_bound
tff(fact_9840_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_9841_isCont__inverse__function,axiom,
    ! [D: real,Xb: real,G: fun_real_real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D)
     => ( ! [Z3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Z3),Xb))),D)
           => ( aa_real_real(G,aa_real_real(F,Z3)) = Z3 ) )
       => ( ! [Z3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Z3),Xb))),D)
             => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Z3,top_top_set_real),F) )
         => topolo4422821103128117721l_real(topolo2177554685111907308n_real(aa_real_real(F,Xb),top_top_set_real),G) ) ) ) ).

% isCont_inverse_function
tff(fact_9842_GMVT_H,axiom,
    ! [Aa: real,Ba: real,F: fun_real_real,G: fun_real_real,G2: fun_real_real,F4: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),Ba)
     => ( ! [Z3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),Z3)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Z3),Ba)
             => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Z3,top_top_set_real),F) ) )
       => ( ! [Z3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),Z3)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Z3),Ba)
               => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Z3,top_top_set_real),G) ) )
         => ( ! [Z3: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),Z3)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),Ba)
                 => has_fi5821293074295781190e_real(G,aa_real_real(G2,Z3),topolo2177554685111907308n_real(Z3,top_top_set_real)) ) )
           => ( ! [Z3: real] :
                  ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),Z3)
                 => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),Ba)
                   => has_fi5821293074295781190e_real(F,aa_real_real(F4,Z3),topolo2177554685111907308n_real(Z3,top_top_set_real)) ) )
             => ? [C3: real] :
                  ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),C3)
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,C3),Ba)
                  & ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(F,Ba)),aa_real_real(F,Aa))),aa_real_real(G2,C3)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(G,Ba)),aa_real_real(G,Aa))),aa_real_real(F4,C3)) ) ) ) ) ) ) ) ).

% GMVT'
tff(fact_9843_bdd__above__nat,axiom,
    ! [X5: set_nat] :
      ( aa_set_nat_o(condit2214826472909112428ve_nat,X5)
    <=> aa_set_nat_o(finite_finite_nat,X5) ) ).

% bdd_above_nat
tff(fact_9844_GMVT,axiom,
    ! [Aa: real,Ba: real,F: fun_real_real,G: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),Ba)
     => ( ! [X: real] :
            ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),X)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Ba) )
           => topolo4422821103128117721l_real(topolo2177554685111907308n_real(X,top_top_set_real),F) )
       => ( ! [X: real] :
              ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),X)
                & aa_real_o(aa_real_fun_real_o(ord_less_real,X),Ba) )
             => differ6690327859849518006l_real(F,topolo2177554685111907308n_real(X,top_top_set_real)) )
         => ( ! [X: real] :
                ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),X)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Ba) )
               => topolo4422821103128117721l_real(topolo2177554685111907308n_real(X,top_top_set_real),G) )
           => ( ! [X: real] :
                  ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),X)
                    & aa_real_o(aa_real_fun_real_o(ord_less_real,X),Ba) )
                 => differ6690327859849518006l_real(G,topolo2177554685111907308n_real(X,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,Aa),C3)
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,C3),Ba)
                  & ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(F,Ba)),aa_real_real(F,Aa))),G_c) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(G,Ba)),aa_real_real(G,Aa))),F_c) ) ) ) ) ) ) ) ).

% GMVT
tff(fact_9845_MVT,axiom,
    ! [Aa: real,Ba: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),Ba)
     => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(Aa,Ba),F)
       => ( ! [X: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),X)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Ba)
               => differ6690327859849518006l_real(F,topolo2177554685111907308n_real(X,top_top_set_real)) ) )
         => ? [L4: real,Z3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),Z3)
              & aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),Ba)
              & has_fi5821293074295781190e_real(F,L4,topolo2177554685111907308n_real(Z3,top_top_set_real))
              & ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(F,Ba)),aa_real_real(F,Aa)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Ba),Aa)),L4) ) ) ) ) ) ).

% MVT
tff(fact_9846_continuous__on__arcosh_H,axiom,
    ! [A3: set_real,F: fun_real_real] :
      ( topolo5044208981011980120l_real(A3,F)
     => ( ! [X: real] :
            ( member_real(X,A3)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(F,X)) )
       => topolo5044208981011980120l_real(A3,aTP_Lamm_aaf(F)) ) ) ).

% continuous_on_arcosh'
tff(fact_9847_eventually__prod__sequentially,axiom,
    ! [P: fun_Pr4796943543331071514_nat_o] :
      ( eventu1038000079068216329at_nat(P,prod_filter_nat_nat(at_top_nat,at_top_nat))
    <=> ? [N8: nat] :
        ! [M6: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N8),M6)
         => ! [N4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N8),N4)
             => aa_Pro6150637604445964947_nat_o(P,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(N4),M6)) ) ) ) ).

% eventually_prod_sequentially
tff(fact_9848_continuous__image__closed__interval,axiom,
    ! [Aa: real,Ba: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),Ba)
     => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(Aa,Ba),F)
       => ? [C3: real,D4: real] :
            ( ( image_real_real(F,set_or1222579329274155063t_real(Aa,Ba)) = set_or1222579329274155063t_real(C3,D4) )
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C3),D4) ) ) ) ).

% continuous_image_closed_interval
tff(fact_9849_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_9850_Rolle__deriv,axiom,
    ! [Aa: real,Ba: real,F: fun_real_real,F4: fun_re7681423839302404990l_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),Ba)
     => ( ( aa_real_real(F,Aa) = aa_real_real(F,Ba) )
       => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(Aa,Ba),F)
         => ( ! [X: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),X)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Ba)
                 => has_de1759254742604945161l_real(F,aa_rea7340605660300375871l_real(F4,X),topolo2177554685111907308n_real(X,top_top_set_real)) ) )
           => ? [Z3: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),Z3)
                & aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),Ba)
                & ! [X3: real] : aa_real_real(aa_rea7340605660300375871l_real(F4,Z3),X3) = zero_zero_real ) ) ) ) ) ).

% Rolle_deriv
tff(fact_9851_mvt,axiom,
    ! [Aa: real,Ba: real,F: fun_real_real,F4: fun_re7681423839302404990l_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),Ba)
     => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(Aa,Ba),F)
       => ( ! [X: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),X)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Ba)
               => has_de1759254742604945161l_real(F,aa_rea7340605660300375871l_real(F4,X),topolo2177554685111907308n_real(X,top_top_set_real)) ) )
         => ~ ! [Xi: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),Xi)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xi),Ba)
                 => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(F,Ba)),aa_real_real(F,Aa)) != aa_real_real(aa_rea7340605660300375871l_real(F4,Xi),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Ba),Aa)) ) ) ) ) ) ) ).

% mvt
tff(fact_9852_DERIV__pos__imp__increasing__open,axiom,
    ! [Aa: real,Ba: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),Ba)
     => ( ! [X: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),X)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Ba)
             => ? [Y4: real] :
                  ( has_fi5821293074295781190e_real(F,Y4,topolo2177554685111907308n_real(X,top_top_set_real))
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y4) ) ) )
       => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(Aa,Ba),F)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,Aa)),aa_real_real(F,Ba)) ) ) ) ).

% DERIV_pos_imp_increasing_open
tff(fact_9853_DERIV__neg__imp__decreasing__open,axiom,
    ! [Aa: real,Ba: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),Ba)
     => ( ! [X: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),X)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Ba)
             => ? [Y4: real] :
                  ( has_fi5821293074295781190e_real(F,Y4,topolo2177554685111907308n_real(X,top_top_set_real))
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,Y4),zero_zero_real) ) ) )
       => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(Aa,Ba),F)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,Ba)),aa_real_real(F,Aa)) ) ) ) ).

% DERIV_neg_imp_decreasing_open
tff(fact_9854_DERIV__isconst__end,axiom,
    ! [Aa: real,Ba: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),Ba)
     => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(Aa,Ba),F)
       => ( ! [X: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),X)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Ba)
               => has_fi5821293074295781190e_real(F,zero_zero_real,topolo2177554685111907308n_real(X,top_top_set_real)) ) )
         => ( aa_real_real(F,Ba) = aa_real_real(F,Aa) ) ) ) ) ).

% DERIV_isconst_end
tff(fact_9855_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_9856_DERIV__isconst2,axiom,
    ! [Aa: real,Ba: real,F: fun_real_real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),Ba)
     => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(Aa,Ba),F)
       => ( ! [X: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),X)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Ba)
               => has_fi5821293074295781190e_real(F,zero_zero_real,topolo2177554685111907308n_real(X,top_top_set_real)) ) )
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Aa),Xb)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Ba)
             => ( aa_real_real(F,Xb) = aa_real_real(F,Aa) ) ) ) ) ) ) ).

% DERIV_isconst2
tff(fact_9857_Rolle,axiom,
    ! [Aa: real,Ba: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),Ba)
     => ( ( aa_real_real(F,Aa) = aa_real_real(F,Ba) )
       => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(Aa,Ba),F)
         => ( ! [X: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),X)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X),Ba)
                 => differ6690327859849518006l_real(F,topolo2177554685111907308n_real(X,top_top_set_real)) ) )
           => ? [Z3: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,Aa),Z3)
                & aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),Ba)
                & has_fi5821293074295781190e_real(F,zero_zero_real,topolo2177554685111907308n_real(Z3,top_top_set_real)) ) ) ) ) ) ).

% Rolle
tff(fact_9858_take__bit__num__simps_I1_J,axiom,
    ! [M: num] : bit_take_bit_num(zero_zero_nat,M) = none_num ).

% take_bit_num_simps(1)
tff(fact_9859_take__bit__num__simps_I2_J,axiom,
    ! [Na: nat] : bit_take_bit_num(aa_nat_nat(suc,Na),one) = some_num(one) ).

% take_bit_num_simps(2)
tff(fact_9860_take__bit__num__simps_I3_J,axiom,
    ! [Na: nat,M: num] : bit_take_bit_num(aa_nat_nat(suc,Na),aa_num_num(bit0,M)) = case_o6005452278849405969um_num(none_num,aTP_Lamm_aag,bit_take_bit_num(Na,M)) ).

% take_bit_num_simps(3)
tff(fact_9861_take__bit__num__simps_I4_J,axiom,
    ! [Na: nat,M: num] : bit_take_bit_num(aa_nat_nat(suc,Na),aa_num_num(bit1,M)) = some_num(case_option_num_num(one,bit1,bit_take_bit_num(Na,M))) ).

% take_bit_num_simps(4)
tff(fact_9862_mono__Suc,axiom,
    order_mono_nat_nat(suc) ).

% mono_Suc
tff(fact_9863_mono__times__nat,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => order_mono_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na)) ) ).

% mono_times_nat
tff(fact_9864_incseq__bounded,axiom,
    ! [X5: fun_nat_real,B4: real] :
      ( order_mono_nat_real(X5)
     => ( ! [I2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(X5,I2)),B4)
       => bfun_nat_real(X5,at_top_nat) ) ) ).

% incseq_bounded
tff(fact_9865_incseq__convergent,axiom,
    ! [X5: fun_nat_real,B4: real] :
      ( order_mono_nat_real(X5)
     => ( ! [I2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(X5,I2)),B4)
       => ~ ! [L6: real] :
              ( filterlim_nat_real(X5,topolo2815343760600316023s_real(L6),at_top_nat)
             => ~ ! [I4: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(X5,I4)),L6) ) ) ) ).

% incseq_convergent
tff(fact_9866_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_aah,K)) ) ).

% mono_ge2_power_minus_self
tff(fact_9867_take__bit__num__def,axiom,
    ! [Na: nat,M: num] :
      bit_take_bit_num(Na,M) = $ite(aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),aa_num_nat(numeral_numeral_nat,M)) = zero_zero_nat,none_num,some_num(aa_nat_num(num_of_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Na),aa_num_nat(numeral_numeral_nat,M))))) ).

% take_bit_num_def
tff(fact_9868_less__eq,axiom,
    ! [M: nat,Na: nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(M),Na),transi6264000038957366511cl_nat(pred_nat))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Na) ) ).

% less_eq
tff(fact_9869_nonneg__incseq__Bseq__subseq__iff,axiom,
    ! [F: fun_nat_real,G: fun_nat_nat] :
      ( ! [X: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,X))
     => ( order_mono_nat_real(F)
       => ( order_5726023648592871131at_nat(G)
         => ( bfun_nat_real(aa_fun1538629612289296340t_real(aTP_Lamm_aai(F),G),at_top_nat)
          <=> bfun_nat_real(F,at_top_nat) ) ) ) ) ).

% nonneg_incseq_Bseq_subseq_iff
tff(fact_9870_infinite__enumerate,axiom,
    ! [S2: set_nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,S2)
     => ? [R3: fun_nat_nat] :
          ( order_5726023648592871131at_nat(R3)
          & ! [N5: nat] : aa_set_nat_o(member_nat(aa_nat_nat(R3,N5)),S2) ) ) ).

% infinite_enumerate
tff(fact_9871_strict__mono__imp__increasing,axiom,
    ! [F: fun_nat_nat,Na: nat] :
      ( order_5726023648592871131at_nat(F)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),aa_nat_nat(F,Na)) ) ).

% strict_mono_imp_increasing
tff(fact_9872_pos__deriv__imp__strict__mono,axiom,
    ! [F: fun_real_real,F4: fun_real_real] :
      ( ! [X: real] : has_fi5821293074295781190e_real(F,aa_real_real(F4,X),topolo2177554685111907308n_real(X,top_top_set_real))
     => ( ! [X: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(F4,X))
       => order_7092887310737990675l_real(F) ) ) ).

% pos_deriv_imp_strict_mono
tff(fact_9873_inj__sgn__power,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => inj_on_real_real(aa_nat_fun_real_real(aTP_Lamm_yp,Na),top_top_set_real) ) ).

% inj_sgn_power
tff(fact_9874_min__Suc__Suc,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,Na)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M),Na)) ).

% min_Suc_Suc
tff(fact_9875_min__0R,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Na),zero_zero_nat) = zero_zero_nat ).

% min_0R
tff(fact_9876_min__0L,axiom,
    ! [Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,zero_zero_nat),Na) = zero_zero_nat ).

% min_0L
tff(fact_9877_min__numeral__Suc,axiom,
    ! [K: num,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(suc,Na)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_num_nat(pred_numeral,K)),Na)) ).

% min_numeral_Suc
tff(fact_9878_min__Suc__numeral,axiom,
    ! [Na: nat,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(suc,Na)),aa_num_nat(numeral_numeral_nat,K)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Na),aa_num_nat(pred_numeral,K))) ).

% min_Suc_numeral
tff(fact_9879_inf__nat__def,axiom,
    inf_inf_nat = ord_min_nat ).

% inf_nat_def
tff(fact_9880_min__diff,axiom,
    ! [M: nat,I: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),I)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),I)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M),Na)),I) ).

% min_diff
tff(fact_9881_nat__mult__min__right,axiom,
    ! [M: nat,Na: nat,Q4: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Na),Q4)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Na)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Q4)) ).

% nat_mult_min_right
tff(fact_9882_nat__mult__min__left,axiom,
    ! [M: nat,Na: nat,Q4: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M),Na)),Q4) = aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Q4)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Na),Q4)) ).

% nat_mult_min_left
tff(fact_9883_min__Suc1,axiom,
    ! [Na: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(suc,Na)),M) = aa_nat_nat(case_nat_nat(zero_zero_nat,aa_nat_fun_nat_nat(aTP_Lamm_aaj,Na)),M) ).

% min_Suc1
tff(fact_9884_min__Suc2,axiom,
    ! [M: nat,Na: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M),aa_nat_nat(suc,Na)) = aa_nat_nat(case_nat_nat(zero_zero_nat,aa_nat_fun_nat_nat(aTP_Lamm_aak,Na)),M) ).

% min_Suc2
tff(fact_9885_log__inj,axiom,
    ! [Ba: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Ba)
     => inj_on_real_real(aa_rea7340605660300375871l_real(log,Ba),set_or5849166863359141190n_real(zero_zero_real)) ) ).

% log_inj
tff(fact_9886_int__of__integer__min,axiom,
    ! [K: code_integer,L: code_integer] : aa_Code_integer_int(code_int_of_integer,ord_min_Code_integer(K,L)) = aa_int_int(ord_min_int(aa_Code_integer_int(code_int_of_integer,K)),aa_Code_integer_int(code_int_of_integer,L)) ).

% int_of_integer_min
tff(fact_9887_inj__on__set__encode,axiom,
    inj_on_set_nat_nat(nat_set_encode,collect_set_nat(finite_finite_nat)) ).

% inj_on_set_encode
tff(fact_9888_inj__Suc,axiom,
    ! [N3: set_nat] : aa_set_nat_o(inj_on_nat_nat(suc),N3) ).

% inj_Suc
tff(fact_9889_inj__on__diff__nat,axiom,
    ! [N3: set_nat,K: nat] :
      ( ! [N: nat] :
          ( aa_set_nat_o(member_nat(N),N3)
         => 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_yl,K)),N3) ) ).

% inj_on_diff_nat
tff(fact_9890_summable__reindex,axiom,
    ! [F: fun_nat_real,G: fun_nat_nat] :
      ( summable_real(F)
     => ( aa_set_nat_o(inj_on_nat_nat(G),top_top_set_nat)
       => ( ! [X: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,X))
         => summable_real(comp_nat_real_nat(F,G)) ) ) ) ).

% summable_reindex
tff(fact_9891_suminf__reindex__mono,axiom,
    ! [F: fun_nat_real,G: fun_nat_nat] :
      ( summable_real(F)
     => ( aa_set_nat_o(inj_on_nat_nat(G),top_top_set_nat)
       => ( ! [X: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,X))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,suminf_real(comp_nat_real_nat(F,G))),suminf_real(F)) ) ) ) ).

% suminf_reindex_mono
tff(fact_9892_suminf__reindex,axiom,
    ! [F: fun_nat_real,G: fun_nat_nat] :
      ( summable_real(F)
     => ( aa_set_nat_o(inj_on_nat_nat(G),top_top_set_nat)
       => ( ! [X: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,X))
         => ( ! [X: nat] :
                ( ~ aa_set_nat_o(member_nat(X),image_nat_nat(G,top_top_set_nat))
               => ( aa_nat_real(F,X) = zero_zero_real ) )
           => ( suminf_real(comp_nat_real_nat(F,G)) = suminf_real(F) ) ) ) ) ) ).

% suminf_reindex
tff(fact_9893_pred__nat__trancl__eq__le,axiom,
    ! [M: nat,Na: nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(M),Na),transi2905341329935302413cl_nat(pred_nat))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na) ) ).

% pred_nat_trancl_eq_le
tff(fact_9894_powr__real__of__int_H,axiom,
    ! [Xb: real,Na: 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),Na) )
       => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),aa_int_real(ring_1_of_int_real,Na)) = aa_int_real(power_int_real(Xb),Na) ) ) ) ).

% powr_real_of_int'
tff(fact_9895_Rats__eq__int__div__nat,axiom,
    field_5140801741446780682s_real = collect_real(aTP_Lamm_aal) ).

% Rats_eq_int_div_nat
tff(fact_9896_Rats__no__top__le,axiom,
    ! [Xb: real] :
    ? [X: real] :
      ( member_real(X,field_5140801741446780682s_real)
      & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),X) ) ).

% Rats_no_top_le
tff(fact_9897_Rats__no__bot__less,axiom,
    ! [Xb: real] :
    ? [X: real] :
      ( member_real(X,field_5140801741446780682s_real)
      & aa_real_o(aa_real_fun_real_o(ord_less_real,X),Xb) ) ).

% Rats_no_bot_less
tff(fact_9898_Rats__dense__in__real,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
     => ? [X: real] :
          ( member_real(X,field_5140801741446780682s_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),X)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,X),Y) ) ) ).

% Rats_dense_in_real
tff(fact_9899_positive__rat,axiom,
    ! [Aa: int,Ba: int] :
      ( aa_rat_o(positive,aa_int_rat(fract(Aa),Ba))
    <=> 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,Aa),Ba)) ) ).

% positive_rat
tff(fact_9900_less__rat__def,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
    <=> aa_rat_o(positive,aa_rat_rat(minus_minus_rat(Y),Xb)) ) ).

% less_rat_def
tff(fact_9901_upt__rec__numeral,axiom,
    ! [M: num,Na: num] :
      upt(aa_num_nat(numeral_numeral_nat,M),aa_num_nat(numeral_numeral_nat,Na)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,Na)),cons_nat(aa_num_nat(numeral_numeral_nat,M),upt(aa_nat_nat(suc,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,Na))),nil_nat) ).

% upt_rec_numeral
tff(fact_9902_Arg__bounded,axiom,
    ! [Z: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,pi)),aa_complex_real(arg,Z))
      & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(arg,Z)),pi) ) ).

% Arg_bounded
tff(fact_9903_tl__upt,axiom,
    ! [M: nat,Na: nat] : tl_nat(upt(M,Na)) = upt(aa_nat_nat(suc,M),Na) ).

% tl_upt
tff(fact_9904_hd__upt,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => ( hd_nat(upt(I,J)) = I ) ) ).

% hd_upt
tff(fact_9905_take__upt,axiom,
    ! [I: nat,M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),M)),Na)
     => ( take_nat(M,upt(I,Na)) = upt(I,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),M)) ) ) ).

% take_upt
tff(fact_9906_upt__conv__Nil,axiom,
    ! [J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),I)
     => ( upt(I,J) = nil_nat ) ) ).

% upt_conv_Nil
tff(fact_9907_upt__eq__Nil__conv,axiom,
    ! [I: nat,J: nat] :
      ( ( upt(I,J) = nil_nat )
    <=> ( ( J = zero_zero_nat )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),I) ) ) ).

% upt_eq_Nil_conv
tff(fact_9908_nth__upt,axiom,
    ! [I: nat,K: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),J)
     => ( aa_nat_nat(nth_nat(upt(I,J)),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K) ) ) ).

% nth_upt
tff(fact_9909_upt__conv__Cons__Cons,axiom,
    ! [M: nat,Na: nat,Ns: list_nat,Q4: nat] :
      ( ( cons_nat(M,cons_nat(Na,Ns)) = upt(M,Q4) )
    <=> ( cons_nat(Na,Ns) = upt(aa_nat_nat(suc,M),Q4) ) ) ).

% upt_conv_Cons_Cons
tff(fact_9910_greaterThanLessThan__upt,axiom,
    ! [Na: nat,M: nat] : set_or5834768355832116004an_nat(Na,M) = set_nat2(upt(aa_nat_nat(suc,Na),M)) ).

% greaterThanLessThan_upt
tff(fact_9911_atLeast__upt,axiom,
    ! [Na: nat] : aa_nat_set_nat(set_ord_lessThan_nat,Na) = set_nat2(upt(zero_zero_nat,Na)) ).

% atLeast_upt
tff(fact_9912_atLeastAtMost__upt,axiom,
    ! [Na: nat,M: nat] : set_or1269000886237332187st_nat(Na,M) = set_nat2(upt(Na,aa_nat_nat(suc,M))) ).

% atLeastAtMost_upt
tff(fact_9913_greaterThanAtMost__upt,axiom,
    ! [Na: nat,M: nat] : set_or6659071591806873216st_nat(Na,M) = set_nat2(upt(aa_nat_nat(suc,Na),aa_nat_nat(suc,M))) ).

% greaterThanAtMost_upt
tff(fact_9914_upt__0,axiom,
    ! [I: nat] : upt(I,zero_zero_nat) = nil_nat ).

% upt_0
tff(fact_9915_atMost__upto,axiom,
    ! [Na: nat] : aa_nat_set_nat(set_ord_atMost_nat,Na) = set_nat2(upt(zero_zero_nat,aa_nat_nat(suc,Na))) ).

% atMost_upto
tff(fact_9916_upt__conv__Cons,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => ( upt(I,J) = cons_nat(I,upt(aa_nat_nat(suc,I),J)) ) ) ).

% upt_conv_Cons
tff(fact_9917_upt__add__eq__append,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( upt(I,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K)) = append_nat(upt(I,J),upt(J,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K))) ) ) ).

% upt_add_eq_append
tff(fact_9918_upt__eq__Cons__conv,axiom,
    ! [I: nat,J: nat,Xb: nat,Xs: list_nat] :
      ( ( upt(I,J) = cons_nat(Xb,Xs) )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
        & ( I = Xb )
        & ( upt(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),one_one_nat),J) = Xs ) ) ) ).

% upt_eq_Cons_conv
tff(fact_9919_upt__rec,axiom,
    ! [I: nat,J: nat] :
      upt(I,J) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J),cons_nat(I,upt(aa_nat_nat(suc,I),J)),nil_nat) ).

% upt_rec
tff(fact_9920_upt__Suc,axiom,
    ! [I: nat,J: nat] :
      upt(I,aa_nat_nat(suc,J)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J),append_nat(upt(I,J),cons_nat(J,nil_nat)),nil_nat) ).

% upt_Suc
tff(fact_9921_upt__Suc__append,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( upt(I,aa_nat_nat(suc,J)) = append_nat(upt(I,J),cons_nat(J,nil_nat)) ) ) ).

% upt_Suc_append
tff(fact_9922_Arg__correct,axiom,
    ! [Z: complex] :
      ( ( Z != zero_zero_complex )
     => ( ( aa_complex_complex(sgn_sgn_complex,Z) = aa_real_complex(cis,aa_complex_real(arg,Z)) )
        & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,pi)),aa_complex_real(arg,Z))
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(arg,Z)),pi) ) ) ).

% Arg_correct
tff(fact_9923_cis__Arg__unique,axiom,
    ! [Z: complex,Xb: real] :
      ( ( aa_complex_complex(sgn_sgn_complex,Z) = aa_real_complex(cis,Xb) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,pi)),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),pi)
         => ( aa_complex_real(arg,Z) = Xb ) ) ) ) ).

% cis_Arg_unique
tff(fact_9924_bij__betw__roots__unity,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => aa_set_complex_o(bij_betw_nat_complex(aTP_Lamm_aam(Na),aa_nat_set_nat(set_ord_lessThan_nat,Na)),collect_complex(aa_nat_fun_complex_o(aTP_Lamm_cs,Na))) ) ).

% bij_betw_roots_unity
tff(fact_9925_bij__betw__nth__root__unity,axiom,
    ! [C: complex,Na: nat] :
      ( ( C != zero_zero_complex )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
       => aa_set_complex_o(bij_be1856998921033663316omplex(times_times_complex(aa_complex_complex(times_times_complex(aa_real_complex(real_V4546457046886955230omplex,aa_real_real(aa_nat_fun_real_real(root,Na),aa_complex_real(real_V1022390504157884413omplex,C)))),aa_real_complex(cis,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(arg,C)),aa_nat_real(semiri5074537144036343181t_real,Na))))),collect_complex(aa_nat_fun_complex_o(aTP_Lamm_cs,Na))),collect_complex(aa_nat_fun_complex_o(aTP_Lamm_yc(C),Na))) ) ) ).

% bij_betw_nth_root_unity
tff(fact_9926_map__Suc__upt,axiom,
    ! [M: nat,Na: nat] : map_nat_nat(suc,upt(M,Na)) = upt(aa_nat_nat(suc,M),aa_nat_nat(suc,Na)) ).

% map_Suc_upt
tff(fact_9927_map__add__upt,axiom,
    ! [Na: nat,M: nat] : map_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_aan,Na),upt(zero_zero_nat,M)) = upt(Na,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Na)) ).

% map_add_upt
tff(fact_9928_map__decr__upt,axiom,
    ! [M: nat,Na: nat] : map_nat_nat(aTP_Lamm_aao,upt(aa_nat_nat(suc,M),aa_nat_nat(suc,Na))) = upt(M,Na) ).

% map_decr_upt
tff(fact_9929_bij__betw__Suc,axiom,
    ! [M7: set_nat,N3: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(bij_betw_nat_nat(suc),M7),N3)
    <=> ( image_nat_nat(suc,M7) = N3 ) ) ).

% bij_betw_Suc
tff(fact_9930_Arg__def,axiom,
    ! [Z: complex] :
      aa_complex_real(arg,Z) = $ite(Z = zero_zero_complex,zero_zero_real,fChoice_real(aTP_Lamm_aap(Z))) ).

% Arg_def
tff(fact_9931_sum__list__upt,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( groups4561878855575611511st_nat(upt(M,Na)) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_cc),set_or4665077453230672383an_nat(M,Na)) ) ) ).

% sum_list_upt
tff(fact_9932_card__length__sum__list__rec,axiom,
    ! [M: nat,N3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),M)
     => ( finite_card_list_nat(collect_list_nat(aa_nat5883842100386203267_nat_o(aTP_Lamm_aaq(M),N3))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,finite_card_list_nat(collect_list_nat(aa_nat5883842100386203267_nat_o(aTP_Lamm_aar(M),N3)))),finite_card_list_nat(collect_list_nat(aa_nat5883842100386203267_nat_o(aTP_Lamm_aas(M),N3)))) ) ) ).

% card_length_sum_list_rec
tff(fact_9933_card__length__sum__list,axiom,
    ! [M: nat,N3: nat] : finite_card_list_nat(collect_list_nat(aa_nat5883842100386203267_nat_o(aTP_Lamm_aaq(M),N3))) = 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,N3),M)),one_one_nat)),N3) ).

% card_length_sum_list
tff(fact_9934_sorted__upt,axiom,
    ! [M: nat,Na: nat] : aa_list_nat_o(sorted_wrt_nat(ord_less_eq_nat),upt(M,Na)) ).

% sorted_upt
tff(fact_9935_sorted__wrt__upt,axiom,
    ! [M: nat,Na: nat] : aa_list_nat_o(sorted_wrt_nat(ord_less_nat),upt(M,Na)) ).

% sorted_wrt_upt
tff(fact_9936_sorted__wrt__less__idx,axiom,
    ! [Ns: list_nat,I: nat] :
      ( aa_list_nat_o(sorted_wrt_nat(ord_less_nat),Ns)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_nat(Ns))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),aa_nat_nat(nth_nat(Ns),I)) ) ) ).

% sorted_wrt_less_idx
tff(fact_9937_sorted__upto,axiom,
    ! [M: int,Na: int] : sorted_wrt_int(ord_less_eq_int,upto(M,Na)) ).

% sorted_upto
tff(fact_9938_sorted__wrt__upto,axiom,
    ! [I: int,J: int] : sorted_wrt_int(ord_less_int,upto(I,J)) ).

% sorted_wrt_upto
tff(fact_9939_fst__divmod__integer,axiom,
    ! [K: code_integer,L: code_integer] : produc8508995932063986495nteger(code_divmod_integer(K,L)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,K),L) ).

% fst_divmod_integer
tff(fact_9940_fst__divmod__abs,axiom,
    ! [K: code_integer,L: code_integer] : produc8508995932063986495nteger(code_divmod_abs(K,L)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(divide6298287555418463151nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,K)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,L)) ).

% fst_divmod_abs
tff(fact_9941_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_9942_bezw_Oelims,axiom,
    ! [Xb: nat,Xaa: nat,Y: product_prod_int_int] :
      ( ( bezw(Xb,Xaa) = Y )
     => ( Y = $ite(Xaa = zero_zero_nat,product_Pair_int_int(one_one_int,zero_zero_int),product_Pair_int_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,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,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_9943_bezw_Osimps,axiom,
    ! [Xb: nat,Y: nat] :
      bezw(Xb,Y) = $ite(Y = zero_zero_nat,product_Pair_int_int(one_one_int,zero_zero_int),product_Pair_int_int(product_snd_int_int(bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Y))),aa_int_int(aa_int_fun_int_int(minus_minus_int,product_fst_int_int(bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Y)))),aa_int_int(aa_int_fun_int_int(times_times_int,product_snd_int_int(bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Y)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xb),Y)))))) ).

% bezw.simps
tff(fact_9944_bezw__non__0,axiom,
    ! [Y: nat,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Y)
     => ( bezw(Xb,Y) = product_Pair_int_int(product_snd_int_int(bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Y))),aa_int_int(aa_int_fun_int_int(minus_minus_int,product_fst_int_int(bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Y)))),aa_int_int(aa_int_fun_int_int(times_times_int,product_snd_int_int(bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Y)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xb),Y))))) ) ) ).

% bezw_non_0
tff(fact_9945_bezw_Opelims,axiom,
    ! [Xb: nat,Xaa: nat,Y: product_prod_int_int] :
      ( ( bezw(Xb,Xaa) = Y )
     => ( aa_Pro6150637604445964947_nat_o(accp_P4275260045618599050at_nat(bezw_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xb),Xaa))
       => ~ ( ( Y = $ite(Xaa = zero_zero_nat,product_Pair_int_int(one_one_int,zero_zero_int),product_Pair_int_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,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,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_9946_snd__divmod__integer,axiom,
    ! [K: code_integer,L: code_integer] : produc6174133586879617921nteger(code_divmod_integer(K,L)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,K),L) ).

% snd_divmod_integer
tff(fact_9947_snd__divmod__abs,axiom,
    ! [K: code_integer,L: code_integer] : produc6174133586879617921nteger(code_divmod_abs(K,L)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(modulo364778990260209775nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,K)),aa_Cod1783974652533396728nteger(abs_abs_Code_integer,L)) ).

% snd_divmod_abs
tff(fact_9948_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_9949_Rat_Opositive_Orep__eq,axiom,
    ! [Xb: rat] :
      ( aa_rat_o(positive,Xb)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,product_fst_int_int(rep_Rat(Xb))),product_snd_int_int(rep_Rat(Xb)))) ) ).

% Rat.positive.rep_eq
tff(fact_9950_normalize__def,axiom,
    ! [P3: product_prod_int_int] :
      normalize(P3) = $ite(
        aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),product_snd_int_int(P3)),
        $let(
          a2: int,
          a2:= aa_int_int(gcd_gcd_int(product_fst_int_int(P3)),product_snd_int_int(P3)),
          product_Pair_int_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,product_fst_int_int(P3)),a2),aa_int_int(aa_int_fun_int_int(divide_divide_int,product_snd_int_int(P3)),a2)) ),
        $ite(
          product_snd_int_int(P3) = zero_zero_int,
          product_Pair_int_int(zero_zero_int,one_one_int),
          $let(
            a2: int,
            a2:= aa_int_int(uminus_uminus_int,aa_int_int(gcd_gcd_int(product_fst_int_int(P3)),product_snd_int_int(P3))),
            product_Pair_int_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,product_fst_int_int(P3)),a2),aa_int_int(aa_int_fun_int_int(divide_divide_int,product_snd_int_int(P3)),a2)) ) ) ) ).

% normalize_def
tff(fact_9951_gcd__pos__int,axiom,
    ! [M: int,Na: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(gcd_gcd_int(M),Na))
    <=> ( ( M != zero_zero_int )
        | ( Na != zero_zero_int ) ) ) ).

% gcd_pos_int
tff(fact_9952_gcd__ge__0__int,axiom,
    ! [Xb: int,Y: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(gcd_gcd_int(Xb),Y)) ).

% gcd_ge_0_int
tff(fact_9953_gcd__le2__int,axiom,
    ! [Ba: int,Aa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Ba)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(gcd_gcd_int(Aa),Ba)),Ba) ) ).

% gcd_le2_int
tff(fact_9954_gcd__le1__int,axiom,
    ! [Aa: int,Ba: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Aa)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(gcd_gcd_int(Aa),Ba)),Aa) ) ).

% gcd_le1_int
tff(fact_9955_gcd__cases__int,axiom,
    ! [Xb: int,Y: 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),Y)
         => aa_int_o(P,aa_int_int(gcd_gcd_int(Xb),Y)) ) )
     => ( ( 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,Y),zero_zero_int)
           => aa_int_o(P,aa_int_int(gcd_gcd_int(Xb),aa_int_int(uminus_uminus_int,Y))) ) )
       => ( ( 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),Y)
             => aa_int_o(P,aa_int_int(gcd_gcd_int(aa_int_int(uminus_uminus_int,Xb)),Y)) ) )
         => ( ( 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,Y),zero_zero_int)
               => aa_int_o(P,aa_int_int(gcd_gcd_int(aa_int_int(uminus_uminus_int,Xb)),aa_int_int(uminus_uminus_int,Y))) ) )
           => aa_int_o(P,aa_int_int(gcd_gcd_int(Xb),Y)) ) ) ) ) ).

% gcd_cases_int
tff(fact_9956_gcd__unique__int,axiom,
    ! [D: int,Aa: int,Ba: 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),Aa)
        & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),Ba)
        & ! [E3: int] :
            ( ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,E3),Aa)
              & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,E3),Ba) )
           => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,E3),D) ) )
    <=> ( D = aa_int_int(gcd_gcd_int(Aa),Ba) ) ) ).

% gcd_unique_int
tff(fact_9957_gcd__non__0__int,axiom,
    ! [Y: int,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Y)
     => ( aa_int_int(gcd_gcd_int(Xb),Y) = aa_int_int(gcd_gcd_int(Y),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Xb),Y)) ) ) ).

% gcd_non_0_int
tff(fact_9958_gcd__nat_Oeq__neutr__iff,axiom,
    ! [Aa: nat,Ba: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Aa),Ba) = zero_zero_nat )
    <=> ( ( Aa = zero_zero_nat )
        & ( Ba = zero_zero_nat ) ) ) ).

% gcd_nat.eq_neutr_iff
tff(fact_9959_gcd__nat_Oleft__neutral,axiom,
    ! [Aa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,zero_zero_nat),Aa) = Aa ).

% gcd_nat.left_neutral
tff(fact_9960_gcd__nat_Oneutr__eq__iff,axiom,
    ! [Aa: nat,Ba: nat] :
      ( ( zero_zero_nat = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Aa),Ba) )
    <=> ( ( Aa = zero_zero_nat )
        & ( Ba = zero_zero_nat ) ) ) ).

% gcd_nat.neutr_eq_iff
tff(fact_9961_gcd__nat_Oright__neutral,axiom,
    ! [Aa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Aa),zero_zero_nat) = Aa ).

% gcd_nat.right_neutral
tff(fact_9962_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_9963_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_9964_gcd__1__nat,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,M),one_one_nat) = one_one_nat ).

% gcd_1_nat
tff(fact_9965_gcd__Suc__0,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,M),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(suc,zero_zero_nat) ).

% gcd_Suc_0
tff(fact_9966_gcd__pos__nat,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,M),Na))
    <=> ( ( M != zero_zero_nat )
        | ( Na != zero_zero_nat ) ) ) ).

% gcd_pos_nat
tff(fact_9967_gcd__nat_Oelims,axiom,
    ! [Xb: nat,Xaa: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Xb),Xaa) = Y )
     => ( Y = $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_9968_gcd__nat_Osimps,axiom,
    ! [Xb: nat,Y: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Xb),Y) = $ite(Y = zero_zero_nat,Xb,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Y))) ).

% gcd_nat.simps
tff(fact_9969_gcd__non__0__nat,axiom,
    ! [Y: nat,Xb: nat] :
      ( ( Y != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Xb),Y) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Y)) ) ) ).

% gcd_non_0_nat
tff(fact_9970_Gcd__in,axiom,
    ! [A3: set_nat] :
      ( ! [A2: nat,B2: nat] :
          ( aa_set_nat_o(member_nat(A2),A3)
         => ( aa_set_nat_o(member_nat(B2),A3)
           => aa_set_nat_o(member_nat(aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A2),B2)),A3) ) )
     => ( ( A3 != bot_bot_set_nat )
       => aa_set_nat_o(member_nat(aa_set_nat_nat(gcd_Gcd_nat,A3)),A3) ) ) ).

% Gcd_in
tff(fact_9971_gcd__diff2__nat,axiom,
    ! [M: nat,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),M)),Na) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,M),Na) ) ) ).

% gcd_diff2_nat
tff(fact_9972_gcd__diff1__nat,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),M)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Na)),Na) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,M),Na) ) ) ).

% gcd_diff1_nat
tff(fact_9973_gcd__le1__nat,axiom,
    ! [Aa: nat,Ba: nat] :
      ( ( Aa != 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,Aa),Ba)),Aa) ) ).

% gcd_le1_nat
tff(fact_9974_gcd__le2__nat,axiom,
    ! [Ba: nat,Aa: nat] :
      ( ( Ba != 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,Aa),Ba)),Ba) ) ).

% gcd_le2_nat
tff(fact_9975_bezout__nat,axiom,
    ! [Aa: nat,Ba: nat] :
      ( ( Aa != zero_zero_nat )
     => ? [X: nat,Y3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),X) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ba),Y3)),aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Aa),Ba)) ) ).

% bezout_nat
tff(fact_9976_bezout__gcd__nat_H,axiom,
    ! [Ba: nat,Aa: nat] :
    ? [X: nat,Y3: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ba),Y3)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),X))
        & ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),X)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ba),Y3)) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Aa),Ba) ) )
      | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),Y3)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ba),X))
        & ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ba),X)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Aa),Y3)) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Aa),Ba) ) ) ) ).

% bezout_gcd_nat'
tff(fact_9977_gcd__nat_Osemilattice__neutr__order__axioms,axiom,
    semila1623282765462674594er_nat(gcd_gcd_nat,zero_zero_nat,dvd_dvd_nat,aTP_Lamm_aat) ).

% gcd_nat.semilattice_neutr_order_axioms
tff(fact_9978_gcd__is__Max__divisors__nat,axiom,
    ! [Na: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,M),Na) = aa_set_nat_nat(lattic8265883725875713057ax_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_aau(Na),M))) ) ) ).

% gcd_is_Max_divisors_nat
tff(fact_9979_gcd__nat_Opelims,axiom,
    ! [Xb: nat,Xaa: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Xb),Xaa) = Y )
     => ( aa_Pro6150637604445964947_nat_o(accp_P4275260045618599050at_nat(gcd_nat_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xb),Xaa))
       => ~ ( ( Y = $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_9980_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_aaw),rep_Integ(Xb)),rep_Integ(Xaa)) ) ).

% less_eq_int.rep_eq
tff(fact_9981_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_aay),rep_Integ(Xb)),rep_Integ(Xaa)) ) ).

% less_int.rep_eq
tff(fact_9982_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,abs_Integ(Xaa)),abs_Integ(Xb))
    <=> aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(aTP_Lamm_aaw),Xaa),Xb) ) ).

% less_eq_int.abs_eq
tff(fact_9983_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,abs_Integ(Xaa)),abs_Integ(Xb))
    <=> aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(aTP_Lamm_aay),Xaa),Xb) ) ).

% less_int.abs_eq
tff(fact_9984_zero__int__def,axiom,
    zero_zero_int = abs_Integ(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),zero_zero_nat)) ).

% zero_int_def
tff(fact_9985_int__def,axiom,
    ! [Na: nat] : aa_nat_int(semiri1314217659103216013at_int,Na) = abs_Integ(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Na),zero_zero_nat)) ).

% int_def
tff(fact_9986_one__int__def,axiom,
    one_one_int = abs_Integ(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(one_one_nat),zero_zero_nat)) ).

% one_int_def
tff(fact_9987_Gcd__nat__set__eq__fold,axiom,
    ! [Xs: list_nat] : aa_set_nat_nat(gcd_Gcd_nat,set_nat2(Xs)) = aa_nat_nat(fold_nat_nat(gcd_gcd_nat,Xs),zero_zero_nat) ).

% Gcd_nat_set_eq_fold
tff(fact_9988_Field__natLeq__on,axiom,
    ! [Na: nat] : field_nat(collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_aaz(Na)))) = collect_nat(aa_nat_fun_nat_o(aTP_Lamm_au,Na)) ).

% Field_natLeq_on
tff(fact_9989_natLess__def,axiom,
    bNF_Ca8459412986667044542atLess = collec3392354462482085612at_nat(produc6081775807080527818_nat_o(ord_less_nat)) ).

% natLess_def
tff(fact_9990_wf__less,axiom,
    wf_nat(collec3392354462482085612at_nat(produc6081775807080527818_nat_o(ord_less_nat))) ).

% wf_less
tff(fact_9991_cauchy__def,axiom,
    ! [X5: fun_nat_rat] :
      ( cauchy(X5)
    <=> ! [R5: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),R5)
         => ? [K3: nat] :
            ! [M6: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K3),M6)
             => ! [N4: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K3),N4)
                 => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(aa_nat_rat(X5,M6)),aa_nat_rat(X5,N4)))),R5) ) ) ) ) ).

% cauchy_def
tff(fact_9992_cauchyI,axiom,
    ! [X5: fun_nat_rat] :
      ( ! [R3: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),R3)
         => ? [K6: nat] :
            ! [M3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K6),M3)
             => ! [N: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K6),N)
                 => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(aa_nat_rat(X5,M3)),aa_nat_rat(X5,N)))),R3) ) ) )
     => cauchy(X5) ) ).

% cauchyI
tff(fact_9993_cauchy__imp__bounded,axiom,
    ! [X5: fun_nat_rat] :
      ( cauchy(X5)
     => ? [B2: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B2)
          & ! [N5: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_nat_rat(X5,N5))),B2) ) ) ).

% cauchy_imp_bounded
tff(fact_9994_cauchyD,axiom,
    ! [X5: fun_nat_rat,R2: rat] :
      ( cauchy(X5)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),R2)
       => ? [K2: nat] :
          ! [M2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),M2)
           => ! [N5: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),N5)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(aa_nat_rat(X5,M2)),aa_nat_rat(X5,N5)))),R2) ) ) ) ) ).

% cauchyD
tff(fact_9995_le__Real,axiom,
    ! [X5: fun_nat_rat,Y6: fun_nat_rat] :
      ( cauchy(X5)
     => ( cauchy(Y6)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,real2(X5)),real2(Y6))
        <=> ! [R5: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),R5)
             => ? [K3: nat] :
                ! [N4: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K3),N4)
                 => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(X5,N4)),aa_rat_rat(plus_plus_rat(aa_nat_rat(Y6,N4)),R5)) ) ) ) ) ) ).

% le_Real
tff(fact_9996_cauchy__not__vanishes,axiom,
    ! [X5: fun_nat_rat] :
      ( cauchy(X5)
     => ( ~ vanishes(X5)
       => ? [B2: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B2)
            & ? [K2: nat] :
              ! [N5: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),N5)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),aa_rat_rat(abs_abs_rat,aa_nat_rat(X5,N5))) ) ) ) ) ).

% cauchy_not_vanishes
tff(fact_9997_vanishes__mult__bounded,axiom,
    ! [X5: fun_nat_rat,Y6: fun_nat_rat] :
      ( ? [A6: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A6)
          & ! [N: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_nat_rat(X5,N))),A6) )
     => ( vanishes(Y6)
       => vanishes(aa_fun765758790203053448at_rat(aTP_Lamm_aba(X5),Y6)) ) ) ).

% vanishes_mult_bounded
tff(fact_9998_vanishes__def,axiom,
    ! [X5: fun_nat_rat] :
      ( vanishes(X5)
    <=> ! [R5: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),R5)
         => ? [K3: nat] :
            ! [N4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K3),N4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_nat_rat(X5,N4))),R5) ) ) ) ).

% vanishes_def
tff(fact_9999_vanishesI,axiom,
    ! [X5: fun_nat_rat] :
      ( ! [R3: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),R3)
         => ? [K6: nat] :
            ! [N: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K6),N)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_nat_rat(X5,N))),R3) ) )
     => vanishes(X5) ) ).

% vanishesI
tff(fact_10000_vanishesD,axiom,
    ! [X5: fun_nat_rat,R2: rat] :
      ( vanishes(X5)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),R2)
       => ? [K2: nat] :
          ! [N5: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),N5)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_nat_rat(X5,N5))),R2) ) ) ) ).

% vanishesD
tff(fact_10001_cauchy__not__vanishes__cases,axiom,
    ! [X5: fun_nat_rat] :
      ( cauchy(X5)
     => ( ~ vanishes(X5)
       => ? [B2: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B2)
            & ? [K2: nat] :
                ( ! [N5: nat] :
                    ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),N5)
                   => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),aa_rat_rat(uminus_uminus_rat,aa_nat_rat(X5,N5))) )
                | ! [N5: nat] :
                    ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),N5)
                   => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),aa_nat_rat(X5,N5)) ) ) ) ) ) ).

% cauchy_not_vanishes_cases
tff(fact_10002_not__positive__Real,axiom,
    ! [X5: fun_nat_rat] :
      ( cauchy(X5)
     => ( ~ aa_real_o(positive2,real2(X5))
      <=> ! [R5: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),R5)
           => ? [K3: nat] :
              ! [N4: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K3),N4)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(X5,N4)),R5) ) ) ) ) ).

% not_positive_Real
tff(fact_10003_positive__Real,axiom,
    ! [X5: fun_nat_rat] :
      ( cauchy(X5)
     => ( aa_real_o(positive2,real2(X5))
      <=> ? [R5: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),R5)
            & ? [K3: nat] :
              ! [N4: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K3),N4)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,R5),aa_nat_rat(X5,N4)) ) ) ) ) ).

% positive_Real
tff(fact_10004_less__real__def,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
    <=> aa_real_o(positive2,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y),Xb)) ) ).

% less_real_def
tff(fact_10005_Real_Opositive_Orep__eq,axiom,
    ! [Xb: real] :
      ( aa_real_o(positive2,Xb)
    <=> ? [R5: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),R5)
          & ? [K3: nat] :
            ! [N4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K3),N4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,R5),aa_nat_rat(rep_real(Xb),N4)) ) ) ) ).

% Real.positive.rep_eq
tff(fact_10006_le__enumerate,axiom,
    ! [S2: set_nat,Na: nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,S2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),aa_nat_nat(infini8530281810654367211te_nat(S2),Na)) ) ).

% le_enumerate
tff(fact_10007_enumerate__Ex,axiom,
    ! [S2: set_nat,S3: nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,S2)
     => ( aa_set_nat_o(member_nat(S3),S2)
       => ? [N: nat] : aa_nat_nat(infini8530281810654367211te_nat(S2),N) = S3 ) ) ).

% enumerate_Ex
tff(fact_10008_strict__mono__enumerate,axiom,
    ! [S2: set_nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,S2)
     => order_5726023648592871131at_nat(infini8530281810654367211te_nat(S2)) ) ).

% strict_mono_enumerate
tff(fact_10009_range__enumerate,axiom,
    ! [S2: set_nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,S2)
     => ( image_nat_nat(infini8530281810654367211te_nat(S2),top_top_set_nat) = S2 ) ) ).

% range_enumerate
tff(fact_10010_finite__le__enumerate,axiom,
    ! [S2: set_nat,Na: nat] :
      ( aa_set_nat_o(finite_finite_nat,S2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Na),aa_set_nat_nat(finite_card_nat,S2))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Na),aa_nat_nat(infini8530281810654367211te_nat(S2),Na)) ) ) ).

% finite_le_enumerate
tff(fact_10011_bij__enumerate,axiom,
    ! [S2: set_nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,S2)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(bij_betw_nat_nat(infini8530281810654367211te_nat(S2)),top_top_set_nat),S2) ) ).

% bij_enumerate
tff(fact_10012_Least__eq__0,axiom,
    ! [P: fun_nat_o] :
      ( aa_nat_o(P,zero_zero_nat)
     => ( ord_Least_nat(P) = zero_zero_nat ) ) ).

% Least_eq_0
tff(fact_10013_Least__Suc2,axiom,
    ! [P: fun_nat_o,Na: nat,Q: fun_nat_o,M: nat] :
      ( aa_nat_o(P,Na)
     => ( aa_nat_o(Q,M)
       => ( ~ aa_nat_o(P,zero_zero_nat)
         => ( ! [K2: nat] :
                ( aa_nat_o(P,aa_nat_nat(suc,K2))
              <=> aa_nat_o(Q,K2) )
           => ( ord_Least_nat(P) = aa_nat_nat(suc,ord_Least_nat(Q)) ) ) ) ) ) ).

% Least_Suc2
tff(fact_10014_Least__Suc,axiom,
    ! [P: fun_nat_o,Na: nat] :
      ( aa_nat_o(P,Na)
     => ( ~ aa_nat_o(P,zero_zero_nat)
       => ( ord_Least_nat(P) = aa_nat_nat(suc,ord_Least_nat(aTP_Lamm_zv(P))) ) ) ) ).

% Least_Suc
tff(fact_10015_Sup__real__def,axiom,
    ! [X5: set_real] : comple1385675409528146559p_real(X5) = ord_Least_real(aTP_Lamm_abb(X5)) ).

% Sup_real_def
tff(fact_10016_rat__less__eq__code,axiom,
    ! [P3: rat,Q4: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,P3),Q4)
    <=> aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(aTP_Lamm_abd(Q4)),quotient_of(P3)) ) ).

% rat_less_eq_code
tff(fact_10017_quotient__of__denom__pos,axiom,
    ! [R2: rat,P3: int,Q4: int] :
      ( ( quotient_of(R2) = product_Pair_int_int(P3,Q4) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Q4) ) ).

% quotient_of_denom_pos
tff(fact_10018_quotient__of__denom__pos_H,axiom,
    ! [R2: rat] : aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),product_snd_int_int(quotient_of(R2))) ).

% quotient_of_denom_pos'
tff(fact_10019_rat__less__code,axiom,
    ! [P3: rat,Q4: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,P3),Q4)
    <=> aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(aTP_Lamm_abf(Q4)),quotient_of(P3)) ) ).

% rat_less_code
tff(fact_10020_quotient__of__def,axiom,
    ! [Xb: rat] : quotient_of(Xb) = the_Pr4378521158711661632nt_int(aTP_Lamm_abg(Xb)) ).

% quotient_of_def
tff(fact_10021_of__real__sqrt,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_complex(real_V4546457046886955230omplex,aa_real_real(sqrt,Xb)) = aa_complex_complex(csqrt,aa_real_complex(real_V4546457046886955230omplex,Xb)) ) ) ).

% of_real_sqrt
tff(fact_10022_coprime__abs__left__iff,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(algebr932160517623751201me_int,aa_int_int(abs_abs_int,K)),L)
    <=> aa_int_o(aa_int_fun_int_o(algebr932160517623751201me_int,K),L) ) ).

% coprime_abs_left_iff
tff(fact_10023_coprime__abs__right__iff,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(algebr932160517623751201me_int,K),aa_int_int(abs_abs_int,L))
    <=> aa_int_o(aa_int_fun_int_o(algebr932160517623751201me_int,K),L) ) ).

% coprime_abs_right_iff
tff(fact_10024_normalize__stable,axiom,
    ! [Q4: int,P3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Q4)
     => ( aa_int_o(aa_int_fun_int_o(algebr932160517623751201me_int,P3),Q4)
       => ( normalize(product_Pair_int_int(P3,Q4)) = product_Pair_int_int(P3,Q4) ) ) ) ).

% normalize_stable
tff(fact_10025_coprime__common__divisor__int,axiom,
    ! [Aa: int,Ba: int,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(algebr932160517623751201me_int,Aa),Ba)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xb),Aa)
       => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xb),Ba)
         => ( aa_int_int(abs_abs_int,Xb) = one_one_int ) ) ) ) ).

% coprime_common_divisor_int
tff(fact_10026_Rat__cases,axiom,
    ! [Q4: rat] :
      ~ ! [A2: int,B2: int] :
          ( ( Q4 = aa_int_rat(fract(A2),B2) )
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
           => ~ aa_int_o(aa_int_fun_int_o(algebr932160517623751201me_int,A2),B2) ) ) ).

% Rat_cases
tff(fact_10027_Rat__induct,axiom,
    ! [P: fun_rat_o,Q4: rat] :
      ( ! [A2: int,B2: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
         => ( aa_int_o(aa_int_fun_int_o(algebr932160517623751201me_int,A2),B2)
           => aa_rat_o(P,aa_int_rat(fract(A2),B2)) ) )
     => aa_rat_o(P,Q4) ) ).

% Rat_induct
tff(fact_10028_Rat__cases__nonzero,axiom,
    ! [Q4: rat] :
      ( ! [A2: int,B2: int] :
          ( ( Q4 = aa_int_rat(fract(A2),B2) )
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
           => ( ( A2 != zero_zero_int )
             => ~ aa_int_o(aa_int_fun_int_o(algebr932160517623751201me_int,A2),B2) ) ) )
     => ( Q4 = zero_zero_rat ) ) ).

% Rat_cases_nonzero
tff(fact_10029_quotient__of__unique,axiom,
    ! [R2: rat] :
    ? [X: product_prod_int_int] :
      ( ( R2 = aa_int_rat(fract(product_fst_int_int(X)),product_snd_int_int(X)) )
      & aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),product_snd_int_int(X))
      & aa_int_o(aa_int_fun_int_o(algebr932160517623751201me_int,product_fst_int_int(X)),product_snd_int_int(X))
      & ! [Y4: product_prod_int_int] :
          ( ( ( R2 = aa_int_rat(fract(product_fst_int_int(Y4)),product_snd_int_int(Y4)) )
            & aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),product_snd_int_int(Y4))
            & aa_int_o(aa_int_fun_int_o(algebr932160517623751201me_int,product_fst_int_int(Y4)),product_snd_int_int(Y4)) )
         => ( Y4 = X ) ) ) ).

% quotient_of_unique
tff(fact_10030_coprime__int__iff,axiom,
    ! [M: nat,Na: nat] :
      ( aa_int_o(aa_int_fun_int_o(algebr932160517623751201me_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,Na))
    <=> aa_nat_o(aa_nat_fun_nat_o(algebr934650988132801477me_nat,M),Na) ) ).

% coprime_int_iff
tff(fact_10031_coprime__nat__abs__left__iff,axiom,
    ! [K: int,Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(algebr934650988132801477me_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),Na)
    <=> aa_int_o(aa_int_fun_int_o(algebr932160517623751201me_int,K),aa_nat_int(semiri1314217659103216013at_int,Na)) ) ).

% coprime_nat_abs_left_iff
tff(fact_10032_coprime__nat__abs__right__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(algebr934650988132801477me_nat,Na),aa_int_nat(nat2,aa_int_int(abs_abs_int,K)))
    <=> aa_int_o(aa_int_fun_int_o(algebr932160517623751201me_int,aa_nat_int(semiri1314217659103216013at_int,Na)),K) ) ).

% coprime_nat_abs_right_iff
tff(fact_10033_coprime__common__divisor__nat,axiom,
    ! [Aa: nat,Ba: nat,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(algebr934650988132801477me_nat,Aa),Ba)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Xb),Aa)
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Xb),Ba)
         => ( Xb = one_one_nat ) ) ) ) ).

% coprime_common_divisor_nat
tff(fact_10034_coprime__Suc__right__nat,axiom,
    ! [Na: nat] : aa_nat_o(aa_nat_fun_nat_o(algebr934650988132801477me_nat,Na),aa_nat_nat(suc,Na)) ).

% coprime_Suc_right_nat
tff(fact_10035_coprime__Suc__left__nat,axiom,
    ! [Na: nat] : aa_nat_o(aa_nat_fun_nat_o(algebr934650988132801477me_nat,aa_nat_nat(suc,Na)),Na) ).

% coprime_Suc_left_nat
tff(fact_10036_coprime__Suc__0__right,axiom,
    ! [Na: nat] : aa_nat_o(aa_nat_fun_nat_o(algebr934650988132801477me_nat,Na),aa_nat_nat(suc,zero_zero_nat)) ).

% coprime_Suc_0_right
tff(fact_10037_coprime__Suc__0__left,axiom,
    ! [Na: nat] : aa_nat_o(aa_nat_fun_nat_o(algebr934650988132801477me_nat,aa_nat_nat(suc,zero_zero_nat)),Na) ).

% coprime_Suc_0_left
tff(fact_10038_coprime__diff__one__right__nat,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => aa_nat_o(aa_nat_fun_nat_o(algebr934650988132801477me_nat,Na),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat)) ) ).

% coprime_diff_one_right_nat
tff(fact_10039_coprime__diff__one__left__nat,axiom,
    ! [Na: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Na)
     => aa_nat_o(aa_nat_fun_nat_o(algebr934650988132801477me_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Na),one_one_nat)),Na) ) ).

% coprime_diff_one_left_nat
tff(fact_10040_Rats__abs__nat__div__natE,axiom,
    ! [Xb: real] :
      ( member_real(Xb,field_5140801741446780682s_real)
     => ~ ! [M3: nat,N: nat] :
            ( ( N != zero_zero_nat )
           => ( ( aa_real_real(abs_abs_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,M3)),aa_nat_real(semiri5074537144036343181t_real,N)) )
             => ~ aa_nat_o(aa_nat_fun_nat_o(algebr934650988132801477me_nat,M3),N) ) ) ) ).

% Rats_abs_nat_div_natE
tff(fact_10041_last__upt,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => ( last_nat(upt(I,J)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),one_one_nat) ) ) ).

% last_upt
tff(fact_10042_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_10043_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_10044_not__integer_Oabs__eq,axiom,
    ! [Xb: int] : aa_Cod1783974652533396728nteger(bit_ri7632146776885996613nteger,aa_int_Code_integer(code_integer_of_int,Xb)) = aa_int_Code_integer(code_integer_of_int,aa_int_int(bit_ri7919022796975470100ot_int,Xb)) ).

% not_integer.abs_eq
tff(fact_10045_not__integer_Orep__eq,axiom,
    ! [Xb: code_integer] : aa_Code_integer_int(code_int_of_integer,aa_Cod1783974652533396728nteger(bit_ri7632146776885996613nteger,Xb)) = aa_int_int(bit_ri7919022796975470100ot_int,aa_Code_integer_int(code_int_of_integer,Xb)) ).

% not_integer.rep_eq
tff(fact_10046_Real_Opositive_Oabs__eq,axiom,
    ! [Xb: fun_nat_rat] :
      ( aa_fun_nat_rat_o(aa_fun6961092367326905321_rat_o(realrel,Xb),Xb)
     => ( aa_real_o(positive2,real2(Xb))
      <=> ? [R5: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),R5)
            & ? [K3: nat] :
              ! [N4: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K3),N4)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,R5),aa_nat_rat(Xb,N4)) ) ) ) ) ).

% Real.positive.abs_eq
tff(fact_10047_Real_Opositive_Orsp,axiom,
    bNF_re7504242029904815401at_o_o(realrel,fequal_o,aTP_Lamm_abh,aTP_Lamm_abh) ).

% Real.positive.rsp
tff(fact_10048_sqr_Osimps_I3_J,axiom,
    ! [Na: num] : aa_num_num(sqr,aa_num_num(bit1,Na)) = aa_num_num(bit1,aa_num_num(bit0,aa_num_num(aa_num_fun_num_num(plus_plus_num,aa_num_num(sqr,Na)),Na))) ).

% sqr.simps(3)
tff(fact_10049_sqr_Osimps_I1_J,axiom,
    aa_num_num(sqr,one) = one ).

% sqr.simps(1)
tff(fact_10050_sqr_Osimps_I2_J,axiom,
    ! [Na: num] : aa_num_num(sqr,aa_num_num(bit0,Na)) = aa_num_num(bit0,aa_num_num(bit0,aa_num_num(sqr,Na))) ).

% sqr.simps(2)
tff(fact_10051_sqr__conv__mult,axiom,
    ! [Xb: num] : aa_num_num(sqr,Xb) = aa_num_num(aa_num_fun_num_num(times_times_num,Xb),Xb) ).

% sqr_conv_mult
tff(fact_10052_less__eq__natural_Orsp,axiom,
    bNF_re1448341924010609215_nat_o(fequal_nat,bNF_re4705727531993890431at_o_o(fequal_nat,fequal_o),ord_less_eq_nat,ord_less_eq_nat) ).

% less_eq_natural.rsp
tff(fact_10053_less__eq__integer_Orsp,axiom,
    bNF_re8654322613993568759_int_o(fequal_int,bNF_re5089333283451836215nt_o_o(fequal_int,fequal_o),ord_less_eq_int,ord_less_eq_int) ).

% less_eq_integer.rsp
tff(fact_10054_num__of__integer_Orsp,axiom,
    bNF_re7626690874201225453um_num(fequal_int,fequal_num,comp_nat_num_int(num_of_nat,nat2),comp_nat_num_int(num_of_nat,nat2)) ).

% num_of_integer.rsp
tff(fact_10055_sgn__integer_Orsp,axiom,
    aa_fun_int_int_o(aa_fun2545150047610057353_int_o(bNF_re4712519889275205905nt_int(fequal_int,fequal_int),sgn_sgn_int),sgn_sgn_int) ).

% sgn_integer.rsp
tff(fact_10056_less__natural_Orsp,axiom,
    bNF_re1448341924010609215_nat_o(fequal_nat,bNF_re4705727531993890431at_o_o(fequal_nat,fequal_o),ord_less_nat,ord_less_nat) ).

% less_natural.rsp
tff(fact_10057_less__integer_Orsp,axiom,
    bNF_re8654322613993568759_int_o(fequal_int,bNF_re5089333283451836215nt_o_o(fequal_int,fequal_o),ord_less_int,ord_less_int) ).

% less_integer.rsp
tff(fact_10058_integer__of__natural_Orsp,axiom,
    bNF_re6650684261131312217nt_int(fequal_nat,fequal_int,semiri1314217659103216013at_int,semiri1314217659103216013at_int) ).

% integer_of_natural.rsp
tff(fact_10059_drop__bit__natural_Orsp,axiom,
    bNF_re5162689864697376059at_nat(fequal_nat,bNF_re5653821019739307937at_nat(fequal_nat,fequal_nat),bit_se8570568707652914677it_nat,bit_se8570568707652914677it_nat) ).

% drop_bit_natural.rsp
tff(fact_10060_drop__bit__integer_Orsp,axiom,
    bNF_re5651586111134833323nt_int(fequal_nat,bNF_re4712519889275205905nt_int(fequal_int,fequal_int),bit_se8568078237143864401it_int,bit_se8568078237143864401it_int) ).

% drop_bit_integer.rsp
tff(fact_10061_dup_Orsp,axiom,
    aa_fun_int_int_o(aa_fun2545150047610057353_int_o(bNF_re4712519889275205905nt_int(fequal_int,fequal_int),aTP_Lamm_abi),aTP_Lamm_abi) ).

% dup.rsp
tff(fact_10062_and__integer_Orsp,axiom,
    bNF_re5717476210804479843nt_int(fequal_int,bNF_re4712519889275205905nt_int(fequal_int,fequal_int),bit_se725231765392027082nd_int,bit_se725231765392027082nd_int) ).

% and_integer.rsp
tff(fact_10063_and__natural_Orsp,axiom,
    bNF_re5162689864697376059at_nat(fequal_nat,bNF_re5653821019739307937at_nat(fequal_nat,fequal_nat),bit_se727722235901077358nd_nat,bit_se727722235901077358nd_nat) ).

% and_natural.rsp
tff(fact_10064_push__bit__natural_Orsp,axiom,
    bNF_re5162689864697376059at_nat(fequal_nat,bNF_re5653821019739307937at_nat(fequal_nat,fequal_nat),bit_se547839408752420682it_nat,bit_se547839408752420682it_nat) ).

% push_bit_natural.rsp
tff(fact_10065_push__bit__integer_Orsp,axiom,
    bNF_re5651586111134833323nt_int(fequal_nat,bNF_re4712519889275205905nt_int(fequal_int,fequal_int),bit_se545348938243370406it_int,bit_se545348938243370406it_int) ).

% push_bit_integer.rsp
tff(fact_10066_abs__integer_Orsp,axiom,
    aa_fun_int_int_o(aa_fun2545150047610057353_int_o(bNF_re4712519889275205905nt_int(fequal_int,fequal_int),abs_abs_int),abs_abs_int) ).

% abs_integer.rsp
tff(fact_10067_flip__bit__natural_Orsp,axiom,
    bNF_re5162689864697376059at_nat(fequal_nat,bNF_re5653821019739307937at_nat(fequal_nat,fequal_nat),bit_se2161824704523386999it_nat,bit_se2161824704523386999it_nat) ).

% flip_bit_natural.rsp
tff(fact_10068_flip__bit__integer_Orsp,axiom,
    bNF_re5651586111134833323nt_int(fequal_nat,bNF_re4712519889275205905nt_int(fequal_int,fequal_int),bit_se2159334234014336723it_int,bit_se2159334234014336723it_int) ).

% flip_bit_integer.rsp
tff(fact_10069_set__bit__natural_Orsp,axiom,
    bNF_re5162689864697376059at_nat(fequal_nat,bNF_re5653821019739307937at_nat(fequal_nat,fequal_nat),bit_se7882103937844011126it_nat,bit_se7882103937844011126it_nat) ).

% set_bit_natural.rsp
tff(fact_10070_set__bit__integer_Orsp,axiom,
    bNF_re5651586111134833323nt_int(fequal_nat,bNF_re4712519889275205905nt_int(fequal_int,fequal_int),bit_se7879613467334960850it_int,bit_se7879613467334960850it_int) ).

% set_bit_integer.rsp
tff(fact_10071_plus__natural_Orsp,axiom,
    bNF_re5162689864697376059at_nat(fequal_nat,bNF_re5653821019739307937at_nat(fequal_nat,fequal_nat),plus_plus_nat,plus_plus_nat) ).

% plus_natural.rsp
tff(fact_10072_plus__integer_Orsp,axiom,
    bNF_re5717476210804479843nt_int(fequal_int,bNF_re4712519889275205905nt_int(fequal_int,fequal_int),plus_plus_int,plus_plus_int) ).

% plus_integer.rsp
tff(fact_10073_uminus__integer_Orsp,axiom,
    aa_fun_int_int_o(aa_fun2545150047610057353_int_o(bNF_re4712519889275205905nt_int(fequal_int,fequal_int),uminus_uminus_int),uminus_uminus_int) ).

% uminus_integer.rsp
tff(fact_10074_bit__integer_Orsp,axiom,
    bNF_re8347998229424194295_nat_o(fequal_int,fequal_fun_nat_o,bit_se1146084159140164899it_int,bit_se1146084159140164899it_int) ).

% bit_integer.rsp
tff(fact_10075_bit__natural_Orsp,axiom,
    bNF_re1448341924010609215_nat_o(fequal_nat,fequal_fun_nat_o,bit_se1148574629649215175it_nat,bit_se1148574629649215175it_nat) ).

% bit_natural.rsp
tff(fact_10076_xor__integer_Orsp,axiom,
    bNF_re5717476210804479843nt_int(fequal_int,bNF_re4712519889275205905nt_int(fequal_int,fequal_int),bit_se6526347334894502574or_int,bit_se6526347334894502574or_int) ).

% xor_integer.rsp
tff(fact_10077_xor__natural_Orsp,axiom,
    bNF_re5162689864697376059at_nat(fequal_nat,bNF_re5653821019739307937at_nat(fequal_nat,fequal_nat),bit_se6528837805403552850or_nat,bit_se6528837805403552850or_nat) ).

% xor_natural.rsp
tff(fact_10078_division__segment__natural_Orsp,axiom,
    aa_fun_nat_nat_o(aa_fun2341495900663639033_nat_o(bNF_re5653821019739307937at_nat(fequal_nat,fequal_nat),euclid3398187327856392827nt_nat),euclid3398187327856392827nt_nat) ).

% division_segment_natural.rsp
tff(fact_10079_division__segment__integer_Orsp,axiom,
    aa_fun_int_int_o(aa_fun2545150047610057353_int_o(bNF_re4712519889275205905nt_int(fequal_int,fequal_int),euclid3395696857347342551nt_int),euclid3395696857347342551nt_int) ).

% division_segment_integer.rsp
tff(fact_10080_take__bit__natural_Orsp,axiom,
    bNF_re5162689864697376059at_nat(fequal_nat,bNF_re5653821019739307937at_nat(fequal_nat,fequal_nat),bit_se2925701944663578781it_nat,bit_se2925701944663578781it_nat) ).

% take_bit_natural.rsp
tff(fact_10081_take__bit__integer_Orsp,axiom,
    bNF_re5651586111134833323nt_int(fequal_nat,bNF_re4712519889275205905nt_int(fequal_int,fequal_int),bit_se2923211474154528505it_int,bit_se2923211474154528505it_int) ).

% take_bit_integer.rsp
tff(fact_10082_unset__bit__integer_Orsp,axiom,
    bNF_re5651586111134833323nt_int(fequal_nat,bNF_re4712519889275205905nt_int(fequal_int,fequal_int),bit_se4203085406695923979it_int,bit_se4203085406695923979it_int) ).

% unset_bit_integer.rsp
tff(fact_10083_Suc_Orsp,axiom,
    aa_fun_nat_nat_o(aa_fun2341495900663639033_nat_o(bNF_re5653821019739307937at_nat(fequal_nat,fequal_nat),suc),suc) ).

% Suc.rsp
tff(fact_10084_divide__integer_Orsp,axiom,
    bNF_re5717476210804479843nt_int(fequal_int,bNF_re4712519889275205905nt_int(fequal_int,fequal_int),divide_divide_int,divide_divide_int) ).

% divide_integer.rsp
tff(fact_10085_divide__natural_Orsp,axiom,
    bNF_re5162689864697376059at_nat(fequal_nat,bNF_re5653821019739307937at_nat(fequal_nat,fequal_nat),divide_divide_nat,divide_divide_nat) ).

% divide_natural.rsp
tff(fact_10086_modulo__natural_Orsp,axiom,
    bNF_re5162689864697376059at_nat(fequal_nat,bNF_re5653821019739307937at_nat(fequal_nat,fequal_nat),modulo_modulo_nat,modulo_modulo_nat) ).

% modulo_natural.rsp
tff(fact_10087_modulo__integer_Orsp,axiom,
    bNF_re5717476210804479843nt_int(fequal_int,bNF_re4712519889275205905nt_int(fequal_int,fequal_int),modulo_modulo_int,modulo_modulo_int) ).

% modulo_integer.rsp
tff(fact_10088_natural__of__integer_Orsp,axiom,
    bNF_re3715656647883201625at_nat(fequal_int,fequal_nat,nat2,nat2) ).

% natural_of_integer.rsp
tff(fact_10089_mask__natural_Orsp,axiom,
    aa_fun_nat_nat_o(aa_fun2341495900663639033_nat_o(bNF_re5653821019739307937at_nat(fequal_nat,fequal_nat),bit_se2002935070580805687sk_nat),bit_se2002935070580805687sk_nat) ).

% mask_natural.rsp
tff(fact_10090_mask__integer_Orsp,axiom,
    bNF_re6650684261131312217nt_int(fequal_nat,fequal_int,bit_se2000444600071755411sk_int,bit_se2000444600071755411sk_int) ).

% mask_integer.rsp
tff(fact_10091_euclidean__size__integer_Orsp,axiom,
    bNF_re3715656647883201625at_nat(fequal_int,fequal_nat,euclid4774559944035922753ze_int,euclid4774559944035922753ze_int) ).

% euclidean_size_integer.rsp
tff(fact_10092_or__integer_Orsp,axiom,
    bNF_re5717476210804479843nt_int(fequal_int,bNF_re4712519889275205905nt_int(fequal_int,fequal_int),bit_se1409905431419307370or_int,bit_se1409905431419307370or_int) ).

% or_integer.rsp
tff(fact_10093_or__natural_Orsp,axiom,
    bNF_re5162689864697376059at_nat(fequal_nat,bNF_re5653821019739307937at_nat(fequal_nat,fequal_nat),bit_se1412395901928357646or_nat,bit_se1412395901928357646or_nat) ).

% or_natural.rsp
tff(fact_10094_minus__integer_Orsp,axiom,
    bNF_re5717476210804479843nt_int(fequal_int,bNF_re4712519889275205905nt_int(fequal_int,fequal_int),minus_minus_int,minus_minus_int) ).

% minus_integer.rsp
tff(fact_10095_minus__natural_Orsp,axiom,
    bNF_re5162689864697376059at_nat(fequal_nat,bNF_re5653821019739307937at_nat(fequal_nat,fequal_nat),minus_minus_nat,minus_minus_nat) ).

% minus_natural.rsp
tff(fact_10096_sub_Orsp,axiom,
    bNF_re6378542964492772123um_int(fequal_num,bNF_re1822329894187522285nt_int(fequal_num,fequal_int),aTP_Lamm_abj,aTP_Lamm_abj) ).

% sub.rsp
tff(fact_10097_times__natural_Orsp,axiom,
    bNF_re5162689864697376059at_nat(fequal_nat,bNF_re5653821019739307937at_nat(fequal_nat,fequal_nat),times_times_nat,times_times_nat) ).

% times_natural.rsp
tff(fact_10098_times__integer_Orsp,axiom,
    bNF_re5717476210804479843nt_int(fequal_int,bNF_re4712519889275205905nt_int(fequal_int,fequal_int),times_times_int,times_times_int) ).

% times_integer.rsp
tff(fact_10099_not__integer_Orsp,axiom,
    aa_fun_int_int_o(aa_fun2545150047610057353_int_o(bNF_re4712519889275205905nt_int(fequal_int,fequal_int),bit_ri7919022796975470100ot_int),bit_ri7919022796975470100ot_int) ).

% not_integer.rsp
tff(fact_10100_Real_Opositive_Otransfer,axiom,
    bNF_re3218747500659316912al_o_o(pcr_real,fequal_o,aTP_Lamm_abh,positive2) ).

% Real.positive.transfer
tff(fact_10101_euclidean__size__natural_Orsp,axiom,
    aa_fun_nat_nat_o(aa_fun2341495900663639033_nat_o(bNF_re5653821019739307937at_nat(fequal_nat,fequal_nat),euclid4777050414544973029ze_nat),euclid4777050414544973029ze_nat) ).

% euclidean_size_natural.rsp
tff(fact_10102_unset__bit__natural_Orsp,axiom,
    bNF_re5162689864697376059at_nat(fequal_nat,bNF_re5653821019739307937at_nat(fequal_nat,fequal_nat),bit_se4205575877204974255it_nat,bit_se4205575877204974255it_nat) ).

% unset_bit_natural.rsp
tff(fact_10103_Rat_Opositive_Otransfer,axiom,
    bNF_re1494630372529172596at_o_o(pcr_rat,fequal_o,aTP_Lamm_abk,positive) ).

% Rat.positive.transfer
tff(fact_10104_pow_Osimps_I3_J,axiom,
    ! [Xb: num,Y: num] : aa_num_num(aa_num_fun_num_num(pow,Xb),aa_num_num(bit1,Y)) = aa_num_num(aa_num_fun_num_num(times_times_num,aa_num_num(sqr,aa_num_num(aa_num_fun_num_num(pow,Xb),Y))),Xb) ).

% pow.simps(3)
tff(fact_10105_pow_Osimps_I1_J,axiom,
    ! [Xb: num] : aa_num_num(aa_num_fun_num_num(pow,Xb),one) = Xb ).

% pow.simps(1)
tff(fact_10106_pow_Osimps_I2_J,axiom,
    ! [Xb: num,Y: num] : aa_num_num(aa_num_fun_num_num(pow,Xb),aa_num_num(bit0,Y)) = aa_num_num(sqr,aa_num_num(aa_num_fun_num_num(pow,Xb),Y)) ).

% pow.simps(2)
tff(fact_10107_less__eq__int_Otransfer,axiom,
    bNF_re4195927319312321177_int_o(pcr_int,bNF_re6644619430987730960nt_o_o(pcr_int,fequal_o),produc6395135316500328296_nat_o(aTP_Lamm_aaw),ord_less_eq_int) ).

% less_eq_int.transfer
tff(fact_10108_zero__int_Otransfer,axiom,
    aa_int_o(aa_Pro2912763917461432128_int_o(pcr_int,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),zero_zero_nat)),zero_zero_int) ).

% zero_int.transfer
tff(fact_10109_int__transfer,axiom,
    bNF_re6830278522597306478at_int(fequal_nat,pcr_int,aTP_Lamm_abl,semiri1314217659103216013at_int) ).

% int_transfer
tff(fact_10110_one__int_Otransfer,axiom,
    aa_int_o(aa_Pro2912763917461432128_int_o(pcr_int,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(one_one_nat),zero_zero_nat)),one_one_int) ).

% one_int.transfer
tff(fact_10111_less__int_Otransfer,axiom,
    bNF_re4195927319312321177_int_o(pcr_int,bNF_re6644619430987730960nt_o_o(pcr_int,fequal_o),produc6395135316500328296_nat_o(aTP_Lamm_aay),ord_less_int) ).

% less_int.transfer
tff(fact_10112_Rat_Opositive_Orsp,axiom,
    bNF_re8699439704749558557nt_o_o(ratrel,fequal_o,aTP_Lamm_abk,aTP_Lamm_abk) ).

% Rat.positive.rsp
tff(fact_10113_Rat_Opositive_Oabs__eq,axiom,
    ! [Xb: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_Pro3952998155787061599_int_o(ratrel,Xb),Xb)
     => ( aa_rat_o(positive,abs_Rat(Xb))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,product_fst_int_int(Xb)),product_snd_int_int(Xb))) ) ) ).

% Rat.positive.abs_eq
tff(fact_10114_vimage__Suc__insert__0,axiom,
    ! [A3: set_nat] : vimage_nat_nat(suc,insert_nat(zero_zero_nat,A3)) = vimage_nat_nat(suc,A3) ).

% vimage_Suc_insert_0
tff(fact_10115_vimage__Suc__insert__Suc,axiom,
    ! [Na: nat,A3: set_nat] : vimage_nat_nat(suc,insert_nat(aa_nat_nat(suc,Na),A3)) = insert_nat(Na,vimage_nat_nat(suc,A3)) ).

% vimage_Suc_insert_Suc
tff(fact_10116_finite__vimage__Suc__iff,axiom,
    ! [F3: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,vimage_nat_nat(suc,F3))
    <=> aa_set_nat_o(finite_finite_nat,F3) ) ).

% finite_vimage_Suc_iff
tff(fact_10117_natLeq__on__wo__rel,axiom,
    ! [Na: nat] : bNF_We3818239936649020644el_nat(collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_aaz(Na)))) ).

% natLeq_on_wo_rel
tff(fact_10118_set__decode__div__2,axiom,
    ! [Xb: 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)))) = vimage_nat_nat(suc,nat_set_decode(Xb)) ).

% set_decode_div_2
tff(fact_10119_set__encode__vimage__Suc,axiom,
    ! [A3: set_nat] : aa_set_nat_nat(nat_set_encode,vimage_nat_nat(suc,A3)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_set_nat_nat(nat_set_encode,A3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% set_encode_vimage_Suc
tff(fact_10120_pairs__le__eq__Sigma,axiom,
    ! [M: nat] : collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_ti(M))) = produc457027306803732586at_nat(aa_nat_set_nat(set_ord_atMost_nat,M),aTP_Lamm_abm(M)) ).

% pairs_le_eq_Sigma
tff(fact_10121_Bseq__monoseq__convergent_H__inc,axiom,
    ! [F: fun_nat_real,M7: nat] :
      ( bfun_nat_real(aa_nat_fun_nat_real(aTP_Lamm_ji(F),M7),at_top_nat)
     => ( ! [M3: nat,N: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M7),M3)
           => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M3),N)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,M3)),aa_nat_real(F,N)) ) )
       => topolo7531315842566124627t_real(F) ) ) ).

% Bseq_monoseq_convergent'_inc
tff(fact_10122_Bseq__mono__convergent,axiom,
    ! [X5: fun_nat_real] :
      ( bfun_nat_real(X5,at_top_nat)
     => ( ! [M3: nat,N: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M3),N)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(X5,M3)),aa_nat_real(X5,N)) )
       => topolo7531315842566124627t_real(X5) ) ) ).

% Bseq_mono_convergent
tff(fact_10123_convergent__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)
       => topolo7531315842566124627t_real(aa_real_fun_nat_real(power_power_real,Xb)) ) ) ).

% convergent_realpow
tff(fact_10124_Bseq__monoseq__convergent_H__dec,axiom,
    ! [F: fun_nat_real,M7: nat] :
      ( bfun_nat_real(aa_nat_fun_nat_real(aTP_Lamm_ji(F),M7),at_top_nat)
     => ( ! [M3: nat,N: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M7),M3)
           => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M3),N)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,N)),aa_nat_real(F,M3)) ) )
       => topolo7531315842566124627t_real(F) ) ) ).

% Bseq_monoseq_convergent'_dec
tff(fact_10125_Restr__natLeq,axiom,
    ! [Na: nat] : inf_in2572325071724192079at_nat(bNF_Ca8665028551170535155natLeq,produc457027306803732586at_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_au,Na)),aTP_Lamm_abn(Na))) = collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_aaz(Na))) ).

% Restr_natLeq
tff(fact_10126_natLeq__def,axiom,
    bNF_Ca8665028551170535155natLeq = collec3392354462482085612at_nat(produc6081775807080527818_nat_o(ord_less_eq_nat)) ).

% natLeq_def
tff(fact_10127_Restr__natLeq2,axiom,
    ! [Na: nat] : inf_in2572325071724192079at_nat(bNF_Ca8665028551170535155natLeq,produc457027306803732586at_nat(order_underS_nat(bNF_Ca8665028551170535155natLeq,Na),aTP_Lamm_abo(Na))) = collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_aaz(Na))) ).

% Restr_natLeq2
tff(fact_10128_natLeq__underS__less,axiom,
    ! [Na: nat] : order_underS_nat(bNF_Ca8665028551170535155natLeq,Na) = collect_nat(aa_nat_fun_nat_o(aTP_Lamm_au,Na)) ).

% natLeq_underS_less
tff(fact_10129_gcd__nat_Oordering__top__axioms,axiom,
    aa_nat_o(ordering_top_nat(dvd_dvd_nat,aTP_Lamm_aat),zero_zero_nat) ).

% gcd_nat.ordering_top_axioms
tff(fact_10130_bot__nat__0_Oordering__top__axioms,axiom,
    aa_nat_o(ordering_top_nat(aTP_Lamm_av,aTP_Lamm_au),zero_zero_nat) ).

% bot_nat_0.ordering_top_axioms
tff(fact_10131_less__eq__int_Orsp,axiom,
    bNF_re4103222659376994899_nat_o(intrel,bNF_re3666534408544137501at_o_o(intrel,fequal_o),produc6395135316500328296_nat_o(aTP_Lamm_aaw),produc6395135316500328296_nat_o(aTP_Lamm_aaw)) ).

% less_eq_int.rsp
tff(fact_10132_zero__int_Orsp,axiom,
    aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(intrel,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),zero_zero_nat)),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),zero_zero_nat)) ).

% zero_int.rsp
tff(fact_10133_one__int_Orsp,axiom,
    aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(intrel,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(one_one_nat),zero_zero_nat)),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(one_one_nat),zero_zero_nat)) ).

% one_int.rsp
tff(fact_10134_less__int_Orsp,axiom,
    bNF_re4103222659376994899_nat_o(intrel,bNF_re3666534408544137501at_o_o(intrel,fequal_o),produc6395135316500328296_nat_o(aTP_Lamm_aay),produc6395135316500328296_nat_o(aTP_Lamm_aay)) ).

% less_int.rsp
tff(fact_10135_less__eq__enat__def,axiom,
    ! [M: extended_enat,Na: extended_enat] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(M),Na)
    <=> aa_Extended_enat_o(extended_case_enat_o(aTP_Lamm_abp(M),$true),Na) ) ).

% less_eq_enat_def
tff(fact_10136_less__enat__def,axiom,
    ! [M: extended_enat,Na: extended_enat] :
      ( aa_Extended_enat_o(ord_le72135733267957522d_enat(M),Na)
    <=> aa_Extended_enat_o(extended_case_enat_o(aTP_Lamm_abq(Na),$false),M) ) ).

% less_enat_def
tff(fact_10137_division__segment__integer__def,axiom,
    euclid6289375185220004616nteger = aa_fun8964553180794676938nteger(map_fu2599414010547811884nteger(code_int_of_integer,code_integer_of_int),euclid3395696857347342551nt_int) ).

% division_segment_integer_def
tff(fact_10138_uminus__integer__def,axiom,
    uminus1351360451143612070nteger = aa_fun8964553180794676938nteger(map_fu2599414010547811884nteger(code_int_of_integer,code_integer_of_int),uminus_uminus_int) ).

% uminus_integer_def
tff(fact_10139_abs__integer__def,axiom,
    abs_abs_Code_integer = aa_fun8964553180794676938nteger(map_fu2599414010547811884nteger(code_int_of_integer,code_integer_of_int),abs_abs_int) ).

% abs_integer_def
tff(fact_10140_sgn__integer__def,axiom,
    sgn_sgn_Code_integer = aa_fun8964553180794676938nteger(map_fu2599414010547811884nteger(code_int_of_integer,code_integer_of_int),sgn_sgn_int) ).

% sgn_integer_def
tff(fact_10141_not__integer__def,axiom,
    bit_ri7632146776885996613nteger = aa_fun8964553180794676938nteger(map_fu2599414010547811884nteger(code_int_of_integer,code_integer_of_int),bit_ri7919022796975470100ot_int) ).

% not_integer_def
tff(fact_10142_Code__Numeral_Odup__def,axiom,
    code_dup = aa_fun8964553180794676938nteger(map_fu2599414010547811884nteger(code_int_of_integer,code_integer_of_int),aTP_Lamm_abi) ).

% Code_Numeral.dup_def
tff(fact_10143_xor__integer__def,axiom,
    bit_se3222712562003087583nteger = map_fu1034890151506530863nteger(code_int_of_integer,map_fu2599414010547811884nteger(code_int_of_integer,code_integer_of_int),bit_se6526347334894502574or_int) ).

% xor_integer_def
tff(fact_10144_Code__Numeral_Odup__code_I1_J,axiom,
    aa_Cod1783974652533396728nteger(code_dup,zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% Code_Numeral.dup_code(1)
tff(fact_10145_dup_Orep__eq,axiom,
    ! [Xb: code_integer] : aa_Code_integer_int(code_int_of_integer,aa_Cod1783974652533396728nteger(code_dup,Xb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_Code_integer_int(code_int_of_integer,Xb)),aa_Code_integer_int(code_int_of_integer,Xb)) ).

% dup.rep_eq
tff(fact_10146_dup_Oabs__eq,axiom,
    ! [Xb: int] : aa_Cod1783974652533396728nteger(code_dup,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,Xb),Xb)) ).

% dup.abs_eq
tff(fact_10147_plus__integer__def,axiom,
    plus_p5714425477246183910nteger = map_fu1034890151506530863nteger(code_int_of_integer,map_fu2599414010547811884nteger(code_int_of_integer,code_integer_of_int),plus_plus_int) ).

% plus_integer_def
tff(fact_10148_times__integer__def,axiom,
    times_3573771949741848930nteger = map_fu1034890151506530863nteger(code_int_of_integer,map_fu2599414010547811884nteger(code_int_of_integer,code_integer_of_int),times_times_int) ).

% times_integer_def
tff(fact_10149_minus__integer__def,axiom,
    minus_8373710615458151222nteger = map_fu1034890151506530863nteger(code_int_of_integer,map_fu2599414010547811884nteger(code_int_of_integer,code_integer_of_int),minus_minus_int) ).

% minus_integer_def
tff(fact_10150_divide__integer__def,axiom,
    divide6298287555418463151nteger = map_fu1034890151506530863nteger(code_int_of_integer,map_fu2599414010547811884nteger(code_int_of_integer,code_integer_of_int),divide_divide_int) ).

% divide_integer_def
tff(fact_10151_modulo__integer__def,axiom,
    modulo364778990260209775nteger = map_fu1034890151506530863nteger(code_int_of_integer,map_fu2599414010547811884nteger(code_int_of_integer,code_integer_of_int),modulo_modulo_int) ).

% modulo_integer_def
tff(fact_10152_and__integer__def,axiom,
    bit_se3949692690581998587nteger = map_fu1034890151506530863nteger(code_int_of_integer,map_fu2599414010547811884nteger(code_int_of_integer,code_integer_of_int),bit_se725231765392027082nd_int) ).

% and_integer_def
tff(fact_10153_or__integer__def,axiom,
    bit_se1080825931792720795nteger = map_fu1034890151506530863nteger(code_int_of_integer,map_fu2599414010547811884nteger(code_int_of_integer,code_integer_of_int),bit_se1409905431419307370or_int) ).

% or_integer_def
tff(fact_10154_Code__Numeral_Osub__code_I9_J,axiom,
    ! [M: num,Na: num] : aa_num_Code_integer(aa_num5808785979904431436nteger(code_sub,aa_num_num(bit0,M)),aa_num_num(bit1,Na)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(code_dup,aa_num_Code_integer(aa_num5808785979904431436nteger(code_sub,M),Na))),one_one_Code_integer) ).

% Code_Numeral.sub_code(9)
tff(fact_10155_Code__Numeral_Osub__code_I8_J,axiom,
    ! [M: num,Na: num] : aa_num_Code_integer(aa_num5808785979904431436nteger(code_sub,aa_num_num(bit1,M)),aa_num_num(bit0,Na)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(code_dup,aa_num_Code_integer(aa_num5808785979904431436nteger(code_sub,M),Na))),one_one_Code_integer) ).

% Code_Numeral.sub_code(8)
tff(fact_10156_Code__Numeral_Osub__code_I7_J,axiom,
    ! [M: num,Na: num] : aa_num_Code_integer(aa_num5808785979904431436nteger(code_sub,aa_num_num(bit1,M)),aa_num_num(bit1,Na)) = aa_Cod1783974652533396728nteger(code_dup,aa_num_Code_integer(aa_num5808785979904431436nteger(code_sub,M),Na)) ).

% Code_Numeral.sub_code(7)
tff(fact_10157_Code__Numeral_Osub__code_I6_J,axiom,
    ! [M: num,Na: num] : aa_num_Code_integer(aa_num5808785979904431436nteger(code_sub,aa_num_num(bit0,M)),aa_num_num(bit0,Na)) = aa_Cod1783974652533396728nteger(code_dup,aa_num_Code_integer(aa_num5808785979904431436nteger(code_sub,M),Na)) ).

% Code_Numeral.sub_code(6)
tff(fact_10158_Code__Numeral_Osub__code_I1_J,axiom,
    aa_num_Code_integer(aa_num5808785979904431436nteger(code_sub,one),one) = zero_z3403309356797280102nteger ).

% Code_Numeral.sub_code(1)
tff(fact_10159_sub_Orep__eq,axiom,
    ! [Xb: num,Xaa: num] : aa_Code_integer_int(code_int_of_integer,aa_num_Code_integer(aa_num5808785979904431436nteger(code_sub,Xb),Xaa)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,Xb)),aa_num_int(numeral_numeral_int,Xaa)) ).

% sub.rep_eq
tff(fact_10160_sub_Oabs__eq,axiom,
    ! [Xaa: num,Xb: num] : aa_num_Code_integer(aa_num5808785979904431436nteger(code_sub,Xaa),Xb) = aa_int_Code_integer(code_integer_of_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,Xaa)),aa_num_int(numeral_numeral_int,Xb))) ).

% sub.abs_eq
tff(fact_10161_less__RealD,axiom,
    ! [Y6: fun_nat_rat,Xb: real] :
      ( cauchy(Y6)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),real2(Y6))
       => ? [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_rat_real(field_7254667332652039916t_real,aa_nat_rat(Y6,N))) ) ) ).

% less_RealD
tff(fact_10162_of__rat__dense,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
     => ? [Q3: rat] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_rat_real(field_7254667332652039916t_real,Q3))
          & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(field_7254667332652039916t_real,Q3)),Y) ) ) ).

% of_rat_dense
tff(fact_10163_le__RealI,axiom,
    ! [Y6: fun_nat_rat,Xb: real] :
      ( cauchy(Y6)
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_rat_real(field_7254667332652039916t_real,aa_nat_rat(Y6,N)))
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),real2(Y6)) ) ) ).

% le_RealI
tff(fact_10164_Real__leI,axiom,
    ! [X5: fun_nat_rat,Y: real] :
      ( cauchy(X5)
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_rat_real(field_7254667332652039916t_real,aa_nat_rat(X5,N))),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,real2(X5)),Y) ) ) ).

% Real_leI
tff(fact_10165_num__of__integer_Otransfer,axiom,
    bNF_re6718328864250387230um_num(code_pcr_integer,fequal_num,comp_nat_num_int(num_of_nat,nat2),code_num_of_integer) ).

% num_of_integer.transfer
tff(fact_10166_not__integer_Otransfer,axiom,
    aa_fun774133186966548707eger_o(aa_fun6864876441520360999eger_o(bNF_re3379532845092657523nteger(code_pcr_integer,code_pcr_integer),bit_ri7919022796975470100ot_int),bit_ri7632146776885996613nteger) ).

% not_integer.transfer
tff(fact_10167_times__integer_Otransfer,axiom,
    bNF_re9130850084932126070nteger(code_pcr_integer,bNF_re3379532845092657523nteger(code_pcr_integer,code_pcr_integer),times_times_int,times_3573771949741848930nteger) ).

% times_integer.transfer
tff(fact_10168_minus__integer_Otransfer,axiom,
    bNF_re9130850084932126070nteger(code_pcr_integer,bNF_re3379532845092657523nteger(code_pcr_integer,code_pcr_integer),minus_minus_int,minus_8373710615458151222nteger) ).

% minus_integer.transfer
tff(fact_10169_nat__of__integer_Otransfer,axiom,
    bNF_re2807294637932363402at_nat(code_pcr_integer,fequal_nat,nat2,code_nat_of_integer) ).

% nat_of_integer.transfer
tff(fact_10170_or__integer_Otransfer,axiom,
    bNF_re9130850084932126070nteger(code_pcr_integer,bNF_re3379532845092657523nteger(code_pcr_integer,code_pcr_integer),bit_se1409905431419307370or_int,bit_se1080825931792720795nteger) ).

% or_integer.transfer
tff(fact_10171_euclidean__size__integer_Otransfer,axiom,
    bNF_re2807294637932363402at_nat(code_pcr_integer,fequal_nat,euclid4774559944035922753ze_int,euclid6377331345833325938nteger) ).

% euclidean_size_integer.transfer
tff(fact_10172_mask__integer_Otransfer,axiom,
    bNF_re4153400068438556298nteger(fequal_nat,code_pcr_integer,bit_se2000444600071755411sk_int,bit_se2119862282449309892nteger) ).

% mask_integer.transfer
tff(fact_10173_modulo__integer_Otransfer,axiom,
    bNF_re9130850084932126070nteger(code_pcr_integer,bNF_re3379532845092657523nteger(code_pcr_integer,code_pcr_integer),modulo_modulo_int,modulo364778990260209775nteger) ).

% modulo_integer.transfer
tff(fact_10174_divide__integer_Otransfer,axiom,
    bNF_re9130850084932126070nteger(code_pcr_integer,bNF_re3379532845092657523nteger(code_pcr_integer,code_pcr_integer),divide_divide_int,divide6298287555418463151nteger) ).

% divide_integer.transfer
tff(fact_10175_unset__bit__integer_Otransfer,axiom,
    bNF_re4578097776928471693nteger(fequal_nat,bNF_re3379532845092657523nteger(code_pcr_integer,code_pcr_integer),bit_se4203085406695923979it_int,bit_se8260200283734997820nteger) ).

% unset_bit_integer.transfer
tff(fact_10176_take__bit__integer_Otransfer,axiom,
    bNF_re4578097776928471693nteger(fequal_nat,bNF_re3379532845092657523nteger(code_pcr_integer,code_pcr_integer),bit_se2923211474154528505it_int,bit_se1745604003318907178nteger) ).

% take_bit_integer.transfer
tff(fact_10177_division__segment__integer_Otransfer,axiom,
    aa_fun774133186966548707eger_o(aa_fun6864876441520360999eger_o(bNF_re3379532845092657523nteger(code_pcr_integer,code_pcr_integer),euclid3395696857347342551nt_int),euclid6289375185220004616nteger) ).

% division_segment_integer.transfer
tff(fact_10178_xor__integer_Otransfer,axiom,
    bNF_re9130850084932126070nteger(code_pcr_integer,bNF_re3379532845092657523nteger(code_pcr_integer,code_pcr_integer),bit_se6526347334894502574or_int,bit_se3222712562003087583nteger) ).

% xor_integer.transfer
tff(fact_10179_bit__integer_Otransfer,axiom,
    bNF_re6821560923358799272_nat_o(code_pcr_integer,fequal_fun_nat_o,bit_se1146084159140164899it_int,bit_se9216721137139052372nteger) ).

% bit_integer.transfer
tff(fact_10180_uminus__integer_Otransfer,axiom,
    aa_fun774133186966548707eger_o(aa_fun6864876441520360999eger_o(bNF_re3379532845092657523nteger(code_pcr_integer,code_pcr_integer),uminus_uminus_int),uminus1351360451143612070nteger) ).

% uminus_integer.transfer
tff(fact_10181_plus__integer_Otransfer,axiom,
    bNF_re9130850084932126070nteger(code_pcr_integer,bNF_re3379532845092657523nteger(code_pcr_integer,code_pcr_integer),plus_plus_int,plus_p5714425477246183910nteger) ).

% plus_integer.transfer
tff(fact_10182_set__bit__integer_Otransfer,axiom,
    bNF_re4578097776928471693nteger(fequal_nat,bNF_re3379532845092657523nteger(code_pcr_integer,code_pcr_integer),bit_se7879613467334960850it_int,bit_se2793503036327961859nteger) ).

% set_bit_integer.transfer
tff(fact_10183_flip__bit__integer_Otransfer,axiom,
    bNF_re4578097776928471693nteger(fequal_nat,bNF_re3379532845092657523nteger(code_pcr_integer,code_pcr_integer),bit_se2159334234014336723it_int,bit_se1345352211410354436nteger) ).

% flip_bit_integer.transfer
tff(fact_10184_abs__integer_Otransfer,axiom,
    aa_fun774133186966548707eger_o(aa_fun6864876441520360999eger_o(bNF_re3379532845092657523nteger(code_pcr_integer,code_pcr_integer),abs_abs_int),abs_abs_Code_integer) ).

% abs_integer.transfer
tff(fact_10185_push__bit__integer_Otransfer,axiom,
    bNF_re4578097776928471693nteger(fequal_nat,bNF_re3379532845092657523nteger(code_pcr_integer,code_pcr_integer),bit_se545348938243370406it_int,bit_se7788150548672797655nteger) ).

% push_bit_integer.transfer
tff(fact_10186_and__integer_Otransfer,axiom,
    bNF_re9130850084932126070nteger(code_pcr_integer,bNF_re3379532845092657523nteger(code_pcr_integer,code_pcr_integer),bit_se725231765392027082nd_int,bit_se3949692690581998587nteger) ).

% and_integer.transfer
tff(fact_10187_integer_Oid__abs__transfer,axiom,
    bNF_re982302072995117890nteger(fequal_int,code_pcr_integer,aTP_Lamm_cd,code_integer_of_int) ).

% integer.id_abs_transfer
tff(fact_10188_drop__bit__integer_Otransfer,axiom,
    bNF_re4578097776928471693nteger(fequal_nat,bNF_re3379532845092657523nteger(code_pcr_integer,code_pcr_integer),bit_se8568078237143864401it_int,bit_se3928097537394005634nteger) ).

% drop_bit_integer.transfer
tff(fact_10189_integer_Orep__transfer,axiom,
    bNF_re3804157879324367682nt_int(code_pcr_integer,fequal_int,aTP_Lamm_cd,code_int_of_integer) ).

% integer.rep_transfer
tff(fact_10190_less__integer_Otransfer,axiom,
    bNF_re5122946609719563319eger_o(code_pcr_integer,bNF_re6574881592172037608er_o_o(code_pcr_integer,fequal_o),ord_less_int,ord_le6747313008572928689nteger) ).

% less_integer.transfer
tff(fact_10191_sgn__integer_Otransfer,axiom,
    aa_fun774133186966548707eger_o(aa_fun6864876441520360999eger_o(bNF_re3379532845092657523nteger(code_pcr_integer,code_pcr_integer),sgn_sgn_int),sgn_sgn_Code_integer) ).

% sgn_integer.transfer
tff(fact_10192_less__eq__integer_Otransfer,axiom,
    bNF_re5122946609719563319eger_o(code_pcr_integer,bNF_re6574881592172037608er_o_o(code_pcr_integer,fequal_o),ord_less_eq_int,ord_le3102999989581377725nteger) ).

% less_eq_integer.transfer
tff(fact_10193_integer_Obi__total,axiom,
    bi_tot1331153423839324337nteger(code_pcr_integer) ).

% integer.bi_total
tff(fact_10194_one__integer_Otransfer,axiom,
    aa_Code_integer_o(aa_int5993532643906289898eger_o(code_pcr_integer,one_one_int),one_one_Code_integer) ).

% one_integer.transfer
tff(fact_10195_zero__integer_Otransfer,axiom,
    aa_Code_integer_o(aa_int5993532643906289898eger_o(code_pcr_integer,zero_zero_int),zero_z3403309356797280102nteger) ).

% zero_integer.transfer
tff(fact_10196_dup_Otransfer,axiom,
    aa_fun774133186966548707eger_o(aa_fun6864876441520360999eger_o(bNF_re3379532845092657523nteger(code_pcr_integer,code_pcr_integer),aTP_Lamm_abi),code_dup) ).

% dup.transfer
tff(fact_10197_sub_Otransfer,axiom,
    bNF_re4520166952630509772nteger(fequal_num,bNF_re6501075790457514782nteger(fequal_num,code_pcr_integer),aTP_Lamm_abj,code_sub) ).

% sub.transfer
tff(fact_10198_UNIV__char__of__nat,axiom,
    top_top_set_char = image_nat_char(unique3096191561947761185of_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_10199_inj__on__char__of__nat,axiom,
    aa_set_nat_o(inj_on_nat_char(unique3096191561947761185of_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_10200_range__nat__of__char,axiom,
    image_char_nat(comm_s629917340098488124ar_nat,top_top_set_char) = 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_10201_char_Osize_I2_J,axiom,
    ! [X15: $o,X2: $o,X33: $o,X42: $o,X52: $o,X62: $o,X72: $o,X8: $o] : size_size_char(char2((X15),(X2),(X33),(X42),(X52),(X62),(X72),(X8))) = zero_zero_nat ).

% char.size(2)
tff(fact_10202_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_10203_char_Osize__gen,axiom,
    ! [X15: $o,X2: $o,X33: $o,X42: $o,X52: $o,X62: $o,X72: $o,X8: $o] : size_char(char2((X15),(X2),(X33),(X42),(X52),(X62),(X72),(X8))) = zero_zero_nat ).

% char.size_gen
tff(fact_10204_integer__of__nat_Otransfer,axiom,
    bNF_re4153400068438556298nteger(fequal_nat,code_pcr_integer,semiri1314217659103216013at_int,code_integer_of_nat) ).

% integer_of_nat.transfer
tff(fact_10205_nat__of__integer__integer__of__nat,axiom,
    ! [Na: nat] : aa_Code_integer_nat(code_nat_of_integer,aa_nat_Code_integer(code_integer_of_nat,Na)) = Na ).

% nat_of_integer_integer_of_nat
tff(fact_10206_int__of__integer__integer__of__nat,axiom,
    ! [Na: nat] : aa_Code_integer_int(code_int_of_integer,aa_nat_Code_integer(code_integer_of_nat,Na)) = aa_nat_int(semiri1314217659103216013at_int,Na) ).

% int_of_integer_integer_of_nat
tff(fact_10207_integer__of__nat_Orep__eq,axiom,
    ! [Xb: nat] : aa_Code_integer_int(code_int_of_integer,aa_nat_Code_integer(code_integer_of_nat,Xb)) = aa_nat_int(semiri1314217659103216013at_int,Xb) ).

% integer_of_nat.rep_eq
tff(fact_10208_integer__of__nat__eq__of__nat,axiom,
    code_integer_of_nat = semiri4939895301339042750nteger ).

% integer_of_nat_eq_of_nat
tff(fact_10209_integer__of__nat__0,axiom,
    aa_nat_Code_integer(code_integer_of_nat,zero_zero_nat) = zero_z3403309356797280102nteger ).

% integer_of_nat_0
tff(fact_10210_integer__of__nat_Oabs__eq,axiom,
    ! [Xb: nat] : aa_nat_Code_integer(code_integer_of_nat,Xb) = aa_int_Code_integer(code_integer_of_int,aa_nat_int(semiri1314217659103216013at_int,Xb)) ).

% integer_of_nat.abs_eq
tff(fact_10211_integer__of__nat__numeral,axiom,
    ! [Na: num] : aa_nat_Code_integer(code_integer_of_nat,aa_num_nat(numeral_numeral_nat,Na)) = aa_num_Code_integer(numera6620942414471956472nteger,Na) ).

% integer_of_nat_numeral
tff(fact_10212_integer__of__nat__1,axiom,
    aa_nat_Code_integer(code_integer_of_nat,one_one_nat) = one_one_Code_integer ).

% integer_of_nat_1
tff(fact_10213_compute__powr__real,axiom,
    ! [Ba: real,I: real] :
      aa_real_real(aa_rea7340605660300375871l_real(powr_real2,Ba),I) = $ite(
        aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ba),zero_zero_real),
        abort_real(literal2($false,$false,$false,$false,$true,$true,$true,literal2($true,$true,$true,$true,$false,$true,$true,literal2($true,$true,$true,$false,$true,$true,$true,literal2($false,$true,$false,$false,$true,$true,$true,literal2($true,$true,$true,$true,$true,$false,$true,literal2($false,$true,$false,$false,$true,$true,$true,literal2($true,$false,$true,$false,$false,$true,$true,literal2($true,$false,$false,$false,$false,$true,$true,literal2($false,$false,$true,$true,$false,$true,$true,literal2($false,$false,$false,$false,$false,$true,$false,literal2($true,$true,$true,$false,$true,$true,$true,literal2($true,$false,$false,$true,$false,$true,$true,literal2($false,$false,$true,$false,$true,$true,$true,literal2($false,$false,$false,$true,$false,$true,$true,literal2($false,$false,$false,$false,$false,$true,$false,literal2($false,$true,$true,$true,$false,$true,$true,literal2($true,$true,$true,$true,$false,$true,$true,literal2($false,$true,$true,$true,$false,$true,$true,literal2($false,$false,$false,$false,$true,$true,$true,literal2($true,$true,$true,$true,$false,$true,$true,literal2($true,$true,$false,$false,$true,$true,$true,literal2($true,$false,$false,$true,$false,$true,$true,literal2($false,$false,$true,$false,$true,$true,$true,literal2($true,$false,$false,$true,$false,$true,$true,literal2($false,$true,$true,$false,$true,$true,$true,literal2($true,$false,$true,$false,$false,$true,$true,literal2($false,$false,$false,$false,$false,$true,$false,literal2($false,$true,$false,$false,$false,$true,$true,literal2($true,$false,$false,$false,$false,$true,$true,literal2($true,$true,$false,$false,$true,$true,$true,literal2($true,$false,$true,$false,$false,$true,$true,zero_zero_literal))))))))))))))))))))))))))))))),aa_rea7683167438103758690t_real(aTP_Lamm_abr(Ba),I)),
        $ite(
          aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,I)) = I,
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),I),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ba),aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,I))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ba),aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,aa_real_real(uminus_uminus_real,I)))))),
          abort_real(literal2($false,$false,$false,$false,$true,$true,$true,literal2($true,$true,$true,$true,$false,$true,$true,literal2($true,$true,$true,$false,$true,$true,$true,literal2($false,$true,$false,$false,$true,$true,$true,literal2($true,$true,$true,$true,$true,$false,$true,literal2($false,$true,$false,$false,$true,$true,$true,literal2($true,$false,$true,$false,$false,$true,$true,literal2($true,$false,$false,$false,$false,$true,$true,literal2($false,$false,$true,$true,$false,$true,$true,literal2($false,$false,$false,$false,$false,$true,$false,literal2($true,$true,$true,$false,$true,$true,$true,literal2($true,$false,$false,$true,$false,$true,$true,literal2($false,$false,$true,$false,$true,$true,$true,literal2($false,$false,$false,$true,$false,$true,$true,literal2($false,$false,$false,$false,$false,$true,$false,literal2($false,$true,$true,$true,$false,$true,$true,literal2($true,$true,$true,$true,$false,$true,$true,literal2($false,$true,$true,$true,$false,$true,$true,literal2($true,$false,$true,$true,$false,$true,$false,literal2($true,$false,$false,$true,$false,$true,$true,literal2($false,$true,$true,$true,$false,$true,$true,literal2($false,$false,$true,$false,$true,$true,$true,literal2($true,$false,$true,$false,$false,$true,$true,literal2($true,$true,$true,$false,$false,$true,$true,literal2($true,$false,$true,$false,$false,$true,$true,literal2($false,$true,$false,$false,$true,$true,$true,literal2($false,$false,$false,$false,$false,$true,$false,literal2($true,$false,$true,$false,$false,$true,$true,literal2($false,$false,$false,$true,$true,$true,$true,literal2($false,$false,$false,$false,$true,$true,$true,literal2($true,$true,$true,$true,$false,$true,$true,literal2($false,$true,$true,$true,$false,$true,$true,literal2($true,$false,$true,$false,$false,$true,$true,literal2($false,$true,$true,$true,$false,$true,$true,literal2($false,$false,$true,$false,$true,$true,$true,zero_zero_literal))))))))))))))))))))))))))))))))))),aa_rea7683167438103758690t_real(aTP_Lamm_abr(Ba),I)) ) ) ).

% compute_powr_real
tff(fact_10214_ATP_Olambda__1,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_pv,Uu) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),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_10215_ATP_Olambda__2,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_xk,Uu) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_nat(suc,Uu)) ).

% ATP.lambda_2
tff(fact_10216_ATP_Olambda__3,axiom,
    ! [Uu: real] :
      ( aa_real_o(aTP_Lamm_xs,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_10217_ATP_Olambda__4,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_wa,Uu) = aa_real_real(aa_rea7340605660300375871l_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_10218_ATP_Olambda__5,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_aao,Uu) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),aa_nat_nat(suc,zero_zero_nat)) ).

% ATP.lambda_5
tff(fact_10219_ATP_Olambda__6,axiom,
    ! [Uu: int] : aa_int_int(aTP_Lamm_abi,Uu) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu),Uu) ).

% ATP.lambda_6
tff(fact_10220_ATP_Olambda__7,axiom,
    ! [Uu: nat] : aa_nat7603221163087113075at_nat(aTP_Lamm_abl,Uu) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uu),zero_zero_nat) ).

% ATP.lambda_7
tff(fact_10221_ATP_Olambda__8,axiom,
    ! [Uu: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aTP_Lamm_abk,Uu)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,product_fst_int_int(Uu)),product_snd_int_int(Uu))) ) ).

% ATP.lambda_8
tff(fact_10222_ATP_Olambda__9,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_zf,Uu) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_li,Uu)) ).

% ATP.lambda_9
tff(fact_10223_ATP_Olambda__10,axiom,
    ! [Uu: nat] : aa_nat_set_nat(aTP_Lamm_zx,Uu) = collect_nat(aa_nat_fun_nat_o(aTP_Lamm_cf,Uu)) ).

% ATP.lambda_10
tff(fact_10224_ATP_Olambda__11,axiom,
    ! [Uu: real] : aa_rea5117059003799598681omplex(aTP_Lamm_aae,Uu) = princi3496590319149328850omplex(collec8663557070575231912omplex(produc6771430404735790350plex_o(aTP_Lamm_aad(Uu)))) ).

% ATP.lambda_11
tff(fact_10225_ATP_Olambda__12,axiom,
    ! [Uu: real] : aa_rea5428163979925208277l_real(aTP_Lamm_aac,Uu) = princi6114159922880469582l_real(collec3799799289383736868l_real(produc5414030515140494994real_o(aTP_Lamm_aab(Uu)))) ).

% ATP.lambda_12
tff(fact_10226_ATP_Olambda__13,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_zm,Uu) = aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uu))) ).

% ATP.lambda_13
tff(fact_10227_ATP_Olambda__14,axiom,
    ! [Uu: num] : aa_num_option_num(aTP_Lamm_aag,Uu) = some_num(aa_num_num(bit0,Uu)) ).

% ATP.lambda_14
tff(fact_10228_ATP_Olambda__15,axiom,
    ! [Uu: nat] : aa_nat8803788312118491388at_nat(aTP_Lamm_fa,Uu) = product_Pair_nat_nat(aa_nat_nat(suc,Uu)) ).

% ATP.lambda_15
tff(fact_10229_ATP_Olambda__16,axiom,
    ! [Uu: fun_nat_rat] :
      ( aa_fun_nat_rat_o(aTP_Lamm_abh,Uu)
    <=> ? [R5: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),R5)
          & ? [K3: nat] :
            ! [N4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K3),N4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,R5),aa_nat_rat(Uu,N4)) ) ) ) ).

% ATP.lambda_16
tff(fact_10230_ATP_Olambda__17,axiom,
    ! [Uu: real] :
      ( aa_real_o(aTP_Lamm_aal,Uu)
    <=> ? [I3: int,N4: nat] :
          ( ( Uu = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,I3)),aa_nat_real(semiri5074537144036343181t_real,N4)) )
          & ( N4 != zero_zero_nat ) ) ) ).

% ATP.lambda_17
tff(fact_10231_ATP_Olambda__18,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_Code_integer(aTP_Lamm_qk(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_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua)),zero_z3403309356797280102nteger) ).

% ATP.lambda_18
tff(fact_10232_ATP_Olambda__19,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_complex(aTP_Lamm_qh(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_19
tff(fact_10233_ATP_Olambda__20,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_ql,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_20
tff(fact_10234_ATP_Olambda__21,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_rat(aTP_Lamm_qi(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_21
tff(fact_10235_ATP_Olambda__22,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_qj,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_22
tff(fact_10236_ATP_Olambda__23,axiom,
    ! [Uu: fun_nat_real,Uua: nat] :
      aa_nat_real(aTP_Lamm_xl(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_23
tff(fact_10237_ATP_Olambda__24,axiom,
    ! [Uu: extended_enat,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_abp(Uu),Uua)
    <=> aa_Extended_enat_o(extended_case_enat_o(aa_nat_fun_nat_o(aTP_Lamm_av,Uua),$false),Uu) ) ).

% ATP.lambda_24
tff(fact_10238_ATP_Olambda__25,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_Code_integer(aTP_Lamm_qf(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_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua)),zero_z3403309356797280102nteger) ).

% ATP.lambda_25
tff(fact_10239_ATP_Olambda__26,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_complex(aTP_Lamm_qc(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_26
tff(fact_10240_ATP_Olambda__27,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_qg,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_27
tff(fact_10241_ATP_Olambda__28,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_rat(aTP_Lamm_qd(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_28
tff(fact_10242_ATP_Olambda__29,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_qe,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_29
tff(fact_10243_ATP_Olambda__30,axiom,
    ! [Uu: extended_enat,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_abq(Uu),Uua)
    <=> aa_Extended_enat_o(extended_case_enat_o(aa_nat_fun_nat_o(ord_less_nat,Uua),$true),Uu) ) ).

% ATP.lambda_30
tff(fact_10244_ATP_Olambda__31,axiom,
    ! [Uu: fun_nat_fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_sf(Uu),Uua) = groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_se(Uu),Uua),aa_nat_set_nat(set_ord_atMost_nat,Uua)) ).

% ATP.lambda_31
tff(fact_10245_ATP_Olambda__32,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_sd(Uu),Uua) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_rz(Uu),Uua)),aa_nat_set_nat(set_ord_atMost_nat,Uua)) ).

% ATP.lambda_32
tff(fact_10246_ATP_Olambda__33,axiom,
    ! [Uu: fun_nat_fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_sc(Uu),Uua) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_sb(Uu),Uua)),aa_nat_set_nat(set_ord_atMost_nat,Uua)) ).

% ATP.lambda_33
tff(fact_10247_ATP_Olambda__34,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_sa(Uu),Uua) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_rz(Uu),Uua)),aa_nat_set_nat(set_ord_atMost_nat,Uua)) ).

% ATP.lambda_34
tff(fact_10248_ATP_Olambda__35,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_lh,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),one_one_nat))))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Uu),one_one_real)),aa_nat_nat(suc,Uua))) ).

% ATP.lambda_35
tff(fact_10249_ATP_Olambda__36,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_xc,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),one_one_nat)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,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_36
tff(fact_10250_ATP_Olambda__37,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_Code_integer(aTP_Lamm_vr(Uu),Uua) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Uua)),aa_nat_Code_integer(semiri4939895301339042750nteger,Uua))),aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua))) ).

% ATP.lambda_37
tff(fact_10251_ATP_Olambda__38,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_complex(aTP_Lamm_vo(Uu),Uua) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Uua)),aa_nat_complex(semiri8010041392384452111omplex,Uua))),aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua))) ).

% ATP.lambda_38
tff(fact_10252_ATP_Olambda__39,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_vs,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(semiri5074537144036343181t_real,Uua))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua))) ).

% ATP.lambda_39
tff(fact_10253_ATP_Olambda__40,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_rat(aTP_Lamm_vp(Uu),Uua) = aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Uua)),aa_nat_rat(semiri681578069525770553at_rat,Uua))),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua))) ).

% ATP.lambda_40
tff(fact_10254_ATP_Olambda__41,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_vq,Uu),Uua) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),Uua)),aa_nat_int(semiri1314217659103216013at_int,Uua))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua))) ).

% ATP.lambda_41
tff(fact_10255_ATP_Olambda__42,axiom,
    ! [Uu: real,Uua: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_xq,Uu),Uua)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),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(aa_rea7340605660300375871l_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_42
tff(fact_10256_ATP_Olambda__43,axiom,
    ! [Uu: real,Uua: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_xp,Uu),Uua)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),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(aa_rea7340605660300375871l_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_43
tff(fact_10257_ATP_Olambda__44,axiom,
    ! [Uu: code_integer,Uua: code_integer] :
      aa_Code_integer_int(aa_Cod1242632294114111009er_int(aTP_Lamm_yi,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))),aa_Code_integer_int(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_44
tff(fact_10258_ATP_Olambda__45,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_Code_integer(aa_nat2010205248571845816nteger(aTP_Lamm_lc,Uu),Uua) = $let(
        m2: code_integer,
        m2:= aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),aa_nat_Code_integer(semiri4939895301339042750nteger,Uu)),
        $ite(Uua = zero_zero_nat,m2,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,m2),one_one_Code_integer)) ) ).

% ATP.lambda_45
tff(fact_10259_ATP_Olambda__46,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_Extended_enat(aa_nat2025370463774033377d_enat(aTP_Lamm_kz,Uu),Uua) = $let(
        m2: extended_enat,
        m2:= times_7803423173614009249d_enat(numera1916890842035813515d_enat(aa_num_num(bit0,one)),aa_nat_Extended_enat(semiri4216267220026989637d_enat,Uu)),
        $ite(Uua = zero_zero_nat,m2,plus_p3455044024723400733d_enat(m2,one_on7984719198319812577d_enat)) ) ).

% ATP.lambda_46
tff(fact_10260_ATP_Olambda__47,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_kw,Uu),Uua) = $let(
        m2: complex,
        m2:= aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_nat_complex(semiri8010041392384452111omplex,Uu)),
        $ite(Uua = zero_zero_nat,m2,aa_complex_complex(plus_plus_complex(m2),one_one_complex)) ) ).

% ATP.lambda_47
tff(fact_10261_ATP_Olambda__48,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_la,Uu),Uua) = $let(
        m2: real,
        m2:= aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Uu)),
        $ite(Uua = zero_zero_nat,m2,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,m2),one_one_real)) ) ).

% ATP.lambda_48
tff(fact_10262_ATP_Olambda__49,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_rat(aa_nat_fun_nat_rat(aTP_Lamm_kx,Uu),Uua) = $let(
        m2: rat,
        m2:= 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,m2,aa_rat_rat(plus_plus_rat(m2),one_one_rat)) ) ).

% ATP.lambda_49
tff(fact_10263_ATP_Olambda__50,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ky,Uu),Uua) = $let(
        m2: nat,
        m2:= 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,m2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,m2),one_one_nat)) ) ).

% ATP.lambda_50
tff(fact_10264_ATP_Olambda__51,axiom,
    ! [Uu: nat,Uua: nat] :
      aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_lb,Uu),Uua) = $let(
        m2: int,
        m2:= 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,m2,aa_int_int(aa_int_fun_int_int(plus_plus_int,m2),one_one_int)) ) ).

% ATP.lambda_51
tff(fact_10265_ATP_Olambda__52,axiom,
    ! [Uu: complex,Uua: real] :
      ( aa_real_o(aTP_Lamm_aap(Uu),Uua)
    <=> ( ( aa_complex_complex(sgn_sgn_complex,Uu) = aa_real_complex(cis,Uua) )
        & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,pi)),Uua)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Uua),pi) ) ) ).

% ATP.lambda_52
tff(fact_10266_ATP_Olambda__53,axiom,
    ! [Uu: real,Uua: int] :
      ( aa_int_o(aa_real_fun_int_o(aTP_Lamm_xt,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_53
tff(fact_10267_ATP_Olambda__54,axiom,
    ! [Uu: rat,Uua: int] :
      ( aa_int_o(aTP_Lamm_xu(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_54
tff(fact_10268_ATP_Olambda__55,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_li,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,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_55
tff(fact_10269_ATP_Olambda__56,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_zg,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,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_56
tff(fact_10270_ATP_Olambda__57,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_zh(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(Uu,Uua)) ).

% ATP.lambda_57
tff(fact_10271_ATP_Olambda__58,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_Code_integer(aTP_Lamm_vx(Uu),Uua) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Uua)),aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua))) ).

% ATP.lambda_58
tff(fact_10272_ATP_Olambda__59,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_complex(aTP_Lamm_vu(Uu),Uua) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Uua)),aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua))) ).

% ATP.lambda_59
tff(fact_10273_ATP_Olambda__60,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_vy,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua))) ).

% ATP.lambda_60
tff(fact_10274_ATP_Olambda__61,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_rat(aTP_Lamm_vv(Uu),Uua) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Uua)),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua))) ).

% ATP.lambda_61
tff(fact_10275_ATP_Olambda__62,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_vw,Uu),Uua) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),Uua)),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua))) ).

% ATP.lambda_62
tff(fact_10276_ATP_Olambda__63,axiom,
    ! [Uu: rat,Uua: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aTP_Lamm_abg(Uu),Uua)
    <=> ( ( Uu = aa_int_rat(fract(product_fst_int_int(Uua)),product_snd_int_int(Uua)) )
        & aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),product_snd_int_int(Uua))
        & aa_int_o(aa_int_fun_int_o(algebr932160517623751201me_int,product_fst_int_int(Uua)),product_snd_int_int(Uua)) ) ) ).

% ATP.lambda_63
tff(fact_10277_ATP_Olambda__64,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_wf(Uu),Uua) = aa_nat_complex(gbinomial_complex(aa_complex_complex(plus_plus_complex(Uu),aa_nat_complex(semiri8010041392384452111omplex,Uua))),Uua) ).

% ATP.lambda_64
tff(fact_10278_ATP_Olambda__65,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_wh,Uu),Uua) = aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),aa_nat_real(semiri5074537144036343181t_real,Uua))),Uua) ).

% ATP.lambda_65
tff(fact_10279_ATP_Olambda__66,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_wg(Uu),Uua) = aa_nat_rat(gbinomial_rat(aa_rat_rat(plus_plus_rat(Uu),aa_nat_rat(semiri681578069525770553at_rat,Uua))),Uua) ).

% ATP.lambda_66
tff(fact_10280_ATP_Olambda__67,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_wc(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_67
tff(fact_10281_ATP_Olambda__68,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_we,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(gbinomial_real,Uu),Uua)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Uu),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,Uua))) ).

% ATP.lambda_68
tff(fact_10282_ATP_Olambda__69,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_wd(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_69
tff(fact_10283_ATP_Olambda__70,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_wi(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_70
tff(fact_10284_ATP_Olambda__71,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_wk,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_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_71
tff(fact_10285_ATP_Olambda__72,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_wj(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_72
tff(fact_10286_ATP_Olambda__73,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_aat,Uu),Uua)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Uu),Uua)
        & ( Uu != Uua ) ) ) ).

% ATP.lambda_73
tff(fact_10287_ATP_Olambda__74,axiom,
    ! [Uu: set_int,Uua: int] :
      ( aa_int_o(aTP_Lamm_aaa(Uu),Uua)
    <=> ( aa_set_int_o(member_int(Uua),Uu)
        & ! [X4: int] :
            ( aa_set_int_o(member_int(X4),Uu)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X4),Uua) ) ) ) ).

% ATP.lambda_74
tff(fact_10288_ATP_Olambda__75,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_vi,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_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_75
tff(fact_10289_ATP_Olambda__76,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_aah,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_76
tff(fact_10290_ATP_Olambda__77,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_th,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_77
tff(fact_10291_ATP_Olambda__78,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_tg,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_78
tff(fact_10292_ATP_Olambda__79,axiom,
    ! [Uu: nat,Uua: complex] :
      ( aa_complex_o(aa_nat_fun_complex_o(aTP_Lamm_cs,Uu),Uua)
    <=> ( aa_nat_complex(power_power_complex(Uua),Uu) = one_one_complex ) ) ).

% ATP.lambda_79
tff(fact_10293_ATP_Olambda__80,axiom,
    ! [Uu: nat,Uua: real] :
      ( aa_real_o(aa_nat_fun_real_o(aTP_Lamm_cr,Uu),Uua)
    <=> ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uu) = one_one_real ) ) ).

% ATP.lambda_80
tff(fact_10294_ATP_Olambda__81,axiom,
    ! [Uu: real,Uua: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_es,Uu),Uua)
    <=> ( member_real(Uua,ring_1_Ints_real)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Uua)),Uu) ) ) ).

% ATP.lambda_81
tff(fact_10295_ATP_Olambda__82,axiom,
    ! [Uu: rat,Uua: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(aTP_Lamm_et,Uu),Uua)
    <=> ( member_rat(Uua,ring_1_Ints_rat)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,Uua)),Uu) ) ) ).

% ATP.lambda_82
tff(fact_10296_ATP_Olambda__83,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ld,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat)))),aa_nat_real(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_83
tff(fact_10297_ATP_Olambda__84,axiom,
    ! [Uu: real,Uua: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_xr,Uu),Uua)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Uua)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Uua),pi)
        & ( aa_real_real(cos_real,Uua) = Uu ) ) ) ).

% ATP.lambda_84
tff(fact_10298_ATP_Olambda__85,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_ox(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))),aa_nat_real(Uu,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))) ).

% ATP.lambda_85
tff(fact_10299_ATP_Olambda__86,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_oy(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_86
tff(fact_10300_ATP_Olambda__87,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_oz(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_87
tff(fact_10301_ATP_Olambda__88,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_pa(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_88
tff(fact_10302_ATP_Olambda__89,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_ko(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))),aa_nat_real(Uu,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))) ).

% ATP.lambda_89
tff(fact_10303_ATP_Olambda__90,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_kl(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_90
tff(fact_10304_ATP_Olambda__91,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_kn(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_91
tff(fact_10305_ATP_Olambda__92,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_km(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_92
tff(fact_10306_ATP_Olambda__93,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_ka(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu,aa_nat_nat(suc,Uua))),aa_nat_real(Uu,Uua)) ).

% ATP.lambda_93
tff(fact_10307_ATP_Olambda__94,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_jy(Uu),Uua) = aa_rat_rat(minus_minus_rat(aa_nat_rat(Uu,aa_nat_nat(suc,Uua))),aa_nat_rat(Uu,Uua)) ).

% ATP.lambda_94
tff(fact_10308_ATP_Olambda__95,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_jz(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_95
tff(fact_10309_ATP_Olambda__96,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat_complex(aTP_Lamm_iu(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_96
tff(fact_10310_ATP_Olambda__97,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_iw(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_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_97
tff(fact_10311_ATP_Olambda__98,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_iv(Uu),Uua) = aa_rat_rat(times_times_rat(aa_nat_rat(Uu,Uua)),aa_nat_rat(power_power_rat(zero_zero_rat),Uua)) ).

% ATP.lambda_98
tff(fact_10312_ATP_Olambda__99,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_sg(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_99
tff(fact_10313_ATP_Olambda__100,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_kk(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu,Uua)),aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),one_one_nat))) ).

% ATP.lambda_100
tff(fact_10314_ATP_Olambda__101,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_ki(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_101
tff(fact_10315_ATP_Olambda__102,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_kj(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_102
tff(fact_10316_ATP_Olambda__103,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_kh(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu,Uua)),aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),one_one_nat))) ).

% ATP.lambda_103
tff(fact_10317_ATP_Olambda__104,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_kf(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_104
tff(fact_10318_ATP_Olambda__105,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_kg(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_105
tff(fact_10319_ATP_Olambda__106,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_sx(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu,Uua)),aa_nat_real(Uu,aa_nat_nat(suc,Uua))) ).

% ATP.lambda_106
tff(fact_10320_ATP_Olambda__107,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_sv(Uu),Uua) = aa_rat_rat(minus_minus_rat(aa_nat_rat(Uu,Uua)),aa_nat_rat(Uu,aa_nat_nat(suc,Uua))) ).

% ATP.lambda_107
tff(fact_10321_ATP_Olambda__108,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_sw(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_108
tff(fact_10322_ATP_Olambda__109,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_rn(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_109
tff(fact_10323_ATP_Olambda__110,axiom,
    ! [Uu: fun_nat_fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_rp(Uu),Uua) = groups705719431365010083at_int(aa_nat_fun_nat_int(Uu,Uua),aa_nat_set_nat(set_ord_lessThan_nat,Uua)) ).

% ATP.lambda_110
tff(fact_10324_ATP_Olambda__111,axiom,
    ! [Uu: fun_nat_fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_rk(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_111
tff(fact_10325_ATP_Olambda__112,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_rh(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_112
tff(fact_10326_ATP_Olambda__113,axiom,
    ! [Uu: fun_complex_real,Uua: complex] :
      ( aa_complex_o(aTP_Lamm_ic(Uu),Uua)
    <=> ( aa_complex_real(Uu,Uua) = zero_zero_real ) ) ).

% ATP.lambda_113
tff(fact_10327_ATP_Olambda__114,axiom,
    ! [Uu: fun_complex_rat,Uua: complex] :
      ( aa_complex_o(aTP_Lamm_if(Uu),Uua)
    <=> ( aa_complex_rat(Uu,Uua) = zero_zero_rat ) ) ).

% ATP.lambda_114
tff(fact_10328_ATP_Olambda__115,axiom,
    ! [Uu: fun_complex_nat,Uua: complex] :
      ( aa_complex_o(aTP_Lamm_ii(Uu),Uua)
    <=> ( aa_complex_nat(Uu,Uua) = zero_zero_nat ) ) ).

% ATP.lambda_115
tff(fact_10329_ATP_Olambda__116,axiom,
    ! [Uu: fun_real_real,Uua: real] :
      ( aa_real_o(aTP_Lamm_ia(Uu),Uua)
    <=> ( aa_real_real(Uu,Uua) = zero_zero_real ) ) ).

% ATP.lambda_116
tff(fact_10330_ATP_Olambda__117,axiom,
    ! [Uu: fun_real_rat,Uua: real] :
      ( aa_real_o(aTP_Lamm_id(Uu),Uua)
    <=> ( aa_real_rat(Uu,Uua) = zero_zero_rat ) ) ).

% ATP.lambda_117
tff(fact_10331_ATP_Olambda__118,axiom,
    ! [Uu: fun_real_nat,Uua: real] :
      ( aa_real_o(aTP_Lamm_ig(Uu),Uua)
    <=> ( aa_real_nat(Uu,Uua) = zero_zero_nat ) ) ).

% ATP.lambda_118
tff(fact_10332_ATP_Olambda__119,axiom,
    ! [Uu: fun_real_int,Uua: real] :
      ( aa_real_o(aTP_Lamm_ij(Uu),Uua)
    <=> ( aa_real_int(Uu,Uua) = zero_zero_int ) ) ).

% ATP.lambda_119
tff(fact_10333_ATP_Olambda__120,axiom,
    ! [Uu: fun_int_real,Uua: int] :
      ( aa_int_o(aTP_Lamm_ib(Uu),Uua)
    <=> ( aa_int_real(Uu,Uua) = zero_zero_real ) ) ).

% ATP.lambda_120
tff(fact_10334_ATP_Olambda__121,axiom,
    ! [Uu: fun_int_rat,Uua: int] :
      ( aa_int_o(aTP_Lamm_ie(Uu),Uua)
    <=> ( aa_int_rat(Uu,Uua) = zero_zero_rat ) ) ).

% ATP.lambda_121
tff(fact_10335_ATP_Olambda__122,axiom,
    ! [Uu: fun_int_nat,Uua: int] :
      ( aa_int_o(aTP_Lamm_ih(Uu),Uua)
    <=> ( aa_int_nat(Uu,Uua) = zero_zero_nat ) ) ).

% ATP.lambda_122
tff(fact_10336_ATP_Olambda__123,axiom,
    ! [Uu: fun_complex_complex,Uua: complex] :
      ( aa_complex_o(aTP_Lamm_nv(Uu),Uua)
    <=> ( aa_complex_complex(Uu,Uua) = one_one_complex ) ) ).

% ATP.lambda_123
tff(fact_10337_ATP_Olambda__124,axiom,
    ! [Uu: fun_complex_real,Uua: complex] :
      ( aa_complex_o(aTP_Lamm_ny(Uu),Uua)
    <=> ( aa_complex_real(Uu,Uua) = one_one_real ) ) ).

% ATP.lambda_124
tff(fact_10338_ATP_Olambda__125,axiom,
    ! [Uu: fun_complex_rat,Uua: complex] :
      ( aa_complex_o(aTP_Lamm_ob(Uu),Uua)
    <=> ( aa_complex_rat(Uu,Uua) = one_one_rat ) ) ).

% ATP.lambda_125
tff(fact_10339_ATP_Olambda__126,axiom,
    ! [Uu: fun_real_complex,Uua: real] :
      ( aa_real_o(aTP_Lamm_nt(Uu),Uua)
    <=> ( aa_real_complex(Uu,Uua) = one_one_complex ) ) ).

% ATP.lambda_126
tff(fact_10340_ATP_Olambda__127,axiom,
    ! [Uu: fun_real_real,Uua: real] :
      ( aa_real_o(aTP_Lamm_nw(Uu),Uua)
    <=> ( aa_real_real(Uu,Uua) = one_one_real ) ) ).

% ATP.lambda_127
tff(fact_10341_ATP_Olambda__128,axiom,
    ! [Uu: fun_real_rat,Uua: real] :
      ( aa_real_o(aTP_Lamm_nz(Uu),Uua)
    <=> ( aa_real_rat(Uu,Uua) = one_one_rat ) ) ).

% ATP.lambda_128
tff(fact_10342_ATP_Olambda__129,axiom,
    ! [Uu: fun_real_nat,Uua: real] :
      ( aa_real_o(aTP_Lamm_oc(Uu),Uua)
    <=> ( aa_real_nat(Uu,Uua) = one_one_nat ) ) ).

% ATP.lambda_129
tff(fact_10343_ATP_Olambda__130,axiom,
    ! [Uu: fun_int_complex,Uua: int] :
      ( aa_int_o(aTP_Lamm_nu(Uu),Uua)
    <=> ( aa_int_complex(Uu,Uua) = one_one_complex ) ) ).

% ATP.lambda_130
tff(fact_10344_ATP_Olambda__131,axiom,
    ! [Uu: fun_int_real,Uua: int] :
      ( aa_int_o(aTP_Lamm_nx(Uu),Uua)
    <=> ( aa_int_real(Uu,Uua) = one_one_real ) ) ).

% ATP.lambda_131
tff(fact_10345_ATP_Olambda__132,axiom,
    ! [Uu: fun_int_rat,Uua: int] :
      ( aa_int_o(aTP_Lamm_oa(Uu),Uua)
    <=> ( aa_int_rat(Uu,Uua) = one_one_rat ) ) ).

% ATP.lambda_132
tff(fact_10346_ATP_Olambda__133,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_zt(Uu),Uua) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_zh(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_133
tff(fact_10347_ATP_Olambda__134,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_zs(Uu),Uua) = aa_set_nat_real(groups6591440286371151544t_real(aTP_Lamm_zh(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_134
tff(fact_10348_ATP_Olambda__135,axiom,
    ! [Uu: nat,Uua: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_yp,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,Uua)),Uu)) ).

% ATP.lambda_135
tff(fact_10349_ATP_Olambda__136,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_vz,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(sin_coeff,Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uua)) ).

% ATP.lambda_136
tff(fact_10350_ATP_Olambda__137,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_wb,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(cos_coeff,Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uua)) ).

% ATP.lambda_137
tff(fact_10351_ATP_Olambda__138,axiom,
    ! [Uu: num,Uua: num] : aa_num_int(aa_num_fun_num_int(aTP_Lamm_abj,Uu),Uua) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,Uu)),aa_num_int(numeral_numeral_int,Uua)) ).

% ATP.lambda_138
tff(fact_10352_ATP_Olambda__139,axiom,
    ! [Uu: code_integer,Uua: code_integer] :
      aa_Code_integer_num(aa_Cod6128727250572145423er_num(aTP_Lamm_yk,Uu),Uua) = $let(
        l2: num,
        l2:= aa_Code_integer_num(code_num_of_integer,Uu),
        $let(
          l3: num,
          l3:= aa_num_num(aa_num_fun_num_num(plus_plus_num,l2),l2),
          $ite(Uua = zero_z3403309356797280102nteger,l3,aa_num_num(aa_num_fun_num_num(plus_plus_num,l3),one)) ) ) ).

% ATP.lambda_139
tff(fact_10353_ATP_Olambda__140,axiom,
    ! [Uu: code_integer,Uua: code_integer] :
      aa_Code_integer_nat(aa_Cod5420483313623307717er_nat(aTP_Lamm_yh,Uu),Uua) = $let(
        l2: nat,
        l2:= aa_Code_integer_nat(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_140
tff(fact_10354_ATP_Olambda__141,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_complex(aTP_Lamm_wl(Uu),Uua) = aa_nat_complex(gbinomial_complex(aa_nat_complex(semiri8010041392384452111omplex,Uua)),Uu) ).

% ATP.lambda_141
tff(fact_10355_ATP_Olambda__142,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_wn,Uu),Uua) = aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_nat_real(semiri5074537144036343181t_real,Uua)),Uu) ).

% ATP.lambda_142
tff(fact_10356_ATP_Olambda__143,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_rat(aTP_Lamm_wm(Uu),Uua) = aa_nat_rat(gbinomial_rat(aa_nat_rat(semiri681578069525770553at_rat,Uua)),Uu) ).

% ATP.lambda_143
tff(fact_10357_ATP_Olambda__144,axiom,
    ! [Uu: code_integer,Uua: code_integer] : aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_ev,Uu),Uua) = produc1086072967326762835nteger(Uu,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Uua)),one_one_Code_integer)) ).

% ATP.lambda_144
tff(fact_10358_ATP_Olambda__145,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_ew,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_145
tff(fact_10359_ATP_Olambda__146,axiom,
    ! [Uu: int,Uua: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_eu,Uu),Uua) = product_Pair_int_int(Uu,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Uua)),one_one_int)) ).

% ATP.lambda_146
tff(fact_10360_ATP_Olambda__147,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_le(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_147
tff(fact_10361_ATP_Olambda__148,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_lg,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,Uua)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ).

% ATP.lambda_148
tff(fact_10362_ATP_Olambda__149,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_lf(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_149
tff(fact_10363_ATP_Olambda__150,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_vt,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_150
tff(fact_10364_ATP_Olambda__151,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_zr,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Uu),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(uminus_uminus_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uua)))))) ).

% ATP.lambda_151
tff(fact_10365_ATP_Olambda__152,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_zq,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),aa_real_real(uminus_uminus_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uua))))) ).

% ATP.lambda_152
tff(fact_10366_ATP_Olambda__153,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_zn,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uua)))) ).

% ATP.lambda_153
tff(fact_10367_ATP_Olambda__154,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_yo(Uu),Uua) = aa_set_nat_real(groups6591440286371151544t_real(Uu),aa_nat_set_nat(set_ord_lessThan_nat,Uua)) ).

% ATP.lambda_154
tff(fact_10368_ATP_Olambda__155,axiom,
    ! [Uu: code_integer,Uua: nat] : aa_nat_Code_integer(aTP_Lamm_wp(Uu),Uua) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,Uu),aa_nat_Code_integer(semiri4939895301339042750nteger,Uua)) ).

% ATP.lambda_155
tff(fact_10369_ATP_Olambda__156,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_wq,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Uu),aa_nat_real(semiri5074537144036343181t_real,Uua)) ).

% ATP.lambda_156
tff(fact_10370_ATP_Olambda__157,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_wo(Uu),Uua) = aa_rat_rat(minus_minus_rat(Uu),aa_nat_rat(semiri681578069525770553at_rat,Uua)) ).

% ATP.lambda_157
tff(fact_10371_ATP_Olambda__158,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_wr,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),aa_nat_nat(semiri1316708129612266289at_nat,Uua)) ).

% ATP.lambda_158
tff(fact_10372_ATP_Olambda__159,axiom,
    ! [Uu: int,Uua: nat] : aa_nat_int(aTP_Lamm_ws(Uu),Uua) = aa_int_int(aa_int_fun_int_int(minus_minus_int,Uu),aa_nat_int(semiri1314217659103216013at_int,Uua)) ).

% ATP.lambda_159
tff(fact_10373_ATP_Olambda__160,axiom,
    ! [Uu: code_integer,Uua: nat] : aa_nat_Code_integer(aTP_Lamm_op(Uu),Uua) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Uu),aa_nat_Code_integer(semiri4939895301339042750nteger,Uua)) ).

% ATP.lambda_160
tff(fact_10374_ATP_Olambda__161,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_oo,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),aa_nat_real(semiri5074537144036343181t_real,Uua)) ).

% ATP.lambda_161
tff(fact_10375_ATP_Olambda__162,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_on(Uu),Uua) = aa_rat_rat(plus_plus_rat(Uu),aa_nat_rat(semiri681578069525770553at_rat,Uua)) ).

% ATP.lambda_162
tff(fact_10376_ATP_Olambda__163,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_oq,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),aa_nat_nat(semiri1316708129612266289at_nat,Uua)) ).

% ATP.lambda_163
tff(fact_10377_ATP_Olambda__164,axiom,
    ! [Uu: int,Uua: nat] : aa_nat_int(aTP_Lamm_or(Uu),Uua) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu),aa_nat_int(semiri1314217659103216013at_int,Uua)) ).

% ATP.lambda_164
tff(fact_10378_ATP_Olambda__165,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_yj,Uu),Uua)
    <=> ( Uua = aa_nat_nat(suc,Uu) ) ) ).

% ATP.lambda_165
tff(fact_10379_ATP_Olambda__166,axiom,
    ! [Uu: real,Uua: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_rd,Uu),Uua)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Uua),Uu) ) ).

% ATP.lambda_166
tff(fact_10380_ATP_Olambda__167,axiom,
    ! [Uu: set_complex,Uua: set_complex] :
      ( aa_set_complex_o(aTP_Lamm_as(Uu),Uua)
    <=> aa_set_complex_o(ord_le211207098394363844omplex(Uua),Uu) ) ).

% ATP.lambda_167
tff(fact_10381_ATP_Olambda__168,axiom,
    ! [Uu: set_nat,Uua: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(aTP_Lamm_ar,Uu),Uua)
    <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Uua),Uu) ) ).

% ATP.lambda_168
tff(fact_10382_ATP_Olambda__169,axiom,
    ! [Uu: set_int,Uua: set_int] :
      ( aa_set_int_o(aTP_Lamm_at(Uu),Uua)
    <=> aa_set_int_o(ord_less_eq_set_int(Uua),Uu) ) ).

% ATP.lambda_169
tff(fact_10383_ATP_Olambda__170,axiom,
    ! [Uu: rat,Uua: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(aTP_Lamm_re,Uu),Uua)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Uua),Uu) ) ).

% ATP.lambda_170
tff(fact_10384_ATP_Olambda__171,axiom,
    ! [Uu: num,Uua: num] :
      ( aa_num_o(aa_num_fun_num_o(aTP_Lamm_rf,Uu),Uua)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Uua),Uu) ) ).

% ATP.lambda_171
tff(fact_10385_ATP_Olambda__172,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_av,Uu),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uua),Uu) ) ).

% ATP.lambda_172
tff(fact_10386_ATP_Olambda__173,axiom,
    ! [Uu: int,Uua: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_rg,Uu),Uua)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Uua),Uu) ) ).

% ATP.lambda_173
tff(fact_10387_ATP_Olambda__174,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_yn,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Uua),Uu) ).

% ATP.lambda_174
tff(fact_10388_ATP_Olambda__175,axiom,
    ! [Uu: real,Uua: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_rc,Uu),Uua)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Uua),Uu) ) ).

% ATP.lambda_175
tff(fact_10389_ATP_Olambda__176,axiom,
    ! [Uu: set_nat,Uua: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(aTP_Lamm_qy,Uu),Uua)
    <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,Uua),Uu) ) ).

% ATP.lambda_176
tff(fact_10390_ATP_Olambda__177,axiom,
    ! [Uu: rat,Uua: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(aTP_Lamm_qz,Uu),Uua)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Uua),Uu) ) ).

% ATP.lambda_177
tff(fact_10391_ATP_Olambda__178,axiom,
    ! [Uu: num,Uua: num] :
      ( aa_num_o(aa_num_fun_num_o(aTP_Lamm_ra,Uu),Uua)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Uua),Uu) ) ).

% ATP.lambda_178
tff(fact_10392_ATP_Olambda__179,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_au,Uu),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uua),Uu) ) ).

% ATP.lambda_179
tff(fact_10393_ATP_Olambda__180,axiom,
    ! [Uu: int,Uua: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_rb,Uu),Uua)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Uua),Uu) ) ).

% ATP.lambda_180
tff(fact_10394_ATP_Olambda__181,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_zi,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),Uu) ).

% ATP.lambda_181
tff(fact_10395_ATP_Olambda__182,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_yl,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uu) ).

% ATP.lambda_182
tff(fact_10396_ATP_Olambda__183,axiom,
    ! [Uu: nat,Uua: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_yq,Uu),Uua) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uu) ).

% ATP.lambda_183
tff(fact_10397_ATP_Olambda__184,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_aan,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uu) ).

% ATP.lambda_184
tff(fact_10398_ATP_Olambda__185,axiom,
    ! [Uu: int,Uua: int] : aa_int_int(aTP_Lamm_ym(Uu),Uua) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uua),Uu) ).

% ATP.lambda_185
tff(fact_10399_ATP_Olambda__186,axiom,
    ! [Uu: real,Uua: real] : aa_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_ys,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,Uua),Uu) ).

% ATP.lambda_186
tff(fact_10400_ATP_Olambda__187,axiom,
    ! [Uu: code_integer,Uua: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(aTP_Lamm_cg,Uu),Uua)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Uua),Uu) ) ).

% ATP.lambda_187
tff(fact_10401_ATP_Olambda__188,axiom,
    ! [Uu: real,Uua: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_ce,Uu),Uua)
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Uua),Uu) ) ).

% ATP.lambda_188
tff(fact_10402_ATP_Olambda__189,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_cf,Uu),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Uua),Uu) ) ).

% ATP.lambda_189
tff(fact_10403_ATP_Olambda__190,axiom,
    ! [Uu: int,Uua: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_ch,Uu),Uua)
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Uua),Uu) ) ).

% ATP.lambda_190
tff(fact_10404_ATP_Olambda__191,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_sm,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uua),Uu) ).

% ATP.lambda_191
tff(fact_10405_ATP_Olambda__192,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_zl,Uu),Uua) = aa_real_real(aa_nat_fun_real_real(root,Uua),Uu) ).

% ATP.lambda_192
tff(fact_10406_ATP_Olambda__193,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat,Uua: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aTP_Lamm_a(Uu),Uua)
    <=> member8440522571783428010at_nat(Uua,Uu) ) ).

% ATP.lambda_193
tff(fact_10407_ATP_Olambda__194,axiom,
    ! [Uu: set_real,Uua: real] :
      ( aa_real_o(aTP_Lamm_aa(Uu),Uua)
    <=> member_real(Uua,Uu) ) ).

% ATP.lambda_194
tff(fact_10408_ATP_Olambda__195,axiom,
    ! [Uu: set_list_nat,Uua: list_nat] :
      ( aa_list_nat_o(aTP_Lamm_ab(Uu),Uua)
    <=> member_list_nat(Uua,Uu) ) ).

% ATP.lambda_195
tff(fact_10409_ATP_Olambda__196,axiom,
    ! [Uu: set_set_nat,Uua: set_nat] :
      ( aa_set_nat_o(aTP_Lamm_ac(Uu),Uua)
    <=> member_set_nat(Uua,Uu) ) ).

% ATP.lambda_196
tff(fact_10410_ATP_Olambda__197,axiom,
    ! [Uu: set_nat,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_ad(Uu),Uua)
    <=> aa_set_nat_o(member_nat(Uua),Uu) ) ).

% ATP.lambda_197
tff(fact_10411_ATP_Olambda__198,axiom,
    ! [Uu: set_int,Uua: int] :
      ( aa_int_o(aTP_Lamm_ae(Uu),Uua)
    <=> aa_set_int_o(member_int(Uua),Uu) ) ).

% ATP.lambda_198
tff(fact_10412_ATP_Olambda__199,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_set_nat(aTP_Lamm_abo(Uu),Uua) = order_underS_nat(bNF_Ca8665028551170535155natLeq,Uu) ).

% ATP.lambda_199
tff(fact_10413_ATP_Olambda__200,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_vh(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_200
tff(fact_10414_ATP_Olambda__201,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_vg(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_201
tff(fact_10415_ATP_Olambda__202,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_jg(Uu),Uua) = aa_nat_real(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_202
tff(fact_10416_ATP_Olambda__203,axiom,
    ! [Uu: fun_nat_o,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_zv(Uu),Uua)
    <=> aa_nat_o(Uu,aa_nat_nat(suc,Uua)) ) ).

% ATP.lambda_203
tff(fact_10417_ATP_Olambda__204,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_jw(Uu),Uua) = aa_nat_rat(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_204
tff(fact_10418_ATP_Olambda__205,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_jf(Uu),Uua) = aa_nat_nat(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_205
tff(fact_10419_ATP_Olambda__206,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_jx(Uu),Uua) = aa_nat_int(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_206
tff(fact_10420_ATP_Olambda__207,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(aTP_Lamm_aay,Uu),Uua) = produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aTP_Lamm_aax(Uu),Uua)) ).

% ATP.lambda_207
tff(fact_10421_ATP_Olambda__208,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(aTP_Lamm_aaw,Uu),Uua) = produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aTP_Lamm_aav(Uu),Uua)) ).

% ATP.lambda_208
tff(fact_10422_ATP_Olambda__209,axiom,
    ! [Uu: fun_nat_real,Uua: real] : aa_real_real(aTP_Lamm_yy(Uu),Uua) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_yx(Uu),Uua)) ).

% ATP.lambda_209
tff(fact_10423_ATP_Olambda__210,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_complex(aTP_Lamm_aam(Uu),Uua) = aa_real_complex(cis,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,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_210
tff(fact_10424_ATP_Olambda__211,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat3829984526574172007omplex(aTP_Lamm_wz(Uu),Uua) = times_times_complex(aa_complex_complex(minus_minus_complex(Uu),aa_nat_complex(semiri8010041392384452111omplex,Uua))) ).

% ATP.lambda_211
tff(fact_10425_ATP_Olambda__212,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_fun_real_real(aTP_Lamm_xb(Uu),Uua) = aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Uu),aa_nat_real(semiri5074537144036343181t_real,Uua))) ).

% ATP.lambda_212
tff(fact_10426_ATP_Olambda__213,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_fun_rat_rat(aTP_Lamm_xa(Uu),Uua) = times_times_rat(aa_rat_rat(minus_minus_rat(Uu),aa_nat_rat(semiri681578069525770553at_rat,Uua))) ).

% ATP.lambda_213
tff(fact_10427_ATP_Olambda__214,axiom,
    ! [Uu: code_integer,Uua: nat] : aa_nat6383147911832638789nteger(aTP_Lamm_kt(Uu),Uua) = aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Uu),aa_nat_Code_integer(semiri4939895301339042750nteger,Uua))) ).

% ATP.lambda_214
tff(fact_10428_ATP_Olambda__215,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat3829984526574172007omplex(aTP_Lamm_kp(Uu),Uua) = times_times_complex(aa_complex_complex(plus_plus_complex(Uu),aa_nat_complex(semiri8010041392384452111omplex,Uua))) ).

% ATP.lambda_215
tff(fact_10429_ATP_Olambda__216,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_fun_real_real(aTP_Lamm_kr(Uu),Uua) = aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),aa_nat_real(semiri5074537144036343181t_real,Uua))) ).

% ATP.lambda_216
tff(fact_10430_ATP_Olambda__217,axiom,
    ! [Uu: rat,Uua: nat] : aa_nat_fun_rat_rat(aTP_Lamm_kq(Uu),Uua) = times_times_rat(aa_rat_rat(plus_plus_rat(Uu),aa_nat_rat(semiri681578069525770553at_rat,Uua))) ).

% ATP.lambda_217
tff(fact_10431_ATP_Olambda__218,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_fun_nat_nat(aTP_Lamm_ku(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_218
tff(fact_10432_ATP_Olambda__219,axiom,
    ! [Uu: int,Uua: nat] : aa_nat_fun_int_int(aTP_Lamm_ks(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_219
tff(fact_10433_ATP_Olambda__220,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_set_nat(aTP_Lamm_abm(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_220
tff(fact_10434_ATP_Olambda__221,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_zp,Uu),Uua) = aa_real_real(inverse_inverse_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uua)) ).

% ATP.lambda_221
tff(fact_10435_ATP_Olambda__222,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_aaj,Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Uu),Uua)) ).

% ATP.lambda_222
tff(fact_10436_ATP_Olambda__223,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_aak,Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Uua),Uu)) ).

% ATP.lambda_223
tff(fact_10437_ATP_Olambda__224,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_yb,Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Uu),Uua)) ).

% ATP.lambda_224
tff(fact_10438_ATP_Olambda__225,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ya,Uu),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Uua),Uu)) ).

% ATP.lambda_225
tff(fact_10439_ATP_Olambda__226,axiom,
    ! [Uu: fun_complex_complex,Uua: complex] : aa_complex_real(aTP_Lamm_pj(Uu),Uua) = aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(Uu,Uua)) ).

% ATP.lambda_226
tff(fact_10440_ATP_Olambda__227,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat_real(aTP_Lamm_pi(Uu),Uua) = aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(Uu,Uua)) ).

% ATP.lambda_227
tff(fact_10441_ATP_Olambda__228,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_pk(Uu),Uua) = aa_real_real(real_V7735802525324610683m_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_228
tff(fact_10442_ATP_Olambda__229,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_zk(Uu),Uua) = aa_real_real(inverse_inverse_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_229
tff(fact_10443_ATP_Olambda__230,axiom,
    ! [Uu: fun_Pr672294278902479040t_real,Uua: product_prod_nat_nat] : aa_Pro6168199777849092593t_real(aTP_Lamm_pc(Uu),Uua) = aa_real_real(ln_ln_real,aa_Pro6168199777849092593t_real(Uu,Uua)) ).

% ATP.lambda_230
tff(fact_10444_ATP_Olambda__231,axiom,
    ! [Uu: fun_complex_real,Uua: complex] : aa_complex_real(aTP_Lamm_pg(Uu),Uua) = aa_real_real(ln_ln_real,aa_complex_real(Uu,Uua)) ).

% ATP.lambda_231
tff(fact_10445_ATP_Olambda__232,axiom,
    ! [Uu: fun_real_real,Uua: real] : aa_real_real(aTP_Lamm_pd(Uu),Uua) = aa_real_real(ln_ln_real,aa_real_real(Uu,Uua)) ).

% ATP.lambda_232
tff(fact_10446_ATP_Olambda__233,axiom,
    ! [Uu: fun_set_nat_real,Uua: set_nat] : aa_set_nat_real(aTP_Lamm_pe(Uu),Uua) = aa_real_real(ln_ln_real,aa_set_nat_real(Uu,Uua)) ).

% ATP.lambda_233
tff(fact_10447_ATP_Olambda__234,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_ph(Uu),Uua) = aa_real_real(ln_ln_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_234
tff(fact_10448_ATP_Olambda__235,axiom,
    ! [Uu: fun_int_real,Uua: int] : aa_int_real(aTP_Lamm_pf(Uu),Uua) = aa_real_real(ln_ln_real,aa_int_real(Uu,Uua)) ).

% ATP.lambda_235
tff(fact_10449_ATP_Olambda__236,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat3829984526574172007omplex(aTP_Lamm_oi(Uu),Uua) = times_times_complex(aa_nat_complex(Uu,Uua)) ).

% ATP.lambda_236
tff(fact_10450_ATP_Olambda__237,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_fun_real_real(aTP_Lamm_oj(Uu),Uua) = aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_237
tff(fact_10451_ATP_Olambda__238,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_fun_rat_rat(aTP_Lamm_ok(Uu),Uua) = times_times_rat(aa_nat_rat(Uu,Uua)) ).

% ATP.lambda_238
tff(fact_10452_ATP_Olambda__239,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_fun_nat_nat(aTP_Lamm_ol(Uu),Uua) = aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu,Uua)) ).

% ATP.lambda_239
tff(fact_10453_ATP_Olambda__240,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_fun_int_int(aTP_Lamm_om(Uu),Uua) = aa_int_fun_int_int(times_times_int,aa_nat_int(Uu,Uua)) ).

% ATP.lambda_240
tff(fact_10454_ATP_Olambda__241,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_fun_real_real(aTP_Lamm_ke(Uu),Uua) = aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_241
tff(fact_10455_ATP_Olambda__242,axiom,
    ! [Uu: fun_nat_rat,Uua: nat] : aa_nat_fun_rat_rat(aTP_Lamm_kb(Uu),Uua) = plus_plus_rat(aa_nat_rat(Uu,Uua)) ).

% ATP.lambda_242
tff(fact_10456_ATP_Olambda__243,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] : aa_nat_fun_nat_nat(aTP_Lamm_kd(Uu),Uua) = aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(Uu,Uua)) ).

% ATP.lambda_243
tff(fact_10457_ATP_Olambda__244,axiom,
    ! [Uu: fun_nat_int,Uua: nat] : aa_nat_fun_int_int(aTP_Lamm_kc(Uu),Uua) = aa_int_fun_int_int(plus_plus_int,aa_nat_int(Uu,Uua)) ).

% ATP.lambda_244
tff(fact_10458_ATP_Olambda__245,axiom,
    ! [Uu: fun_real_real,Uua: real] : aa_real_real(aTP_Lamm_aaf(Uu),Uua) = aa_real_real(arcosh_real,aa_real_real(Uu,Uua)) ).

% ATP.lambda_245
tff(fact_10459_ATP_Olambda__246,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_fy(Uu),Uua) = aa_real_real(abs_abs_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_246
tff(fact_10460_ATP_Olambda__247,axiom,
    ! [Uu: fun_int_int,Uua: int] : aa_int_int(aTP_Lamm_fx(Uu),Uua) = aa_int_int(abs_abs_int,aa_int_int(Uu,Uua)) ).

% ATP.lambda_247
tff(fact_10461_ATP_Olambda__248,axiom,
    ! [Uu: fun_complex_complex,Uua: complex] : aa_complex_complex(aTP_Lamm_of(Uu),Uua) = aa_complex_complex(exp_complex,aa_complex_complex(Uu,Uua)) ).

% ATP.lambda_248
tff(fact_10462_ATP_Olambda__249,axiom,
    ! [Uu: fun_complex_real,Uua: complex] : aa_complex_real(aTP_Lamm_oe(Uu),Uua) = aa_real_real(exp_real,aa_complex_real(Uu,Uua)) ).

% ATP.lambda_249
tff(fact_10463_ATP_Olambda__250,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_og(Uu),Uua) = aa_real_real(exp_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_250
tff(fact_10464_ATP_Olambda__251,axiom,
    ! [Uu: fun_int_real,Uua: int] : aa_int_real(aTP_Lamm_od(Uu),Uua) = aa_real_real(exp_real,aa_int_real(Uu,Uua)) ).

% ATP.lambda_251
tff(fact_10465_ATP_Olambda__252,axiom,
    ! [Uu: fun_re4238438383274954274t_real,Uua: real] : aa_real_real(aTP_Lamm_yt(Uu),Uua) = suminf_real(aa_real_fun_nat_real(Uu,Uua)) ).

% ATP.lambda_252
tff(fact_10466_ATP_Olambda__253,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_set_nat(aTP_Lamm_abn(Uu),Uua) = collect_nat(aa_nat_fun_nat_o(aTP_Lamm_au,Uu)) ).

% ATP.lambda_253
tff(fact_10467_ATP_Olambda__254,axiom,
    ! [Uu: set_real,Uua: real] :
      ( aa_real_o(aTP_Lamm_abb(Uu),Uua)
    <=> ! [X4: real] :
          ( member_real(X4,Uu)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X4),Uua) ) ) ).

% ATP.lambda_254
tff(fact_10468_ATP_Olambda__255,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] :
      aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_xm(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_255
tff(fact_10469_ATP_Olambda__256,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] :
      aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_vf(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_256
tff(fact_10470_ATP_Olambda__257,axiom,
    ! [Uu: num,Uua: code_integer,Uub: code_integer] :
      aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_ex(Uu),Uua),Uub) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_num_Code_integer(numera6620942414471956472nteger,Uu)),Uub),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Uua)),one_one_Code_integer),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,Uub),aa_num_Code_integer(numera6620942414471956472nteger,Uu))),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_num_Code_integer(numera6620942414471956472nteger,aa_num_num(bit0,one))),Uua),Uub)) ).

% ATP.lambda_257
tff(fact_10471_ATP_Olambda__258,axiom,
    ! [Uu: num,Uua: nat,Uub: nat] :
      aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_ey(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_258
tff(fact_10472_ATP_Olambda__259,axiom,
    ! [Uu: num,Uua: int,Uub: int] :
      aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_ez(Uu),Uua),Uub) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Uu)),Uub),product_Pair_int_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Uua)),one_one_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,Uub),aa_num_int(numeral_numeral_int,Uu))),product_Pair_int_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Uua),Uub)) ).

% ATP.lambda_259
tff(fact_10473_ATP_Olambda__260,axiom,
    ! [Uu: set_nat,Uua: fun_nat_real,Uub: nat] :
      aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_qr(Uu),Uua),Uub) = $ite(aa_set_nat_o(member_nat(Uub),Uu),aa_nat_real(Uua,Uub),zero_zero_real) ).

% ATP.lambda_260
tff(fact_10474_ATP_Olambda__261,axiom,
    ! [Uu: set_nat,Uua: fun_nat_nat,Uub: nat] :
      aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_qs(Uu),Uua),Uub) = $ite(aa_set_nat_o(member_nat(Uub),Uu),aa_nat_nat(Uua,Uub),zero_zero_nat) ).

% ATP.lambda_261
tff(fact_10475_ATP_Olambda__262,axiom,
    ! [Uu: set_nat,Uua: fun_nat_int,Uub: nat] :
      aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_qt(Uu),Uua),Uub) = $ite(aa_set_nat_o(member_nat(Uub),Uu),aa_nat_int(Uua,Uub),zero_zero_int) ).

% ATP.lambda_262
tff(fact_10476_ATP_Olambda__263,axiom,
    ! [Uu: complex,Uua: fun_complex_real,Uub: complex] :
      aa_complex_real(aa_fun8933909628229591020x_real(aTP_Lamm_fp(Uu),Uua),Uub) = $ite(Uu = Uub,aa_complex_real(Uua,Uub),zero_zero_real) ).

% ATP.lambda_263
tff(fact_10477_ATP_Olambda__264,axiom,
    ! [Uu: complex,Uua: fun_complex_rat,Uub: complex] :
      aa_complex_rat(aa_fun7622890401989837636ex_rat(aTP_Lamm_ft(Uu),Uua),Uub) = $ite(Uu = Uub,aa_complex_rat(Uua,Uub),zero_zero_rat) ).

% ATP.lambda_264
tff(fact_10478_ATP_Olambda__265,axiom,
    ! [Uu: complex,Uua: fun_complex_nat,Uub: complex] :
      aa_complex_nat(aa_fun2465681967280144180ex_nat(aTP_Lamm_fw(Uu),Uua),Uub) = $ite(Uu = Uub,aa_complex_nat(Uua,Uub),zero_zero_nat) ).

% ATP.lambda_265
tff(fact_10479_ATP_Olambda__266,axiom,
    ! [Uu: real,Uua: fun_real_real,Uub: real] :
      aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_fn(Uu),Uua),Uub) = $ite(Uu = Uub,aa_real_real(Uua,Uub),zero_zero_real) ).

% ATP.lambda_266
tff(fact_10480_ATP_Olambda__267,axiom,
    ! [Uu: real,Uua: fun_real_rat,Uub: real] :
      aa_real_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_fq(Uu),Uua),Uub) = $ite(Uu = Uub,aa_real_rat(Uua,Uub),zero_zero_rat) ).

% ATP.lambda_267
tff(fact_10481_ATP_Olambda__268,axiom,
    ! [Uu: real,Uua: fun_real_nat,Uub: real] :
      aa_real_nat(aa_fun211602392931233840al_nat(aTP_Lamm_fu(Uu),Uua),Uub) = $ite(Uu = Uub,aa_real_nat(Uua,Uub),zero_zero_nat) ).

% ATP.lambda_268
tff(fact_10482_ATP_Olambda__269,axiom,
    ! [Uu: nat,Uua: fun_nat_rat,Uub: nat] :
      aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_fr(Uu),Uua),Uub) = $ite(Uu = Uub,aa_nat_rat(Uua,Uub),zero_zero_rat) ).

% ATP.lambda_269
tff(fact_10483_ATP_Olambda__270,axiom,
    ! [Uu: int,Uua: fun_int_real,Uub: int] :
      aa_int_real(aa_fun3252798400008884712t_real(aTP_Lamm_fo(Uu),Uua),Uub) = $ite(Uu = Uub,aa_int_real(Uua,Uub),zero_zero_real) ).

% ATP.lambda_270
tff(fact_10484_ATP_Olambda__271,axiom,
    ! [Uu: int,Uua: fun_int_rat,Uub: int] :
      aa_int_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_fs(Uu),Uua),Uub) = $ite(Uu = Uub,aa_int_rat(Uua,Uub),zero_zero_rat) ).

% ATP.lambda_271
tff(fact_10485_ATP_Olambda__272,axiom,
    ! [Uu: int,Uua: fun_int_nat,Uub: int] :
      aa_int_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_fv(Uu),Uua),Uub) = $ite(Uu = Uub,aa_int_nat(Uua,Uub),zero_zero_nat) ).

% ATP.lambda_272
tff(fact_10486_ATP_Olambda__273,axiom,
    ! [Uu: complex,Uua: fun_complex_complex,Uub: complex] :
      aa_complex_complex(aa_fun3187328805393345264omplex(aTP_Lamm_lp(Uu),Uua),Uub) = $ite(Uu = Uub,aa_complex_complex(Uua,Uub),one_one_complex) ).

% ATP.lambda_273
tff(fact_10487_ATP_Olambda__274,axiom,
    ! [Uu: complex,Uua: fun_complex_real,Uub: complex] :
      aa_complex_real(aa_fun8933909628229591020x_real(aTP_Lamm_lt(Uu),Uua),Uub) = $ite(Uu = Uub,aa_complex_real(Uua,Uub),one_one_real) ).

% ATP.lambda_274
tff(fact_10488_ATP_Olambda__275,axiom,
    ! [Uu: real,Uua: fun_real_complex,Uub: real] :
      aa_real_complex(aa_fun2233226938935547116omplex(aTP_Lamm_lm(Uu),Uua),Uub) = $ite(Uu = Uub,aa_real_complex(Uua,Uub),one_one_complex) ).

% ATP.lambda_275
tff(fact_10489_ATP_Olambda__276,axiom,
    ! [Uu: real,Uua: fun_real_real,Uub: real] :
      aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_lq(Uu),Uua),Uub) = $ite(Uu = Uub,aa_real_real(Uua,Uub),one_one_real) ).

% ATP.lambda_276
tff(fact_10490_ATP_Olambda__277,axiom,
    ! [Uu: real,Uua: fun_real_rat,Uub: real] :
      aa_real_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_lu(Uu),Uua),Uub) = $ite(Uu = Uub,aa_real_rat(Uua,Uub),one_one_rat) ).

% ATP.lambda_277
tff(fact_10491_ATP_Olambda__278,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: nat] :
      aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_ln(Uu),Uua),Uub) = $ite(Uu = Uub,aa_nat_complex(Uua,Uub),one_one_complex) ).

% ATP.lambda_278
tff(fact_10492_ATP_Olambda__279,axiom,
    ! [Uu: nat,Uua: fun_nat_real,Uub: nat] :
      aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_lr(Uu),Uua),Uub) = $ite(Uu = Uub,aa_nat_real(Uua,Uub),one_one_real) ).

% ATP.lambda_279
tff(fact_10493_ATP_Olambda__280,axiom,
    ! [Uu: nat,Uua: fun_nat_rat,Uub: nat] :
      aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_lv(Uu),Uua),Uub) = $ite(Uu = Uub,aa_nat_rat(Uua,Uub),one_one_rat) ).

% ATP.lambda_280
tff(fact_10494_ATP_Olambda__281,axiom,
    ! [Uu: int,Uua: fun_int_complex,Uub: int] :
      aa_int_complex(aa_fun3088484340408535532omplex(aTP_Lamm_lo(Uu),Uua),Uub) = $ite(Uu = Uub,aa_int_complex(Uua,Uub),one_one_complex) ).

% ATP.lambda_281
tff(fact_10495_ATP_Olambda__282,axiom,
    ! [Uu: int,Uua: fun_int_real,Uub: int] :
      aa_int_real(aa_fun3252798400008884712t_real(aTP_Lamm_ls(Uu),Uua),Uub) = $ite(Uu = Uub,aa_int_real(Uua,Uub),one_one_real) ).

% ATP.lambda_282
tff(fact_10496_ATP_Olambda__283,axiom,
    ! [Uu: complex,Uua: fun_complex_real,Uub: complex] :
      aa_complex_real(aa_fun8933909628229591020x_real(aTP_Lamm_ff(Uu),Uua),Uub) = $ite(Uub = Uu,aa_complex_real(Uua,Uub),zero_zero_real) ).

% ATP.lambda_283
tff(fact_10497_ATP_Olambda__284,axiom,
    ! [Uu: complex,Uua: fun_complex_rat,Uub: complex] :
      aa_complex_rat(aa_fun7622890401989837636ex_rat(aTP_Lamm_fj(Uu),Uua),Uub) = $ite(Uub = Uu,aa_complex_rat(Uua,Uub),zero_zero_rat) ).

% ATP.lambda_284
tff(fact_10498_ATP_Olambda__285,axiom,
    ! [Uu: complex,Uua: fun_complex_nat,Uub: complex] :
      aa_complex_nat(aa_fun2465681967280144180ex_nat(aTP_Lamm_fm(Uu),Uua),Uub) = $ite(Uub = Uu,aa_complex_nat(Uua,Uub),zero_zero_nat) ).

% ATP.lambda_285
tff(fact_10499_ATP_Olambda__286,axiom,
    ! [Uu: real,Uua: fun_real_real,Uub: real] :
      aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_fd(Uu),Uua),Uub) = $ite(Uub = Uu,aa_real_real(Uua,Uub),zero_zero_real) ).

% ATP.lambda_286
tff(fact_10500_ATP_Olambda__287,axiom,
    ! [Uu: real,Uua: fun_real_rat,Uub: real] :
      aa_real_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_fg(Uu),Uua),Uub) = $ite(Uub = Uu,aa_real_rat(Uua,Uub),zero_zero_rat) ).

% ATP.lambda_287
tff(fact_10501_ATP_Olambda__288,axiom,
    ! [Uu: real,Uua: fun_real_nat,Uub: real] :
      aa_real_nat(aa_fun211602392931233840al_nat(aTP_Lamm_fk(Uu),Uua),Uub) = $ite(Uub = Uu,aa_real_nat(Uua,Uub),zero_zero_nat) ).

% ATP.lambda_288
tff(fact_10502_ATP_Olambda__289,axiom,
    ! [Uu: nat,Uua: fun_nat_real,Uub: nat] :
      aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_qm(Uu),Uua),Uub) = $ite(Uub = Uu,aa_nat_real(Uua,Uub),zero_zero_real) ).

% ATP.lambda_289
tff(fact_10503_ATP_Olambda__290,axiom,
    ! [Uu: nat,Uua: fun_nat_rat,Uub: nat] :
      aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_fh(Uu),Uua),Uub) = $ite(Uub = Uu,aa_nat_rat(Uua,Uub),zero_zero_rat) ).

% ATP.lambda_290
tff(fact_10504_ATP_Olambda__291,axiom,
    ! [Uu: nat,Uua: fun_nat_nat,Uub: nat] :
      aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_qn(Uu),Uua),Uub) = $ite(Uub = Uu,aa_nat_nat(Uua,Uub),zero_zero_nat) ).

% ATP.lambda_291
tff(fact_10505_ATP_Olambda__292,axiom,
    ! [Uu: nat,Uua: fun_nat_int,Uub: nat] :
      aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_qo(Uu),Uua),Uub) = $ite(Uub = Uu,aa_nat_int(Uua,Uub),zero_zero_int) ).

% ATP.lambda_292
tff(fact_10506_ATP_Olambda__293,axiom,
    ! [Uu: int,Uua: fun_int_real,Uub: int] :
      aa_int_real(aa_fun3252798400008884712t_real(aTP_Lamm_fe(Uu),Uua),Uub) = $ite(Uub = Uu,aa_int_real(Uua,Uub),zero_zero_real) ).

% ATP.lambda_293
tff(fact_10507_ATP_Olambda__294,axiom,
    ! [Uu: int,Uua: fun_int_rat,Uub: int] :
      aa_int_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_fi(Uu),Uua),Uub) = $ite(Uub = Uu,aa_int_rat(Uua,Uub),zero_zero_rat) ).

% ATP.lambda_294
tff(fact_10508_ATP_Olambda__295,axiom,
    ! [Uu: int,Uua: fun_int_nat,Uub: int] :
      aa_int_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_fl(Uu),Uua),Uub) = $ite(Uub = Uu,aa_int_nat(Uua,Uub),zero_zero_nat) ).

% ATP.lambda_295
tff(fact_10509_ATP_Olambda__296,axiom,
    ! [Uu: complex,Uua: fun_complex_complex,Uub: complex] :
      aa_complex_complex(aa_fun3187328805393345264omplex(aTP_Lamm_lz(Uu),Uua),Uub) = $ite(Uub = Uu,aa_complex_complex(Uua,Uub),one_one_complex) ).

% ATP.lambda_296
tff(fact_10510_ATP_Olambda__297,axiom,
    ! [Uu: complex,Uua: fun_complex_real,Uub: complex] :
      aa_complex_real(aa_fun8933909628229591020x_real(aTP_Lamm_md(Uu),Uua),Uub) = $ite(Uub = Uu,aa_complex_real(Uua,Uub),one_one_real) ).

% ATP.lambda_297
tff(fact_10511_ATP_Olambda__298,axiom,
    ! [Uu: real,Uua: fun_real_complex,Uub: real] :
      aa_real_complex(aa_fun2233226938935547116omplex(aTP_Lamm_lw(Uu),Uua),Uub) = $ite(Uub = Uu,aa_real_complex(Uua,Uub),one_one_complex) ).

% ATP.lambda_298
tff(fact_10512_ATP_Olambda__299,axiom,
    ! [Uu: real,Uua: fun_real_real,Uub: real] :
      aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_ma(Uu),Uua),Uub) = $ite(Uub = Uu,aa_real_real(Uua,Uub),one_one_real) ).

% ATP.lambda_299
tff(fact_10513_ATP_Olambda__300,axiom,
    ! [Uu: real,Uua: fun_real_rat,Uub: real] :
      aa_real_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_me(Uu),Uua),Uub) = $ite(Uub = Uu,aa_real_rat(Uua,Uub),one_one_rat) ).

% ATP.lambda_300
tff(fact_10514_ATP_Olambda__301,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: nat] :
      aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_lx(Uu),Uua),Uub) = $ite(Uub = Uu,aa_nat_complex(Uua,Uub),one_one_complex) ).

% ATP.lambda_301
tff(fact_10515_ATP_Olambda__302,axiom,
    ! [Uu: nat,Uua: fun_nat_real,Uub: nat] :
      aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_mb(Uu),Uua),Uub) = $ite(Uub = Uu,aa_nat_real(Uua,Uub),one_one_real) ).

% ATP.lambda_302
tff(fact_10516_ATP_Olambda__303,axiom,
    ! [Uu: nat,Uua: fun_nat_rat,Uub: nat] :
      aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_mf(Uu),Uua),Uub) = $ite(Uub = Uu,aa_nat_rat(Uua,Uub),one_one_rat) ).

% ATP.lambda_303
tff(fact_10517_ATP_Olambda__304,axiom,
    ! [Uu: int,Uua: fun_int_complex,Uub: int] :
      aa_int_complex(aa_fun3088484340408535532omplex(aTP_Lamm_ly(Uu),Uua),Uub) = $ite(Uub = Uu,aa_int_complex(Uua,Uub),one_one_complex) ).

% ATP.lambda_304
tff(fact_10518_ATP_Olambda__305,axiom,
    ! [Uu: int,Uua: fun_int_real,Uub: int] :
      aa_int_real(aa_fun3252798400008884712t_real(aTP_Lamm_mc(Uu),Uua),Uub) = $ite(Uub = Uu,aa_int_real(Uua,Uub),one_one_real) ).

% ATP.lambda_305
tff(fact_10519_ATP_Olambda__306,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] :
      aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_ye(Uu),Uua),Uub) = $ite(Uub = zero_z3403309356797280102nteger,produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),zero_z3403309356797280102nteger),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua)),one_one_Code_integer),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uu)),Uub))) ).

% ATP.lambda_306
tff(fact_10520_ATP_Olambda__307,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] :
      aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_yg(Uu),Uua),Uub) = $ite(Uub = zero_z3403309356797280102nteger,produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),zero_z3403309356797280102nteger),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua)),one_one_Code_integer),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(abs_abs_Code_integer,Uu)),Uub))) ).

% ATP.lambda_307
tff(fact_10521_ATP_Olambda__308,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] :
      aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_yd(Uu),Uua),Uub) = $ite(Uub = zero_z3403309356797280102nteger,produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),zero_z3403309356797280102nteger),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua)),one_one_Code_integer),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,Uu),Uub))) ).

% ATP.lambda_308
tff(fact_10522_ATP_Olambda__309,axiom,
    ! [Uu: fun_nat_o,Uua: fun_nat_real,Uub: nat] :
      aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_qu(Uu),Uua),Uub) = $ite(aa_nat_o(Uu,Uub),aa_nat_real(Uua,Uub),zero_zero_real) ).

% ATP.lambda_309
tff(fact_10523_ATP_Olambda__310,axiom,
    ! [Uu: fun_nat_o,Uua: fun_nat_nat,Uub: nat] :
      aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_qv(Uu),Uua),Uub) = $ite(aa_nat_o(Uu,Uub),aa_nat_nat(Uua,Uub),zero_zero_nat) ).

% ATP.lambda_310
tff(fact_10524_ATP_Olambda__311,axiom,
    ! [Uu: fun_nat_o,Uua: fun_nat_int,Uub: nat] :
      aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_qw(Uu),Uua),Uub) = $ite(aa_nat_o(Uu,Uub),aa_nat_int(Uua,Uub),zero_zero_int) ).

% ATP.lambda_311
tff(fact_10525_ATP_Olambda__312,axiom,
    ! [Uu: fun_complex_real,Uua: fun_complex_o,Uub: complex] :
      aa_complex_real(aa_fun8261057609414086622x_real(aTP_Lamm_hs(Uu),Uua),Uub) = $ite(aa_complex_o(Uua,Uub),aa_complex_real(Uu,Uub),zero_zero_real) ).

% ATP.lambda_312
tff(fact_10526_ATP_Olambda__313,axiom,
    ! [Uu: fun_complex_rat,Uua: fun_complex_o,Uub: complex] :
      aa_complex_rat(aa_fun4010016270681809034ex_rat(aTP_Lamm_hw(Uu),Uua),Uub) = $ite(aa_complex_o(Uua,Uub),aa_complex_rat(Uu,Uub),zero_zero_rat) ).

% ATP.lambda_313
tff(fact_10527_ATP_Olambda__314,axiom,
    ! [Uu: fun_complex_nat,Uua: fun_complex_o,Uub: complex] :
      aa_complex_nat(aa_fun7518244829075205762ex_nat(aTP_Lamm_hz(Uu),Uua),Uub) = $ite(aa_complex_o(Uua,Uub),aa_complex_nat(Uu,Uub),zero_zero_nat) ).

% ATP.lambda_314
tff(fact_10528_ATP_Olambda__315,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_o,Uub: real] :
      aa_real_real(aa_fun4448911596306054750l_real(aTP_Lamm_hp(Uu),Uua),Uub) = $ite(aa_real_o(Uua,Uub),aa_real_real(Uu,Uub),zero_zero_real) ).

% ATP.lambda_315
tff(fact_10529_ATP_Olambda__316,axiom,
    ! [Uu: fun_real_rat,Uua: fun_real_o,Uub: real] :
      aa_real_rat(aa_fun2004904883527618314al_rat(aTP_Lamm_ht(Uu),Uua),Uub) = $ite(aa_real_o(Uua,Uub),aa_real_rat(Uu,Uub),zero_zero_rat) ).

% ATP.lambda_316
tff(fact_10530_ATP_Olambda__317,axiom,
    ! [Uu: fun_real_nat,Uua: fun_real_o,Uub: real] :
      aa_real_nat(aa_fun5513133441921015042al_nat(aTP_Lamm_hx(Uu),Uua),Uub) = $ite(aa_real_o(Uua,Uub),aa_real_nat(Uu,Uub),zero_zero_nat) ).

% ATP.lambda_317
tff(fact_10531_ATP_Olambda__318,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_nat_o,Uub: nat] :
      aa_nat_rat(aa_fun696632391729880074at_rat(aTP_Lamm_hu(Uu),Uua),Uub) = $ite(aa_nat_o(Uua,Uub),aa_nat_rat(Uu,Uub),zero_zero_rat) ).

% ATP.lambda_318
tff(fact_10532_ATP_Olambda__319,axiom,
    ! [Uu: fun_int_real,Uua: fun_int_o,Uub: int] :
      aa_int_real(aa_fun7506669883404309598t_real(aTP_Lamm_hq(Uu),Uua),Uub) = $ite(aa_int_o(Uua,Uub),aa_int_real(Uu,Uub),zero_zero_real) ).

% ATP.lambda_319
tff(fact_10533_ATP_Olambda__320,axiom,
    ! [Uu: fun_int_rat,Uua: fun_int_o,Uub: int] :
      aa_int_rat(aa_fun268837275264525578nt_rat(aTP_Lamm_hv(Uu),Uua),Uub) = $ite(aa_int_o(Uua,Uub),aa_int_rat(Uu,Uub),zero_zero_rat) ).

% ATP.lambda_320
tff(fact_10534_ATP_Olambda__321,axiom,
    ! [Uu: fun_int_nat,Uua: fun_int_o,Uub: int] :
      aa_int_nat(aa_fun3777065833657922306nt_nat(aTP_Lamm_hy(Uu),Uua),Uub) = $ite(aa_int_o(Uua,Uub),aa_int_nat(Uu,Uub),zero_zero_nat) ).

% ATP.lambda_321
tff(fact_10535_ATP_Olambda__322,axiom,
    ! [Uu: fun_complex_complex,Uua: fun_complex_o,Uub: complex] :
      aa_complex_complex(aa_fun3681620399087435360omplex(aTP_Lamm_nl(Uu),Uua),Uub) = $ite(aa_complex_o(Uua,Uub),aa_complex_complex(Uu,Uub),one_one_complex) ).

% ATP.lambda_322
tff(fact_10536_ATP_Olambda__323,axiom,
    ! [Uu: fun_complex_real,Uua: fun_complex_o,Uub: complex] :
      aa_complex_real(aa_fun8261057609414086622x_real(aTP_Lamm_np(Uu),Uua),Uub) = $ite(aa_complex_o(Uua,Uub),aa_complex_real(Uu,Uub),one_one_real) ).

% ATP.lambda_323
tff(fact_10537_ATP_Olambda__324,axiom,
    ! [Uu: fun_real_complex,Uua: fun_real_o,Uub: real] :
      aa_real_complex(aa_fun1098856492951973088omplex(aTP_Lamm_ni(Uu),Uua),Uub) = $ite(aa_real_o(Uua,Uub),aa_real_complex(Uu,Uub),one_one_complex) ).

% ATP.lambda_324
tff(fact_10538_ATP_Olambda__325,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_o,Uub: real] :
      aa_real_real(aa_fun4448911596306054750l_real(aTP_Lamm_nm(Uu),Uua),Uub) = $ite(aa_real_o(Uua,Uub),aa_real_real(Uu,Uub),one_one_real) ).

% ATP.lambda_325
tff(fact_10539_ATP_Olambda__326,axiom,
    ! [Uu: fun_real_rat,Uua: fun_real_o,Uub: real] :
      aa_real_rat(aa_fun2004904883527618314al_rat(aTP_Lamm_nq(Uu),Uua),Uub) = $ite(aa_real_o(Uua,Uub),aa_real_rat(Uu,Uub),one_one_rat) ).

% ATP.lambda_326
tff(fact_10540_ATP_Olambda__327,axiom,
    ! [Uu: fun_nat_complex,Uua: fun_nat_o,Uub: nat] :
      aa_nat_complex(aa_fun1400689748890728928omplex(aTP_Lamm_nj(Uu),Uua),Uub) = $ite(aa_nat_o(Uua,Uub),aa_nat_complex(Uu,Uub),one_one_complex) ).

% ATP.lambda_327
tff(fact_10541_ATP_Olambda__328,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_o,Uub: nat] :
      aa_nat_real(aa_fun5510454223943551838t_real(aTP_Lamm_nn(Uu),Uua),Uub) = $ite(aa_nat_o(Uua,Uub),aa_nat_real(Uu,Uub),one_one_real) ).

% ATP.lambda_328
tff(fact_10542_ATP_Olambda__329,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_nat_o,Uub: nat] :
      aa_nat_rat(aa_fun696632391729880074at_rat(aTP_Lamm_nr(Uu),Uua),Uub) = $ite(aa_nat_o(Uua,Uub),aa_nat_rat(Uu,Uub),one_one_rat) ).

% ATP.lambda_329
tff(fact_10543_ATP_Olambda__330,axiom,
    ! [Uu: fun_int_complex,Uua: fun_int_o,Uub: int] :
      aa_int_complex(aa_fun6439487526662521568omplex(aTP_Lamm_nk(Uu),Uua),Uub) = $ite(aa_int_o(Uua,Uub),aa_int_complex(Uu,Uub),one_one_complex) ).

% ATP.lambda_330
tff(fact_10544_ATP_Olambda__331,axiom,
    ! [Uu: fun_int_real,Uua: fun_int_o,Uub: int] :
      aa_int_real(aa_fun7506669883404309598t_real(aTP_Lamm_no(Uu),Uua),Uub) = $ite(aa_int_o(Uua,Uub),aa_int_real(Uu,Uub),one_one_real) ).

% ATP.lambda_331
tff(fact_10545_ATP_Olambda__332,axiom,
    ! [Uu: fun_nat_fun_nat_real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_sb(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_332
tff(fact_10546_ATP_Olambda__333,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_rz(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_333
tff(fact_10547_ATP_Olambda__334,axiom,
    ! [Uu: fun_nat_fun_nat_int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_se(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_334
tff(fact_10548_ATP_Olambda__335,axiom,
    ! [Uu: fun_co636942507509095880at_nat,Uua: nat,Uub: complex] : aa_complex_nat(aa_nat3391284541783913865ex_nat(aTP_Lamm_hi(Uu),Uua),Uub) = aa_nat_nat(aa_com6383720667567721865at_nat(Uu,Uub),Uua) ).

% ATP.lambda_335
tff(fact_10549_ATP_Olambda__336,axiom,
    ! [Uu: fun_co7370231399432144932at_int,Uua: nat,Uub: complex] : aa_complex_int(aa_nat8436805559129492965ex_int(aTP_Lamm_mq(Uu),Uua),Uub) = aa_nat_int(aa_com2205869648058525157at_int(Uu,Uub),Uua) ).

% ATP.lambda_336
tff(fact_10550_ATP_Olambda__337,axiom,
    ! [Uu: fun_co333780937270390912nt_int,Uua: int,Uub: complex] : aa_complex_int(aa_int6126987627036542017ex_int(aTP_Lamm_gj(Uu),Uua),Uub) = aa_int_int(aa_com3205223359959579713nt_int(Uu,Uub),Uua) ).

% ATP.lambda_337
tff(fact_10551_ATP_Olambda__338,axiom,
    ! [Uu: fun_re7208826350674946562omplex,Uua: complex,Uub: real] : aa_real_complex(aa_com4766338864777664963omplex(aTP_Lamm_go(Uu),Uua),Uub) = aa_complex_complex(aa_rea2618301233319580611omplex(Uu,Uub),Uua) ).

% ATP.lambda_338
tff(fact_10552_ATP_Olambda__339,axiom,
    ! [Uu: fun_re4238438383274954274t_real,Uua: nat,Uub: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_hl(Uu),Uua),Uub) = aa_nat_real(aa_real_fun_nat_real(Uu,Uub),Uua) ).

% ATP.lambda_339
tff(fact_10553_ATP_Olambda__340,axiom,
    ! [Uu: fun_real_fun_nat_nat,Uua: nat,Uub: real] : aa_real_nat(aa_nat_fun_real_nat(aTP_Lamm_ha(Uu),Uua),Uub) = aa_nat_nat(aa_real_fun_nat_nat(Uu,Uub),Uua) ).

% ATP.lambda_340
tff(fact_10554_ATP_Olambda__341,axiom,
    ! [Uu: fun_real_fun_nat_int,Uua: nat,Uub: real] : aa_real_int(aa_nat_fun_real_int(aTP_Lamm_mn(Uu),Uua),Uub) = aa_nat_int(aa_real_fun_nat_int(Uu,Uub),Uua) ).

% ATP.lambda_341
tff(fact_10555_ATP_Olambda__342,axiom,
    ! [Uu: fun_real_fun_int_int,Uua: int,Uub: real] : aa_real_int(aa_int_fun_real_int(aTP_Lamm_gb(Uu),Uua),Uub) = aa_int_int(aa_real_fun_int_int(Uu,Uub),Uua) ).

% ATP.lambda_342
tff(fact_10556_ATP_Olambda__343,axiom,
    ! [Uu: fun_na1591161823130732966omplex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_gs(Uu),Uua),Uub) = aa_complex_complex(aa_nat3829984526574172007omplex(Uu,Uub),Uua) ).

% ATP.lambda_343
tff(fact_10557_ATP_Olambda__344,axiom,
    ! [Uu: fun_na8457291176974909128ex_nat,Uua: complex,Uub: nat] : aa_nat_nat(aa_com6383720667567721865at_nat(aTP_Lamm_nf(Uu),Uua),Uub) = aa_complex_nat(aa_nat3391284541783913865ex_nat(Uu,Uub),Uua) ).

% ATP.lambda_344
tff(fact_10558_ATP_Olambda__345,axiom,
    ! [Uu: fun_nat_fun_real_nat,Uua: real,Uub: nat] : aa_nat_nat(aa_real_fun_nat_nat(aTP_Lamm_my(Uu),Uua),Uub) = aa_real_nat(aa_nat_fun_real_nat(Uu,Uub),Uua) ).

% ATP.lambda_345
tff(fact_10559_ATP_Olambda__346,axiom,
    ! [Uu: fun_nat_fun_nat_real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_rl(Uu),Uua),Uub) = aa_nat_real(aa_nat_fun_nat_real(Uu,Uub),Uua) ).

% ATP.lambda_346
tff(fact_10560_ATP_Olambda__347,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ri(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(Uu,Uub),Uua) ).

% ATP.lambda_347
tff(fact_10561_ATP_Olambda__348,axiom,
    ! [Uu: fun_nat_fun_nat_int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_rq(Uu),Uua),Uub) = aa_nat_int(aa_nat_fun_nat_int(Uu,Uub),Uua) ).

% ATP.lambda_348
tff(fact_10562_ATP_Olambda__349,axiom,
    ! [Uu: fun_nat_fun_int_nat,Uua: int,Uub: nat] : aa_nat_nat(aa_int_fun_nat_nat(aTP_Lamm_nc(Uu),Uua),Uub) = aa_int_nat(aa_nat_fun_int_nat(Uu,Uub),Uua) ).

% ATP.lambda_349
tff(fact_10563_ATP_Olambda__350,axiom,
    ! [Uu: fun_nat_fun_int_int,Uua: int,Uub: nat] : aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_gf(Uu),Uua),Uub) = aa_int_int(aa_nat_fun_int_int(Uu,Uub),Uua) ).

% ATP.lambda_350
tff(fact_10564_ATP_Olambda__351,axiom,
    ! [Uu: fun_in1307110748760623106omplex,Uua: complex,Uub: int] : aa_int_complex(aa_com9156536935073639875omplex(aTP_Lamm_gw(Uu),Uua),Uub) = aa_complex_complex(aa_int3545933452204062147omplex(Uu,Uub),Uua) ).

% ATP.lambda_351
tff(fact_10565_ATP_Olambda__352,axiom,
    ! [Uu: fun_int_fun_nat_nat,Uua: nat,Uub: int] : aa_int_nat(aa_nat_fun_int_nat(aTP_Lamm_he(Uu),Uua),Uub) = aa_nat_nat(aa_int_fun_nat_nat(Uu,Uub),Uua) ).

% ATP.lambda_352
tff(fact_10566_ATP_Olambda__353,axiom,
    ! [Uu: nat,Uua: complex,Uub: nat] :
      aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_xf(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_353
tff(fact_10567_ATP_Olambda__354,axiom,
    ! [Uu: nat,Uua: real,Uub: nat] :
      aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_xg(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_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_354
tff(fact_10568_ATP_Olambda__355,axiom,
    ! [Uu: nat,Uua: int,Uub: nat] :
      aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_xh(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_355
tff(fact_10569_ATP_Olambda__356,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] :
      aa_Cod3168588076859124107eger_o(aa_Cod1096751537565085541eger_o(aTP_Lamm_xw(Uu),Uua),Uub) = produc6677183202524767010eger_o($ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),Uu),Uua,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(minus_8373710615458151222nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua)),Uub)),Uub = one_one_Code_integer) ).

% ATP.lambda_356
tff(fact_10570_ATP_Olambda__357,axiom,
    ! [Uu: fun_nat_fun_nat_int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_rr(Uu),Uua),Uub) = groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_rq(Uu),Uub),set_or1269000886237332187st_nat(aa_nat_nat(suc,Uub),Uua)) ).

% ATP.lambda_357
tff(fact_10571_ATP_Olambda__358,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ro(Uu),Uua),Uub) = aa_set_nat_nat(groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ri(Uu),Uub)),set_or1269000886237332187st_nat(aa_nat_nat(suc,Uub),Uua)) ).

% ATP.lambda_358
tff(fact_10572_ATP_Olambda__359,axiom,
    ! [Uu: fun_nat_fun_nat_real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_rm(Uu),Uua),Uub) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aTP_Lamm_rl(Uu),Uub)),set_or1269000886237332187st_nat(aa_nat_nat(suc,Uub),Uua)) ).

% ATP.lambda_359
tff(fact_10573_ATP_Olambda__360,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_rj(Uu),Uua),Uub) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ri(Uu),Uub)),set_or1269000886237332187st_nat(aa_nat_nat(suc,Uub),Uua)) ).

% ATP.lambda_360
tff(fact_10574_ATP_Olambda__361,axiom,
    ! [Uu: rat,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_abf(Uu),Uua),Uub)
    <=> aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(aa_int2464330468366073224_int_o(aTP_Lamm_abe(Uua),Uub)),quotient_of(Uu)) ) ).

% ATP.lambda_361
tff(fact_10575_ATP_Olambda__362,axiom,
    ! [Uu: rat,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_abd(Uu),Uua),Uub)
    <=> aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(aa_int2464330468366073224_int_o(aTP_Lamm_abc(Uua),Uub)),quotient_of(Uu)) ) ).

% ATP.lambda_362
tff(fact_10576_ATP_Olambda__363,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_yz(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Uu,Uub),zero_zero_real)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uub)) ).

% ATP.lambda_363
tff(fact_10577_ATP_Olambda__364,axiom,
    ! [Uu: real,Uua: fun_na2180513548729900188l_real,Uub: nat] : aa_nat_real(aa_fun4674562053777771865t_real(aTP_Lamm_ui(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_literal_real(aa_nat8303615512603269253l_real(Uua,Uub),zero_zero_literal)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uub)) ).

% ATP.lambda_364
tff(fact_10578_ATP_Olambda__365,axiom,
    ! [Uu: real,Uua: fun_na6724076748692969250l_real,Uub: nat] : aa_nat_real(aa_fun1113820866546176217t_real(aTP_Lamm_uj(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Uua,Uub),zero_zero_real)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uub)) ).

% ATP.lambda_365
tff(fact_10579_ATP_Olambda__366,axiom,
    ! [Uu: real,Uua: fun_nat_fun_rat_real,Uub: nat] : aa_nat_real(aa_fun507281639146497029t_real(aTP_Lamm_uk(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_rat_real(aa_nat_fun_rat_real(Uua,Uub),zero_zero_rat)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uub)) ).

% ATP.lambda_366
tff(fact_10580_ATP_Olambda__367,axiom,
    ! [Uu: real,Uua: fun_nat_fun_nat_real,Uub: nat] : aa_nat_real(aa_fun7039663905532832253t_real(aTP_Lamm_ul(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_nat_fun_nat_real(Uua,Uub),zero_zero_nat)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uub)) ).

% ATP.lambda_367
tff(fact_10581_ATP_Olambda__368,axiom,
    ! [Uu: real,Uua: fun_nat_fun_int_real,Uub: nat] : aa_nat_real(aa_fun241848180832411225t_real(aTP_Lamm_um(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(aa_nat_fun_int_real(Uua,Uub),zero_zero_int)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uub)) ).

% ATP.lambda_368
tff(fact_10582_ATP_Olambda__369,axiom,
    ! [Uu: fun_nat_complex,Uua: nat,Uub: complex] :
      ( aa_complex_o(aa_nat_fun_complex_o(aTP_Lamm_tm(Uu),Uua),Uub)
    <=> ( aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_ru(Uu),Uub)),aa_nat_set_nat(set_ord_atMost_nat,Uua)) = zero_zero_complex ) ) ).

% ATP.lambda_369
tff(fact_10583_ATP_Olambda__370,axiom,
    ! [Uu: fun_nat_real,Uua: nat,Uub: real] :
      ( aa_real_o(aa_nat_fun_real_o(aTP_Lamm_tn(Uu),Uua),Uub)
    <=> ( aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_rs(Uu),Uub)),aa_nat_set_nat(set_ord_atMost_nat,Uua)) = zero_zero_real ) ) ).

% ATP.lambda_370
tff(fact_10584_ATP_Olambda__371,axiom,
    ! [Uu: fun_nat_complex,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_ix(Uu),Uua),Uub) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(aa_nat_complex(Uu,Uub)),aa_nat_complex(power_power_complex(zero_zero_complex),Uub))),aa_nat_complex(Uua,Uub)) ).

% ATP.lambda_371
tff(fact_10585_ATP_Olambda__372,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_iz(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,Uub)),aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),Uub))),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_372
tff(fact_10586_ATP_Olambda__373,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_nat_rat,Uub: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_iy(Uu),Uua),Uub) = aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(aa_nat_rat(Uu,Uub)),aa_nat_rat(power_power_rat(zero_zero_rat),Uub))),aa_nat_rat(Uua,Uub)) ).

% ATP.lambda_373
tff(fact_10587_ATP_Olambda__374,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_yw(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,Uub)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uub)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uub)) ).

% ATP.lambda_374
tff(fact_10588_ATP_Olambda__375,axiom,
    ! [Uu: real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_uq(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_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_375
tff(fact_10589_ATP_Olambda__376,axiom,
    ! [Uu: nat,Uua: nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_nat5883842100386203267_nat_o(aTP_Lamm_aar(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_376
tff(fact_10590_ATP_Olambda__377,axiom,
    ! [Uu: nat,Uua: nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_nat5883842100386203267_nat_o(aTP_Lamm_aas(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_377
tff(fact_10591_ATP_Olambda__378,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: nat,Uub: list_VEBT_VEBT] :
      ( aa_list_VEBT_VEBT_o(aa_nat1274389614848952367VEBT_o(aTP_Lamm_cz(Uu),Uua),Uub)
    <=> ( ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(Uub),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_s6755466524823107622T_VEBT(Uub)),Uua) ) ) ).

% ATP.lambda_378
tff(fact_10592_ATP_Olambda__379,axiom,
    ! [Uu: set_complex,Uua: nat,Uub: list_complex] :
      ( aa_list_complex_o(aa_nat5940423528545770789plex_o(aTP_Lamm_cy(Uu),Uua),Uub)
    <=> ( aa_set_complex_o(ord_le211207098394363844omplex(set_complex2(Uub)),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_s3451745648224563538omplex(Uub)),Uua) ) ) ).

% ATP.lambda_379
tff(fact_10593_ATP_Olambda__380,axiom,
    ! [Uu: set_o,Uua: nat,Uub: list_o] :
      ( aa_list_o_o(aa_nat_fun_list_o_o(aTP_Lamm_da(Uu),Uua),Uub)
    <=> ( ord_less_eq_set_o(set_o2(Uub),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_o(Uub)),Uua) ) ) ).

% ATP.lambda_380
tff(fact_10594_ATP_Olambda__381,axiom,
    ! [Uu: set_nat,Uua: nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_nat5883842100386203267_nat_o(aTP_Lamm_db(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,set_nat2(Uub)),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_nat(Uub)),Uua) ) ) ).

% ATP.lambda_381
tff(fact_10595_ATP_Olambda__382,axiom,
    ! [Uu: set_int,Uua: nat,Uub: list_int] :
      ( aa_list_int_o(aa_nat3638697132715757351_int_o(aTP_Lamm_dc(Uu),Uua),Uub)
    <=> ( aa_set_int_o(ord_less_eq_set_int(set_int2(Uub)),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_int(Uub)),Uua) ) ) ).

% ATP.lambda_382
tff(fact_10596_ATP_Olambda__383,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: nat,Uub: list_VEBT_VEBT] :
      ( aa_list_VEBT_VEBT_o(aa_nat1274389614848952367VEBT_o(aTP_Lamm_cu(Uu),Uua),Uub)
    <=> ( ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(Uub),Uu)
        & ( size_s6755466524823107622T_VEBT(Uub) = Uua ) ) ) ).

% ATP.lambda_383
tff(fact_10597_ATP_Olambda__384,axiom,
    ! [Uu: set_complex,Uua: nat,Uub: list_complex] :
      ( aa_list_complex_o(aa_nat5940423528545770789plex_o(aTP_Lamm_ct(Uu),Uua),Uub)
    <=> ( aa_set_complex_o(ord_le211207098394363844omplex(set_complex2(Uub)),Uu)
        & ( size_s3451745648224563538omplex(Uub) = Uua ) ) ) ).

% ATP.lambda_384
tff(fact_10598_ATP_Olambda__385,axiom,
    ! [Uu: set_o,Uua: nat,Uub: list_o] :
      ( aa_list_o_o(aa_nat_fun_list_o_o(aTP_Lamm_cv(Uu),Uua),Uub)
    <=> ( ord_less_eq_set_o(set_o2(Uub),Uu)
        & ( size_size_list_o(Uub) = Uua ) ) ) ).

% ATP.lambda_385
tff(fact_10599_ATP_Olambda__386,axiom,
    ! [Uu: set_nat,Uua: nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_nat5883842100386203267_nat_o(aTP_Lamm_cw(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,set_nat2(Uub)),Uu)
        & ( size_size_list_nat(Uub) = Uua ) ) ) ).

% ATP.lambda_386
tff(fact_10600_ATP_Olambda__387,axiom,
    ! [Uu: set_int,Uua: nat,Uub: list_int] :
      ( aa_list_int_o(aa_nat3638697132715757351_int_o(aTP_Lamm_cx(Uu),Uua),Uub)
    <=> ( aa_set_int_o(ord_less_eq_set_int(set_int2(Uub)),Uu)
        & ( size_size_list_int(Uub) = Uua ) ) ) ).

% ATP.lambda_387
tff(fact_10601_ATP_Olambda__388,axiom,
    ! [Uu: nat,Uua: nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_nat5883842100386203267_nat_o(aTP_Lamm_aaq(Uu),Uua),Uub)
    <=> ( ( size_size_list_nat(Uub) = Uu )
        & ( groups4561878855575611511st_nat(Uub) = Uua ) ) ) ).

% ATP.lambda_388
tff(fact_10602_ATP_Olambda__389,axiom,
    ! [Uu: set_nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_xy(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_389
tff(fact_10603_ATP_Olambda__390,axiom,
    ! [Uu: nat,Uua: nat,Uub: set_nat] :
      ( aa_set_nat_o(aa_nat_fun_set_nat_o(aTP_Lamm_yf(Uu),Uua),Uub)
    <=> ( member_set_nat(Uub,pow_nat(set_or4665077453230672383an_nat(zero_zero_nat,Uu)))
        & ( aa_set_nat_nat(finite_card_nat,Uub) = Uua ) ) ) ).

% ATP.lambda_390
tff(fact_10604_ATP_Olambda__391,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_aaz(Uu),Uua),Uub)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uua),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uub),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uua),Uub) ) ) ).

% ATP.lambda_391
tff(fact_10605_ATP_Olambda__392,axiom,
    ! [Uu: set_nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_xz(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_392
tff(fact_10606_ATP_Olambda__393,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_tj(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_393
tff(fact_10607_ATP_Olambda__394,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_xn(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_394
tff(fact_10608_ATP_Olambda__395,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_aw(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_395
tff(fact_10609_ATP_Olambda__396,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_xo(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_396
tff(fact_10610_ATP_Olambda__397,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_az(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_397
tff(fact_10611_ATP_Olambda__398,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_ay(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_398
tff(fact_10612_ATP_Olambda__399,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_ax(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_399
tff(fact_10613_ATP_Olambda__400,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_aau(Uu),Uua),Uub)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Uub),Uua)
        & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Uub),Uu) ) ) ).

% ATP.lambda_400
tff(fact_10614_ATP_Olambda__401,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat,Uua: fun_Pr4796943543331071514_nat_o,Uub: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_fun359341089595923442_nat_o(aTP_Lamm_bk(Uu),Uua),Uub)
    <=> ( member8440522571783428010at_nat(Uub,Uu)
        & aa_Pro6150637604445964947_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_401
tff(fact_10615_ATP_Olambda__402,axiom,
    ! [Uu: set_complex,Uua: fun_complex_o,Uub: complex] :
      ( aa_complex_o(aa_fun2514317545224092796plex_o(aTP_Lamm_hr(Uu),Uua),Uub)
    <=> ( aa_set_complex_o(member_complex(Uub),Uu)
        & aa_complex_o(Uua,Uub) ) ) ).

% ATP.lambda_402
tff(fact_10616_ATP_Olambda__403,axiom,
    ! [Uu: set_real,Uua: fun_real_o,Uub: real] :
      ( aa_real_o(aa_fun5868358341814070780real_o(aTP_Lamm_bl(Uu),Uua),Uub)
    <=> ( member_real(Uub,Uu)
        & aa_real_o(Uua,Uub) ) ) ).

% ATP.lambda_403
tff(fact_10617_ATP_Olambda__404,axiom,
    ! [Uu: set_list_nat,Uua: fun_list_nat_o,Uub: list_nat] :
      ( aa_list_nat_o(aa_fun6155608546934989052_nat_o(aTP_Lamm_bm(Uu),Uua),Uub)
    <=> ( member_list_nat(Uub,Uu)
        & aa_list_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_404
tff(fact_10618_ATP_Olambda__405,axiom,
    ! [Uu: set_set_nat,Uua: fun_set_nat_o,Uub: set_nat] :
      ( aa_set_nat_o(aa_fun751667557118977660_nat_o(aTP_Lamm_bn(Uu),Uua),Uub)
    <=> ( member_set_nat(Uub,Uu)
        & aa_set_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_405
tff(fact_10619_ATP_Olambda__406,axiom,
    ! [Uu: set_nat,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aTP_Lamm_bo(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu)
        & aa_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_406
tff(fact_10620_ATP_Olambda__407,axiom,
    ! [Uu: set_int,Uua: fun_int_o,Uub: int] :
      ( aa_int_o(aa_fun6243517787477317628_int_o(aTP_Lamm_bp(Uu),Uua),Uub)
    <=> ( aa_set_int_o(member_int(Uub),Uu)
        & aa_int_o(Uua,Uub) ) ) ).

% ATP.lambda_407
tff(fact_10621_ATP_Olambda__408,axiom,
    ! [Uu: set_complex,Uua: fun_complex_real,Uub: complex] :
      ( aa_complex_o(aa_fun6195079535267300846plex_o(aTP_Lamm_ee(Uu),Uua),Uub)
    <=> ( aa_set_complex_o(member_complex(Uub),Uu)
        & ( aa_complex_real(Uua,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_408
tff(fact_10622_ATP_Olambda__409,axiom,
    ! [Uu: set_complex,Uua: fun_complex_rat,Uub: complex] :
      ( aa_complex_o(aa_fun5099116602557847362plex_o(aTP_Lamm_em(Uu),Uua),Uub)
    <=> ( aa_set_complex_o(member_complex(Uub),Uu)
        & ( aa_complex_rat(Uua,Uub) != zero_zero_rat ) ) ) ).

% ATP.lambda_409
tff(fact_10623_ATP_Olambda__410,axiom,
    ! [Uu: set_real,Uua: fun_real_real,Uub: real] :
      ( aa_real_o(aa_fun2895569901277098738real_o(aTP_Lamm_dy(Uu),Uua),Uub)
    <=> ( member_real(Uub,Uu)
        & ( aa_real_real(Uua,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_410
tff(fact_10624_ATP_Olambda__411,axiom,
    ! [Uu: set_real,Uua: fun_real_rat,Uub: real] :
      ( aa_real_o(aa_fun3209485225100452678real_o(aTP_Lamm_eg(Uu),Uua),Uub)
    <=> ( member_real(Uub,Uu)
        & ( aa_real_rat(Uua,Uub) != zero_zero_rat ) ) ) ).

% ATP.lambda_411
tff(fact_10625_ATP_Olambda__412,axiom,
    ! [Uu: set_real,Uua: fun_real_nat,Uub: real] :
      ( aa_real_o(aa_fun7775777430665565006real_o(aTP_Lamm_eo(Uu),Uua),Uub)
    <=> ( member_real(Uub,Uu)
        & ( aa_real_nat(Uua,Uub) != zero_zero_nat ) ) ) ).

% ATP.lambda_412
tff(fact_10626_ATP_Olambda__413,axiom,
    ! [Uu: set_nat,Uua: fun_nat_real,Uub: nat] :
      ( aa_nat_o(aa_fun6456149469358400426_nat_o(aTP_Lamm_ea(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu)
        & ( aa_nat_real(Uua,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_413
tff(fact_10627_ATP_Olambda__414,axiom,
    ! [Uu: set_nat,Uua: fun_nat_rat,Uub: nat] :
      ( aa_nat_o(aa_fun1118977351124560894_nat_o(aTP_Lamm_ei(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu)
        & ( aa_nat_rat(Uua,Uub) != zero_zero_rat ) ) ) ).

% ATP.lambda_414
tff(fact_10628_ATP_Olambda__415,axiom,
    ! [Uu: set_nat,Uua: fun_nat_nat,Uub: nat] :
      ( aa_nat_o(aa_fun8673969331427483654_nat_o(aTP_Lamm_eq(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu)
        & ( aa_nat_nat(Uua,Uub) != zero_zero_nat ) ) ) ).

% ATP.lambda_415
tff(fact_10629_ATP_Olambda__416,axiom,
    ! [Uu: set_int,Uua: fun_int_real,Uub: int] :
      ( aa_int_o(aa_fun4444401208693448434_int_o(aTP_Lamm_ec(Uu),Uua),Uub)
    <=> ( aa_set_int_o(member_int(Uub),Uu)
        & ( aa_int_real(Uua,Uub) != zero_zero_real ) ) ) ).

% ATP.lambda_416
tff(fact_10630_ATP_Olambda__417,axiom,
    ! [Uu: set_int,Uua: fun_int_rat,Uub: int] :
      ( aa_int_o(aa_fun7094158214180239686_int_o(aTP_Lamm_ek(Uu),Uua),Uub)
    <=> ( aa_set_int_o(member_int(Uub),Uu)
        & ( aa_int_rat(Uua,Uub) != zero_zero_rat ) ) ) ).

% ATP.lambda_417
tff(fact_10631_ATP_Olambda__418,axiom,
    ! [Uu: set_complex,Uua: fun_complex_complex,Uub: complex] :
      ( aa_complex_o(aa_fun573566726631320556plex_o(aTP_Lamm_dk(Uu),Uua),Uub)
    <=> ( aa_set_complex_o(member_complex(Uub),Uu)
        & ( aa_complex_complex(Uua,Uub) != one_one_complex ) ) ) ).

% ATP.lambda_418
tff(fact_10632_ATP_Olambda__419,axiom,
    ! [Uu: set_complex,Uua: fun_complex_real,Uub: complex] :
      ( aa_complex_o(aa_fun6195079535267300846plex_o(aTP_Lamm_ds(Uu),Uua),Uub)
    <=> ( aa_set_complex_o(member_complex(Uub),Uu)
        & ( aa_complex_real(Uua,Uub) != one_one_real ) ) ) ).

% ATP.lambda_419
tff(fact_10633_ATP_Olambda__420,axiom,
    ! [Uu: set_real,Uua: fun_real_complex,Uub: real] :
      ( aa_real_o(aa_fun5218151893725926128real_o(aTP_Lamm_de(Uu),Uua),Uub)
    <=> ( member_real(Uub,Uu)
        & ( aa_real_complex(Uua,Uub) != one_one_complex ) ) ) ).

% ATP.lambda_420
tff(fact_10634_ATP_Olambda__421,axiom,
    ! [Uu: set_real,Uua: fun_real_real,Uub: real] :
      ( aa_real_o(aa_fun2895569901277098738real_o(aTP_Lamm_dm(Uu),Uua),Uub)
    <=> ( member_real(Uub,Uu)
        & ( aa_real_real(Uua,Uub) != one_one_real ) ) ) ).

% ATP.lambda_421
tff(fact_10635_ATP_Olambda__422,axiom,
    ! [Uu: set_real,Uua: fun_real_rat,Uub: real] :
      ( aa_real_o(aa_fun3209485225100452678real_o(aTP_Lamm_du(Uu),Uua),Uub)
    <=> ( member_real(Uub,Uu)
        & ( aa_real_rat(Uua,Uub) != one_one_rat ) ) ) ).

% ATP.lambda_422
tff(fact_10636_ATP_Olambda__423,axiom,
    ! [Uu: set_nat,Uua: fun_nat_complex,Uub: nat] :
      ( aa_nat_o(aa_fun3019658233935452328_nat_o(aTP_Lamm_dg(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu)
        & ( aa_nat_complex(Uua,Uub) != one_one_complex ) ) ) ).

% ATP.lambda_423
tff(fact_10637_ATP_Olambda__424,axiom,
    ! [Uu: set_nat,Uua: fun_nat_real,Uub: nat] :
      ( aa_nat_o(aa_fun6456149469358400426_nat_o(aTP_Lamm_do(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu)
        & ( aa_nat_real(Uua,Uub) != one_one_real ) ) ) ).

% ATP.lambda_424
tff(fact_10638_ATP_Olambda__425,axiom,
    ! [Uu: set_nat,Uua: fun_nat_rat,Uub: nat] :
      ( aa_nat_o(aa_fun1118977351124560894_nat_o(aTP_Lamm_dw(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu)
        & ( aa_nat_rat(Uua,Uub) != one_one_rat ) ) ) ).

% ATP.lambda_425
tff(fact_10639_ATP_Olambda__426,axiom,
    ! [Uu: set_int,Uua: fun_int_complex,Uub: int] :
      ( aa_int_o(aa_fun5261878201068379632_int_o(aTP_Lamm_di(Uu),Uua),Uub)
    <=> ( aa_set_int_o(member_int(Uub),Uu)
        & ( aa_int_complex(Uua,Uub) != one_one_complex ) ) ) ).

% ATP.lambda_426
tff(fact_10640_ATP_Olambda__427,axiom,
    ! [Uu: set_int,Uua: fun_int_real,Uub: int] :
      ( aa_int_o(aa_fun4444401208693448434_int_o(aTP_Lamm_dq(Uu),Uua),Uub)
    <=> ( aa_set_int_o(member_int(Uub),Uu)
        & ( aa_int_real(Uua,Uub) != one_one_real ) ) ) ).

% ATP.lambda_427
tff(fact_10641_ATP_Olambda__428,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_zy(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uub),Uua)),Uu) ) ).

% ATP.lambda_428
tff(fact_10642_ATP_Olambda__429,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_ti(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_429
tff(fact_10643_ATP_Olambda__430,axiom,
    ! [Uu: real,Uua: complex,Uub: complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(aTP_Lamm_aad(Uu),Uua),Uub)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V3694042436643373181omplex(Uua),Uub)),Uu) ) ).

% ATP.lambda_430
tff(fact_10644_ATP_Olambda__431,axiom,
    ! [Uu: real,Uua: real,Uub: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_aab(Uu),Uua),Uub)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(real_V975177566351809787t_real,Uua),Uub)),Uu) ) ).

% ATP.lambda_431
tff(fact_10645_ATP_Olambda__432,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_ry(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_432
tff(fact_10646_ATP_Olambda__433,axiom,
    ! [Uu: set_Pr4811707699266497531nteger,Uua: code_integer,Uub: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(aTP_Lamm_ci(Uu),Uua),Uub)
    <=> member157494554546826820nteger(produc1086072967326762835nteger(Uua,Uub),Uu) ) ).

% ATP.lambda_433
tff(fact_10647_ATP_Olambda__434,axiom,
    ! [Uu: set_Pr448751882837621926eger_o,Uua: code_integer,Uub: $o] :
      ( aa_o_o(aa_Cod6379977994291289310un_o_o(aTP_Lamm_cj(Uu),Uua),(Uub))
    <=> member1379723562493234055eger_o(produc6677183202524767010eger_o(Uua,(Uub)),Uu) ) ).

% ATP.lambda_434
tff(fact_10648_ATP_Olambda__435,axiom,
    ! [Uu: set_Pr8693737435421807431at_nat,Uua: product_prod_nat_nat,Uub: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(aTP_Lamm_ck(Uu),Uua),Uub)
    <=> member8206827879206165904at_nat(produc6161850002892822231at_nat(Uua,Uub),Uu) ) ).

% ATP.lambda_435
tff(fact_10649_ATP_Olambda__436,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_cl(Uu),Uua),Uub)
    <=> member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Uua),Uub),Uu) ) ).

% ATP.lambda_436
tff(fact_10650_ATP_Olambda__437,axiom,
    ! [Uu: set_Pr958786334691620121nt_int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_cm(Uu),Uua),Uub)
    <=> member5262025264175285858nt_int(product_Pair_int_int(Uua,Uub),Uu) ) ).

% ATP.lambda_437
tff(fact_10651_ATP_Olambda__438,axiom,
    ! [Uu: nat,Uua: complex,Uub: complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(aTP_Lamm_dd(Uu),Uua),Uub)
    <=> ( aa_nat_complex(power_power_complex(Uub),Uu) = Uua ) ) ).

% ATP.lambda_438
tff(fact_10652_ATP_Olambda__439,axiom,
    ! [Uu: complex,Uua: nat,Uub: complex] :
      ( aa_complex_o(aa_nat_fun_complex_o(aTP_Lamm_yc(Uu),Uua),Uub)
    <=> ( aa_nat_complex(power_power_complex(Uub),Uua) = Uu ) ) ).

% ATP.lambda_439
tff(fact_10653_ATP_Olambda__440,axiom,
    ! [Uu: real,Uua: real,Uub: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_cp(Uu),Uua),Uub)
    <=> ( member_real(Uub,ring_1_Ints_real)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Uu),Uub)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Uub),Uua) ) ) ).

% ATP.lambda_440
tff(fact_10654_ATP_Olambda__441,axiom,
    ! [Uu: rat,Uua: rat,Uub: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(aTP_Lamm_cq(Uu),Uua),Uub)
    <=> ( member_rat(Uub,ring_1_Ints_rat)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Uu),Uub)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Uub),Uua) ) ) ).

% ATP.lambda_441
tff(fact_10655_ATP_Olambda__442,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_sh(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_442
tff(fact_10656_ATP_Olambda__443,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_si(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_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_443
tff(fact_10657_ATP_Olambda__444,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_yx(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_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_444
tff(fact_10658_ATP_Olambda__445,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_ru(Uu),Uua),Uub) = aa_complex_complex(times_times_complex(aa_nat_complex(Uu,Uub)),aa_nat_complex(power_power_complex(Uua),Uub)) ).

% ATP.lambda_445
tff(fact_10659_ATP_Olambda__446,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_rs(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,Uub)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uub)) ).

% ATP.lambda_446
tff(fact_10660_ATP_Olambda__447,axiom,
    ! [Uu: fun_nat_rat,Uua: rat,Uub: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aTP_Lamm_rw(Uu),Uua),Uub) = aa_rat_rat(times_times_rat(aa_nat_rat(Uu,Uub)),aa_nat_rat(power_power_rat(Uua),Uub)) ).

% ATP.lambda_447
tff(fact_10661_ATP_Olambda__448,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_un(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_448
tff(fact_10662_ATP_Olambda__449,axiom,
    ! [Uu: fun_nat_int,Uua: int,Uub: nat] : aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_rx(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_449
tff(fact_10663_ATP_Olambda__450,axiom,
    ! [Uu: fun_nat_o,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_cn(Uu),Uua),Uub)
    <=> ( aa_nat_o(Uu,Uub)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uub),Uua) ) ) ).

% ATP.lambda_450
tff(fact_10664_ATP_Olambda__451,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_nat_rat,Uub: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_aba(Uu),Uua),Uub) = aa_rat_rat(times_times_rat(aa_nat_rat(Uu,Uub)),aa_nat_rat(Uua,Uub)) ).

% ATP.lambda_451
tff(fact_10665_ATP_Olambda__452,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_xv(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_452
tff(fact_10666_ATP_Olambda__453,axiom,
    ! [Uu: fun_re6978948838062957830nteger,Uua: fun_re6978948838062957830nteger,Uub: real] : aa_real_Code_integer(aa_fun7923231501384339402nteger(aTP_Lamm_ik(Uu),Uua),Uub) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_real_Code_integer(Uua,Uub)),aa_real_Code_integer(Uu,Uub)) ).

% ATP.lambda_453
tff(fact_10667_ATP_Olambda__454,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_in(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(Uua,Uub)),aa_real_real(Uu,Uub)) ).

% ATP.lambda_454
tff(fact_10668_ATP_Olambda__455,axiom,
    ! [Uu: fun_real_rat,Uua: fun_real_rat,Uub: real] : aa_real_rat(aa_fun2468082234287226944al_rat(aTP_Lamm_ip(Uu),Uua),Uub) = aa_rat_rat(times_times_rat(aa_real_rat(Uua,Uub)),aa_real_rat(Uu,Uub)) ).

% ATP.lambda_455
tff(fact_10669_ATP_Olambda__456,axiom,
    ! [Uu: fun_real_int,Uua: fun_real_int,Uub: real] : aa_real_int(aa_fun8270430325945277160al_int(aTP_Lamm_is(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_real_int(Uua,Uub)),aa_real_int(Uu,Uub)) ).

% ATP.lambda_456
tff(fact_10670_ATP_Olambda__457,axiom,
    ! [Uu: fun_nat_Code_integer,Uua: fun_nat_Code_integer,Uub: nat] : aa_nat_Code_integer(aa_fun589076786913273746nteger(aTP_Lamm_il(Uu),Uua),Uub) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(Uua,Uub)),aa_nat_Code_integer(Uu,Uub)) ).

% ATP.lambda_457
tff(fact_10671_ATP_Olambda__458,axiom,
    ! [Uu: fun_nat_rat,Uua: fun_nat_rat,Uub: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_iq(Uu),Uua),Uub) = aa_rat_rat(times_times_rat(aa_nat_rat(Uua,Uub)),aa_nat_rat(Uu,Uub)) ).

% ATP.lambda_458
tff(fact_10672_ATP_Olambda__459,axiom,
    ! [Uu: fun_nat_int,Uua: fun_nat_int,Uub: nat] : aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_it(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(Uua,Uub)),aa_nat_int(Uu,Uub)) ).

% ATP.lambda_459
tff(fact_10673_ATP_Olambda__460,axiom,
    ! [Uu: fun_int_Code_integer,Uua: fun_int_Code_integer,Uub: int] : aa_int_Code_integer(aa_fun5304498221708291146nteger(aTP_Lamm_im(Uu),Uua),Uub) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_int_Code_integer(Uua,Uub)),aa_int_Code_integer(Uu,Uub)) ).

% ATP.lambda_460
tff(fact_10674_ATP_Olambda__461,axiom,
    ! [Uu: fun_int_real,Uua: fun_int_real,Uub: int] : aa_int_real(aa_fun3252798400008884712t_real(aTP_Lamm_io(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(Uua,Uub)),aa_int_real(Uu,Uub)) ).

% ATP.lambda_461
tff(fact_10675_ATP_Olambda__462,axiom,
    ! [Uu: fun_int_rat,Uua: fun_int_rat,Uub: int] : aa_int_rat(aa_fun1481061427733998144nt_rat(aTP_Lamm_ir(Uu),Uua),Uub) = aa_rat_rat(times_times_rat(aa_int_rat(Uua,Uub)),aa_int_rat(Uu,Uub)) ).

% ATP.lambda_462
tff(fact_10676_ATP_Olambda__463,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_zj(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu,Uub)),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_463
tff(fact_10677_ATP_Olambda__464,axiom,
    ! [Uu: fun_Pr2602117396471556196at_nat,Uua: fun_Pr2602117396471556196at_nat,Uub: product_prod_nat_nat] : aa_Pro4243173770044397973at_nat(aa_fun2351638177499875132at_nat(aTP_Lamm_ja(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_Pro4243173770044397973at_nat(Uua,Uub)),aa_Pro4243173770044397973at_nat(Uu,Uub)) ).

% ATP.lambda_464
tff(fact_10678_ATP_Olambda__465,axiom,
    ! [Uu: fun_real_nat,Uua: fun_real_nat,Uub: real] : aa_real_nat(aa_fun211602392931233840al_nat(aTP_Lamm_jb(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_465
tff(fact_10679_ATP_Olambda__466,axiom,
    ! [Uu: fun_set_nat_nat,Uua: fun_set_nat_nat,Uub: set_nat] : aa_set_nat_nat(aa_fun3386010016858254564at_nat(aTP_Lamm_jc(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_466
tff(fact_10680_ATP_Olambda__467,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_xi(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uua,Uub)),aa_nat_real(Uu,Uub)) ).

% ATP.lambda_467
tff(fact_10681_ATP_Olambda__468,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_je(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_468
tff(fact_10682_ATP_Olambda__469,axiom,
    ! [Uu: fun_int_nat,Uua: fun_int_nat,Uub: int] : aa_int_nat(aa_fun6506752150744536624nt_nat(aTP_Lamm_jd(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_469
tff(fact_10683_ATP_Olambda__470,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_yv(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(Uu,Uub)),aa_real_real(Uua,Uub)) ).

% ATP.lambda_470
tff(fact_10684_ATP_Olambda__471,axiom,
    ! [Uu: fun_complex_o,Uua: fun_complex_o,Uub: complex] :
      ( aa_complex_o(aa_fun2514317545224092796plex_o(aTP_Lamm_ak(Uu),Uua),Uub)
    <=> ( aa_complex_o(Uu,Uub)
        | aa_complex_o(Uua,Uub) ) ) ).

% ATP.lambda_471
tff(fact_10685_ATP_Olambda__472,axiom,
    ! [Uu: fun_real_o,Uua: fun_real_o,Uub: real] :
      ( aa_real_o(aa_fun5868358341814070780real_o(aTP_Lamm_af(Uu),Uua),Uub)
    <=> ( aa_real_o(Uu,Uub)
        | aa_real_o(Uua,Uub) ) ) ).

% ATP.lambda_472
tff(fact_10686_ATP_Olambda__473,axiom,
    ! [Uu: fun_list_nat_o,Uua: fun_list_nat_o,Uub: list_nat] :
      ( aa_list_nat_o(aa_fun6155608546934989052_nat_o(aTP_Lamm_ag(Uu),Uua),Uub)
    <=> ( aa_list_nat_o(Uu,Uub)
        | aa_list_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_473
tff(fact_10687_ATP_Olambda__474,axiom,
    ! [Uu: fun_set_nat_o,Uua: fun_set_nat_o,Uub: set_nat] :
      ( aa_set_nat_o(aa_fun751667557118977660_nat_o(aTP_Lamm_ah(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(Uu,Uub)
        | aa_set_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_474
tff(fact_10688_ATP_Olambda__475,axiom,
    ! [Uu: fun_nat_o,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aTP_Lamm_ai(Uu),Uua),Uub)
    <=> ( aa_nat_o(Uu,Uub)
        | aa_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_475
tff(fact_10689_ATP_Olambda__476,axiom,
    ! [Uu: fun_int_o,Uua: fun_int_o,Uub: int] :
      ( aa_int_o(aa_fun6243517787477317628_int_o(aTP_Lamm_aj(Uu),Uua),Uub)
    <=> ( aa_int_o(Uu,Uub)
        | aa_int_o(Uua,Uub) ) ) ).

% ATP.lambda_476
tff(fact_10690_ATP_Olambda__477,axiom,
    ! [Uu: fun_complex_o,Uua: fun_complex_o,Uub: complex] :
      ( aa_complex_o(aa_fun2514317545224092796plex_o(aTP_Lamm_aq(Uu),Uua),Uub)
    <=> ( aa_complex_o(Uu,Uub)
        & aa_complex_o(Uua,Uub) ) ) ).

% ATP.lambda_477
tff(fact_10691_ATP_Olambda__478,axiom,
    ! [Uu: fun_real_o,Uua: fun_real_o,Uub: real] :
      ( aa_real_o(aa_fun5868358341814070780real_o(aTP_Lamm_al(Uu),Uua),Uub)
    <=> ( aa_real_o(Uu,Uub)
        & aa_real_o(Uua,Uub) ) ) ).

% ATP.lambda_478
tff(fact_10692_ATP_Olambda__479,axiom,
    ! [Uu: fun_list_nat_o,Uua: fun_list_nat_o,Uub: list_nat] :
      ( aa_list_nat_o(aa_fun6155608546934989052_nat_o(aTP_Lamm_am(Uu),Uua),Uub)
    <=> ( aa_list_nat_o(Uu,Uub)
        & aa_list_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_479
tff(fact_10693_ATP_Olambda__480,axiom,
    ! [Uu: fun_set_nat_o,Uua: fun_set_nat_o,Uub: set_nat] :
      ( aa_set_nat_o(aa_fun751667557118977660_nat_o(aTP_Lamm_an(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(Uu,Uub)
        & aa_set_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_480
tff(fact_10694_ATP_Olambda__481,axiom,
    ! [Uu: fun_nat_o,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aTP_Lamm_ao(Uu),Uua),Uub)
    <=> ( aa_nat_o(Uu,Uub)
        & aa_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_481
tff(fact_10695_ATP_Olambda__482,axiom,
    ! [Uu: fun_int_o,Uua: fun_int_o,Uub: int] :
      ( aa_int_o(aa_fun6243517787477317628_int_o(aTP_Lamm_ap(Uu),Uua),Uub)
    <=> ( aa_int_o(Uu,Uub)
        & aa_int_o(Uua,Uub) ) ) ).

% ATP.lambda_482
tff(fact_10696_ATP_Olambda__483,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_co(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(Uu,Uub)),Uua) ) ).

% ATP.lambda_483
tff(fact_10697_ATP_Olambda__484,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ho(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(Uu,Uub)),Uua) ).

% ATP.lambda_484
tff(fact_10698_ATP_Olambda__485,axiom,
    ! [Uu: fun_nat_int,Uua: int,Uub: nat] : aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_nh(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_nat_int(Uu,Uub)),Uua) ).

% ATP.lambda_485
tff(fact_10699_ATP_Olambda__486,axiom,
    ! [Uu: fun_int_int,Uua: int,Uub: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_hn(Uu),Uua),Uub) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(Uu,Uub)),Uua) ).

% ATP.lambda_486
tff(fact_10700_ATP_Olambda__487,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_qq(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(Uu,Uub)),Uua) ).

% ATP.lambda_487
tff(fact_10701_ATP_Olambda__488,axiom,
    ! [Uu: real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_xd(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uua,Uub)),Uu) ).

% ATP.lambda_488
tff(fact_10702_ATP_Olambda__489,axiom,
    ! [Uu: fun_real_real,Uua: nat,Uub: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_yr(Uu),Uua),Uub) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(Uu,Uub)),Uua) ).

% ATP.lambda_489
tff(fact_10703_ATP_Olambda__490,axiom,
    ! [Uu: nat,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_zu(Uu),Uua),Uub) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(Uua,Uub)),Uu) ).

% ATP.lambda_490
tff(fact_10704_ATP_Olambda__491,axiom,
    ! [Uu: fun_real_real,Uua: real,Uub: real] : aa_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_yu(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(Uu,Uub)),Uua) ).

% ATP.lambda_491
tff(fact_10705_ATP_Olambda__492,axiom,
    ! [Uu: real,Uua: fun_nat_complex,Uub: nat] : aa_nat_real(aa_fun7054779078796324018t_real(aTP_Lamm_tl(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(Uua,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uub)) ).

% ATP.lambda_492
tff(fact_10706_ATP_Olambda__493,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: nat] : aa_nat_Code_integer(aa_Cod1578151519121547077nteger(aTP_Lamm_jt(Uu),Uua),Uub) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Uu),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Uub)),Uua)) ).

% ATP.lambda_493
tff(fact_10707_ATP_Olambda__494,axiom,
    ! [Uu: extended_enat,Uua: extended_enat,Uub: nat] : aa_nat_Extended_enat(aa_Ext1337733060012393797d_enat(aTP_Lamm_jr(Uu),Uua),Uub) = plus_p3455044024723400733d_enat(Uu,times_7803423173614009249d_enat(aa_nat_Extended_enat(semiri4216267220026989637d_enat,Uub),Uua)) ).

% ATP.lambda_494
tff(fact_10708_ATP_Olambda__495,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_jp(Uu),Uua),Uub) = aa_complex_complex(plus_plus_complex(Uu),aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,Uub)),Uua)) ).

% ATP.lambda_495
tff(fact_10709_ATP_Olambda__496,axiom,
    ! [Uu: real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_jv(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Uub)),Uua)) ).

% ATP.lambda_496
tff(fact_10710_ATP_Olambda__497,axiom,
    ! [Uu: rat,Uua: rat,Uub: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aTP_Lamm_jq(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_497
tff(fact_10711_ATP_Olambda__498,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ju(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_498
tff(fact_10712_ATP_Olambda__499,axiom,
    ! [Uu: int,Uua: int,Uub: nat] : aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_js(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_499
tff(fact_10713_ATP_Olambda__500,axiom,
    ! [Uu: complex,Uua: nat,Uub: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_ue(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_500
tff(fact_10714_ATP_Olambda__501,axiom,
    ! [Uu: real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_uh(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_501
tff(fact_10715_ATP_Olambda__502,axiom,
    ! [Uu: rat,Uua: nat,Uub: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aTP_Lamm_uf(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_502
tff(fact_10716_ATP_Olambda__503,axiom,
    ! [Uu: int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_ug(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_503
tff(fact_10717_ATP_Olambda__504,axiom,
    ! [Uu: real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_zo(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Uua),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uub)) ).

% ATP.lambda_504
tff(fact_10718_ATP_Olambda__505,axiom,
    ! [Uu: complex,Uua: nat,Uub: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_jj(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_505
tff(fact_10719_ATP_Olambda__506,axiom,
    ! [Uu: real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_jm(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_506
tff(fact_10720_ATP_Olambda__507,axiom,
    ! [Uu: rat,Uua: nat,Uub: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aTP_Lamm_jk(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_507
tff(fact_10721_ATP_Olambda__508,axiom,
    ! [Uu: int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_jl(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_508
tff(fact_10722_ATP_Olambda__509,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_kv(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_509
tff(fact_10723_ATP_Olambda__510,axiom,
    ! [Uu: real,Uua: real,Uub: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_zw(Uu),Uua),Uub)
    <=> member_real(Uub,set_or1633881224788618240n_real(Uu,Uua)) ) ).

% ATP.lambda_510
tff(fact_10724_ATP_Olambda__511,axiom,
    ! [Uu: real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_qp(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Uu),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_511
tff(fact_10725_ATP_Olambda__512,axiom,
    ! [Uu: code_integer,Uua: nat,Uub: nat] : aa_nat_Code_integer(aa_nat2010205248571845816nteger(aTP_Lamm_ou(Uu),Uua),Uub) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Uu),aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uub))) ).

% ATP.lambda_512
tff(fact_10726_ATP_Olambda__513,axiom,
    ! [Uu: real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_ot(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uub))) ).

% ATP.lambda_513
tff(fact_10727_ATP_Olambda__514,axiom,
    ! [Uu: rat,Uua: nat,Uub: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aTP_Lamm_os(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_514
tff(fact_10728_ATP_Olambda__515,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ov(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_515
tff(fact_10729_ATP_Olambda__516,axiom,
    ! [Uu: int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_ow(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_516
tff(fact_10730_ATP_Olambda__517,axiom,
    ! [Uu: real,Uua: real,Uub: product_unit] : aa_Product_unit_real(aa_rea7683167438103758690t_real(aTP_Lamm_abr(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(powr_real2,Uu),Uua) ).

% ATP.lambda_517
tff(fact_10731_ATP_Olambda__518,axiom,
    ! [Uu: fun_nat_real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_sk(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_518
tff(fact_10732_ATP_Olambda__519,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_sj(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_519
tff(fact_10733_ATP_Olambda__520,axiom,
    ! [Uu: fun_nat_int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_sl(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_520
tff(fact_10734_ATP_Olambda__521,axiom,
    ! [Uu: fun_nat_complex,Uua: nat,Uub: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_tk(Uu),Uua),Uub) = aa_nat_complex(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ).

% ATP.lambda_521
tff(fact_10735_ATP_Olambda__522,axiom,
    ! [Uu: fun_nat_real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_ji(Uu),Uua),Uub) = aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ).

% ATP.lambda_522
tff(fact_10736_ATP_Olambda__523,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_jh(Uu),Uua),Uub) = aa_nat_nat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ).

% ATP.lambda_523
tff(fact_10737_ATP_Olambda__524,axiom,
    ! [Uu: fun_nat_int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_ns(Uu),Uua),Uub) = aa_nat_int(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ).

% ATP.lambda_524
tff(fact_10738_ATP_Olambda__525,axiom,
    ! [Uu: nat,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_xe(Uu),Uua),Uub) = aa_nat_real(Uua,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uu)) ).

% ATP.lambda_525
tff(fact_10739_ATP_Olambda__526,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_nat,Uub: nat] : aa_nat_real(aa_fun1538629612289296340t_real(aTP_Lamm_aai(Uu),Uua),Uub) = aa_nat_real(Uu,aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_526
tff(fact_10740_ATP_Olambda__527,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_real(aa_com20334259897932645t_real(aTP_Lamm_rv(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_527
tff(fact_10741_ATP_Olambda__528,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_rt(Uu),Uua),Uub) = aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,Uub)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uub))) ).

% ATP.lambda_528
tff(fact_10742_ATP_Olambda__529,axiom,
    ! [Uu: fun_Pr4050773816208733250omplex,Uua: fun_Pr4050773816208733250omplex,Uub: product_prod_nat_nat] : aa_Pro6168199777849092593t_real(aa_fun6819693891783611194t_real(aTP_Lamm_pp(Uu),Uua),Uub) = aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(aa_Pro82899191565270131omplex(Uu,Uub)),aa_Pro82899191565270131omplex(Uua,Uub))) ).

% ATP.lambda_529
tff(fact_10743_ATP_Olambda__530,axiom,
    ! [Uu: fun_Pr672294278902479040t_real,Uua: fun_Pr672294278902479040t_real,Uub: product_prod_nat_nat] : aa_Pro6168199777849092593t_real(aa_fun8712866722335801276t_real(aTP_Lamm_pl(Uu),Uua),Uub) = aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_Pro6168199777849092593t_real(Uu,Uub)),aa_Pro6168199777849092593t_real(Uua,Uub))) ).

% ATP.lambda_530
tff(fact_10744_ATP_Olambda__531,axiom,
    ! [Uu: fun_real_complex,Uua: fun_real_complex,Uub: real] : aa_real_real(aa_fun2786211375485122666l_real(aTP_Lamm_pq(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_531
tff(fact_10745_ATP_Olambda__532,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_pm(Uu),Uua),Uub) = aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(Uu,Uub)),aa_real_real(Uua,Uub))) ).

% ATP.lambda_532
tff(fact_10746_ATP_Olambda__533,axiom,
    ! [Uu: fun_set_nat_complex,Uua: fun_set_nat_complex,Uub: set_nat] : aa_set_nat_real(aa_fun5767801665285160478t_real(aTP_Lamm_pr(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_533
tff(fact_10747_ATP_Olambda__534,axiom,
    ! [Uu: fun_set_nat_real,Uua: fun_set_nat_real,Uub: set_nat] : aa_set_nat_real(aa_fun3569254276315353500t_real(aTP_Lamm_pn(Uu),Uua),Uub) = aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_set_nat_real(Uu,Uub)),aa_set_nat_real(Uua,Uub))) ).

% ATP.lambda_534
tff(fact_10748_ATP_Olambda__535,axiom,
    ! [Uu: fun_nat_complex,Uua: fun_nat_complex,Uub: nat] : aa_nat_real(aa_fun7054779078796324018t_real(aTP_Lamm_pu(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_535
tff(fact_10749_ATP_Olambda__536,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_pt(Uu),Uua),Uub) = aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu,Uub)),aa_nat_real(Uua,Uub))) ).

% ATP.lambda_536
tff(fact_10750_ATP_Olambda__537,axiom,
    ! [Uu: fun_int_complex,Uua: fun_int_complex,Uub: int] : aa_int_real(aa_fun1985759105522019690t_real(aTP_Lamm_ps(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_537
tff(fact_10751_ATP_Olambda__538,axiom,
    ! [Uu: fun_int_real,Uua: fun_int_real,Uub: int] : aa_int_real(aa_fun3252798400008884712t_real(aTP_Lamm_po(Uu),Uua),Uub) = aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(Uu,Uub)),aa_int_real(Uua,Uub))) ).

% ATP.lambda_538
tff(fact_10752_ATP_Olambda__539,axiom,
    ! [Uu: nat,Uua: fun_nat_real,Uub: fun_nat_real,Uuc: nat] :
      aa_nat_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_ux(Uu),Uua),Uub),Uuc) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu),
        aa_nat_real(Uua,Uuc),
        $ite(Uuc = Uu,zero_zero_real,aa_nat_real(Uub,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),aa_nat_nat(suc,zero_zero_nat)))) ) ).

% ATP.lambda_539
tff(fact_10753_ATP_Olambda__540,axiom,
    ! [Uu: nat,Uua: fun_nat_rat,Uub: fun_nat_rat,Uuc: nat] :
      aa_nat_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_ur(Uu),Uua),Uub),Uuc) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu),
        aa_nat_rat(Uua,Uuc),
        $ite(Uuc = Uu,zero_zero_rat,aa_nat_rat(Uub,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),aa_nat_nat(suc,zero_zero_nat)))) ) ).

% ATP.lambda_540
tff(fact_10754_ATP_Olambda__541,axiom,
    ! [Uu: nat,Uua: fun_nat_nat,Uub: fun_nat_nat,Uuc: nat] :
      aa_nat_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_uv(Uu),Uua),Uub),Uuc) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu),
        aa_nat_nat(Uua,Uuc),
        $ite(Uuc = Uu,zero_zero_nat,aa_nat_nat(Uub,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),aa_nat_nat(suc,zero_zero_nat)))) ) ).

% ATP.lambda_541
tff(fact_10755_ATP_Olambda__542,axiom,
    ! [Uu: nat,Uua: fun_nat_int,Uub: fun_nat_int,Uuc: nat] :
      aa_nat_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_ut(Uu),Uua),Uub),Uuc) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu),
        aa_nat_int(Uua,Uuc),
        $ite(Uuc = Uu,zero_zero_int,aa_nat_int(Uub,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),aa_nat_nat(suc,zero_zero_nat)))) ) ).

% ATP.lambda_542
tff(fact_10756_ATP_Olambda__543,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: fun_nat_complex,Uuc: nat] :
      aa_nat_complex(aa_fun2955221450537884980omplex(aa_fun7497765718855568684omplex(aTP_Lamm_uz(Uu),Uua),Uub),Uuc) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu),
        aa_nat_complex(Uua,Uuc),
        $ite(Uuc = Uu,one_one_complex,aa_nat_complex(Uub,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),aa_nat_nat(suc,zero_zero_nat)))) ) ).

% ATP.lambda_543
tff(fact_10757_ATP_Olambda__544,axiom,
    ! [Uu: nat,Uua: fun_nat_real,Uub: fun_nat_real,Uuc: nat] :
      aa_nat_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_vb(Uu),Uua),Uub),Uuc) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu),
        aa_nat_real(Uua,Uuc),
        $ite(Uuc = Uu,one_one_real,aa_nat_real(Uub,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),aa_nat_nat(suc,zero_zero_nat)))) ) ).

% ATP.lambda_544
tff(fact_10758_ATP_Olambda__545,axiom,
    ! [Uu: nat,Uua: fun_nat_rat,Uub: fun_nat_rat,Uuc: nat] :
      aa_nat_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_vc(Uu),Uua),Uub),Uuc) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu),
        aa_nat_rat(Uua,Uuc),
        $ite(Uuc = Uu,one_one_rat,aa_nat_rat(Uub,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),aa_nat_nat(suc,zero_zero_nat)))) ) ).

% ATP.lambda_545
tff(fact_10759_ATP_Olambda__546,axiom,
    ! [Uu: nat,Uua: fun_nat_nat,Uub: fun_nat_nat,Uuc: nat] :
      aa_nat_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_vd(Uu),Uua),Uub),Uuc) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu),
        aa_nat_nat(Uua,Uuc),
        $ite(Uuc = Uu,one_one_nat,aa_nat_nat(Uub,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),aa_nat_nat(suc,zero_zero_nat)))) ) ).

% ATP.lambda_546
tff(fact_10760_ATP_Olambda__547,axiom,
    ! [Uu: nat,Uua: fun_nat_int,Uub: fun_nat_int,Uuc: nat] :
      aa_nat_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_ve(Uu),Uua),Uub),Uuc) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu),
        aa_nat_int(Uua,Uuc),
        $ite(Uuc = Uu,one_one_int,aa_nat_int(Uub,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),aa_nat_nat(suc,zero_zero_nat)))) ) ).

% ATP.lambda_547
tff(fact_10761_ATP_Olambda__548,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: fun_nat_complex,Uuc: nat] :
      aa_nat_complex(aa_fun2955221450537884980omplex(aa_fun7497765718855568684omplex(aTP_Lamm_va(Uu),Uua),Uub),Uuc) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu),aa_nat_complex(Uua,Uuc),aa_nat_complex(Uub,Uuc)) ).

% ATP.lambda_548
tff(fact_10762_ATP_Olambda__549,axiom,
    ! [Uu: nat,Uua: fun_nat_real,Uub: fun_nat_real,Uuc: nat] :
      aa_nat_real(aa_fun1839206346889086768t_real(aa_fun8674848580393814182t_real(aTP_Lamm_uy(Uu),Uua),Uub),Uuc) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu),aa_nat_real(Uua,Uuc),aa_nat_real(Uub,Uuc)) ).

% ATP.lambda_549
tff(fact_10763_ATP_Olambda__550,axiom,
    ! [Uu: nat,Uua: fun_nat_rat,Uub: fun_nat_rat,Uuc: nat] :
      aa_nat_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_us(Uu),Uua),Uub),Uuc) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu),aa_nat_rat(Uua,Uuc),aa_nat_rat(Uub,Uuc)) ).

% ATP.lambda_550
tff(fact_10764_ATP_Olambda__551,axiom,
    ! [Uu: nat,Uua: fun_nat_nat,Uub: fun_nat_nat,Uuc: nat] :
      aa_nat_nat(aa_fun5791449513213591928at_nat(aa_fun6854695580826382482at_nat(aTP_Lamm_uw(Uu),Uua),Uub),Uuc) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu),aa_nat_nat(Uua,Uuc),aa_nat_nat(Uub,Uuc)) ).

% ATP.lambda_551
tff(fact_10765_ATP_Olambda__552,axiom,
    ! [Uu: nat,Uua: fun_nat_int,Uub: fun_nat_int,Uuc: nat] :
      aa_nat_int(aa_fun5565043122120104496at_int(aa_fun5140708168904641446at_int(aTP_Lamm_uu(Uu),Uua),Uub),Uuc) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuc),Uu),aa_nat_int(Uua,Uuc),aa_nat_int(Uub,Uuc)) ).

% ATP.lambda_552
tff(fact_10766_ATP_Olambda__553,axiom,
    ! [Uu: fun_nat_real,Uua: set_nat,Uub: fun_nat_real,Uuc: nat] :
      aa_nat_real(aa_fun1839206346889086768t_real(aa_set1316517696749040435t_real(aTP_Lamm_xj(Uu),Uua),Uub),Uuc) = $ite(aa_set_nat_o(member_nat(Uuc),Uua),aa_nat_real(Uub,Uuc),aa_nat_real(Uu,Uuc)) ).

% ATP.lambda_553
tff(fact_10767_ATP_Olambda__554,axiom,
    ! [Uu: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_qb(Uu),Uua),Uub),Uuc) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_qa(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_554
tff(fact_10768_ATP_Olambda__555,axiom,
    ! [Uu: rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_pz(Uu),Uua),Uub),Uuc) = groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_py(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_555
tff(fact_10769_ATP_Olambda__556,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_px(Uu),Uua),Uub),Uuc) = aa_set_nat_complex(groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_pw(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_556
tff(fact_10770_ATP_Olambda__557,axiom,
    ! [Uu: nat,Uua: code_integer,Uub: code_integer,Uuc: nat] :
      aa_nat_Code_integer(aa_Cod1578151519121547077nteger(aa_Cod6397266357770349983nteger(aTP_Lamm_vl(Uu),Uua),Uub),Uuc) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,
          $ite(
            Uuc = zero_zero_nat,
            aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uub),
            $ite(Uuc = Uu,one_one_Code_integer,zero_z3403309356797280102nteger) )),aa_nat_Code_integer(power_8256067586552552935nteger(Uua),Uuc)) ).

% ATP.lambda_557
tff(fact_10771_ATP_Olambda__558,axiom,
    ! [Uu: nat,Uua: complex,Uub: complex,Uuc: nat] :
      aa_nat_complex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aTP_Lamm_vj(Uu),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex($ite(
            Uuc = zero_zero_nat,
            aa_complex_complex(uminus1482373934393186551omplex,Uub),
            $ite(Uuc = Uu,one_one_complex,zero_zero_complex) )),aa_nat_complex(power_power_complex(Uua),Uuc)) ).

% ATP.lambda_558
tff(fact_10772_ATP_Olambda__559,axiom,
    ! [Uu: nat,Uua: real,Uub: real,Uuc: nat] :
      aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_vn(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,
          $ite(
            Uuc = zero_zero_nat,
            aa_real_real(uminus_uminus_real,Uub),
            $ite(Uuc = Uu,one_one_real,zero_zero_real) )),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uuc)) ).

% ATP.lambda_559
tff(fact_10773_ATP_Olambda__560,axiom,
    ! [Uu: nat,Uua: rat,Uub: rat,Uuc: nat] :
      aa_nat_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aTP_Lamm_vm(Uu),Uua),Uub),Uuc) = aa_rat_rat(times_times_rat($ite(
            Uuc = zero_zero_nat,
            aa_rat_rat(uminus_uminus_rat,Uub),
            $ite(Uuc = Uu,one_one_rat,zero_zero_rat) )),aa_nat_rat(power_power_rat(Uua),Uuc)) ).

% ATP.lambda_560
tff(fact_10774_ATP_Olambda__561,axiom,
    ! [Uu: nat,Uua: int,Uub: int,Uuc: nat] :
      aa_nat_int(aa_int_fun_nat_int(aa_int5842867502692063756at_int(aTP_Lamm_vk(Uu),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,
          $ite(
            Uuc = zero_zero_nat,
            aa_int_int(uminus_uminus_int,Uub),
            $ite(Uuc = Uu,one_one_int,zero_zero_int) )),aa_nat_int(power_power_int(Uua),Uuc)) ).

% ATP.lambda_561
tff(fact_10775_ATP_Olambda__562,axiom,
    ! [Uu: set_nat,Uua: fun_na8457291176974909128ex_nat,Uub: fun_na1554462459066888140plex_o,Uuc: complex] : aa_complex_nat(aa_fun1649587401935388427ex_nat(aa_fun3336366540202158548ex_nat(aTP_Lamm_ng(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups708209901874060359at_nat(aa_com6383720667567721865at_nat(aTP_Lamm_nf(Uua),Uuc)),collect_nat(aa_complex_fun_nat_o(aa_fun1708641150486296380_nat_o(aTP_Lamm_bf(Uu),Uub),Uuc))) ).

% ATP.lambda_562
tff(fact_10776_ATP_Olambda__563,axiom,
    ! [Uu: set_nat,Uua: fun_nat_fun_int_nat,Uub: fun_nat_fun_int_o,Uuc: int] : aa_int_nat(aa_fun442955204921438347nt_nat(aa_fun5585961000942689878nt_nat(aTP_Lamm_nd(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups708209901874060359at_nat(aa_int_fun_nat_nat(aTP_Lamm_nc(Uua),Uuc)),collect_nat(aa_int_fun_nat_o(aa_fun7393466555336238524_nat_o(aTP_Lamm_be(Uu),Uub),Uuc))) ).

% ATP.lambda_563
tff(fact_10777_ATP_Olambda__564,axiom,
    ! [Uu: set_nat,Uua: fun_nat_fun_real_nat,Uub: fun_nat_fun_real_o,Uuc: real] : aa_real_nat(aa_fun1455215566758156043al_nat(aa_fun1180509458258223190al_nat(aTP_Lamm_na(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups708209901874060359at_nat(aa_real_fun_nat_nat(aTP_Lamm_my(Uua),Uuc)),collect_nat(aa_real_fun_nat_o(aa_fun7270066119382671164_nat_o(aTP_Lamm_mz(Uu),Uub),Uuc))) ).

% ATP.lambda_564
tff(fact_10778_ATP_Olambda__565,axiom,
    ! [Uu: set_complex,Uua: fun_co7370231399432144932at_int,Uub: fun_co5676208753217392844_nat_o,Uuc: nat] : aa_nat_int(aa_fun1349975166126329225at_int(aa_fun7503395680861046006at_int(aTP_Lamm_mr(Uu),Uua),Uub),Uuc) = groups858564598930262913ex_int(aa_nat8436805559129492965ex_int(aTP_Lamm_mq(Uua),Uuc),collect_complex(aa_nat_fun_complex_o(aa_fun4610599729497257276plex_o(aTP_Lamm_bj(Uu),Uub),Uuc))) ).

% ATP.lambda_565
tff(fact_10779_ATP_Olambda__566,axiom,
    ! [Uu: set_real,Uua: fun_real_fun_nat_int,Uub: fun_real_fun_nat_o,Uuc: nat] : aa_nat_int(aa_fun3188195345082886411at_int(aa_fun8006801000864960506at_int(aTP_Lamm_mo(Uu),Uua),Uub),Uuc) = groups4694064378042380927al_int(aa_nat_fun_real_int(aTP_Lamm_mn(Uua),Uuc),collect_real(aa_nat_fun_real_o(aa_fun6917832400337718588real_o(aTP_Lamm_ba(Uu),Uub),Uuc))) ).

% ATP.lambda_566
tff(fact_10780_ATP_Olambda__567,axiom,
    ! [Uu: set_complex,Uua: fun_co636942507509095880at_nat,Uub: fun_co5676208753217392844_nat_o,Uuc: nat] : aa_nat_nat(aa_fun5527826185635525933at_nat(aa_fun4894236392271416310at_nat(aTP_Lamm_ml(Uu),Uua),Uub),Uuc) = groups861055069439313189ex_nat(aa_nat3391284541783913865ex_nat(aTP_Lamm_hi(Uua),Uuc),collect_complex(aa_nat_fun_complex_o(aa_fun4610599729497257276plex_o(aTP_Lamm_bj(Uu),Uub),Uuc))) ).

% ATP.lambda_567
tff(fact_10781_ATP_Olambda__568,axiom,
    ! [Uu: set_int,Uua: fun_int_fun_nat_nat,Uub: fun_int_fun_nat_o,Uuc: nat] : aa_nat_nat(aa_fun3651443648018535471at_nat(aa_fun6116586148035635706at_nat(aTP_Lamm_mj(Uu),Uua),Uub),Uuc) = groups1707563613775114915nt_nat(aa_nat_fun_int_nat(aTP_Lamm_he(Uua),Uuc),collect_int(aa_nat_fun_int_o(aa_fun5439836533389078972_int_o(aTP_Lamm_bg(Uu),Uub),Uuc))) ).

% ATP.lambda_568
tff(fact_10782_ATP_Olambda__569,axiom,
    ! [Uu: set_real,Uua: fun_real_fun_nat_nat,Uub: fun_real_fun_nat_o,Uuc: nat] : aa_nat_nat(aa_fun7366046364592083119at_nat(aa_fun7718511627822915834at_nat(aTP_Lamm_mh(Uu),Uua),Uub),Uuc) = groups4696554848551431203al_nat(aa_nat_fun_real_nat(aTP_Lamm_ha(Uua),Uuc),collect_real(aa_nat_fun_real_o(aa_fun6917832400337718588real_o(aTP_Lamm_ba(Uu),Uub),Uuc))) ).

% ATP.lambda_569
tff(fact_10783_ATP_Olambda__570,axiom,
    ! [Uu: set_complex,Uua: fun_co333780937270390912nt_int,Uub: fun_co3431063785546946928_int_o,Uuc: int] : aa_int_int(aa_fun2158622962795119753nt_int(aa_fun607866698265288346nt_int(aTP_Lamm_mv(Uu),Uua),Uub),Uuc) = groups858564598930262913ex_int(aa_int6126987627036542017ex_int(aTP_Lamm_gj(Uua),Uuc),collect_complex(aa_int_fun_complex_o(aa_fun7279530946696969788plex_o(aTP_Lamm_gk(Uu),Uub),Uuc))) ).

% ATP.lambda_570
tff(fact_10784_ATP_Olambda__571,axiom,
    ! [Uu: set_real,Uua: fun_real_fun_int_int,Uub: fun_real_fun_int_o,Uuc: int] : aa_int_int(aa_fun3996843141751676939nt_int(aa_fun129049606478405534nt_int(aTP_Lamm_mt(Uu),Uua),Uub),Uuc) = groups4694064378042380927al_int(aa_int_fun_real_int(aTP_Lamm_gb(Uua),Uuc),collect_real(aa_int_fun_real_o(aa_fun2391495421969208892real_o(aTP_Lamm_bb(Uu),Uub),Uuc))) ).

% ATP.lambda_571
tff(fact_10785_ATP_Olambda__572,axiom,
    ! [Uu: set_real,Uua: fun_re4238438383274954274t_real,Uub: fun_real_fun_nat_o,Uuc: nat] : aa_nat_real(aa_fun6247686994378451851t_real(aa_fun7274880274332377722t_real(aTP_Lamm_hm(Uu),Uua),Uub),Uuc) = groups8097168146408367636l_real(aa_nat_fun_real_real(aTP_Lamm_hl(Uua),Uuc),collect_real(aa_nat_fun_real_o(aa_fun6917832400337718588real_o(aTP_Lamm_ba(Uu),Uub),Uuc))) ).

% ATP.lambda_572
tff(fact_10786_ATP_Olambda__573,axiom,
    ! [Uu: set_complex,Uua: fun_co636942507509095880at_nat,Uub: fun_co5676208753217392844_nat_o,Uuc: nat] : aa_nat_nat(aa_fun5527826185635525933at_nat(aa_fun4894236392271416310at_nat(aTP_Lamm_hj(Uu),Uua),Uub),Uuc) = groups5693394587270226106ex_nat(aa_nat3391284541783913865ex_nat(aTP_Lamm_hi(Uua),Uuc),collect_complex(aa_nat_fun_complex_o(aa_fun4610599729497257276plex_o(aTP_Lamm_bj(Uu),Uub),Uuc))) ).

% ATP.lambda_573
tff(fact_10787_ATP_Olambda__574,axiom,
    ! [Uu: set_int,Uua: fun_int_fun_nat_nat,Uub: fun_int_fun_nat_o,Uuc: nat] : aa_nat_nat(aa_fun3651443648018535471at_nat(aa_fun6116586148035635706at_nat(aTP_Lamm_hf(Uu),Uua),Uub),Uuc) = groups4541462559716669496nt_nat(aa_nat_fun_int_nat(aTP_Lamm_he(Uua),Uuc),collect_int(aa_nat_fun_int_o(aa_fun5439836533389078972_int_o(aTP_Lamm_bg(Uu),Uub),Uuc))) ).

% ATP.lambda_574
tff(fact_10788_ATP_Olambda__575,axiom,
    ! [Uu: set_real,Uua: fun_real_fun_nat_nat,Uub: fun_real_fun_nat_o,Uuc: nat] : aa_nat_nat(aa_fun7366046364592083119at_nat(aa_fun7718511627822915834at_nat(aTP_Lamm_hb(Uu),Uua),Uub),Uuc) = groups1935376822645274424al_nat(aa_nat_fun_real_nat(aTP_Lamm_ha(Uua),Uuc),collect_real(aa_nat_fun_real_o(aa_fun6917832400337718588real_o(aTP_Lamm_ba(Uu),Uub),Uuc))) ).

% ATP.lambda_575
tff(fact_10789_ATP_Olambda__576,axiom,
    ! [Uu: set_int,Uua: fun_in1307110748760623106omplex,Uub: fun_in5065666251602127472plex_o,Uuc: complex] : aa_complex_complex(aa_fun7450476157521870733omplex(aa_fun6156949752960786844omplex(aTP_Lamm_gx(Uu),Uua),Uub),Uuc) = groups3049146728041665814omplex(aa_com9156536935073639875omplex(aTP_Lamm_gw(Uua),Uuc),collect_int(aa_complex_fun_int_o(aa_fun7455058932602313788_int_o(aTP_Lamm_bi(Uu),Uub),Uuc))) ).

% ATP.lambda_576
tff(fact_10790_ATP_Olambda__577,axiom,
    ! [Uu: set_nat,Uua: fun_na1591161823130732966omplex,Uub: fun_na1554462459066888140plex_o,Uuc: complex] : aa_complex_complex(aa_fun1423577888295648489omplex(aa_fun601428455815558228omplex(aTP_Lamm_gt(Uu),Uua),Uub),Uuc) = aa_set_nat_complex(groups2073611262835488442omplex(aa_com8181001469867462503omplex(aTP_Lamm_gs(Uua),Uuc)),collect_nat(aa_complex_fun_nat_o(aa_fun1708641150486296380_nat_o(aTP_Lamm_bf(Uu),Uub),Uuc))) ).

% ATP.lambda_577
tff(fact_10791_ATP_Olambda__578,axiom,
    ! [Uu: set_real,Uua: fun_re7208826350674946562omplex,Uub: fun_re7319081385389557872plex_o,Uuc: complex] : aa_complex_complex(aa_fun9084337531551611917omplex(aa_fun6739043078600750748omplex(aTP_Lamm_gp(Uu),Uua),Uub),Uuc) = groups5754745047067104278omplex(aa_com4766338864777664963omplex(aTP_Lamm_go(Uua),Uuc),collect_real(aa_com1951324360330529881real_o(aa_fun7692540709613205436real_o(aTP_Lamm_bc(Uu),Uub),Uuc))) ).

% ATP.lambda_578
tff(fact_10792_ATP_Olambda__579,axiom,
    ! [Uu: set_complex,Uua: fun_co333780937270390912nt_int,Uub: fun_co3431063785546946928_int_o,Uuc: int] : aa_int_int(aa_fun2158622962795119753nt_int(aa_fun607866698265288346nt_int(aTP_Lamm_gl(Uu),Uua),Uub),Uuc) = groups5690904116761175830ex_int(aa_int6126987627036542017ex_int(aTP_Lamm_gj(Uua),Uuc),collect_complex(aa_int_fun_complex_o(aa_fun7279530946696969788plex_o(aTP_Lamm_gk(Uu),Uub),Uuc))) ).

% ATP.lambda_579
tff(fact_10793_ATP_Olambda__580,axiom,
    ! [Uu: set_nat,Uua: fun_nat_fun_int_int,Uub: fun_nat_fun_int_o,Uuc: int] : aa_int_int(aa_fun5488476222267017447nt_int(aa_fun3345280138976288598nt_int(aTP_Lamm_gg(Uu),Uua),Uub),Uuc) = groups3539618377306564664at_int(aa_int_fun_nat_int(aTP_Lamm_gf(Uua),Uuc),collect_nat(aa_int_fun_nat_o(aa_fun7393466555336238524_nat_o(aTP_Lamm_be(Uu),Uub),Uuc))) ).

% ATP.lambda_580
tff(fact_10794_ATP_Olambda__581,axiom,
    ! [Uu: set_real,Uua: fun_real_fun_int_int,Uub: fun_real_fun_int_o,Uuc: int] : aa_int_int(aa_fun3996843141751676939nt_int(aa_fun129049606478405534nt_int(aTP_Lamm_gc(Uu),Uua),Uub),Uuc) = groups1932886352136224148al_int(aa_int_fun_real_int(aTP_Lamm_gb(Uua),Uuc),collect_real(aa_int_fun_real_o(aa_fun2391495421969208892real_o(aTP_Lamm_bb(Uu),Uub),Uuc))) ).

% ATP.lambda_581
tff(fact_10795_ATP_Olambda__582,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_up(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_uo(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_582
tff(fact_10796_ATP_Olambda__583,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_ud(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_fun3023522663955195517t_real(aTP_Lamm_uc(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_583
tff(fact_10797_ATP_Olambda__584,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_ub(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_ua(Uu),Uua),Uuc),aa_nat_set_nat(set_ord_atMost_nat,Uuc))),aa_nat_int(power_power_int(Uub),Uuc)) ).

% ATP.lambda_584
tff(fact_10798_ATP_Olambda__585,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_tz(Uu),Uua),Uub),Uuc) = aa_rat_rat(times_times_rat(groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_fun2663889555218712533at_rat(aTP_Lamm_ty(Uu),Uua),Uuc),aa_nat_set_nat(set_ord_atMost_nat,Uuc))),aa_nat_rat(power_power_rat(Uub),Uuc)) ).

% ATP.lambda_585
tff(fact_10799_ATP_Olambda__586,axiom,
    ! [Uu: fun_nat_complex,Uua: fun_nat_complex,Uub: complex,Uuc: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_fun3721018923249493343omplex(aTP_Lamm_tx(Uu),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(aa_set_nat_complex(groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_fun5379041630539201665omplex(aTP_Lamm_tw(Uu),Uua),Uuc)),aa_nat_set_nat(set_ord_atMost_nat,Uuc))),aa_nat_complex(power_power_complex(Uub),Uuc)) ).

% ATP.lambda_586
tff(fact_10800_ATP_Olambda__587,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_ze(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(suc,Uua)),Uuc)),zero_zero_real)),aa_nat_real(semiri2265585572941072030t_real,Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uub),Uuc)) ).

% ATP.lambda_587
tff(fact_10801_ATP_Olambda__588,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_zc(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uuc)),zero_zero_real)),aa_nat_real(semiri2265585572941072030t_real,Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uub),Uuc)) ).

% ATP.lambda_588
tff(fact_10802_ATP_Olambda__589,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: real,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_zb(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Uu,Uuc),Uua)),aa_nat_real(semiri2265585572941072030t_real,Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Uub),Uua)),Uuc)) ).

% ATP.lambda_589
tff(fact_10803_ATP_Olambda__590,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: real,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_za(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Uu,Uuc),Uub)),aa_nat_real(semiri2265585572941072030t_real,Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Uua),Uub)),Uuc)) ).

% ATP.lambda_590
tff(fact_10804_ATP_Olambda__591,axiom,
    ! [Uu: complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_pw(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_591
tff(fact_10805_ATP_Olambda__592,axiom,
    ! [Uu: real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_qa(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uua),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_592
tff(fact_10806_ATP_Olambda__593,axiom,
    ! [Uu: rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_py(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_593
tff(fact_10807_ATP_Olambda__594,axiom,
    ! [Uu: complex,Uua: nat,Uub: complex,Uuc: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_ts(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_594
tff(fact_10808_ATP_Olambda__595,axiom,
    ! [Uu: real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_tv(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Uub),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),aa_nat_nat(suc,Uuc)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uuc)) ).

% ATP.lambda_595
tff(fact_10809_ATP_Olambda__596,axiom,
    ! [Uu: rat,Uua: nat,Uub: rat,Uuc: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aa_nat298952458717073160at_rat(aTP_Lamm_tt(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_596
tff(fact_10810_ATP_Olambda__597,axiom,
    ! [Uu: int,Uua: nat,Uub: int,Uuc: nat] : aa_nat_int(aa_int_fun_nat_int(aa_nat6704147566042551216at_int(aTP_Lamm_tu(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_597
tff(fact_10811_ATP_Olambda__598,axiom,
    ! [Uu: set_complex,Uua: fun_re7319081385389557872plex_o,Uub: real,Uuc: complex] :
      ( aa_complex_o(aa_rea2382570592068293209plex_o(aa_fun789424914597169340plex_o(aTP_Lamm_gm(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_complex_o(member_complex(Uuc),Uu)
        & aa_complex_o(aa_rea2382570592068293209plex_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_598
tff(fact_10812_ATP_Olambda__599,axiom,
    ! [Uu: set_complex,Uua: fun_na1554462459066888140plex_o,Uub: nat,Uuc: complex] :
      ( aa_complex_o(aa_nat_fun_complex_o(aa_fun6810266893190567484plex_o(aTP_Lamm_gq(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_complex_o(member_complex(Uuc),Uu)
        & aa_complex_o(aa_nat_fun_complex_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_599
tff(fact_10813_ATP_Olambda__600,axiom,
    ! [Uu: set_complex,Uua: fun_in5065666251602127472plex_o,Uub: int,Uuc: complex] :
      ( aa_complex_o(aa_int_fun_complex_o(aa_fun9089661398657494332plex_o(aTP_Lamm_gu(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_complex_o(member_complex(Uuc),Uu)
        & aa_complex_o(aa_int_fun_complex_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_600
tff(fact_10814_ATP_Olambda__601,axiom,
    ! [Uu: set_real,Uua: fun_nat_fun_real_o,Uub: nat,Uuc: real] :
      ( aa_real_o(aa_nat_fun_real_o(aa_fun1719600354486425148real_o(aTP_Lamm_mw(Uu),Uua),Uub),Uuc)
    <=> ( member_real(Uuc,Uu)
        & aa_real_o(aa_nat_fun_real_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_601
tff(fact_10815_ATP_Olambda__602,axiom,
    ! [Uu: set_nat,Uua: fun_co5676208753217392844_nat_o,Uub: complex,Uuc: nat] :
      ( aa_nat_o(aa_complex_fun_nat_o(aa_fun8732346023647761980_nat_o(aTP_Lamm_hg(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu)
        & aa_nat_o(aa_complex_fun_nat_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_602
tff(fact_10816_ATP_Olambda__603,axiom,
    ! [Uu: set_nat,Uua: fun_real_fun_nat_o,Uub: real,Uuc: nat] :
      ( aa_nat_o(aa_real_fun_nat_o(aa_fun3244926128379188796_nat_o(aTP_Lamm_gy(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu)
        & aa_nat_o(aa_real_fun_nat_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_603
tff(fact_10817_ATP_Olambda__604,axiom,
    ! [Uu: set_nat,Uua: fun_int_fun_nat_o,Uub: int,Uuc: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(aa_fun16395659681819580_nat_o(aTP_Lamm_hc(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu)
        & aa_nat_o(aa_int_fun_nat_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_604
tff(fact_10818_ATP_Olambda__605,axiom,
    ! [Uu: set_int,Uua: fun_co3431063785546946928_int_o,Uub: complex,Uuc: int] :
      ( aa_int_o(aa_complex_fun_int_o(aa_fun5644928480641789244_int_o(aTP_Lamm_gh(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_int_o(member_int(Uuc),Uu)
        & aa_int_o(aa_complex_fun_int_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_605
tff(fact_10819_ATP_Olambda__606,axiom,
    ! [Uu: set_int,Uua: fun_real_fun_int_o,Uub: real,Uuc: int] :
      ( aa_int_o(aa_real_fun_int_o(aa_fun2967211649596192572_int_o(aTP_Lamm_fz(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_int_o(member_int(Uuc),Uu)
        & aa_int_o(aa_real_fun_int_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_606
tff(fact_10820_ATP_Olambda__607,axiom,
    ! [Uu: set_int,Uua: fun_nat_fun_int_o,Uub: nat,Uuc: int] :
      ( aa_int_o(aa_nat_fun_int_o(aa_fun3593535392188722108_int_o(aTP_Lamm_gd(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_int_o(member_int(Uuc),Uu)
        & aa_int_o(aa_nat_fun_int_o(Uua,Uub),Uuc) ) ) ).

% ATP.lambda_607
tff(fact_10821_ATP_Olambda__608,axiom,
    ! [Uu: set_complex,Uua: fun_co5676208753217392844_nat_o,Uub: nat,Uuc: complex] :
      ( aa_complex_o(aa_nat_fun_complex_o(aa_fun4610599729497257276plex_o(aTP_Lamm_bj(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_complex_o(member_complex(Uuc),Uu)
        & aa_nat_o(aa_complex_fun_nat_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_608
tff(fact_10822_ATP_Olambda__609,axiom,
    ! [Uu: set_complex,Uua: fun_co3431063785546946928_int_o,Uub: int,Uuc: complex] :
      ( aa_complex_o(aa_int_fun_complex_o(aa_fun7279530946696969788plex_o(aTP_Lamm_gk(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_complex_o(member_complex(Uuc),Uu)
        & aa_int_o(aa_complex_fun_int_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_609
tff(fact_10823_ATP_Olambda__610,axiom,
    ! [Uu: set_real,Uua: fun_re7319081385389557872plex_o,Uub: complex,Uuc: real] :
      ( aa_real_o(aa_com1951324360330529881real_o(aa_fun7692540709613205436real_o(aTP_Lamm_bc(Uu),Uua),Uub),Uuc)
    <=> ( member_real(Uuc,Uu)
        & aa_complex_o(aa_rea2382570592068293209plex_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_610
tff(fact_10824_ATP_Olambda__611,axiom,
    ! [Uu: set_real,Uua: fun_real_fun_nat_o,Uub: nat,Uuc: real] :
      ( aa_real_o(aa_nat_fun_real_o(aa_fun6917832400337718588real_o(aTP_Lamm_ba(Uu),Uua),Uub),Uuc)
    <=> ( member_real(Uuc,Uu)
        & aa_nat_o(aa_real_fun_nat_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_611
tff(fact_10825_ATP_Olambda__612,axiom,
    ! [Uu: set_real,Uua: fun_real_fun_int_o,Uub: int,Uuc: real] :
      ( aa_real_o(aa_int_fun_real_o(aa_fun2391495421969208892real_o(aTP_Lamm_bb(Uu),Uua),Uub),Uuc)
    <=> ( member_real(Uuc,Uu)
        & aa_int_o(aa_real_fun_int_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_612
tff(fact_10826_ATP_Olambda__613,axiom,
    ! [Uu: set_nat,Uua: fun_na1554462459066888140plex_o,Uub: complex,Uuc: nat] :
      ( aa_nat_o(aa_complex_fun_nat_o(aa_fun1708641150486296380_nat_o(aTP_Lamm_bf(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu)
        & aa_complex_o(aa_nat_fun_complex_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_613
tff(fact_10827_ATP_Olambda__614,axiom,
    ! [Uu: set_nat,Uua: fun_nat_fun_real_o,Uub: real,Uuc: nat] :
      ( aa_nat_o(aa_real_fun_nat_o(aa_fun7270066119382671164_nat_o(aTP_Lamm_mz(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu)
        & aa_real_o(aa_nat_fun_real_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_614
tff(fact_10828_ATP_Olambda__615,axiom,
    ! [Uu: set_nat,Uua: fun_nat_fun_nat_o,Uub: nat,Uuc: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aa_fun6699960423209655484_nat_o(aTP_Lamm_bd(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_615
tff(fact_10829_ATP_Olambda__616,axiom,
    ! [Uu: set_nat,Uua: fun_nat_fun_int_o,Uub: int,Uuc: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(aa_fun7393466555336238524_nat_o(aTP_Lamm_be(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu)
        & aa_int_o(aa_nat_fun_int_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_616
tff(fact_10830_ATP_Olambda__617,axiom,
    ! [Uu: set_int,Uua: fun_in5065666251602127472plex_o,Uub: complex,Uuc: int] :
      ( aa_int_o(aa_complex_fun_int_o(aa_fun7455058932602313788_int_o(aTP_Lamm_bi(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_int_o(member_int(Uuc),Uu)
        & aa_complex_o(aa_int_fun_complex_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_617
tff(fact_10831_ATP_Olambda__618,axiom,
    ! [Uu: set_int,Uua: fun_int_fun_nat_o,Uub: nat,Uuc: int] :
      ( aa_int_o(aa_nat_fun_int_o(aa_fun5439836533389078972_int_o(aTP_Lamm_bg(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_int_o(member_int(Uuc),Uu)
        & aa_nat_o(aa_int_fun_nat_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_618
tff(fact_10832_ATP_Olambda__619,axiom,
    ! [Uu: set_int,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_set_int_o(member_int(Uuc),Uu)
        & aa_int_o(aa_int_fun_int_o(Uua,Uuc),Uub) ) ) ).

% ATP.lambda_619
tff(fact_10833_ATP_Olambda__620,axiom,
    ! [Uu: complex,Uua: nat,Uub: complex,Uuc: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_nat2301396582920332468omplex(aTP_Lamm_to(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_620
tff(fact_10834_ATP_Olambda__621,axiom,
    ! [Uu: real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_tr(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_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_621
tff(fact_10835_ATP_Olambda__622,axiom,
    ! [Uu: rat,Uua: nat,Uub: rat,Uuc: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aa_nat298952458717073160at_rat(aTP_Lamm_tp(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_622
tff(fact_10836_ATP_Olambda__623,axiom,
    ! [Uu: int,Uua: nat,Uub: int,Uuc: nat] : aa_nat_int(aa_int_fun_nat_int(aa_nat6704147566042551216at_int(aTP_Lamm_tq(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_623
tff(fact_10837_ATP_Olambda__624,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] :
      ( aa_int_o(aa_int_fun_int_o(aa_int2464330468366073224_int_o(aTP_Lamm_abc(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_624
tff(fact_10838_ATP_Olambda__625,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aa_nat8314798964848609180_nat_o(aTP_Lamm_aav(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_625
tff(fact_10839_ATP_Olambda__626,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] :
      ( aa_int_o(aa_int_fun_int_o(aa_int2464330468366073224_int_o(aTP_Lamm_abe(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_626
tff(fact_10840_ATP_Olambda__627,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aa_nat8314798964848609180_nat_o(aTP_Lamm_aax(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_627
tff(fact_10841_ATP_Olambda__628,axiom,
    ! [Uu: set_complex,Uua: fun_complex_complex,Uub: fun_complex_complex,Uuc: complex] :
      ( aa_complex_o(aa_fun573566726631320556plex_o(aa_fun4378733901284544684plex_o(aTP_Lamm_dl(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_complex_o(member_complex(Uuc),Uu)
        & ( aa_complex_complex(times_times_complex(aa_complex_complex(Uua,Uuc)),aa_complex_complex(Uub,Uuc)) != one_one_complex ) ) ) ).

% ATP.lambda_628
tff(fact_10842_ATP_Olambda__629,axiom,
    ! [Uu: set_complex,Uua: fun_complex_real,Uub: fun_complex_real,Uuc: complex] :
      ( aa_complex_o(aa_fun6195079535267300846plex_o(aa_fun805843229300662576plex_o(aTP_Lamm_dt(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_complex_o(member_complex(Uuc),Uu)
        & ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(Uua,Uuc)),aa_complex_real(Uub,Uuc)) != one_one_real ) ) ) ).

% ATP.lambda_629
tff(fact_10843_ATP_Olambda__630,axiom,
    ! [Uu: set_real,Uua: fun_real_complex,Uub: fun_real_complex,Uuc: real] :
      ( aa_real_o(aa_fun5218151893725926128real_o(aa_fun1467772539989157170real_o(aTP_Lamm_df(Uu),Uua),Uub),Uuc)
    <=> ( member_real(Uuc,Uu)
        & ( aa_complex_complex(times_times_complex(aa_real_complex(Uua,Uuc)),aa_real_complex(Uub,Uuc)) != one_one_complex ) ) ) ).

% ATP.lambda_630
tff(fact_10844_ATP_Olambda__631,axiom,
    ! [Uu: set_real,Uua: fun_real_real,Uub: fun_real_real,Uuc: real] :
      ( aa_real_o(aa_fun2895569901277098738real_o(aa_fun3693225987673163702real_o(aTP_Lamm_dn(Uu),Uua),Uub),Uuc)
    <=> ( member_real(Uuc,Uu)
        & ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(Uua,Uuc)),aa_real_real(Uub,Uuc)) != one_one_real ) ) ) ).

% ATP.lambda_631
tff(fact_10845_ATP_Olambda__632,axiom,
    ! [Uu: set_real,Uua: fun_real_rat,Uub: fun_real_rat,Uuc: real] :
      ( aa_real_o(aa_fun3209485225100452678real_o(aa_fun8459995508950063326real_o(aTP_Lamm_dv(Uu),Uua),Uub),Uuc)
    <=> ( member_real(Uuc,Uu)
        & ( aa_rat_rat(times_times_rat(aa_real_rat(Uua,Uuc)),aa_real_rat(Uub,Uuc)) != one_one_rat ) ) ) ).

% ATP.lambda_632
tff(fact_10846_ATP_Olambda__633,axiom,
    ! [Uu: set_nat,Uua: fun_nat_complex,Uub: fun_nat_complex,Uuc: nat] :
      ( aa_nat_o(aa_fun3019658233935452328_nat_o(aa_fun894001447060918982_nat_o(aTP_Lamm_dh(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu)
        & ( aa_complex_complex(times_times_complex(aa_nat_complex(Uua,Uuc)),aa_nat_complex(Uub,Uuc)) != one_one_complex ) ) ) ).

% ATP.lambda_633
tff(fact_10847_ATP_Olambda__634,axiom,
    ! [Uu: set_nat,Uua: fun_nat_real,Uub: fun_nat_real,Uuc: nat] :
      ( aa_nat_o(aa_fun6456149469358400426_nat_o(aa_fun5464864432877286986_nat_o(aTP_Lamm_dp(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu)
        & ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uua,Uuc)),aa_nat_real(Uub,Uuc)) != one_one_real ) ) ) ).

% ATP.lambda_634
tff(fact_10848_ATP_Olambda__635,axiom,
    ! [Uu: set_nat,Uua: fun_nat_rat,Uub: fun_nat_rat,Uuc: nat] :
      ( aa_nat_o(aa_fun1118977351124560894_nat_o(aa_fun8314661095172591474_nat_o(aTP_Lamm_dx(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu)
        & ( aa_rat_rat(times_times_rat(aa_nat_rat(Uua,Uuc)),aa_nat_rat(Uub,Uuc)) != one_one_rat ) ) ) ).

% ATP.lambda_635
tff(fact_10849_ATP_Olambda__636,axiom,
    ! [Uu: set_int,Uua: fun_int_complex,Uub: fun_int_complex,Uuc: int] :
      ( aa_int_o(aa_fun5261878201068379632_int_o(aa_fun2392861216585087154_int_o(aTP_Lamm_dj(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_int_o(member_int(Uuc),Uu)
        & ( aa_complex_complex(times_times_complex(aa_int_complex(Uua,Uuc)),aa_int_complex(Uub,Uuc)) != one_one_complex ) ) ) ).

% ATP.lambda_636
tff(fact_10850_ATP_Olambda__637,axiom,
    ! [Uu: set_int,Uua: fun_int_real,Uub: fun_int_real,Uuc: int] :
      ( aa_int_o(aa_fun4444401208693448434_int_o(aa_fun135101165924023862_int_o(aTP_Lamm_dr(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_int_o(member_int(Uuc),Uu)
        & ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(Uua,Uuc)),aa_int_real(Uub,Uuc)) != one_one_real ) ) ) ).

% ATP.lambda_637
tff(fact_10851_ATP_Olambda__638,axiom,
    ! [Uu: set_complex,Uua: fun_complex_real,Uub: fun_complex_real,Uuc: complex] :
      ( aa_complex_o(aa_fun6195079535267300846plex_o(aa_fun805843229300662576plex_o(aTP_Lamm_ef(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_complex_o(member_complex(Uuc),Uu)
        & ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(Uua,Uuc)),aa_complex_real(Uub,Uuc)) != zero_zero_real ) ) ) ).

% ATP.lambda_638
tff(fact_10852_ATP_Olambda__639,axiom,
    ! [Uu: set_complex,Uua: fun_complex_rat,Uub: fun_complex_rat,Uuc: complex] :
      ( aa_complex_o(aa_fun5099116602557847362plex_o(aa_fun4586712283274990936plex_o(aTP_Lamm_en(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_complex_o(member_complex(Uuc),Uu)
        & ( aa_rat_rat(plus_plus_rat(aa_complex_rat(Uua,Uuc)),aa_complex_rat(Uub,Uuc)) != zero_zero_rat ) ) ) ).

% ATP.lambda_639
tff(fact_10853_ATP_Olambda__640,axiom,
    ! [Uu: set_real,Uua: fun_real_real,Uub: fun_real_real,Uuc: real] :
      ( aa_real_o(aa_fun2895569901277098738real_o(aa_fun3693225987673163702real_o(aTP_Lamm_dz(Uu),Uua),Uub),Uuc)
    <=> ( member_real(Uuc,Uu)
        & ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(Uua,Uuc)),aa_real_real(Uub,Uuc)) != zero_zero_real ) ) ) ).

% ATP.lambda_640
tff(fact_10854_ATP_Olambda__641,axiom,
    ! [Uu: set_real,Uua: fun_real_rat,Uub: fun_real_rat,Uuc: real] :
      ( aa_real_o(aa_fun3209485225100452678real_o(aa_fun8459995508950063326real_o(aTP_Lamm_eh(Uu),Uua),Uub),Uuc)
    <=> ( member_real(Uuc,Uu)
        & ( aa_rat_rat(plus_plus_rat(aa_real_rat(Uua,Uuc)),aa_real_rat(Uub,Uuc)) != zero_zero_rat ) ) ) ).

% ATP.lambda_641
tff(fact_10855_ATP_Olambda__642,axiom,
    ! [Uu: set_real,Uua: fun_real_nat,Uub: fun_real_nat,Uuc: real] :
      ( aa_real_o(aa_fun7775777430665565006real_o(aa_fun1158036852828689646real_o(aTP_Lamm_ep(Uu),Uua),Uub),Uuc)
    <=> ( member_real(Uuc,Uu)
        & ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_real_nat(Uua,Uuc)),aa_real_nat(Uub,Uuc)) != zero_zero_nat ) ) ) ).

% ATP.lambda_642
tff(fact_10856_ATP_Olambda__643,axiom,
    ! [Uu: set_nat,Uua: fun_nat_real,Uub: fun_nat_real,Uuc: nat] :
      ( aa_nat_o(aa_fun6456149469358400426_nat_o(aa_fun5464864432877286986_nat_o(aTP_Lamm_eb(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu)
        & ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(Uua,Uuc)),aa_nat_real(Uub,Uuc)) != zero_zero_real ) ) ) ).

% ATP.lambda_643
tff(fact_10857_ATP_Olambda__644,axiom,
    ! [Uu: set_nat,Uua: fun_nat_rat,Uub: fun_nat_rat,Uuc: nat] :
      ( aa_nat_o(aa_fun1118977351124560894_nat_o(aa_fun8314661095172591474_nat_o(aTP_Lamm_ej(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu)
        & ( aa_rat_rat(plus_plus_rat(aa_nat_rat(Uua,Uuc)),aa_nat_rat(Uub,Uuc)) != zero_zero_rat ) ) ) ).

% ATP.lambda_644
tff(fact_10858_ATP_Olambda__645,axiom,
    ! [Uu: set_nat,Uua: fun_nat_nat,Uub: fun_nat_nat,Uuc: nat] :
      ( aa_nat_o(aa_fun8673969331427483654_nat_o(aa_fun6348600740168882050_nat_o(aTP_Lamm_er(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_nat_o(member_nat(Uuc),Uu)
        & ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(Uua,Uuc)),aa_nat_nat(Uub,Uuc)) != zero_zero_nat ) ) ) ).

% ATP.lambda_645
tff(fact_10859_ATP_Olambda__646,axiom,
    ! [Uu: set_int,Uua: fun_int_real,Uub: fun_int_real,Uuc: int] :
      ( aa_int_o(aa_fun4444401208693448434_int_o(aa_fun135101165924023862_int_o(aTP_Lamm_ed(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_int_o(member_int(Uuc),Uu)
        & ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(Uua,Uuc)),aa_int_real(Uub,Uuc)) != zero_zero_real ) ) ) ).

% ATP.lambda_646
tff(fact_10860_ATP_Olambda__647,axiom,
    ! [Uu: set_int,Uua: fun_int_rat,Uub: fun_int_rat,Uuc: int] :
      ( aa_int_o(aa_fun7094158214180239686_int_o(aa_fun1445893531507182942_int_o(aTP_Lamm_el(Uu),Uua),Uub),Uuc)
    <=> ( aa_set_int_o(member_int(Uuc),Uu)
        & ( aa_rat_rat(plus_plus_rat(aa_int_rat(Uua,Uuc)),aa_int_rat(Uub,Uuc)) != zero_zero_rat ) ) ) ).

% ATP.lambda_647
tff(fact_10861_ATP_Olambda__648,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_sy(Uu),Uua),Uub),Uuc) = aa_complex_complex(times_times_complex(aa_nat_complex(Uu,Uuc)),aa_nat_complex(power_power_complex(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),Uub)),one_one_nat))) ).

% ATP.lambda_648
tff(fact_10862_ATP_Olambda__649,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_te(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_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,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),Uub)),one_one_nat))) ).

% ATP.lambda_649
tff(fact_10863_ATP_Olambda__650,axiom,
    ! [Uu: fun_nat_rat,Uua: rat,Uub: nat,Uuc: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_ta(Uu),Uua),Uub),Uuc) = aa_rat_rat(times_times_rat(aa_nat_rat(Uu,Uuc)),aa_nat_rat(power_power_rat(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),Uub)),one_one_nat))) ).

% ATP.lambda_650
tff(fact_10864_ATP_Olambda__651,axiom,
    ! [Uu: fun_nat_int,Uua: int,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_tc(Uu),Uua),Uub),Uuc) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(Uu,Uuc)),aa_nat_int(power_power_int(Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uuc),Uub)),one_one_nat))) ).

% ATP.lambda_651
tff(fact_10865_ATP_Olambda__652,axiom,
    ! [Uu: fun_nat_complex,Uua: fun_nat_complex,Uub: nat,Uuc: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_fun5379041630539201665omplex(aTP_Lamm_tw(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_652
tff(fact_10866_ATP_Olambda__653,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_uc(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,Uuc)),aa_nat_real(Uua,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_653
tff(fact_10867_ATP_Olambda__654,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_ty(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_654
tff(fact_10868_ATP_Olambda__655,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_uo(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_655
tff(fact_10869_ATP_Olambda__656,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_ua(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_656
tff(fact_10870_ATP_Olambda__657,axiom,
    ! [Uu: set_real,Uua: fun_nat_fun_real_nat,Uub: fun_nat_fun_real_o,Uuc: nat] : aa_nat_nat(aa_fun6477086565944264623at_nat(aa_fun888394026836741882at_nat(aTP_Lamm_mx(Uu),Uua),Uub),Uuc) = groups4696554848551431203al_nat(aa_nat_fun_real_nat(Uua,Uuc),collect_real(aa_nat_fun_real_o(aa_fun1719600354486425148real_o(aTP_Lamm_mw(Uu),Uub),Uuc))) ).

% ATP.lambda_657
tff(fact_10871_ATP_Olambda__658,axiom,
    ! [Uu: set_nat,Uua: fun_co636942507509095880at_nat,Uub: fun_co5676208753217392844_nat_o,Uuc: complex] : aa_complex_nat(aa_fun2265263913897893899ex_nat(aa_fun1127489917733235668ex_nat(aTP_Lamm_mk(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups708209901874060359at_nat(aa_com6383720667567721865at_nat(Uua,Uuc)),collect_nat(aa_complex_fun_nat_o(aa_fun8732346023647761980_nat_o(aTP_Lamm_hg(Uu),Uub),Uuc))) ).

% ATP.lambda_658
tff(fact_10872_ATP_Olambda__659,axiom,
    ! [Uu: set_nat,Uua: fun_co7370231399432144932at_int,Uub: fun_co5676208753217392844_nat_o,Uuc: complex] : aa_complex_int(aa_fun7310784931243472999ex_int(aa_fun7210183792663747796ex_int(aTP_Lamm_mp(Uu),Uua),Uub),Uuc) = groups705719431365010083at_int(aa_com2205869648058525157at_int(Uua,Uuc),collect_nat(aa_complex_fun_nat_o(aa_fun8732346023647761980_nat_o(aTP_Lamm_hg(Uu),Uub),Uuc))) ).

% ATP.lambda_659
tff(fact_10873_ATP_Olambda__660,axiom,
    ! [Uu: set_nat,Uua: fun_int_fun_nat_nat,Uub: fun_int_fun_nat_o,Uuc: int] : aa_int_nat(aa_fun4650797359919590027nt_nat(aa_fun8303507722728657494nt_nat(aTP_Lamm_mi(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups708209901874060359at_nat(aa_int_fun_nat_nat(Uua,Uuc)),collect_nat(aa_int_fun_nat_o(aa_fun16395659681819580_nat_o(aTP_Lamm_hc(Uu),Uub),Uuc))) ).

% ATP.lambda_660
tff(fact_10874_ATP_Olambda__661,axiom,
    ! [Uu: set_nat,Uua: fun_real_fun_nat_nat,Uub: fun_real_fun_nat_o,Uuc: real] : aa_real_nat(aa_fun2272639302901571595al_nat(aa_fun7039802393428226646al_nat(aTP_Lamm_mg(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups708209901874060359at_nat(aa_real_fun_nat_nat(Uua,Uuc)),collect_nat(aa_real_fun_nat_o(aa_fun3244926128379188796_nat_o(aTP_Lamm_gy(Uu),Uub),Uuc))) ).

% ATP.lambda_661
tff(fact_10875_ATP_Olambda__662,axiom,
    ! [Uu: set_nat,Uua: fun_real_fun_nat_int,Uub: fun_real_fun_nat_o,Uuc: real] : aa_real_int(aa_fun7318160320247150695al_int(aa_fun4230461673125498710al_int(aTP_Lamm_mm(Uu),Uua),Uub),Uuc) = groups705719431365010083at_int(aa_real_fun_nat_int(Uua,Uuc),collect_nat(aa_real_fun_nat_o(aa_fun3244926128379188796_nat_o(aTP_Lamm_gy(Uu),Uub),Uuc))) ).

% ATP.lambda_662
tff(fact_10876_ATP_Olambda__663,axiom,
    ! [Uu: set_complex,Uua: fun_na8457291176974909128ex_nat,Uub: fun_na1554462459066888140plex_o,Uuc: nat] : aa_nat_nat(aa_fun1031781741659083821at_nat(aa_fun7518049653624490486at_nat(aTP_Lamm_ne(Uu),Uua),Uub),Uuc) = groups861055069439313189ex_nat(aa_nat3391284541783913865ex_nat(Uua,Uuc),collect_complex(aa_nat_fun_complex_o(aa_fun6810266893190567484plex_o(aTP_Lamm_gq(Uu),Uub),Uuc))) ).

% ATP.lambda_663
tff(fact_10877_ATP_Olambda__664,axiom,
    ! [Uu: set_int,Uua: fun_co333780937270390912nt_int,Uub: fun_co3431063785546946928_int_o,Uuc: complex] : aa_complex_int(aa_fun5055871773053899019ex_int(aa_fun7361901416012365340ex_int(aTP_Lamm_mu(Uu),Uua),Uub),Uuc) = groups1705073143266064639nt_int(aa_com3205223359959579713nt_int(Uua,Uuc),collect_int(aa_complex_fun_int_o(aa_fun5644928480641789244_int_o(aTP_Lamm_gh(Uu),Uub),Uuc))) ).

% ATP.lambda_664
tff(fact_10878_ATP_Olambda__665,axiom,
    ! [Uu: set_int,Uua: fun_nat_fun_int_nat,Uub: fun_nat_fun_int_o,Uuc: nat] : aa_nat_nat(aa_fun8666973529875159599at_nat(aa_fun3399039426249668090at_nat(aTP_Lamm_nb(Uu),Uua),Uub),Uuc) = groups1707563613775114915nt_nat(aa_nat_fun_int_nat(Uua,Uuc),collect_int(aa_nat_fun_int_o(aa_fun3593535392188722108_int_o(aTP_Lamm_gd(Uu),Uub),Uuc))) ).

% ATP.lambda_665
tff(fact_10879_ATP_Olambda__666,axiom,
    ! [Uu: set_int,Uua: fun_real_fun_int_int,Uub: fun_real_fun_int_o,Uuc: real] : aa_real_int(aa_fun7555737461897725707al_int(aa_fun1679075639499870878al_int(aTP_Lamm_ms(Uu),Uua),Uub),Uuc) = groups1705073143266064639nt_int(aa_real_fun_int_int(Uua,Uuc),collect_int(aa_real_fun_int_o(aa_fun2967211649596192572_int_o(aTP_Lamm_fz(Uu),Uub),Uuc))) ).

% ATP.lambda_666
tff(fact_10880_ATP_Olambda__667,axiom,
    ! [Uu: set_nat,Uua: fun_co636942507509095880at_nat,Uub: fun_co5676208753217392844_nat_o,Uuc: complex] : aa_complex_nat(aa_fun2265263913897893899ex_nat(aa_fun1127489917733235668ex_nat(aTP_Lamm_hh(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_com6383720667567721865at_nat(Uua,Uuc)),collect_nat(aa_complex_fun_nat_o(aa_fun8732346023647761980_nat_o(aTP_Lamm_hg(Uu),Uub),Uuc))) ).

% ATP.lambda_667
tff(fact_10881_ATP_Olambda__668,axiom,
    ! [Uu: set_nat,Uua: fun_int_fun_nat_nat,Uub: fun_int_fun_nat_o,Uuc: int] : aa_int_nat(aa_fun4650797359919590027nt_nat(aa_fun8303507722728657494nt_nat(aTP_Lamm_hd(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_int_fun_nat_nat(Uua,Uuc)),collect_nat(aa_int_fun_nat_o(aa_fun16395659681819580_nat_o(aTP_Lamm_hc(Uu),Uub),Uuc))) ).

% ATP.lambda_668
tff(fact_10882_ATP_Olambda__669,axiom,
    ! [Uu: set_nat,Uua: fun_re4238438383274954274t_real,Uub: fun_real_fun_nat_o,Uuc: real] : aa_real_real(aa_fun6320755270648597735l_real(aa_fun6240130701872165846l_real(aTP_Lamm_hk(Uu),Uua),Uub),Uuc) = aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(Uua,Uuc)),collect_nat(aa_real_fun_nat_o(aa_fun3244926128379188796_nat_o(aTP_Lamm_gy(Uu),Uub),Uuc))) ).

% ATP.lambda_669
tff(fact_10883_ATP_Olambda__670,axiom,
    ! [Uu: set_nat,Uua: fun_real_fun_nat_nat,Uub: fun_real_fun_nat_o,Uuc: real] : aa_real_nat(aa_fun2272639302901571595al_nat(aa_fun7039802393428226646al_nat(aTP_Lamm_gz(Uu),Uua),Uub),Uuc) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_real_fun_nat_nat(Uua,Uuc)),collect_nat(aa_real_fun_nat_o(aa_fun3244926128379188796_nat_o(aTP_Lamm_gy(Uu),Uub),Uuc))) ).

% ATP.lambda_670
tff(fact_10884_ATP_Olambda__671,axiom,
    ! [Uu: set_complex,Uua: fun_in1307110748760623106omplex,Uub: fun_in5065666251602127472plex_o,Uuc: int] : aa_int_complex(aa_fun3141780957025892107omplex(aa_fun323909820831547930omplex(aTP_Lamm_gv(Uu),Uua),Uub),Uuc) = groups7754918857620584856omplex(aa_int3545933452204062147omplex(Uua,Uuc),collect_complex(aa_int_fun_complex_o(aa_fun9089661398657494332plex_o(aTP_Lamm_gu(Uu),Uub),Uuc))) ).

% ATP.lambda_671
tff(fact_10885_ATP_Olambda__672,axiom,
    ! [Uu: set_complex,Uua: fun_na1591161823130732966omplex,Uub: fun_na1554462459066888140plex_o,Uuc: nat] : aa_nat_complex(aa_fun4713049822795992587omplex(aa_fun3479084698764006518omplex(aTP_Lamm_gr(Uu),Uua),Uub),Uuc) = groups7754918857620584856omplex(aa_nat3829984526574172007omplex(Uua,Uuc),collect_complex(aa_nat_fun_complex_o(aa_fun6810266893190567484plex_o(aTP_Lamm_gq(Uu),Uub),Uuc))) ).

% ATP.lambda_672
tff(fact_10886_ATP_Olambda__673,axiom,
    ! [Uu: set_complex,Uua: fun_re7208826350674946562omplex,Uub: fun_re7319081385389557872plex_o,Uuc: real] : aa_real_complex(aa_fun401690246322682763omplex(aa_fun7501666206714588442omplex(aTP_Lamm_gn(Uu),Uua),Uub),Uuc) = groups7754918857620584856omplex(aa_rea2618301233319580611omplex(Uua,Uuc),collect_complex(aa_rea2382570592068293209plex_o(aa_fun789424914597169340plex_o(aTP_Lamm_gm(Uu),Uub),Uuc))) ).

% ATP.lambda_673
tff(fact_10887_ATP_Olambda__674,axiom,
    ! [Uu: set_int,Uua: fun_co333780937270390912nt_int,Uub: fun_co3431063785546946928_int_o,Uuc: complex] : aa_complex_int(aa_fun5055871773053899019ex_int(aa_fun7361901416012365340ex_int(aTP_Lamm_gi(Uu),Uua),Uub),Uuc) = groups4538972089207619220nt_int(aa_com3205223359959579713nt_int(Uua,Uuc),collect_int(aa_complex_fun_int_o(aa_fun5644928480641789244_int_o(aTP_Lamm_gh(Uu),Uub),Uuc))) ).

% ATP.lambda_674
tff(fact_10888_ATP_Olambda__675,axiom,
    ! [Uu: set_int,Uua: fun_nat_fun_int_int,Uub: fun_nat_fun_int_o,Uuc: nat] : aa_nat_int(aa_fun4489122510365962891at_int(aa_fun1158358564283266810at_int(aTP_Lamm_ge(Uu),Uua),Uub),Uuc) = groups4538972089207619220nt_int(aa_nat_fun_int_int(Uua,Uuc),collect_int(aa_nat_fun_int_o(aa_fun3593535392188722108_int_o(aTP_Lamm_gd(Uu),Uub),Uuc))) ).

% ATP.lambda_675
tff(fact_10889_ATP_Olambda__676,axiom,
    ! [Uu: set_int,Uua: fun_real_fun_int_int,Uub: fun_real_fun_int_o,Uuc: real] : aa_real_int(aa_fun7555737461897725707al_int(aa_fun1679075639499870878al_int(aTP_Lamm_ga(Uu),Uua),Uub),Uuc) = groups4538972089207619220nt_int(aa_real_fun_int_int(Uua,Uuc),collect_int(aa_real_fun_int_o(aa_fun2967211649596192572_int_o(aTP_Lamm_fz(Uu),Uub),Uuc))) ).

% ATP.lambda_676
tff(fact_10890_ATP_Olambda__677,axiom,
    ! [Uu: fun_nat_real,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_nat4085805313368488148t_real(aTP_Lamm_jo(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_677
tff(fact_10891_ATP_Olambda__678,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_jn(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_678
tff(fact_10892_ATP_Olambda__679,axiom,
    ! [Uu: fun_nat_int,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_nat1293065384339088724at_int(aTP_Lamm_oh(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_679
tff(fact_10893_ATP_Olambda__680,axiom,
    ! [Uu: nat,Uua: fun_nat_real,Uub: real,Uuc: real,Uud: nat] : aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aa_fun2548873334967081986t_real(aTP_Lamm_su(Uu),Uua),Uub),Uuc),Uud) = aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aa_rea6454571821648764761t_real(aTP_Lamm_st(Uua),Uub),Uuc),Uud)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uud))) ).

% ATP.lambda_680
tff(fact_10894_ATP_Olambda__681,axiom,
    ! [Uu: nat,Uua: fun_nat_int,Uub: int,Uuc: int,Uud: nat] : aa_nat_int(aa_int_fun_nat_int(aa_int5842867502692063756at_int(aa_fun7421131748584218370at_int(aTP_Lamm_ss(Uu),Uua),Uub),Uuc),Uud) = groups3539618377306564664at_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aa_int6823550873765277145at_int(aTP_Lamm_sr(Uua),Uub),Uuc),Uud),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uud))) ).

% ATP.lambda_681
tff(fact_10895_ATP_Olambda__682,axiom,
    ! [Uu: nat,Uua: fun_nat_rat,Uub: rat,Uuc: rat,Uud: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aa_fun8528462509625601970at_rat(aTP_Lamm_sq(Uu),Uua),Uub),Uuc),Uud) = groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aa_rat8984911954100663005at_rat(aTP_Lamm_sp(Uua),Uub),Uuc),Uud),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uud))) ).

% ATP.lambda_682
tff(fact_10896_ATP_Olambda__683,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: complex,Uuc: complex,Uud: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_fun5031706661949900810omplex(aTP_Lamm_so(Uu),Uua),Uub),Uuc),Uud) = aa_set_nat_complex(groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aa_com7713930848417683039omplex(aTP_Lamm_sn(Uua),Uub),Uuc),Uud)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uud))) ).

% ATP.lambda_683
tff(fact_10897_ATP_Olambda__684,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_zd(Uu),Uua),Uub),Uuc),Uud) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_nat_fun_real_real(Uua,Uuc),Uud)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_set_nat_real(groups6591440286371151544t_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_zc(Uua),Uuc),Uud)),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uuc)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Uub),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(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_684
tff(fact_10898_ATP_Olambda__685,axiom,
    ! [Uu: nat,Uua: fun_nat_real,Uub: real,Uuc: real,Uud: nat] : aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aa_fun2548873334967081986t_real(aTP_Lamm_tf(Uu),Uua),Uub),Uuc),Uud) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_set_nat_real(groups6591440286371151544t_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aTP_Lamm_te(Uua),Uuc),Uud)),set_or1269000886237332187st_nat(aa_nat_nat(suc,Uud),Uu))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uub),Uud)) ).

% ATP.lambda_685
tff(fact_10899_ATP_Olambda__686,axiom,
    ! [Uu: nat,Uua: fun_nat_int,Uub: int,Uuc: int,Uud: nat] : aa_nat_int(aa_int_fun_nat_int(aa_int5842867502692063756at_int(aa_fun7421131748584218370at_int(aTP_Lamm_td(Uu),Uua),Uub),Uuc),Uud) = aa_int_int(aa_int_fun_int_int(times_times_int,groups3539618377306564664at_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aTP_Lamm_tc(Uua),Uuc),Uud),set_or1269000886237332187st_nat(aa_nat_nat(suc,Uud),Uu))),aa_nat_int(power_power_int(Uub),Uud)) ).

% ATP.lambda_686
tff(fact_10900_ATP_Olambda__687,axiom,
    ! [Uu: nat,Uua: fun_nat_rat,Uub: rat,Uuc: rat,Uud: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aa_fun8528462509625601970at_rat(aTP_Lamm_tb(Uu),Uua),Uub),Uuc),Uud) = aa_rat_rat(times_times_rat(groups2906978787729119204at_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aTP_Lamm_ta(Uua),Uuc),Uud),set_or1269000886237332187st_nat(aa_nat_nat(suc,Uud),Uu))),aa_nat_rat(power_power_rat(Uub),Uud)) ).

% ATP.lambda_687
tff(fact_10901_ATP_Olambda__688,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: complex,Uuc: complex,Uud: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_fun5031706661949900810omplex(aTP_Lamm_sz(Uu),Uua),Uub),Uuc),Uud) = aa_complex_complex(times_times_complex(aa_set_nat_complex(groups2073611262835488442omplex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aTP_Lamm_sy(Uua),Uuc),Uud)),set_or1269000886237332187st_nat(aa_nat_nat(suc,Uud),Uu))),aa_nat_complex(power_power_complex(Uub),Uud)) ).

% ATP.lambda_688
tff(fact_10902_ATP_Olambda__689,axiom,
    ! [Uu: nat,Uua: complex,Uub: complex,Uuc: complex,Uud: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_com177695874307951165omplex(aTP_Lamm_wu(Uu),Uua),Uub),Uuc),Uud) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,Uud)),Uua)),one_one_complex)),Uud)),aa_nat_complex(power_power_complex(Uub),Uud))),aa_nat_complex(power_power_complex(aa_complex_complex(plus_plus_complex(Uub),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uud))) ).

% ATP.lambda_689
tff(fact_10903_ATP_Olambda__690,axiom,
    ! [Uu: nat,Uua: real,Uub: real,Uuc: real,Uud: nat] : aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aa_rea1729780184207679413t_real(aTP_Lamm_wy(Uu),Uua),Uub),Uuc),Uud) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,Uud)),Uua)),one_one_real)),Uud)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uub),Uud))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uub),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uud))) ).

% ATP.lambda_690
tff(fact_10904_ATP_Olambda__691,axiom,
    ! [Uu: nat,Uua: rat,Uub: rat,Uuc: rat,Uud: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aa_rat7467449789483521253at_rat(aTP_Lamm_ww(Uu),Uua),Uub),Uuc),Uud) = aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,Uud)),Uua)),one_one_rat)),Uud)),aa_nat_rat(power_power_rat(Uub),Uud))),aa_nat_rat(power_power_rat(aa_rat_rat(plus_plus_rat(Uub),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uud))) ).

% ATP.lambda_691
tff(fact_10905_ATP_Olambda__692,axiom,
    ! [Uu: nat,Uua: complex,Uub: complex,Uuc: complex,Uud: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aa_com8177640530696437650omplex(aa_com177695874307951165omplex(aTP_Lamm_wt(Uu),Uua),Uub),Uuc),Uud) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_nat_complex(gbinomial_complex(aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,Uu)),Uua)),Uud)),aa_nat_complex(power_power_complex(Uub),Uud))),aa_nat_complex(power_power_complex(Uuc),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uud))) ).

% ATP.lambda_692
tff(fact_10906_ATP_Olambda__693,axiom,
    ! [Uu: nat,Uua: real,Uub: real,Uuc: real,Uud: nat] : aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aa_rea1729780184207679413t_real(aTP_Lamm_wx(Uu),Uua),Uub),Uuc),Uud) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,Uu)),Uua)),Uud)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uub),Uud))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uuc),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uud))) ).

% ATP.lambda_693
tff(fact_10907_ATP_Olambda__694,axiom,
    ! [Uu: nat,Uua: rat,Uub: rat,Uuc: rat,Uud: nat] : aa_nat_rat(aa_rat_fun_nat_rat(aa_rat3146053865535587600at_rat(aa_rat7467449789483521253at_rat(aTP_Lamm_wv(Uu),Uua),Uub),Uuc),Uud) = aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_nat_rat(gbinomial_rat(aa_rat_rat(plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,Uu)),Uua)),Uud)),aa_nat_rat(power_power_rat(Uub),Uud))),aa_nat_rat(power_power_rat(Uuc),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uud))) ).

% ATP.lambda_694
tff(fact_10908_ATP_Olambda__695,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: complex,Uuc: nat,Uud: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aa_com4347742694490471604omplex(aa_com7713930848417683039omplex(aTP_Lamm_sn(Uu),Uua),Uub),Uuc),Uud) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_nat_complex(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uuc),Uud)),one_one_nat))),aa_nat_complex(power_power_complex(Uub),Uud))),aa_nat_complex(power_power_complex(Uua),Uuc)) ).

% ATP.lambda_695
tff(fact_10909_ATP_Olambda__696,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: real,Uuc: nat,Uud: nat] : aa_nat_real(aa_nat_fun_nat_real(aa_rea6853821027689443120t_real(aa_rea6454571821648764761t_real(aTP_Lamm_st(Uu),Uua),Uub),Uuc),Uud) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uuc),Uud)),one_one_nat))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uub),Uud))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uuc)) ).

% ATP.lambda_696
tff(fact_10910_ATP_Olambda__697,axiom,
    ! [Uu: fun_nat_rat,Uua: rat,Uub: rat,Uuc: nat,Uud: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aa_rat250263598611652360at_rat(aa_rat8984911954100663005at_rat(aTP_Lamm_sp(Uu),Uua),Uub),Uuc),Uud) = aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_nat_rat(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uuc),Uud)),one_one_nat))),aa_nat_rat(power_power_rat(Uub),Uud))),aa_nat_rat(power_power_rat(Uua),Uuc)) ).

% ATP.lambda_697
tff(fact_10911_ATP_Olambda__698,axiom,
    ! [Uu: fun_nat_int,Uua: int,Uub: int,Uuc: nat,Uud: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_int431785320988601264at_int(aa_int6823550873765277145at_int(aTP_Lamm_sr(Uu),Uua),Uub),Uuc),Uud) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uuc),Uud)),one_one_nat))),aa_nat_int(power_power_int(Uub),Uud))),aa_nat_int(power_power_int(Uua),Uuc)) ).

% ATP.lambda_698
tff(fact_10912_ATP_Olambda__699,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_zz(Uu),Uua),Uub),Uuc),Uud),Uue),Uuf)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uue),Uuf)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuf),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uu))
        & ! [I3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uud))
           => ( ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(Uua,I3)),X_12)
            <=> aa_nat_o(vEBT_V8194947554948674370ptions(Uub),I3) ) )
        & $ite(
            Uue = Uuf,
            ! [X4: vEBT_VEBT] :
              ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(Uua))
             => ~ ? [X7: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X4),X7) ),
            ( aa_nat_o(vEBT_V5917875025757280293ildren(Uuc,Uua),Uuf)
            & ! [X4: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uu))
               => ( aa_nat_o(vEBT_V5917875025757280293ildren(Uuc,Uua),X4)
                 => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uue),X4)
                    & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X4),Uuf) ) ) ) ) ) ) ) ).

% ATP.lambda_699
tff(fact_10913_ATP_Olambda__700,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_qx,Uu),Uua) = Uu ).

% ATP.lambda_700
tff(fact_10914_ATP_Olambda__701,axiom,
    ! [Uu: complex] : aa_complex_complex(aTP_Lamm_bz,Uu) = Uu ).

% ATP.lambda_701
tff(fact_10915_ATP_Olambda__702,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_ca,Uu) = Uu ).

% ATP.lambda_702
tff(fact_10916_ATP_Olambda__703,axiom,
    ! [Uu: rat] : aa_rat_rat(aTP_Lamm_cb,Uu) = Uu ).

% ATP.lambda_703
tff(fact_10917_ATP_Olambda__704,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_cc,Uu) = Uu ).

% ATP.lambda_704
tff(fact_10918_ATP_Olambda__705,axiom,
    ! [Uu: int] : aa_int_int(aTP_Lamm_cd,Uu) = Uu ).

% ATP.lambda_705
tff(fact_10919_ATP_Olambda__706,axiom,
    ! [Uu: complex] : aa_complex_complex(aTP_Lamm_fb,Uu) = zero_zero_complex ).

% ATP.lambda_706
tff(fact_10920_ATP_Olambda__707,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_bv,Uu) = zero_zero_real ).

% ATP.lambda_707
tff(fact_10921_ATP_Olambda__708,axiom,
    ! [Uu: rat] : aa_rat_rat(aTP_Lamm_bw,Uu) = zero_zero_rat ).

% ATP.lambda_708
tff(fact_10922_ATP_Olambda__709,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_fc,Uu) = zero_zero_real ).

% ATP.lambda_709
tff(fact_10923_ATP_Olambda__710,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_bx,Uu) = zero_zero_nat ).

% ATP.lambda_710
tff(fact_10924_ATP_Olambda__711,axiom,
    ! [Uu: nat] : aa_nat_int(aTP_Lamm_pb,Uu) = zero_zero_int ).

% ATP.lambda_711
tff(fact_10925_ATP_Olambda__712,axiom,
    ! [Uu: int] : aa_int_int(aTP_Lamm_by,Uu) = zero_zero_int ).

% ATP.lambda_712
tff(fact_10926_ATP_Olambda__713,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_lj,Uu) = one_one_nat ).

% ATP.lambda_713
tff(fact_10927_ATP_Olambda__714,axiom,
    ! [Uu: nat] : aa_nat_int(aTP_Lamm_lk,Uu) = one_one_int ).

% ATP.lambda_714
tff(fact_10928_ATP_Olambda__715,axiom,
    ! [Uu: int] : aa_int_int(aTP_Lamm_ll,Uu) = one_one_int ).

% ATP.lambda_715
tff(fact_10929_ATP_Olambda__716,axiom,
    ! [Uu: real] :
      ( aa_real_o(aTP_Lamm_bs,Uu)
    <=> $false ) ).

% ATP.lambda_716
tff(fact_10930_ATP_Olambda__717,axiom,
    ! [Uu: list_nat] :
      ( aa_list_nat_o(aTP_Lamm_bq,Uu)
    <=> $false ) ).

% ATP.lambda_717
tff(fact_10931_ATP_Olambda__718,axiom,
    ! [Uu: set_nat] :
      ( aa_set_nat_o(aTP_Lamm_br,Uu)
    <=> $false ) ).

% ATP.lambda_718
tff(fact_10932_ATP_Olambda__719,axiom,
    ! [Uu: nat] :
      ( aa_nat_o(aTP_Lamm_bt,Uu)
    <=> $false ) ).

% ATP.lambda_719
tff(fact_10933_ATP_Olambda__720,axiom,
    ! [Uu: int] :
      ( aa_int_o(aTP_Lamm_bu,Uu)
    <=> $false ) ).

% ATP.lambda_720
tff(fact_10934_ATP_Olambda__721,axiom,
    ! [Uu: nat] :
      ( aa_nat_o(aTP_Lamm_xx,Uu)
    <=> $true ) ).

% ATP.lambda_721

% Helper facts (11)
tff(help_fequal_2_1_fequal_001_Eo_T,axiom,
    ! [X9: $o,Y: $o] :
      ( ( (X9) != (Y) )
      | aa_o_o(aa_o_fun_o_o(fequal_o,(X9)),(Y)) ) ).

tff(help_fequal_1_1_fequal_001_Eo_T,axiom,
    ! [X9: $o,Y: $o] :
      ( ~ aa_o_o(aa_o_fun_o_o(fequal_o,(X9)),(Y))
      | ( (X9) = (Y) ) ) ).

tff(help_fequal_2_1_fequal_001t__Int__Oint_T,axiom,
    ! [X9: int,Y: int] :
      ( ( X9 != Y )
      | aa_int_o(aa_int_fun_int_o(fequal_int,X9),Y) ) ).

tff(help_fequal_1_1_fequal_001t__Int__Oint_T,axiom,
    ! [X9: int,Y: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(fequal_int,X9),Y)
      | ( X9 = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__Nat__Onat_T,axiom,
    ! [X9: nat,Y: nat] :
      ( ( X9 != Y )
      | aa_nat_o(aa_nat_fun_nat_o(fequal_nat,X9),Y) ) ).

tff(help_fequal_1_1_fequal_001t__Nat__Onat_T,axiom,
    ! [X9: nat,Y: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(fequal_nat,X9),Y)
      | ( X9 = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__Num__Onum_T,axiom,
    ! [X9: num,Y: num] :
      ( ( X9 != Y )
      | aa_num_o(aa_num_fun_num_o(fequal_num,X9),Y) ) ).

tff(help_fequal_1_1_fequal_001t__Num__Onum_T,axiom,
    ! [X9: num,Y: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(fequal_num,X9),Y)
      | ( X9 = Y ) ) ).

tff(help_fChoice_1_1_fChoice_001t__Real__Oreal_T,axiom,
    ! [P: fun_real_o] :
      ( aa_real_o(P,fChoice_real(P))
      = ( ? [X7: real] : aa_real_o(P,X7) ) ) ).

tff(help_fequal_2_1_fequal_001t__fun_It__Nat__Onat_M_Eo_J_T,axiom,
    ! [X9: fun_nat_o,Y: fun_nat_o] :
      ( ( X9 != Y )
      | aa_fun_nat_o_o(aa_fun4840518763754620511at_o_o(fequal_fun_nat_o,X9),Y) ) ).

tff(help_fequal_1_1_fequal_001t__fun_It__Nat__Onat_M_Eo_J_T,axiom,
    ! [X9: fun_nat_o,Y: fun_nat_o] :
      ( ~ aa_fun_nat_o_o(aa_fun4840518763754620511at_o_o(fequal_fun_nat_o,X9),Y)
      | ( X9 = Y ) ) ).

% Conjectures (1)
tff(conj_0,conjecture,
    aa_nat_o(aa_nat_fun_nat_o(vEBT_is_pred_in_set(vEBT_VEBT_set_vebt(vEBT_Leaf(a,b))),xa),one_one_nat) ).

%------------------------------------------------------------------------------